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of properties realizes φ in organism X at time t if and only if the formula A(Y 1 , . . . , Y n ; χ, t) is true of
; and that such an n-tuple uniquely realizes φ in X at t if it and no other n-tuple realizes φ in X at t. 18 But if you want to add that (to my mind unnecessary) requirement, you may: the argument of this section of the
paper will not be affected. The restriction to common sense psychological theories might interfere with a proposal to be made later on: that we explicitly build the postulate of a system of internal representation into φ. At first blush this would seem to conflict with letting φ be a common sense theory, since common sense would not appear to be committed to systems of internal representation. However, Lewis is rather generous in what he regards common sense committed to: he does not regard the fact that a theory postulates possible worlds other than our own as going against the requirement that it say only what is common sense. I believe that systems of internal representation are at least as close to being postulates of common sense as are possible worlds. I have said that I think that the requirement that φ be a common sense theory is unnecessary. I also think that it is harmful. For one thing, it is not at all clear that any of the common sense theories of belief are (or even come close to being) uniquely realized; the only way to get anything like unique realization may be to invoke non-common-sense suppositions. It may well be for instance that only a theory of belief that explicitly postulates a system of internal representation can come anywhere near to being uniquely realized. But I do not want to press this point here: the
argument of this section goes through even if φ itself does not postulate a system of internal representation or in any other way (apart from propositions!) strain the bounds of common sense. 19 Lewis' own account of what realizations are is strictly speaking inaccurate because he does not take the precaution I've recommended: as his account stands, something can serve as a realization of pain only if it realizes pain in all organisms at all times, thus depriving functionalism of its point (cf. Harman 1973, Ch. 3, Section 4). But it is clear that what I have suggested (which is equivalent to what Harman suggests) is what Lewis really had in mind. 20 Here I oversimplify: we might imagine that T1 , . . . , Tn include non-psychological terms that are needed in the theory; and we might also imagine that some psychological terms needed in the theory are not included among T1 , . . . , Tn , because their reference is fixed independently of φ. 21 For reasons not relevant to the present paper, Lewis finds it convenient to imagine that the theory has been
rewritten so that all the psychological terms are names. I have chosen not to do this, and the features of the next paragraph that do not look quite like anything explicitly in Lewis result from this fact. The conclusion I will derive however would have been forthcoming on his procedure as well. end p.47
Now if φ is a psychological theory with n primitive psychological predicates we can use it to define n functional properties. Suppose for instance that T j is a predicate that stands for a 1-place property of organisms, the kind of property (like pain) of which it makes sense to say that the organism has it at one time but not at another. (Lewis calls properties of this sort 'states', 22 but I prefer to reserve this term for a different use.) If T j is a predicate of this sort, then the jth functional property associated with φ is the property ψ defined as follows: (2) X has ψ at t ≡ df there is some 1-place physical23 property P such that (i) P is the jth component of a unique 24 realization of φ in X at t (ii) X has P at t. If ψ is the jth functional property associated with φ, we can then say that a realization of ψ in X at t is simply the jth component of a unique realization of φ in X at t. From this and (2) we derive (2′) X has ψ at t if and only if there is some 1-place physical property P such that (i) P realizes ψ in X at t, and (ii) X has P at t. This machinery enables us to give a precise sense to the general remarks of five paragraphs back. What functionalism about pain claims is that the property of being in pain 25 is a functional property associated with some theory φ p by (2) (or 22 'I take states to be attributes of a special kind: attributes of things at times' (1966: 165). When Lewis says that
'pain' refers to a state he does not of course mean that it refers to what might be called a state-token, that is, an individual inner occurrence (an occurrence of the type that the organism feels when it 'feels pain'). Nor does he mean what might be called a state type: a sortal property of such inner occurrences, a property that might be expressed by the predicate 'is a pain'. Rather, pain is taken to be an attribute of organisms, the attribute that is expressed by 'feels pain' (cf. 1966: n. 1). I think myself that the word 'state' is best reserved for what I've called state-tokens and statetypes; for Lewis' use of the term helps foster a confusion between states in his sense and state-types in mine, and this can lead to a disastrous confusion discussed on pages 33-6 of the original printing of this paper, but deleted here. (Lewis himself does not fall into the confusion, as n. 13 of Lewis 1972 shows, but I think his terminology has led others to make it.) 23 Strictly speaking, it might be better to leave out the word 'physical' here: that way if materialism is only contingently true, then we can allow organisms in those possible worlds in which φ is realized by irreducibly mental properties to have functional properties. This change in the definition of functional property would in no way affect the application of functional properties to organisms in those worlds where materialism is true; in particular, it would not affect the application of functional properties in the actual world, if as I am assuming materialism is indeed true. Since my own interest lies wholly in the actual world, my addition of the word 'physical' does no harm. I chose to put in the word 'physical' partly to remind the reader of the materialist premise, and partly as a convenient way to introduce distinctions of order. In order for the method of defining properties employed in (2) to make sense (and not lead to paradoxes), the functional property being defined cannot be one of the properties in the range of the property-quantifier. If we use the notion of 'physical property' narrowly, so that functional properties like those defined in (2) do not themselves count as physical, then this condition is met. On this narrow use of the term physical, the materialist thesis is that all non-functional properties are physical. 24 The uniqueness requirement is Lewis'. I think that it must be taken with a grain of salt, but I do not have the space
here to explain my reservations or to develop the machinery needed to avoid it. The word 'unique' in (2) cannot merely be dropped. [Note added for this volume: These remarks are explained in the Postscript to Chapter 6.] 25 According to Lewis, 'the property of being in pain' and 'pain' refer to different properties: 'the property of being in pain' refers to a functional property, and 'pain' refers (in the context of discussing a specific organism X at a specific time t) to the non-functional property which realizes the functional property in X at t. I have tried to remain neutral on the question of whether 'pain' refers to the functional property or refers (in a context-dependent way) to a realization of it. end p.48
by the analog of (2) with the word 'physical' replaced by 'non-functional'—see note 23 ). By taking functionalism in this way we can make precise sense of various vague notions appealed to in the general remarks (e.g., the notion of psychological isomorphism), and we can also verify the really important claim that if materialism is true, then for an organism to have the psychological property of being in pain it must have some physical property that realizes that psychological property in the organism. Now let us apply this machinery to the case of believing, where we assume as before that belief is a relation between an organism and a proposition. Belief, then, will be a functional relation associated with some theory φ b in which the term 'believes' occurs, say as the jth psychological term. We cannot of course apply the schema (2) to define such a functional property, since (2) made sense only in defining properties that correspond to 1-place predicates, but it is clear enough how to generalize it to properties of another kind: (3) X bears ψ to p at t ≡ df there is some 2-place physical property R such that (i) R is the jth component of a unique realization of φ in X at t (ii) X bears R to p at t. No other generalization of (2) to the case of 2-place functional properties is reasonable. Now the important thing to note about this is that it existentially quantifies over physical relations between people and propositions. If there is no physical relation of an appropriate sort between a person and a proposition, then according to (3) the person cannot stand in the functional relation ψ to the proposition. The functional relation ψ is not itself a physical relation; but if ψ is to relate an organism to a proposition at a time, there must be some physical relation R(ψ) which realizes ψ and which relates the organism to the proposition. Thus even if we take belief to be a functional relation, we have to solve Brentano's problem: we have to show that there could be physical (non-functional) relations between people and propositions. That is what I meant when I said earlier that functionalism does not either solve or dissolve Brentano's problem. The only thing it says of relevance to that problem is something that probably no one ever doubted anyway: that the physical relation that relates me now to the propositions I believe can differ from the one that relates dogs to the propositions they believe, and even from the one that relates other people to the propositions they believe, and from the one that related me twenty years ago to the propositions I believed then. In other words: it is indeed legitimate to solve Brentano's problem in different ways for different organisms; but this does not remove the need for solving the problem within an organism. I don't mean to downgrade the importance of the observation that we don't need to solve Brentano's problem in the same way in different organisms: that fact is crucial to the plausibility of the model of believing put forth in section 1, 23 Strictly speaking, it might be better to leave out the word 'physical' here: that way if materialism is only contingently
true, then we can allow organisms in those possible worlds in which φ is realized by irreducibly mental properties to have functional properties. This change in the definition of functional property would in no way affect the application of functional properties to organisms in those worlds where materialism is true; in particular, it would not affect the application of functional properties in the actual world, if as I am assuming materialism is indeed true. Since my own interest lies wholly in the actual world, my addition of the word 'physical' does no harm. I chose to put in the word 'physical' partly to remind the reader of the materialist premise, and partly as a convenient way to introduce distinctions of order. In order for the method of defining properties employed in (2) to make sense (and not lead to paradoxes), the functional property being defined cannot be one of the properties in the range of the property-quantifier. If we use the notion of 'physical property' narrowly, so that functional properties like those defined in (2) do not themselves count as physical, then this condition is met. On this narrow use of the term physical, the materialist thesis is that all non-functional properties are physical. end p.49
for no one could plausibly claim that the relation of believing* is physically the same across different species. All I am saying is that to admit the allowability and importance of such variation across organisms does not in any way diminish the force of the tentative argument for the model of believing put forth in section 1. It remains true that if no other model can be given of what a physical relation between people and propositions would be like, then we should tentatively accept the model there proposed.
The argument I have just given is I think an extremely obvious one: one would expect it to be obvious to anyone who thought about functionalism for a moment. Yet I have found in conversation that the conclusion of the argument is one that nearly all functionalists oppose: nearly all of them are convinced that functionalism somehow undermines the argument of section 1. One reason, I suspect, is a tendency to slip from functionalism to an extreme behaviorism according to which nothing inside an organism is relevant to determining its psychological properties. Another reason is perhaps a Leibnizian tendency to regard relations as less real than 1-place properties—not out of conscious doctrine but merely because of the fact that the word 'relation', unlike the word 'property', doubles as a word for a certain kind of set. But there is a third source of error that I now want to consider. It is equally crude, but I think that there are deep and subtle confusions that lead people to implicitly make it. The third source of error is that even among people who explicitly advocate the view that belief is a relation between people and propositions, there seems to be a tendency to sometimes fall into the 'orthographic accident' view: the view that an adequate theory of belief could treat 'X believes that Russell was hairless', 'X believes that either Russell was hairless or snow is white', etc., as primitive 1-place predicates, and do without the 2-place predicate 'X believes that p' entirely. (The fact that the term 'believes that' occurs in both 1-place predicates would then be, from a theoretical view, of no significance, a mere orthographic accident; that fact that both contain 'Russell was hairless' would likewise be an orthographic accident.) It is not easy to take such a view seriously. But let us suppose it were true: what would follow? Well, the account would clearly obviate the need for a physical relation between people and propositions: since it didn't talk of a psychological relation between people and propositions, it is clear that no physical relation between people and propositions would be needed in a realization. But this of course does not refute the point I was making in this section, which is that if you do construe belief relationally, you need a physical realization of the belief relation. In spite of the crudity of this mistake, I think that it is an easy one to make implicitly. In fact, in the opening paragraph of this section, when I tried to motivate the view that functionalism obviated the need of a system of representation, I found myself talking in a way that strongly suggested the orthographic accident view: I said 'a state of an organism is a state of believing that p if that state plays the appropriate role in the organism's psychology'. Now, for this to make any sense, the letter 'p' here must be understood as abbreviating end p.50
a specific sentence—say 'Either Russell was hairless or snow is white'. Now, what is 'the appropriate role' of the state of believing that either Russell was hairless or snow is white? I do not say that no sense can be made of such talk: if we give a functional account of the relation of believing which holds between organisms and propositions, such an account will certainly have implications about the state of believing the particular proposition that Russell was hairless or snow is white. I think however that talk of 'the appropriate role' of the state of believing this particular proposition strongly suggests that we can give a direct functional definition of this particular state. And that strongly suggests that the kind of procedure used for 'pain' can be applied to 'believes that either Russell was hairless or snow is white': in other words, it suggests that believing that either Russell was hairless or snow is white can be regarded as a functional 1-place property defined by schema (2). That however is the orthographic accident view, for it presupposes that the theory φ from which believing that either Russell was hairless or snow is white is to be functionally defined contains a primitive term that represents this property. The moral is that if you want to avoid the orthographic accident view, you should not regard 'X believes that p 0 ' for specific sentences p 0 as functionally definable in the way that 'X is in pain' is: you should regard them as defined non-functionally from a relational predicate 'X believes that p' which is functionally defined by (3). And that means that we must invoke physical relations between organisms and sets of possible worlds in the functional definition. [Note added for this volume: this is a functional definition only in a broad sense: see section 3 of the postscript. In the narrow sense, we can give a functional account of the relation of believing* (and of the internal sentences that are the object of this relation), but the connection between internal sentences and propositions can not be constructed functionally. These points are elaborated in section 4 of this chapter.]
3. Dispensing With Propositions At the beginning of section 2 I quoted an argument of Stalnaker's which suggested that a functional theory of belief would obviate the need for a system of internal representation. I remarked that in order to fully evaluate the argument, it was necessary to get clear on what a functional account of belief would be like. I have done that only in a very minimal sense: I have looked only at those features of a functional account of belief which follow from the nature of functionalism together with the (supposed) fact that belief involves a relation between people and propositions. But much more than this is involved in getting clear about what a functional account of belief would be like. For we have seen that on any functional account of belief, the relation of believing is a functional relation associated with a certain theory φ b . Part of what's involved in getting clear on what a functional account of belief would be like, then, is getting
clear on what φ b would be like. I am strongly inclined end p.51
to think that any adequate φ b would have to explicitly postulate a system of representation, and that this provides a reason to believe the internal representation hypothesis independently of any considerations about Brentano's problem. I will not however try to argue this claim here. Instead I will merely note a converse: that since (as I've argued) we appear to need a system of representation in order to solve Brentano's problem, we shouldn't have too much reluctance about explicitly incorporating such a system into our psychological theory φ b if doing so seems helpful. One advantage of explicitly incorporating a system of representation into one's psychological theory is that it enables us to obtain most of the advantages of regarding belief and desire as relations between people and propositions, without the attendant liabilities. Suppose that for some reason or other we do not want to quantify over propositions; in that case, then apparently we will be unable to say things like (8) There are many things she believes about him, and none of them are at all complimentary. or (9) N Fo one can perceive an object without coming to believe various things about it; for apparently the 'things' quantified over in (8) and (9) are propositions. However I know of no reason why our purposes in uttering (8) and (9) wouldn't be equally well served by quantifying over objects of belief* rather than of belief: e.g. 'No one can perceive an object without coming to believe* various things about it'; here the 'things' quantified over are not propositions, but sentences in an internal system of representation. Of course, we could accept this reformulated version of (9) in our psychological theory only if we incorporated the inner representation hypothesis explicitly into the theory. I think that by now the asterisks have become tiresome: so let us introduce the terminological convention that the word 'believes' is to be used in the way that I have heretofore used 'believes*'. On this way of talking, the objects of belief are sentences or sentence-analogs, and these sentences or sentence-analogs have content or meaning. Contrary to the suggestion in the first paragraph of section 1, this way of talking does not really remove Brentano's problem: that problem rearises as the problem of giving a materialistic account of having content. Unless such an account of content can be given, much of what we say about belief (e.g. that certain beliefs are about Caesar, that certain beliefs are true, and so forth) makes no sense at all. I have however suggested, at the end of section 1, that the problem of giving a materialistic account of content seems manageable: one way to manage it is to give a Tarski-like account of truth, supplemented by a theory of reference. My use of the word 'believes' to mean 'believes*' does not accord very well with the use of the term 'believes' in English: in English 'believes' is pretty much synonymous with 'believes that', and we can say even of organisms who know no English that they believe that snow is white. Let us then introduce a end p.52
new technical term, 'believes that', which will serve the purposes that 'believe' and 'believe that' serve in English: let us say that a person 'believes that p' (where 'p' abbreviates an English sentence S) if that person believes some sentence in his system of internal representation whose translation into English is S. In effect then 'believes that' is to be used for a relation between organisms and English sentences. It must be realized that the notion of translation employed in the definition of 'believes that' is a loose and sloppy one. For even those of us who are hostile to Quine's radical indeterminacy thesis are bound to recognize that translation between languages of very different structure or expressive power is highly indeterminate. (Part of the reason is that a sentence in language L 1 need not have exactly the same meaning as any sentence in language L 2 , so that a translator of L 1 into L 2 has to settle for approximate sameness of meaning; and which approximation one picks depends on complicated pragmatic considerations.) Since the notion of believing that is to be applied to organisms whose system of internal representation is doubtless quite different from ours—e.g., chimpanzees, Martians, and (if Whorf is right) humans whose spoken and written language differs significantly from ours—then it would be absurd not to recognize that the translation involved in the definition of 'believes that' is highly indeterminate. For this reason the notion of 'believing that' defined above is itself a highly indeterminate notion, and consequently it is a notion that we ought to avoid in our psychological theorizing.26 Instead, we should use the notion of believing (i.e., believing*), together perhaps with various semantic notions applied to the sentences believed. (Another reason for avoiding use of 'believing that' (as defined above) in our psychological theories has already been mentioned in section 1—cf. (B) on p. 39.) I have introduced conventions about the use of 'believes' and 'believes that' according to which the first term relates organisms to sentences in their own system of representation and the second relates organisms to sentences in English. These terminological conventions are not intended to rule out the possibility that we need propositions in the
analysis of belief and desire: it may be that we need propositions in our account of meaning for the sentences that are believed and desired. My own view however is (a) that talk of propositions is best avoided (except as a dispensable manner of speaking) unless it can be shown to serve purposes that cannot be served otherwise; and it is doubtful that this condition is met (b) that such talk commits us to semantic theses which (independently of ontological scruples) may well be false. Let's take (a) first. That one should not posit entities needlessly is, I take it, uncontroversial: to do so would be to indulge in 'unfounded ontological speculation'. 26 This is not intended as an argument against using a notion of believing that which relates people to propositions,
but only as an argument against using a notion which relates people to English sentences (or to utterances of English sentences). A theory involving propositions is exempt from the criticism, since if there are propositions at all then presumably there are propositions not expressible in English. (However, even if 'believes that' relates people to propositions, it is not clear that sentences attributing particular beliefs to organisms very unlike us could ever be literally true.) end p.53
And that there is no particular need to introduce propositions—at least, no need to introduce propositions construed in terms of possible worlds—was implicit in my discussion at the end of section 1. There I argued that if one wanted to use the idea of possible worlds, and if one construed a proposition as a set of possible worlds, then the correlation of a particular proposition with a sentence posed no problem as long as we had an adequate truth-theoretic semantics for the language. Another way of putting this point is that talk of propositions adds nothing of semantic interest; everything that is semantically of interest is already there in the truth-theoretic semantics. (I'm speaking of course of a truththeoretic semantics that assigns properties rather than sets to predicates.) In other words, instead of saying that a person is related to the set of possible worlds in which Russell was hairless (or to some fine-grained proposition constructed out of possible worlds), why not say instead that he is related to a sentence that consists of a name that stands for Russell copulated with a predicate that stands for hairlessness? 27 One might try to respond to this question by holding (implausibly, I think) that names are non-rigid designators. But this response, while adequate to the question just asked, does not undercut the point I am trying to make (as has in effect been observed already, in footnote 13 ). For if 'Russell' is non-rigid, then to believe that Russell was hairless is to believe some sentence that contains a definite description, some sentence of the form 'G(ιxFx)' in which 'F' is an individuating description of Russell and 'G' stands for hairlessness. A truth-theoretic analysis of this sentence contains all the semantic information that is contained in a possible world analysis; so why bring in possible worlds? Again, they seem to be excess ontological baggage serving no semantic role. Under (b) there are two points to be made. The first is that if one talks of propositions and also assumes that propositions must be explicable in terms of possible worlds, then in order to get needed distinctions among propositions, one will be driven to postulate that proper names and theoretical terms like 'temperature' are non-rigid designators; and this postulate is controversial at best. The second point is that whether or not one adheres to the possible worlds approach, talk of propositions commits one to a general relation of synonymy, and it is not at all obvious that there is any such general relation. I will develop these points, especially the second one, in section 6. Their upshot is that the postulate of propositions involves not only unfounded ontological speculation but highly controversial semantic speculation as well. To me both the ontological and the semantic claims that underlie the postulate of propositions seem far more dubious than the postulate of a system of internal representation, especially since this last postulate appears to be a necessary one for the solution of Brentano's problem. 27 Some people may feel that there is no ontological gain in quantifying over properties rather than over propositions.
Such a person should read Putnam 1969. Putnam makes a good case (a) that quantification over properties is needed in science, and (b) that properties are quite distinct from meanings, in that two predicates like 'x has temperature 210°C' and 'x has mean molecular energy 10 −20 joules' can turn out to stand for the same property even though they clearly differ in meaning. 13 I have assumed in this paragraph that proper names are always rigid designators. Some people however deny this: they believe that when we use the name 'Bertrand Russell' we associate with this name a property φ which we think picks out Russell uniquely; and that when we say 'Bertrand Russell is hairless', what we say is true at a world w if and only if there is exactly one thing in w with property φ in w, and that thing is hairless in w. (The thing with φ in w need not be the thing with φ in the actual world, i.e., it need not be Russell.) I find this doctrine that names can denote non-rigidly quite implausible; but the point I am making in the paper could be generalized so as to allow for names to be non-rigid designators for anyone who finds that desirable. To see this, reflect on how we would associate a property φ with a name. The only way I can see to do that is to associate with that name some expression (in a natural language or in a system of internal representation) which stands for that property. What the advocate of non-
rigid designation is saying, then, is that when we utter the sentence 'Bertrand Russell is hairless', that sentence merely serves to abbreviate the sentence (or sentence-analog) that we really mean, which is something of the form 'There is exactly one thing x such that ψ(x), and x is hairless'. Truth-at-a-possible-world can then be defined for the sentences (or sentence-analogs) that we really mean, by the process described in the text; and we can say that derivatively, a sentence S containing the name 'Bertrand Russell' is true at w if and only if the associated sentence (or sentence-analog) that we really mean when we utter S is true at w. end p.54
Let us say then that belief and desire are not attitudes toward propositions, but toward meaningful sentences in a system of internal representation. Presumably part of any adequate account of meaning for a system of internal representation is a truth-theoretic semantics; I will suggest in section 6 that there is another aspect of meaning as well. But first I would like to try to clarify the idea of a system of internal representation.
4. Remarks on the Inner Representation Hypothesis I have spoken of belief and desire as involving a system of internal representation; but I have allowed that in the case of organisms that have a genuine language, the system of internal representation might either be the language or include the language as a part. 28 This combination of claims may seem puzzling: if the only representation is in natural language, why call it internal representation? If there is internal representation, what sense does it make to say that the representation is in a natural language? The answer to these questions lies in the distinction between types and tokens. I have talked of an organism as believing (i.e., believing*) sentence-types. But I said in section 1 that (neglecting the complication about core beliefs) a person believes a sentence-type if and only if he employs that sentence-type in an appropriate way in reasoning, deliberating, etc.; and the only way to employ a sentence-type is to employ some of its tokens. Now, it is clear that in order to believe (or core believe) a sentence I can't be required to employ spoken or written tokens of it: no one writes down all of his beliefs (or all of his core beliefs). Consequently, if I believe sentences of my language, what I employ has to include internal tokens of those sentences. That explains why, even if all representation is in natural languages, we have to speak of internal representation: some of the tokens are certainly internal. This brings us to the second question: does it make sense to speak of internal representation as representation in natural language? It makes sense if, and only if, it makes sense to speak of internal tokens as being of the same type as spoken or written tokens. One might argue that any claim of type-identity between internal tokens and spoken or written tokens is highly implausible: after all, it seems pretty absurd to suppose that there is much of a physical resemblance between internal tokens on the one hand and spoken or written tokens on the other. But this of course would be a very bad argument: after all, a spoken token can be said to be of the same type as a written token, and yet spoken and written tokens bear little physical resemblance to each other. My own view is that the issue of whether we speak of internal tokens as type-identical to spoken or written tokens is partly a verbal issue, but that there are 28 I have allowed this by allowing for the possibility that the sentences which are believed be genuine sentences in a
genuine language, rather than sentence analogs. end p.55
interesting empirical questions that underlie it, of which the most important is the extent to which (and the manner in which) linguistic development involves conceptual development. To put the point very vaguely: if (as seems to me quite likely) learning a first language involves extending an initial representational system to include an isomorphic copy of the language being learned, then I think it is quite natural to view the isomorphism as establishing a criterion of typeidentity between internal tokens and spoken or written tokens. However, the issues here are pretty complicated. See Harman 1970 for an interesting discussion which more or less supports the pro-type-identity position, and Fodor 1975 (ch. 2) for an interesting discussion in support of the other side. My sympathies are much more with Harman than with Fodor on this matter, but I will try to remain neutral on the question in the rest of the paper.29 But there are other questions that need to be considered. First, I have talked of 'internal tokens'. What kinds of entities are these? Presumably they are inner occurrences of some kind, but what kind? Second, although I have begged off the question of what it is for an internal token to be of the same type as a spoken or written token, there is also the question of what it is for two internal tokens to be of the same type as each other; and since a psychological theory will clearly need to use the notion of type-identity between different tokens, this question cannot very well be ignored. Third, there is the question of syntactically characterizing internal tokens—what is it for an internal token to belong to a given syntactic category (e.g., what is it for it to be a sentence token)? Again, we will need to appeal to syntactic
characterizations of inner tokens in developing the psychological theory, so this question is a fairly pressing one. There are two possible strategies in dealing with such questions. The first strategy, which I think is the wrong one, is to try to answer the above questions prior to developing the psychological theory. Such a strategy is a bad one because it is hard to see how to carry it out without doing a great deal of neurophysiological speculation: e.g., we would apparently have to specify two neurophysiological properties P 1 and P 2 and a neurophysiological relation R, and say that an inner occurrence is an expression-token if and only if it has P 1 and is a sentence token if in addition it has P 2 , and that two inner occurrences with property P 1 are of the same type if one bears R to the other. The task of specifying P 1 , P 2 and R is certainly not a task we are equipped for in our current state of knowledge, and I don't see how we could ever become equipped for it prior to the development of psychological theory. The second strategy, which is the one I advocate, is to develop the syntax of the system of representations as part of the psychological theory: we can then use the psychological theory to give a functionalist answer to the questions raised above. That this is the right way to proceed seems completely obvious: it is 29 Where neutrality of formulation is difficult to achieve without verbosity, I have frequently sacrificed the neutrality; but
even in these cases it is not difficult to rephrase what I say so as to accord with the position that I am temporarily excluding. end p.56
simply an instance of the general rule that psychological theories ought to be construed functionally. Let me be a bit more explicit about what is involved in this functionalist approach. If one were to write out in detail a theory φ that postulates a system of inner representation, such predicates as 'x is an expression-token', 'x is a sentence-token', and 'x and y are expression-tokens of the same type' would either appear as primitives or be explicitly definable (within set theory or higher order logic) from other such syntactic primitives. Such syntactic primitives are to be included among the psychological primitives when we 'functionalize' φ: that is, a realization of φ is to be an n-tuple consisting of properties and relations corresponding to the syntactic predicates of the theory as well as to the more straightforwardly psychological predicates. If we 'functionalize' the theory in this way then the answers to the questions raised above can be read right off of the theory φ. 30 For instance, suppose for simplicity that 'x and y are tokens of the same type' is a primitive of φ. If we want to know what it is for two internal occurrences in an organism X at a time t to be tokens of the same type, the answer is simple: c and d are tokens of the same type if and only if there is a physical relation R which is the appropriate component of a unique realization of φ in X at t, and c bears R to d.31 , 32 (If 'x and y are tokens of the same type' is a defined term rather than a primitive, the answer is slightly more complicated; but again, it can be read right off of the theory.) The upshot is that there is no need in developing psychological theory to specify what R is; we can leave that to future neurophysiology. (Moreover, we can allow that there are different physical type-identity relations in different organisms.) The fact that we leave the question to future neurophysiology does not imply any unclarity in our theory: in some sense, the theory implicitly specifies what it is for two inner tokens in a system of representation to be tokens of the same type. 30 This assumes that the syntactic theory is formulated in such a way that all syntactic predicates are predicates of
tokens. To assume this is not to assume that the syntax can be given nominalistically, i.e., without quantifying over abstract entities like sets, or even like sequences which can be intuitively regarded as having more or less the role of expression-types. Once this is realized, the task of formulating the syntactic theory in the way required presents no difficulty. Incidentally, I should remark that the syntactic theory can be formulated in such a way as to allow wide diversity in the grammars of the systems of representation of organisms to which the theory is intended to apply. (One way in which it might do this is to postulate a general system of syntactic categories. Different sub-sets of this general system could be instantiated in different organisms to which the theory applies, and in this way a wide syntactic diversity in systems of representation would be compatible with the general theory. See Lewis 1970a for an illustration of the kind of general syntax I have in mind.) 31 Incidentally, R could perfectly well be a disjunctive relation, say of the form 'Either x and y are both occurrences in the left hemisphere of the brain and x bears R 1 to y, or x and y are both occurrences in the right hemisphere and x bears R 2 to y, or one is an occurrence in the left and the other in the right and the one in the left bears R 3 to the one in the right'. Disjunctive realizations are needed for other functional theories—e.g., it is perfectly possible that pain is realized in a given organism at a given time by a disjunctive property like 'is a stimulation of the C-fibers or a firing of the X-neurons'—and there is no reason to rule out the same flexibility in the theory of belief and desire. I make this point because I think that a main reason why many philosophers resist the inner representation hypothesis is that they tend to exaggerate its neurophysiological commitments. 32 This only defines what it is for two tokens in the same organism to be of the same type; but the notion of typeidentity between tokens in one organism and tokens in the other is not needed for psychological theory, and can be
regarded as a meaningless notion. end p.57
There is one final point to be made about theories that postulate a system of internal representation. I have said that the syntax of a system of internal representation should be explicitly stated in a psychological theory of belief and desire. Should the semantics of the system of internal representation also be stated as part of the psychological theory? That depends on what we want psychological theory for. If the task of psychology is to state: (i) the laws by which an organism's beliefs and desires evolve as he is subjected to sensory stimulations, and (ii) the laws by which those beliefs and desires affect his bodily movements, then I think that it is clear we do not need to use the semantics of the system of representation in stating the psychological laws: the sentences in the system of internal representation might as well be meaningless as far as the psychology is concerned. 33 This is not the only way to view a psychological theory—a broader conception of a beliefdesire psychology will be suggested in the next section, and in it semantic notions would play a genuine role. But it is worth stressing the narrow kind of psychology at least momentarily. For we have seen that the syntax and type-identity conditions for a system of internal representation should be regarded as functionally characterized by a psychological theory in which they appear; and we can take that theory to be narrow psychology, that is, the kind of psychology that does not employ any semantic characterizations of the sentences in a system of representation. This is important, for it means that the syntax and conditions of type-identity for the system of representation could in principle be determined independently of any considerations about what the sentences in the system mean.
5. Truth What would a theory of meaning for a system of internal representation be like? In sections 1 and 3 I have hinted at one aspect of my views on this question: a theory of meaning for a system of internal representation must consist in part of a truth-theoretic semantics of a more-or-less Tarskian kind. If we do not give a theory of truth for the system of internal representation, we cannot make sense of the idea that some of our beliefs are true and others are false; and I think we do want to be able to make sense of this idea (for reasons to be sketched shortly). Moreover, the only kind of theory of truth that I have ever heard of which is not obviously deficient is the Tarskian kind. The upshot, then, is that we need to give a Tarski-type semantic theory for the system of internal representation. In a recent article to which I am in most respects sympathetic, Gilbert Harman has questioned this: no reason has been given for a compositional theory of meaning for whatever system of representation we think in, be it Mentalese or English (1975: 286). 33 I will elaborate on this point in the next section. end p.58
Presumably however if the notion of truth makes any sense, then truth and meaning must be related in the following way: the truth of the sentence 'Caesar crossed the Rubicon' should follow from the meaning of the sentence together with the fact that Caesar crossed the Rubicon. In this sense a theory of meaning must include a theory of truthconditions. And as far as I can see, the theory of truth-conditions has to be a compositional theory of roughly the type that Tarski made famous. Harman's critique of compositional semantics is based on an important insight. He points out in the article that there is a serious problem for those philosophers, like Davidson, who regard knowledge of truth-conditions as what is essential in semantics: Davidson would (presumably) say that the speaker understands [the sentence 'Snow is white'] by virtue of the fact that he knows it is true if and only if snow is white. The difficulty . . . is that [for the speaker to know any such thing he] needs some way to represent to himself snow's being white. If the relevant speaker uses the words 'snow is white' to represent in the relevant way that snow is white, . . . Davidson's [theory] would be circular. And, if speakers have available a form of Mentalese in which they can represent that snow is white, so that the [theory avoids] circularity, there is still the problem of meaning for Mentalese (1975: 286). But the moral to be drawn is that knowledge of truth-conditions is not what is important to the semantics of a system of internal representation. The theory must ascribe truth-conditions not knowledge of truth-conditions, to the sentences of English or Mentalese; for if it doesn't ascribe truth-conditions to these sentences it will not have given sense to talk of our beliefs as being true or false. I have assumed that we do want to make sense of the idea that some of our beliefs are true and others are false; I
would also assume that we want to make sense of the idea that some of our beliefs are about Julius Caesar and that other of our beliefs are about quarks. It seems to me however that there is a serious question as to why we should want to make sense of these ideas. Is our desire to do so based on anything other than a naive metaphysics that has no place in a properly scientific account of the world? That this question is not a silly one can be seen from a fact noted at the end of the previous section. If the task of psychology is to state (i) the laws by which an organism's beliefs and desires evolve as he is subjected to sensory stimulations, and (ii) the laws by which those beliefs and desires affect his bodily movements, then semantic characterizations of beliefs and desires are irrelevant to psychology: one can state the laws without saying anything at all about what the believed and desired sentences mean, or what their truth-conditions are or what their subject matter is. For instance, we might imagine a super-crude psychology that contained laws like the following: there is some connective '→' in the system of internal representation such end p.59
that for all sentences S 1 and S 2 in the system, whenever a person believes S 1 → S 2 and desires S 2 then he also desires S 1 . The connective '→' that satisfied this law might mean 'only if', that is, it might obey the truth-table for the conditional; but the fact that it obeys this truth-table is not something we need to say in stating the psychological laws.34 The psychology, since we are imagining it to be super-crude, might also contain laws like this: there is a privileged class of sentences in the system of representation, called the class of observation sentences, with the property that each sentence in the class has associated with it a particular type of sensory stimulation. Whenever a sensory stimulation of the appropriate type occurs, the organism believes the observation sentence. Intuitively, we might expect that if a particular observation sentence is associated in this way with the class of retinal stimulations that are characteristically caused by nearby rabbits, then the observation sentence means something like 'there are rabbits nearby'; but even if this is true, the psychological theory need not say that it is true. Why then do we need to semantically characterize the sentences in the system of inner representation? Why not simply say that belief and desire are relations between people and meaningless sentences? Saying this would preclude us from speaking of beliefs as being true or as being about rabbits, but would anything of scientific value be lost? Here is one answer to this question—not the whole story, I think, but the part of the story that is emphasized in the writings of such philosophers as Quine, Davidson, and Harman (and perhaps Lewis in 1974b). Imagine that we find ourselves with a foreigner whose language we do not understand. A rabbit scurries by, in the foreigner's line of sight, and the foreigner raises his gun in its direction—by now we have rather overwhelming grounds for thinking that he believes that there is a rabbit nearby. But can we say so in the vocabulary of the narrow psychology considered heretofore? We can't say 'He believes the sentence . . . ' (giving the name of the sentence): for we don't know his language. How about if we say instead 'He believes some sentence which is an observation sentence associated with sensory stimulations of type . . . (and which serves as evidence for syntactically related sentences according to laws . . . , and so forth)'. The difficulty is clear: only someone with a great deal of very detailed information about the psychology of our foreigner could fill in the blanks. A third possibility is much better: He believes some sentence of his language which plays approximately the 34 The proposed 'law' involving '→' is of course exceptionally crude: to get anything that looks at all plausible, one
probably has to bring in something like degrees of belief and degrees of desirability. But the point I am making is unaffected: the degrees of belief and degrees of desirability can be regarded as attaching to sentences, and laws can be given relating the degrees of belief and desirability of a sentence of the form S1 → S2 to the degrees of belief and desirability of the component sentences S1 and S2 and of other compound sentences containing S1 and S2 (e.g., if B is degree of belief, one law might be that B(S1 → S2 ) ≥ max{B(S2 ), 1 − B(S1 )}). At no point do the meanings of the component sentences or of the logical connectives need to be mentioned. end p.60
role in his psychology that the sentence 'There's a rabbit nearby' plays in mine. But this, it might be claimed, really involves a semantic notion! For isn't it really just a long-winded way of saying
He believes some sentence of his language that translates into my language as 'There's a rabbit nearby'? And isn't translation a semantic notion? This answer does not satisfy me, for though it does definitely motivate the introduction of a notion of translation, it does not motivate the introduction of any non-translational semantic notions; that is, it motivates the introduction of a moreor-less 35 semantic notion about the relation of one language to another, but it does not motivate the introduction of any semantic notions like 'true' or 'refers' which relate language to the world. No reason has been offered, it seems to me, for regarding another person as having beliefs that are true or about rabbits. The Quinean reply, I would imagine, is that we need such notions as truth in connection with our own language (e.g., to state generalizations like 'Every sentence of the form "p or not p" is true');36 and that we can then use the notion of translation that we have motivated on other grounds to carry over the truth concept to foreign languages. I do not want to discuss this Quinean reply here; but it seems to me rather weak, and I would like to do better. I think that the reason why we need to be able to apply semantic notions like truth and reference to the sentences that people believe and desire is that we hold the theory that people's beliefs are, in many circumstances, reliable indicators about the world; and the only way to state this theory is to use the notion of truth (and probably the notion of reference as well). Moreover, this theory is not a piece of gratuitous metaphysics that could easily be dispensed with: it is central to our getting information about the world, for we are constantly using our opinions about other people's beliefs in forming opinions about the world. The fact that a child believes that he has done something I won't like (a fact that can often be inferred from his behavior) gives good reason to think he has done something I won't like; the fact that most physicists believe that there are gravitational waves (a fact that can be inferred from reading a few physics books) is good reason for me to believe that there are gravitational waves; and so forth. These inferences evidently proceed by means of certain reliability principles, principles that say under what conditions a person's beliefs about certain things are likely to be true. The principles we need are not easy to state: after all, the fact that a child believes in Santa Claus is not good reason for me to believe in Santa Claus, and the fact that most members of a certain religious cult believe that flying saucers will land on a certain farm in 35 I think that there is some question as to whether the notion of translation, used in the very loose and pragmatic
way just explained, ought to count as a genuinely semantic notion; hence the 'more or less'. 36 Cf. Quine 1970: 11. end p.61
Arizona next month is not good enough reason for me to believe that flying saucers will land there. (I don't think that the failure of these inferences is due entirely to independent evidence as to the falsity of the conclusion.) We do evidently have a stock of reliability principles, though we can not explicitly formulate them; and one can imagine that they will someday be systematized into an explicit theory. In a suitably broad sense, we might even regard this 'reliability theory' as part of psychology. What such a reliability theory would look like I do not know. My guess is that it would have to include not only the notion of truth but the notion of reference: for we want to be able to say in the theory that some people have very reliable beliefs about physics but very unreliable beliefs about the state of the economy, and so forth. We might imagine then that 'true', 'refers', etc. are primitives of the theory. If we do that—and if we imagine that the reliability theory is elaborated with such detail and precision that it is uniquely realized in each of the organisms to which the theory is intended to apply—then we could use this theory to give a functional account of truth and reference for systems of internal representation. Such a functional account would of course be desirable, for the reason that functional accounts are always desirable: it would allow for the possibility that the reference relation is realized by different physical relations in different organisms. 37
6. More on the Semantics of Internal Representation Is there more to the semantics of a system of internal representation than is given by truth-theoretic semantics? I think that there is; and I think that this casts considerable doubt on the possible worlds analysis of propositions, even independently of ontological considerations. In fact, I think it casts some doubt (independent of ontological considerations) on whether any notion of proposition is possible. In explaining these matters, I will begin by discussing the semantics of spoken and written languages, since I want to discuss some points that Quine has made in the context of them; but what I say will carry over to systems of internal representation. Many years ago Quine made the following observation: A lexicographer may be concerned with synonymy between forms in one language and forms in another or . . .
he may be concerned with synonymy between forms in the same language. It is an open question how satisfactorily the two cases can be subsumed under a single general formulation of the synonymy concept . . . (1953a: 56). The point that Quine is making here is very relevant to the question of propositions: if one postulates propositions one is assuming a positive answer to 37 Without a functional account, we can in a sense recognize this: we can let 'refers' stand for a certain physical
relation in connection with organisms of one kind (say, organisms that speak some language of a certain general type), and also let 'refers' stand for a different physical relation in connection with organisms of another type; but without a functional account we do not have the means to explain what these two uses of 'refers' have in common. (Similarly for 'true'.) end p.62
Quine's 'open question'. For if sentences mean propositions, then apparently two sentences are synonymous if they mean the same proposition; and this would be a general concept of synonymy, it would apply both intra-linguistically and inter-linguistically. Is Quine's 'open question' really open? Well, at least this much is true: intralinguistic synonymy seems a lot easier to define than interlinguistic synonymy. (Similarly, intraspeaker synonymy is easier to define than interspeaker synonymy, suggesting that there might be still further divergence in the synonymy concept.) That intralinguistic synonymy is easier to define than interlinguistic synonymy at least for what Quine calls occasion sentences is a point that Quine argues in sections 9 and 11 of [1960]: he points out that the difference in meaning between 'Everest' and 'Gaurisanker' (construed as one-word sentences) for a certain speaker is revealed by the fact that different sensory stimulations would prompt him to assent to one than to the other; and that the difference in meaning of these one-word sentences in a given linguistic community is revealed by the fact that these sentences are intra-subjectively non-synonymous for most members of that community. Now, it seems to me that this talk of 'prompting to assent' is much too behavioristic, and leads Quine into unnecessary worries, e.g., about 'stimulations of second intention' (verbal stimulations like 'Assent to one-word sentences that begin with "E" or I'll beat your brains out')—cf. Quine 1960: 48-9. Nevertheless, it seems to me that Quine's general point is correct: we can explain intralinguistic differences of meaning by evidential considerations. I have developed this point elsewhere (Field 1977), using a non-behavioristic (but, I admit, highly idealized) conception of evidence; and in addition to solving such humdrum difficulties with Quine's approach as the second-intention problem, it also obviates the need for restricting the account to occasion sentences. What is interesting about this approach I think is that it gives you differences of meaning where you would intuitively expect them, but where you do not get them on the possible worlds approach (without adopting implausible assumptions about non-rigid designation). For instance, the following pair of sentences come out equivalent in meaning on the possible worlds approach: (10) Everest is Gaurisanker comes out equivalent to (11) Everest is Everest; and (12) Temperature is mean molecular energy comes out equivalent to38 38 Strictly, this equivalence holds only on the coarse-grained possible worlds approach. But we could imagine that a
one-word phrase 'glub' had been introduced into the language by the stipulation that it was to (rigidly) denote the property of mean molecular energy; then 'Temperature is glub' would be equivalent to (13) even on the fine-grained possible worlds approach, but they would clearly be non-synonymous. end p.63
(13) Temperature is temperature. But it should be pretty clear, even without looking up the details of my account, that these equivalences will not hold on any sufficiently sophisticated evidential criterion of intraspeaker synonymy, even for speakers who believe the sentences (10) and (12). We do not need to assume a description theory of reference to get the result. Such evidential considerations (coupled with truth-theoretic considerations if the latter are not redundant 39 ) seem to me to provide a very natural account of intra-speaker synonymy. For inter-speaker synonymy however the situation is quite different, for here it is very hard to formulate any evidential criteria for two words differing in meaning. The source of the difficulty is clear: you and I may disagree about what counts as evidence for a certain sentence, not because that sentence means something different to you than it means to me, but because of differences in the rest of our
beliefs. One might try to find some rule of the form 'If there is an evidential difference of such and such a kind between your sentence and mine, then they differ in meaning'; this is the task that Quine refers to as 'trying to strip away the effects of collateral information'. Quine has cast considerable doubts on the possibility of carrying out this task (1960, section 9), and I think that if you look at the question in terms of the formal model in Field 1977 you will find that Quine's doubts are reinforced. 40 It seems to me that the criteria of inter-linguistic synonymy we actually employ are, almost exclusively, the criteria provided by truth-theoretic semantics. 41 This is not quite true; when there are two sentences S 1 and S 2 in one language that are equivalent from the point of view of truth-theoretic (or possible world) semantics, but which differ evidentially, and when there is a sentence S 3 in another language that is truth-theoretically equivalent to both S 1 and S 2 but is much more similar evidentially to S 1 than to S 2 , then we regard it as definitely a mistake to translate S 3 as S 2 —the translation S 1 seems required. But except in such cases, little if any inter-subjective sameness of evidential role is required; if the Martians have singled out Everest, i.e., Gaurisanker, by their powerful telescopes and have named it 'Schrdlu', we would translate their name 'Schrdlu' by 'Everest' or by 'Gaurisanker' indifferently, however much 39 Whether they are redundant depends on whether one treats evidential considerations in a fine-grained or a coarse-
grained manner; see Field 1977: 396-7. 40 (Note added to this volume:) That is because the relevant evidential differences for intrapersonal synonymy of sentences S1 and S2 include not just different reactions to sensory stimulations, but different reactions to other sentences. Indeed, these other sentences need not be 'less theoretical' than S1 and S2 . If one tries to generalize to an account of synonymy that applies interpersonally as well as intrapersonally, there is a real problem, one that is especially clear in the case where the people involved do not share a language. In that case, the linguistic difference precludes our use of the different ways that the same sentence impacts on S1 and S2 ; but then we seem to need to know whether E1 (in the language of S1 ) is synonymous with E2 (in the language of S2 ) before knowing whether the different impact of E1 on S1 than of E2 on S2 gives reason for thinking that S1 and S2 are nonsynonymous. 41 Our criteria for inter-speaker synonymy between speakers of the same language are less clear: sometimes we
treat this case like the inter-linguistic case, and sometimes we use the common language to extend considerations of intra-speaker synonymy across different speakers. end p.64
we might want to say that 'Everest' and 'Gaurisanker' differ in meaning for us. One might object to this argument by saying that translation is a loose and pragmatic notion: the true situation (one might say) is that 'Schrdlu' differs in meaning both from 'Everest' and from 'Gaurisanker'; we translate the name indifferently by 'Everest' or by 'Gaurisanker' because these approximate 'Schrdlu' about equally well. I sympathize with this response, except for one thing: it assumes that there is some clear notion of inter-linguistic difference in meaning between words that refer to the same thing, and that is what I think needs to be established. My view then is that truth-theoretic semantics—i.e., possible world semantics without the possible worlds—is almost enough, but that there are certain very fine-grained distinctions of meaning that it cannot explain. These fine-grained distinctions of meaning seem to be clearly drawable only intra-linguistically, and that fact appears to make the notion of proposition suspect. It should be clear that these remarks about synonymy between sentences in public languages apply also to synonymy between sentences in systems of internal representation. It should also be clear that to define the evidential differences between sentences in the same system of internal representation, one does not need to employ any concepts that go outside psychology in the narrow sense. Consequently there seems to be no special problem in motivating the introduction of such evidential considerations into a semantic theory for the system of inner representation, in a way that there did seem to be a special problem of motivating the introduction of truth-theoretic concepts. I want to conclude this paper by raising what I regard as an open question: what connections are there between a theory of meaning for a system of internal representation and a theory of meaning for a spoken or written language? 42 According to one influential approach 43 to the theory of meaning for spoken and written languages, the meaning of a sentence in such languages is to be explained in terms of beliefs (or desires, etc.) that are conventionally correlated with the sentence. Roughly, to know the meaning of 'Caesar was egotistical' is to know that this sentence is conventionally correlated with the belief that Caesar was egotistical; and similarly for every other sentence in one's public language. (The conventional correlation must be spelled out recursively, of course.) This approach presupposes that one can explain what it is to believe that Caesar was egotistical without relying at any point on the semantic features of the sentence 'Caesar was egotistical' in one's spoken or written language: for if one relied on the semantic features of the spoken or written 42 Here I am concerned primarily with the truth-theoretic aspects of meaning. Also, the question I am asking does not
really presuppose what the formulation in the text suggests, that a person's system of internal representation is distinct from his public language. If we adopt the opposite presupposition, we can phrase the question as follows: in what ways if any does one need to bring in the fact that a system of internal representation is also used publicly, in giving a complete theory of truth-conditions for the system of internal representation; and in what ways does one need to bring in the fact that a public language is used for thinking, in giving a complete theory of meaning for it. 43 See Schiffer 1972. Lewis' approach in 1975 is quite similar. Harman's suggestion in 1968 that the 'level 2 theory of meaning' presupposes the 'level 1 theory of meaning' but not conversely seems to involve a similar idea. end p.65
sentence in one's account of the belief, then to explain the meaning of the sentence in terms of belief would involve a circularity. So a crucial question is whether that presupposition is correct. Putting the presupposition in terms of internal representation, it is this: that one can explain what it is for a sentence in the internal system of representation to mean that Caesar was egotistical without relying in one's explanation on the fact that certain words and phrases in the public language stand for Caesar or that certain other words and phrases in the public language stand for the property of being egotistical. If this presupposition is correct, then the above approach to the theory of meaning is quite attractive: if worked out, it would reduce all questions about the semantics of the spoken or written language to the corresponding questions about the semantics of the system of internal representation; and those questions could then be answered without further reference to language. I am inclined to doubt however that the presupposition is true. My guess is that in a typical case, part of what makes a symbol in my system of internal representation a symbol that stands for Caesar is that this symbol acquired its role in my system of representation as a result of my acquisition of a name that stands for Caesar in the public language. If something of this sort is true, it would appear to defeat 44 the above approach to a theory of meaning for a public language. I believe however that the issues here are quite complicated, and deserve a great deal of further study. An opposite approach to the theory of meaning would be to try to reduce the semantics of the system of internal representation to the semantics for public language. To do this would be to assume that although dogs, chimpanzees, etc. might have systems of internal representation which played an important role in the explanation of their behavior, these systems of representation would not be ones to which notions like meaning and truth applied: such semantic notions could be applicable only to the representational systems of organisms with a spoken or written language. To me, such a conclusion seems very implausible. It may nevertheless be that there is something to this approach, for it may be that certain aspects of meaning can be explained more directly for public language than for systems of internal representation. The suggestion is not that these aspects of meaning can be explained without reference to beliefs and desires, for two reasons. In the first place, in explaining the fact that the public word 'Caesar' referred to Caesar one could appeal 44 It might not defeat it, if as suggested at the end of section 5 we give a functional account of reference: for then we
could grant that we need to specify the use of the word 'Caesar' in public language in order to specify a realization for the reference-relation as applied to the system of internal representation, but hold that we do not need to mention it in the functional theory itself. However, it does not seem to me at all obvious that we do not need to mention the public use of words in a satisfactory functional theory of truth-conditions for systems of internal representation; it seems to me that no one has developed any such theory in anywhere near enough detail for us to tell. It may well be for instance that a fairly detailed causal account of reference has to be explicitly built into reliability theory (or whatever other kind of broad psychological theory it is in which the notion of reference or of truth-conditions appears), and that such a detailed causal theory will have to explicitly mention acquisition of a public word that refers to something as one of the mechanisms by which people can become causally related to an object in the way that is relevant to having beliefs about the object. end p.66
freely to certain sorts of beliefs and desires, e.g., those beliefs and desires that a languageless organism could possess; for we have granted that such beliefs and desires can be accounted for in a language-independent way. In the second place, there is an important sense in which one could appeal to other beliefs and desires as well in our semantic theory for public language: we could appeal to beliefs and desired construed as attitudes towards internal sentences all we liked; the only restriction would be on which semantic features of those internal sentences we appealed to. I do not want to draw any very definite conclusions from this last discussion. I merely want to say that it may well be necessary to develop the semantic theory for internal languages and the semantic theory for public languages together rather than developing one kind of semantics independently of the other and reducing the other kind to it. The reductionist strategy (particularly the strategy that tried to reduce public semantics to internal semantics) is worth pursuing, but alternative strategies are worth pursuing as well. 45
45 I have greatly benefitted from discussion on these issues with Janet Levin, Brian Loar, and Stephen Schiffer. The
latter two especially, by their relentless criticism of the way I expressed my ideas in conversation, have forced me to write a very much better paper than I would otherwise have written. I am also grateful to Ned Block, Tyler Burge, Michael Devitt, Keith Donellan and David Hills for helpful comments on an earlier version. end p.67
Postscript
1. The Language of Thought Hypothesis I now take more seriously than when I wrote this paper the idea that explanations of the behavior of cats and dogs via their beliefs and desires might be a 'second grade' kind of explanation in Quine's sense: a 'dramatic idiom' not to be taken seriously. If so, there is no reason to take a 'language of thought' hypothesis seriously in the case of cats and dogs. We do of course need a serious psychological model of cats and dogs, but it is not implausible that some kind of connectionist model could be given, one that doesn't serve simply to implement a language of thought model or any other literally representational model but works in a wholly different way. The usual objection to connectionist models that aren't simply means of implementing representational models is that they can't account for certain systematicities in thought (Fodor and Pylyshyn 1988); but in the case of cats and dogs, it is by no means evident that the systematicities are there. Representational models also seem almost inevitable for creatures who show clear evidence of being able to evaluate their own reasoning processes; but again, this wouldn't seem to apply to cats and dogs. In the terminology used in the paper, what I'm now saying is that I was too quick to dismiss instrumentalism about belief- and desire-attributions for such creatures. In the case of adult humans, on the other hand, instrumentalism seems virtually inconceivable: I know perfectly well that I believe that the Earth is not flat, by introspection. My introspective confidence that this is so very likely arises from an awareness of accepting the sentence 'the Earth is not flat'. Doubtless we can introspect beliefs and desires with finer contents than we know how to express in our language, but the kinds of beliefs and desires that seem indubitably more than instrumental devices are those that don't go too far beyond the expressive power of our language. If this is right, it is all the more reason for supposing (as I tentatively did in sections 4 and 6) that an adult human's 'language of thought' is best viewed as just an extension of his or her public language. I didn't really mean to rule out that there is representation of a nonlinguistic kind, e.g. pictorial representation; indeed, such representation would suit the ends of the paper, for it lends itself to a naturalistic (and more or less compositional) account of how the internal representations represent the world. Linguistic representation seems prima facie a bit harder to explain naturalistically; that was the main reason for focusing on it. It would not have altered my overall view much to have held that for lower animals, mental representation might be exclusively pictorial. end p.68
(The possibility of representational systems that are neither linguistic nor pictorial is discussed in the next chapter: see the (somewhat jocular) 'light bulb model'. Such a purely Boolean model seems clearly inadequate to explain the behavior of adult humans, for reasons there given, though I suppose something like it could conceivably work as the basis for a non-instrumental theory of intentionality for some lower animals.)
2. Brentano's Problem (1) I first formulate Brentano's problem as the problem of making naturalistic sense of belief, desire, and so forth, given that they are relations between organisms and intentional entities like propositions. In correspondence about this chapter in the late 1970s David Lewis suggested comparing Brentano's problem to the following problem about numbers: 'Many seemingly physical properties appear to relate physical things to non-physical entities called numbers. What kind of physical relation can a seven-gram stone bear to a non-physical abstract entity: the number seven?' The same comparison has been urged by many others, e.g., Churchland 1979, Dennett 1982, Stalnaker 1984. I believe the comparison to have some, though rather limited, use. One solution in the number case is the nominalistic one: literally speaking there are no numbers, and so literally speaking there are no relations between physical objects and numbers (see Field 1980; Yablo, unpublished). This allows talk of numbers, and relations between stones and numbers, as useful fictions, but if they are so regarded then there is no pressure to solve the analog of Brentano's problem. Similarly, we might regard propositions as useful fictions (or maybe even not-very-useful fictions). In that case there is no pressure to solve Brentano's problem as I
stated it, i.e. in terms of propositions. But this wouldn't dissolve questions about how mental states can stand in referential relations to the world. This points up an important disanalogy between the two problems: there is more to Brentano's problem than the use of abstract entities. I will return to this. Lewis of course did not have the nominalist solution in mind, and there is no need to resort to it in dealing with the numerical analog of Brentano's problem: a solution to the problem has been found by mathematical work in the 'theory of measurement'. (See Krantz et al. 1971.) The idea is that what is explanatorily basic is not relations like x has massin-kg r between physical objects and numbers; rather, what is explanatorily basic is certain 'intrinsic relations' holding among physical objects themselves, e.g., the mass of x is the sum of the masses of y and z. This idea is made precise by laying down axioms governing these intrinsic relations that make no mention of numbers, and proving a 'representation theorem' that shows that whenever these axioms are satisfied there is a mapping of objects into the real numbers that 'preserves structure'. Consequently, assigning real numbers to physical objects can be viewed as just a convenient way of discussing the intrinsic mass-relations that end p.69
those objects have. The numerical analog of Brentano's problem is then solved: this physical object bears to the number seven the relation of being mapped into it by the unique structure-preserving mapping that accords with the convention laid down to determine the scale (i.e., the convention that determines the standard gram). It is initially plausible to suppose that Brentano's problem should be solved somewhat analogously. But two points need to be made. The less important is that taking the analogy seriously seems to involve a commitment to strong assumptions about the internal structure of mental states. The obvious analog to the postulate of a rich array of intrinsic relations among physical objects in the number case is to postulate a rich array of intrinsic relations among internal states inside the believer in the psychological case: the relations among these states must be 'intrinsic' in the sense that they are to be describable independent of propositions. The intrinsic relations and the assumptions about them must be powerful enough to prove a representation theorem that shows that whenever the assumptions are satisfied there is a mapping of internal states into propositions that preserves the kind of structure that is important to propositions. This includes at least Boolean structure, and very likely something more sentential. So the analogy seems to require us to ascribe psychological reality to at least the Boolean structure of propositions if not the sentential, and to think that this psychologically real structure can be described independent of propositions. Such an intrinsic structure, related to the system of propositions via a structure-preserving mapping, would seem to be enough to constitute a system of internal representation.1 (It is compatible with this argument that the intrinsic structure be only the sort of Boolean structure exemplified in the 'light bulb model' in the next chapter. However, for reasons given in that chapter, that kind of psychological structure seems far too impoverished to describe the psychology of creatures with a language.) But there is a much more important point (related to the one made in connection with the nominalistic view). It is that not just any mapping of internal representations to propositions that preserves Boolean or sentential structure will serve the explanatory purposes to which we put propositions: for 1 I put this tentatively, in part because of the vagueness of 'system of internal representation' and in part because one
might demand only a slightly weaker representation theorem. Dennett 1982 (note 2) observes in effect that there is an alternate version of the representation theorem, where numbers are assigned not to physical objects directly but to monadic mass properties, properties of having a particular mass; and that the analog of this in the intentional case invokes not objects or inner occurrences with a Boolean or sentential structure but properties with such a structure. Does this give a way to commit ourselves to less inner structure? If we follow the numerical analogy closely, by supposing that the properties in question are ones that a given object can have at most one of at a given time, then the answer is clearly 'no': we would need a system of inner objects or inner occurrences to instantiate the properties. But maybe what Dennett had in mind is that the properties are ones that apply to organisms as a whole, and which the organism can have a large number of simultaneously. In that case, we need only intrinsic structural relations of the sort appropriate to propositions among the simultaneous states of the organism as a whole; and this may not be what most people have in mind when they speak of a 'system of inner representation'. end p.70
it is easy to see that we could preserve Boolean or sentential structure while assigning the proposition that grass is green to the state associated with 'Snow is white'. For propositions, unlike for numbers, structure is not enough (unless 'structure' is taken in an extraordinarily stretched sense, i.e. as involving a kind of encoding of the physical world not present in the numerical case). This suggests that a significant complication of the strategy used in the numerical case will be required. The analogy of propositions to numbers turns out to be rather limited. If the only point of the analogy is to show that there is nothing mysterious about entities playing a role in explanations other than causal, it succeeds (though nothing in the chapter suggested that the acausal role of propositions was mysterious in itself). But if it is taken
as substantially illuminating the way that propositions play a role in explanations, it fails: the role of propositions is importantly different.
3. Brentano's Problem (2) Horwich 1998 suggests a more drastic way of dismissing Brentano's problem. Horwich (unlike the possibly mythical proponent of the 'orthographic accident' view) grants that terms like 'means' and 'believes' stand for genuine relations between people and propositions; but he claims that there is no reason to suppose that a physicalistic account of meaning or believing (an account of 'what constitutes' meaning something or believing something) would preserve this relational status. Indeed, he introduces the label 'the Constitution Fallacy' for the supposition that an account of what constitutes relational facts must itself be relational. So Horwich's idea is that one can give an account of what constitutes believing that by finding one monadic physical property that constitutes believing that snow is white, another that constitutes believing that Pope Pius X was the brother of Malcolm X, another that constitutes believing that Michel Foucault invented the pendulum, and so on; these distinct monadic properties need have nothing to do with each other, and they certainly don't need to involve a common physical relation. Horwich's claims are in service of a deflationist approach to the theory of meaning with which I am now highly sympathetic; my own version of such a theory is given in Chapters 4 and 5. But trying to achieve deflationism by the means just sketched seems to me extremely ill-advised: I see nothing fallacious about 'the Constitution Fallacy', and think that Horwich's requirements on a physicalistic account are far too weak. Consider ordinary physical relations, like 'has the same temperature as'. Surely we wouldn't count it as an acceptable answer to the question of what constitutes sameness of temperature to say that having the same temperature as b is constituted by one monadic property in the case of one object b 1 , a different monadic property for a different object b 2 (of different temperature), a third monadic property for b 3 , etc., where these monadic properties have nothing in common. If the belief relation between people and propositions doesn't require a physicalistic account that meets the end p.71
same standards as we would impose in the case of other relations, one must say why. It wouldn't help Horwich to point out that the standards of reduction (or constitution) appropriate to relations between purely physical items can't be supposed to apply when one of the items involved is abstract. For in the first place, the measurement-theoretic solution of the numerical analog of Brentano's problem provides a genuine account of relations between objects and numbers; obviously it differs in important ways from a typical reduction of a relation between purely physical items, but nonetheless it does preserve the relational character. (As we've seen, this solution does not carry over to intentional relations. But unless we can give grounds for adopting lower standards on a physicalistic account in the intentional case than the numerical, that doesn't show that there is no need of a genuinely relational account in the intentional case, it only shows that the job is harder there.) In the second place, some of the mental relations for which Horwich wants to avoid giving a relational account hold between physical entities: e.g. 'x has a belief about person p'. To repeat, I am inclined to think that Horwich is right that the present chapter treats such mental relations as 'too much like' ordinary physical relations, but we need an alternative model of what they are like (one that I try to supply in Chapters 4 and 5). One shouldn't simply label as fallacious the problem of trying to provide such a model.
4. Nonintentional Explanations The two main topics of the paper are (i) What explanatory purpose (if any) is served by ascribing truth conditions to our inner representations? (ii) How (if at all) can the relation r has the truth conditions that p (where r ranges over inner representations) be naturalistically explained? On (ii) (which is essentially Brentano's problem), I argue that by taking inner representations as having sentential structure, we can adapt the theory of truth in Chapter 1 to inner representations; also, I make explicit that such an approach naturalistically explains not just truth but truth-conditions (that is, the truth condition relation mentioned in (ii)). As succeeding chapters will make clear, I have become quite doubtful of the need for such an approach. As for (i), a good bit of my discussion was directed toward showing that it is not entirely easy to find an explanatory role for truth conditions: for once one has the system of internal representations, one can do much of one's explaining simply in terms of it, with no reference to the truth conditions that are assigned to the representations. I made a distinction between what I would now call intentional explanations, that make appeal to the truth conditions of states (or
to related semantic features of the components of states, such as their referents), and non-intentional explanations that do not involve any such thing end p.72
but simply rely on the causal laws governing those states. (By 'laws' I don't mean anything very heavy duty: just macroscopic regularities that hold, within limits, for the system.) Unfortunately, I used the term 'narrow' for 'nonintentional' and 'broad' for 'intentional': this was unfortunate because shortly afterwards, Fodor 1980 popularized the use of similar terms for a very different distinction from mine, and many people failed to realize that the two distinctions were not at all the same. Fodor was concerned with the distinction between 'wide explanations' that make appeal to things outside the agent and 'narrow explanations' that do not; mine (to repeat) was between intentional explanations that make appeal to the truth conditions of states (and to the referents of components of states) and non-intentional explanations that do not. The narrow-wide distinction and the intentionalnonintentional distinctions cut across each other. An explanation that appeals to external causes of our internal representations will be wide, even if the causes it appeals to have nothing to do with the truth-conditions of the representations. Conversely, an explanation in which appeal is made to an internal symbol obeying the truth-table for disjunction is an intentional explanation even if the assignment of this truth-table to the symbol doesn't depend on factors external to the agent. In my view, the intentional v. non-intentional distinction is far more important than the distinction of wide v. narrow. I don't think the paper makes clear enough how broad the scope of non-intentional explanations is. In the first place, I should have made clear that we can theorize about an agent at many different levels of detail and idealization, and that appeal to internal representations (described without mention of what they represent) can appear in theories at many different levels. At a highly idealized level, there is Bayesian decision theory construed as a psychological theory, using degrees of belief* and desire* (i.e., 'attitudes toward' sentences instead of toward propositions); or rather, Bayesian decision theory supplemented with laws governing how sensory stimulations affect observation sentences and how decisions lead to motor outputs. We can introduce more and more complications, to get more realistic. If we give straightforward causal laws of how beliefs and desires and other representational states evolve over time (influenced by sensory stimulations and the like), their representational properties (truth conditions, reference etc.) will play no role in the laws; similarly, such properties will play no role in any laws of how such internal states lead to motor outputs. I'm not restricting my claim here to deterministic theories: it seems to me that except in the crudest of idealizations the psychological laws will mostly be indeterministic, but it is still hard to see how representational properties like reference or truth conditions could play a role in the laws. I have described the psychology 'narrowly', i.e., as taking as inputs sensory stimulations rather than external conditions, and analogously at the output end. But nothing hangs on this: using 'wide' inputs and outputs, i.e., inputs and outputs described in external terms, is not enough to make the laws end p.73
intentional. This should be obvious: no talk of reference or truth conditions is involved. Indeed, an input law that says that coming to believe 'There is a rabbit in front of me' is typically caused by the presence of rabbits can be 'factored' into two components, one saying that coming to believe the sentence is typically caused by sensory stimulations in a certain class and the other saying that sensory stimulations in that class are typically caused by rabbits. ('Factor' may not be quite the right word: it suggests that the two resulting laws taken together are fully equivalent to the original law, but actually they introduce a slight improvement of it.) A 'wide' input law is in effect just an existential quantification over conjunctions of 'narrow' input laws and laws about what causes sensory stimulations. (Existential quantification is needed simply because we may not know exactly which class of sensory stimulations it is that is caused by the presence of rabbits and causes the belief in 'There is a rabbit in front of me'.) 2 The kind of non-intentional theories of an agent that I've been considering might be called computational theories, in a broad sense that allows features of the external world to appear in the computational theory if they regularly cause sensory stimulations or are regularly affected by motor outputs. These are the kind of theories of an agent that are used in functionalist models. One of the things I was arguing in the paper was that it is hard to see how we could ever give a functionalist account of belief and desire in the usual sense, i.e., belief and desire as relations to propositions: discussions of how to do this typically treat each belief and desire as a separate state, without getting the interconnections among (the potential infinity of) such states right. What we can hope to give a functionalist account of is of the relations belief* and desire*, which are relations to inner representations. The account of the relation of the inner representations to the propositions (or of the 'singular term components' of inner representations to objects and the 'general term components' of inner representation to sets or properties) has to be given separately from the functional story. This is part of what is argued in Section II of the Chapter. (There is a broad sense in which the relation between inner representations and propositions might be functional: it might be a second order relation, the relation of standing
in some first order relation or other that meets condition Ψ. Indeed, in the paper I use the term 'functional' in this broad sense, a terminological choice I now regret. But my point now is that the condition Ψ won't be the kind of 'broadly computational' conditional that we are familiar with from standard functional theories. The question of what other kind of condition it might be is one way to put the question of finding an explanatory role for propositional content or truthconditions.) A small point: in explaining how there can be a functional theory of belief* and desire*, I offer a functionalist account of what it is to be an inner representation, and of the syntactic features of inner representations. This is still my preferred way of thinking of inner representations, but there is an alternative: 2 In this paragraph I've stated the sample input law very vaguely and crudely, but I believe that the basic point could
be made with more precise and realistic laws. end p.74
we could think of the inner representations as abstract syntactic objects, and use them more or less as we might use numbers in formulating the computational theory. For abstract syntactic objects, the analogy to numbers suggested by Lewis, Churchland and others (mentioned in Section 2 of this postscript) would be quite a good one: by removing the representational aspect of the abstract objects, we have removed what makes the analogy problematic.3 I have said that computational theories (even in the broad sense that allows reference to external causes and effects of internal representations) don't employ representational properties like reference and truth conditions. I haven't said that they don't employ semantic properties, for in a broad sense of 'semantic' there may be semantic properties for which something having the property is constituted by its having a certain computational role. For instance, the paper argues that the right moral to draw from Frege's Hesperus-Phosphorus problem is that yes, these coreferential names differ in some semantic property loosely construed, but (contrary to Frege) the difference is only that the names have different computational roles for most of us: one is more directly connected to beliefs involving the term 'morning' and the other to beliefs involving the term 'evening'. If this computational difference is counted as semantic, then obviously semantic features of representations are not left out of the computational story. But it is the other aspects of semantics, going beyond computational role even in the broad sense, that are left out. And to reiterate, the kind of computational role that the functionalist story allows for includes what has been called 'long-armed conceptual role' (Harman 1982), which includes causal connections to the environment. It may be part of the conceptual role of 'That is a rabbit' to be typically caused by nearby rabbits. Equally, it may have been part of the conceptual role of 'Phlogiston is disappearing from that flask' to have been typically caused by oxygen entering the flask. This latter example makes clear, I hope, that to bring in long-armed conceptual roles is not to bring in representational properties. (Admittedly, there is a heavy overlap between the causal connections to the environment that would be used in wide input laws and those that an advocate of a causal theory of representation would want to build into the truth-condition relation for observation sentences. The explanation is obvious: most of those who believe in a substantive theory of truth-conditions 3 Whichever route one prefers, a point made in the chapter (footnote 32) bears repeating: there is no evident need in
the functional theory for the idea of two 'internal word-tokens' or 'internal sentence-tokens' employed by different thinkers being instances of the same word or the same sentence; an intrapersonal notion of type-identity is sufficient. I believe that Stich 1983 gets his 'syntactic theory of the mind' into needless trouble by failing to realize this: for instance, in his chapter 4. (If anyone thinks that the use of abstract syntactic objects requires interpersonal identifications, see the discussion of local abstract objects in Chapter 5, near the end of Section 6.) Of course, disallowing talk of interpersonal sameness of 'internal words' doesn't rule out semantic comparisons: one might say that a word in you and a word in me both refer to Aristotle without regarding as meaningful the question of whether they are the same word. Also, when speakers (by behavioral criteria) share a public language in which a certain word is used with no intrapersonal ambiguity, this sets up a natural way to correlate those of their internal symbols that are causally associated with the word. end p.75
want it to be a consequence of their theory that observational beliefs tend to be true, and to guarantee this they suppose that by and large the factors involved in the causal input law are involved in constituting truth conditions. But this obviously does nothing to show that causal input laws are themselves intentional; and cases where it is natural to say that our observation reports are laden with a false theory makes this especially evident.) The fact that representational properties are left out of the causal story should be even clearer in the case of other kinds of words. Consider proper names of long-dead people like Aristotle: the factors usually thought relevant to reference include causal chains extending through the past back to Aristotle, but it is a stretch to say that these are involved in the conceptual or functional role of the word 'Aristotle'. (Causal connections to certain portraits might be part of the functional role of 'Aristotle', but this is so even if the portraits turn out to have someone else as their causal
source.) And consider, once again, logical connectives like 'not'. A representational semantics will take this as standing for the truth function that maps truth to falsehood and vice versa. Or if you don't like talk of 'standing for truth functions', the point can be put differently: the representational semantics will say that when attached to a true sentence, 'not' yields a false sentence, and when attached to a false yields a true. But this isn't a fact about the conceptual or functional role of 'not'. The conceptual role of 'not' is relatively easy to specify: it is given in large part by the rules of deductive inference that we take to govern the term. But specifying the conceptual role in this way doesn't seem to say anything about truth functions. Perhaps the last few paragraphs need a qualification. When I said that computational theories (even in the broad sense) don't employ representational properties like reference and truth conditions, maybe all I was entitled to say is that they don't employ representational concepts? Maybe conceptual role concepts and representational concepts are distinct concepts for the same property? I don't think this is a plausible view even in cases like the logical connective case where it is plausible that there is some kind of supervenience of the representational property on the conceptual role property. (For one thing, the representational property seems to have a relational element that isn't part of the conceptual role property.) And in other cases, like the case of names of long-dead people, there doesn't even seem to be such supervenience; in which case the properties certainly can't be identical. But in any case, it wouldn't much matter to my point if we did qualify the previous paragraphs in the way suggested. For the important point is that a great deal of psychological theorizing can be done without representational concepts. And the question is, what additional advantages do we get by employing representational concepts? end p.76
5. The Explanatory Role of Intentional Properties (1) How then do representational concepts enter into psychological explanations? I will be especially concerned with explanations of phenomena described without representational concepts. Obviously we frequently use representational concepts (e.g. beliefs and desires) in explaining other beliefs and desires, or in explaining behavior described in an intentional way ('murdered his wife') that presupposes representational concepts; but it seems to me that an illuminating account of the explanatory role of representational concepts should focus first and foremost on their use in explaining facts not so described. ('Why did her arm move in that way?'; 'Why did she end up in San Francisco?') Once we have motivated the use of representational concepts, it is no surprise that we want to explain facts described in terms of them in addition to facts described without them; but an illuminating account of the explanatory role of the practice requires that we show how it would be explanatorily useful for someone who didn't already have the practice to introduce it. A natural view is that when we use representational concepts or properties to explain facts described in nonrepresentational terms, the representational concepts or properties just code for conceptual or functional role properties: we specify the functional role property by specifying the representational property. This is certainly very plausible, if we don't take it as committed to there being any very uniform account of what conceptual role properties a representational property codes for, and if we don't suppose that even on a given occasion there is a very precise conceptual role property that is coded for. But (as it stands anyway) it leaves unanswered the question of why it is useful to code conceptual role properties by representational properties (or by representational concepts). Additionally, even if it is granted that one function of representational properties or concepts is to code for conceptual role properties, the question arises whether there might be other functions in addition. One obvious fact that needs emphasis is that typically when one offers an explanation in terms of beliefs and desires, one is not in a position to offer a complete explanation in computational terms. In itself this doesn't show much, for the same holds in explaining anything: if we explain the center's goal by citing the wing's perfectly placed pass, we certainly aren't in a position to cite precise laws of physics (or regularities of hockey, or of the system consisting of this particular center and this particular goalie) which would determine probabilities for the goal under the conditions present with and without the perfectly placed pass. What we call 'explanations' are only explanation sketches (indeed, only sketches of partial explanations): sometimes very incomplete sketches that consist of nothing more than mentioning a salient part of the overall cause. No reason so far for supposing the need of any theory in psychology beyond the computational. Still, this point doesn't seem to be enough to account for the fact that in the intentional case, the ideology that we use in our explanation sketches is so far end p.77
removed from the ideology of the computational theories that provide the only kinds of laws we have so far found hope of finding. If there aren't even very rough laws at the intentional level (and from which the intentionality isn't easily 4
eliminable), why is it that the policy of explaining in intentional terms is so useful? One answer that I briefly consider (and deem insufficient) in section 5 is that intentional explanations of another person's behavior are 'projective', in that they involve reference to the explainer's language even though that is not causally relevant to the behavior. When explaining a person's behavior (say the raising of his gun) in terms of his belief that there is a rabbit nearby, what I am in effect doing is explaining the behavior in terms of his believing* a representation that plays a role in his psychology rather similar to the role that 'There are rabbits nearby' (or the mental representation associated with it) plays in mine. (This has some similarity to the 'second grade' explanations mentioned in section 1 of this postscript, but unlike those, this takes the computational belief-desire psychology seriously.) Such an explanation is still basically non-intentional: truth conditions play no real explanatory role. Of course, there is a sense in which my sentence 'There are rabbits' plays an explanatory role here: obviously not as a causal factor in the explanation, but as a device we use in picking out the agent's internal representation (which is a causal factor). Moreover, my sentence, to which his sentence is compared, has truth conditions, and I know what these truth conditions are. But it is my sentence, not its truth conditions, that is playing the direct role in picking out the agent's internal representation. Moreover, even my sentence is playing a very indirect role in explaining the agent's behavior: the agent's internal representation is what is centrally involved in that. So truth conditions are playing a doubly indirect role in the explanation. (And note that it is the assignment of truth conditions to my sentence that is playing this doubly indirect role. If, as I assumed in the paper, the assignment of truth conditions to 4 Stephen Schiffer (in conversation) has raised a second question about the explanation sketch view: if all we are
doing in ordinary explanation is mentioning a cause of the behavior, why insist on causes of the form 'He believes (or desires etc.) some sentence that roughly translates as "p" '? Why not other causes or cause-descriptions, e.g., 'such and such a neuron was firing'? Maybe you could rule out that particular alternative cause-description, perhaps a bit ad hocly, by demanding that the causes described be ones that serve as the agent's reasons as well. But even so, that wouldn't seem to handle 'He is in some belief state that he'd express by S' (S a sentence of his language that we don't understand); that may give his reason (as he would express it), but we wouldn't find it terribly helpful to us. I think that the answer to Schiffer's question involves a connection between explanation and prediction. One of the main reasons that explaining the behavior of individuals is important to us is that it aids us in predicting their behavior on other occasions. (Being a good explainer of one person also improves our ability to predict other people.) And prediction goes better if we can relate the state of the agent in question to one of our sentences than to one of his (or to the firing of one of his neurons). The reason for that is a combination of two factors. Probably the more important is that there is something right in the simulationist account of prediction (Gordon 1986), according to which we typically predict the state of others by imagining that we accept and desire what they do and making an 'off-line decision' ourselves: we need to have beliefs and desires described in our language in order to do this. The other is the role of 'reliability assumptions' in explanation: this component (first clearly brought to my attention by Schiffer 1981) is discussed below. end p.78
the agent's representations is assumed autonomous, i.e., not constituted by their translatability into a sentence of mine with those truth conditions, then the assignment of truth conditions to the agent's states doesn't even play an indirect role in the explanation.) We still have no example where truth conditions play a serious explanatory role. And at this point we might well wonder whether there are such examples. If not, the motivation for a serious theory of the truth conditions of internal representations may be called into question: the assignment of truth conditions to the representations seems explanatorily idle.
6. The Explanatory Role of Intentional Properties (2) At this point in section 5, I introduced an idea from Chapter 1: that we assign truth conditions to the states of others as a means to use their states to find out about the world. Given that they accept 'It's raining here' or a sentence that would normally be translated that way, we have reason to think that it is raining there. (I should have added the converse: given that it is raining there, and that they are outdoors etc., we have reason to think that they accept 'It is raining here' or a corresponding sentence in their language.) I said that in making such inferences we implicitly employ a 'reliability theory' of agents, and that this should be viewed as an extension of our non-intentional psychology of the agents.5 Stephen Stich (1983: 200-4) argues against the idea of such a reliability theory. He first notes that we would need different reliability theories for different people, since we use idiosyncracies of different people in predicting and explaining their behavior. (I have to admit that the end of section 5 suggested the contrary; this was a silly mistake.) He then says it is implausible that we could carry around so many different reliability theories in our head. But of course
there is an obvious story to tell here, namely that we carry around a lot of general assumptions about what things people tend to be reliable about, and that for people we know we carry around information about how the general assumptions are to be expanded and/or modified in their case. (Not just for individual people actually: we also have general assumptions about certain categories of people, like doctors and preachers and politicians.) Of course, the general views about people, as well as the assumptions about specific individuals and specific categories, can evolve over time. Looked at this way, the idea that we have reliability theories that vary from person to person doesn't seem as preposterous as Stich tries to make it sound. And since our computational theories would presumably vary from person to person also—even very idealized computational theories such as Bayesian theories (where various 'caution 5 I think what I had in mind was not simply a theory about when the agents are reliable but also how their reliability
contributes to their success; but the inclusion of the latter was certainly not explicit. end p.79
parameters' and other aspects of inductive procedures would surely vary from one person to the next)—it's hard to see why Stich thinks that there is a special problem about reliability theories. Stich's objection isn't simply to the question of whether our reliability assumptions are sufficiently systematic to deserve to be called a 'theory'; he disagrees with the very idea that we use the notion of truth in learning from others. But as he admits, he has no serious alternative account of how we learn from others on offer, and I can't imagine how there could be an alternative story that doesn't involve a notion of truth. But Stich's larger point was to defend his 'syntactic theory of the mind', and it now seems to me that the defense of this does not require his radical line that truth plays no role in learning from others. For it is hard to see why, in an account of how we learn from others, anything more than a purely disquotational truth predicate is needed. (This will be discussed in section 11 of Chapter 4.) Finding an explanatory role for the assignment of truth conditions to mental states might give reason for believing in the sort of heavy-duty account of truth conditions that the paper suggests, but saying that we need to mention truth or truth conditions in an account of how we learn from others isn't to give truth conditions an explanatory role. But at this point there is a new wrinkle to consider: Stephen Schiffer (1981) argues that the kind of reliability assumptions that we use in learning from others also play a role in explanations. If we were in a position to give a full computational explanation of an agent's behavior, we wouldn't need to use truth conditions, but of course we are never really in a position to do that; and (he claims) in explanations given in ignorance of such details, truth conditions serve a more central role than they have according to the projectivist story of the previous section. To see what Schiffer had in mind, consider an example (a variation on his). Smith is in a fairly large nearby room, into which I can't see; Oscar runs by with gun in hand, yelling 'I'll kill that bastard Smith', seeming deadly serious, and he goes into the room that Smith is in; I hear a shot. Now, I don't know where in the room Smith is: I just have a broad probability distribution. I also have no idea where in the room Oscar is aiming his gun: I just have a broad probability distribution for that too. If these two probability distributions were independent, I wouldn't attribute a very high probability to Smith's having been shot. But of course they are not independent: I think that Oscar is likely to have pointed the gun in Smith's direction. And it seems that the justification for the correlation in the probabilities is that I think that Oscar is likely to have had the correct belief about where in the room Smith is. Correctness—truth—is playing an explanatory role. Of course there are other cases where we will introduce correlations between belief states and external states of affairs not based on truth. (I read Patrick Buchanan's latest rantings, and know that the very same sentences will appear in my uncle's belief box. I see a 6′4″ person in front of my mother, and know she will believe the sentence 'He is at least 6′7″. I see the lightning in front of end p.80
a Zeus-worshipper, and know he will believe 'Zeus is throwing thunderbolts'.) Still, we might have a hard time describing the correlations in the Oscar case without using a notion of truth. Actually this is not entirely clear: after all, we have the causal input laws. Doesn't it follow from the causal input laws that (with high probability) Oscar will believe a sentence of the form 'Smith is at angle α from me' if Smith is at angle α from him? If so, there is no need of the notion of truth here. This is of course a reliability law of sorts, but that is because all causal input laws are in effect reliability laws of sorts; the point is that this sort of reliability law doesn't go beyond the broad computational theory. Still, once one appreciates the idea of using reliability assumptions in explanation, there is no obvious reason to confine them to the causal input laws. By our earlier discussion, we have rather wide-ranging beliefs about the reliability of different people, and different sorts of people, on particular issues: we expect, for instance, that the adults around us will have true beliefs about who is President and what city they are in and whether they are homeless and what they need to do if they want a meal in a restaurant. (That is: we expect that they will believe* a sentence that is appropriate
to translate as a true sentence of the form 'x is President'; etc.) There is no reason that these reliability assumptions can't be used in explanations, and these do go beyond computational psychology even in the broad sense explained earlier. This is a serious use of truth in explanations, going beyond computational psychology. (I will have more to say about this—in particular, about the relation between using truth in this way and 'projective explanations'—near the end of the postscript to Chapter 4.) But this use of the notion of truth in explanations is not enough to motivate the assumptions about the need for a substantive theory of truth that I made in this chapter. (In this I agree with Schiffer: see the final section of his paper.) As I see it, the crucial point is that in the 'reliability laws' and 'reliability assumptions', truth is serving simply as a device of generalization. For each given angle α 0 , we can explain, from Smith's being at angle α 0 from Oscar and Oscar's believing 'Smith is at angle α 0 from me', how the shooting took place. The role of truth is simply to generalize on these explanations. (Similarly for explanations involving reliability assumptions that go beyond the causal input laws.) And as noted in the postscript to the previous chapter, the use of truth as a device of generalization is not enough to motivate the kind of heavy-duty truth theory that this chapter and the previous one assume. Does what I've just said depend essentially on supposing that Oscar speaks my language? At one point I seem to have thought so, but now I don't see that it does. If Oscar speaks a different language, the situation is that for each given angle α 0 , we know how to explain, from Smith's being at angle α 0 from Oscar and Oscar's believing a sentence that we translate as 'Smith is at angle α 0 from me', how the shooting took place. Truth under this translation scheme can then serve as a device of generalization in this case. We don't even need to assume end p.81
that the translation scheme has any privileged role in relating his language to ours: we might employ different translation schemes in different contexts, for different kinds of generalization that proved useful to us. If there is an argument for a heavy-duty notion of truth or truth conditions, we have yet to find it. end p.82
3 Stalnaker on Intentionality Abstract: Argues that there are two reasons for ascribing to mental states, structures more fine-grained than the sets of possible world they represent: first, fine-grained structure enters naturally into the explanation of behaviour; second, fine-grained structure is needed in a theory of how those states represent the sets of possible worlds they represent. In connection with the first point, it is argued that Stalnaker's attempt to use metalinguistic content to obviate the need of fine-grained structure cannot work. In connection with the second point, it is argued that the systematicity in the assignment of content to mental states can be accommodated only by postulating a fine-grained structure for mental states. Also contains some discussion of doing without representational content, understood in the non-deflationary sense that Stalnaker assumes. Keywords: belief, conceptual role, functionalism, language of thought, logical connectives, narrow psychology, representation, semantics, Stalnaker, truth-conditions
Hartry Field Stalnaker's Inquiry (1984) offers us a theory of the deliberation and inquiry of intelligent agents that makes heavy use of believing and desiring, construed in a particular way. Let a Stalnaker-proposition, or S-proposition, be a function from some algebra of possible worlds (not necessarily comprising all the possible worlds) into truth values. (If we ignore the qualification that not all possible worlds need be assigned truth values, we could just as well say that an Sproposition is simply a set of possible worlds.) Stalnaker wants to construe believing and desiring as relations between the cognitive states of the agent and S-propositions. Stalnaker thinks that the S-proposition assigned to an intentional mental state should be viewed as the content of that state, or the object of that state. Stalnaker sees two sorts of advantages to his view of the objects of intentional states over rivals (in particular, rivals that ascribe more fine-grained contents): first, that it better fits a pragmatic picture of the purpose of postulating intentional states (in large part because of the identity conditions on contents that it offers); and second, that only by conceiving of the objects of mental states in this way can we solve the problem of intentionality, that is, the problem of giving a naturalistic account of what gives mental states the content they have. I will be disagreeing with his views on both points. I should say at the outset that the whole idea of 'the object of' or 'the content of' an intentional mental state needs to be treated with care; Stalnaker's book (like virtually all of the literature on this subject, including some of my own previous work) takes this notion too much for granted. I have no doubt that there are useful rough and ready senses in
which a mental state can have content and in which different mental states can have the same content. But our relatively ordinary assertions of contentfulness and of sameness of content seem highly context-dependent— especially assertions of sameness of content between the mental states of different agents. (Even more especially, when those agents do not share a language; still more, when they don't even belong to the same species.) Any view according to which we are to assign entities to mental states that are to serve as their contents (and are then to define having-ofcontent and sameness-of-content in terms of such assigned entities) is clearly ladening our ordinary talk of having-ofcontent and sameness-of-content with a substantial body of theory; and my first point is that the nature of the theory and the motivation for introducing it deserve serious discussion. end p.83
My second point, to be elaborated in sections 2 and 3, is that Stalnaker not only wants to assign entities to mental states to serve as their contents, he wants to assign intrinsically representational entities, i.e., entities for which it is built in that they represent the real universe as being a certain way; the nature of the theoretical role that such an assignment is to play, and the motivation for introducing such a body of theory, deserve discussion still more. And my third point is that even given that we want to assign intrinsically representational entities to mental states to serve as contents, it is by no means obvious that there is anything like a unique sort of intrinsically representational entity to assign. 1 Stalnaker seems to take it for granted that there is a clear issue between a position according to which we assign S-propositions to mental states as their contents and a position according to which we assign finer-grained entities as the contents of mental states; but I do not think that this is a clear issue at all. So, though I will be disagreeing with much that Stalnaker says when he tries to motivate using S-propositions as 'the' contents of mental states, still I wouldn't want to put my disagreement by saying that he has chosen 'the wrong entities' as contents. In fact, I agree that S-propositions are in a sense the core of content (we can call it S-content), and the aspect of content for which the problem of intentionality arises most forcefully. One of my points will be that we need to postulate other 'content-like' features of mental states not only in explaining behavior but even in solving the problem of giving a naturalistic account of S-content; again, whether we call these features part of content is merely a matter of terminology.
1. The Pragmatic Picture and the Linguistic Picture Stalnaker says that there are two very different pictures that can govern people's conceptions of intentional mental states— he calls them the pragmatic picture and the linguistic picture. He advocates the pragmatic picture, and holds that it leads almost inevitably to 1. the view of objects of belief as 'coarse-grained', in the sense that there can not be distinct but logically equivalent objects of belief; and 2. the view that there can be no philosophical point to adopting a language of thought hypothesis. Apparently he holds, then, that advocates of a language of thought, or of finer-grained content, deny the pragmatic picture of intentional states. It seems to me that his discussion of these matters is quite misleading. The pragmatic picture of belief and desire is described in part as follows: Representational mental states should be understood primarily in terms of the role that they play in the characterization of action. . . . One explains why an agent tends to act 1 Also, there is no obvious reason to assume that there is anything like a uniquely best assignment of one entity of a
given kind to a given mental state— cf. Quinean indeterminacy. end p.84
in the way he does in terms of . . . beliefs and attitudes. . . . [O]ur conceptions of belief and of attitude pro and con are conceptions of states which explain why a rational agent does what he does. . . . Linguistic action, according to this picture, has no special status. Speech is just one kind of action which is to be explained and evaluated according to the same pattern. (p. 4) The pragmatic picture described here is most compelling. Stalnaker says that philosophers who have advocated thinking of belief and desire in terms of a 'language of thought' or an 'internal system of representation' have had a different picture, but that does not seem to me to be true: except for the restriction in the above quotation to rational action, the quotation seems to precisely describe the attitude toward belief and desire taken by Fodor (1975) and Harman (1973), and that I took in Chapter 2. Indeed, although Stalnaker begins his characterization of the supposedly
contrasting 'linguistic picture' by saying that on this picture 'rational creatures are essentially speakers' (p. 6), he ends up saying that any view that conceives the intentionality of intentional mental states on analogy with the intentionality of linguistic expressions counts as a version of the linguistic picture (p. 7). But then the pictures do not contrast: the whole point of much of the argumentation in the three works cited was that there was reason to postulate structure resembling linguistic structure in intentional mental states in order to explain non-linguistic as well as linguistic behavior. I'm not here saying that this argumentation is right— in fact, I now think that my claims on this point in Chapter 2 are rather overblown and are stronger than I needed for the points I was primarily concerned to make in that paper— but I think that Stalnaker's long discussion of language of thought views is seriously warped by his suggestion that the point of a language of thought theory is to do something other than explain behavior. Turning now to the issue of 'coarse-grainedness' of content: an almost universal first reaction to a possible worlds view of belief and desire like Stalnaker's is that it can't be adequate, since people can clearly believe that P and disbelieve that Q when P and Q are logically equivalent. Now, Stalnaker has an ingenious reply to this line of objection, which I will consider at the end of the paper, but for now I just want to emphasize that this line of objection against Stalnaker is an initially natural one from the point of view of the pragmatic picture. That is, even when one's task is to explain nonlinguistic behavior, there is often a prima facie need to attribute a belief in one proposition and a disbelief in an equivalent proposition (i.e., there is a prima facie need for a conception of content more fine-grained than S-content). For instance, if I offer someone who doesn't know much mathematics $1000 for an example of a plane map that requires more than four colors to color (according to the usual coloring conventions), he will behave very differently than he would if I had offered $1000 for a trisection of a Euclidean 60 degree angle by straight edge and compass; to explain this, I need to attribute different beliefs and desires to him in the different cases, and it is prima facie difficult to see how I can do this in a relevant way if the desire to do one end p.85
impossible task is identified with the desire to do any other impossible task. Stalnaker thinks (pp. ix, 23-5) that the pragmatic picture leads inevitably or almost inevitably to the coarse-grained conception of the objects of intentional states. Why, given the obviousness of examples like the one above? The argument appears to be that on a pragmatic picture we ought to think of the deliberation that is involved in determining behavior as consisting of the weighing of possibilities (i.e., weighing which possible states of the world are more likely, what the effects of various courses of actions would be in these different states of the world, and how desirable each of these possible effects might be)— see pp. 4-5 and 85. If deliberation did indeed have to literally consist in the weighing of logical possibilities, Stalnaker's conclusion might well follow. But why not say instead that deliberation consists in the weighing of states of affairs which are epistemically possible for the agent— or more accurately, the weighing of state of affair representations which are treated by the agent as consistent? 2 This would certainly handle the case of the $1000 offer. Stalnaker considers such a model, but argues against it as follows: Could we escape the problem of equivalence by individuating [mental states], not by genuine possibilities, but by epistemic possibilities— what the agent takes to be possible? This would avoid imposing implausible identity conditions on [mental states], but unfortunately, it would also introduce intentional notions into the explanation, compromising the strategy for solving the problem of intentionality. If belief and desire are to be explicated in terms of naturalistic relations such as indication and tendency-to-bring-about [i.e. if they are to be explicated directly in information theoretic terms], then the propositions used to individuate [belief- and desire-states] must be the ones that are relevant to these relations, and these clearly must be genuine, and not merely epistemic possibilities. (pp. 24-5) In short, epistemic possibility is an intentional notion that can't be cashed out in the way that he wants intentional relations to be cashed out (namely in terms of 'indication relations' and the like, to be discussed shortly). But as I've said, talk of what is epistemically possible for the agent is really just a loose way of talking about which representations the agent treats as consistent, and one can presumably explain treating a representation as consistent in terms of the employment of that representation in thought (e.g., one treats it as consistent iff one doesn't treat it as implying everything else); no intentional notion of representing a possibility enters the picture at all. Indeed, as I will argue at the end of section 3, there is serious doubt as to whether Stalnaker's strategy for explaining representation in terms of natural relations could work without appeal to antecedent inferential relations among states that would generate a notion of epistemic possibility; so if talk of epistemic possibility is for some reason disallowed, it is the talk of real possibility that is in danger of using intentional notions that can't be cashed out naturalistically. 2 Stalnaker himself agrees that beliefs and desires must have representations (p. 22). He wants to avoid taking a
stand on the nature of those representations (e.g. on how similar they are to linguistic representation), but the suggestion outlined in the text does not require a stand on that.
end p.86
2. The Problem of Intentionality (1) Much of the first two chapters of Inquiry is a discussion of the problem of intentionality— the problem of giving a naturalistic explanation of intentional relations like believing and desiring that apparently relate people to the objects of belief. Stalnaker does not attempt a detailed solution to this problem, but he does make some sketchy remarks about how one might develop a solution that was based on the pragmatic picture. And he argues for the superiority of this sort of solution over two solution-sketches which he regards as anti-pragmatic, namely a solution-sketch that I suggested in Chapters 1 and 2, and a solution-sketch that might be suggested by Davidson's writings. As will emerge, I think that Stalnaker misrepresents the differences between his own approach and the approach I urged in the papers he cites. However, my concern here will not be to give a detailed defense of my views against all his criticisms— these are issues on which I have significantly changed my views anyway. Rather, I will just try to clarify the issues a bit. First I need to develop the distinction, mentioned in my opening remarks, between using 'intrinsically representational' entities as objects of belief and using entities that are not 'intrinsically representational'. Suppose that we think of intentional mental states as follows: most (perhaps all) of them are significant (have-content), and among those that do there is a relation of synonymy or sameness-of-content (applicable between states of different agents as well as between states of the same agent) which is an equivalence relation. Then in one reasonable sense of 'objects of belief', we could use the equivalence classes of states, under the sameness-of-content relation, as objects of belief. I count this as an example of 'objects of belief' that are not 'intrinsically representational': they are simply sets of states, and don't have built into them that they represent the world in any way. Alternatively, if one assumes that the sameness-of-content relation makes clear sense only among states of the same agent, but that among the states of any one agentit is an equivalence relation, then one could instead use equivalence classes of the states of a single agent as the objects of belief for that agent; objects of belief would then be 'local objects' in that no two agents could literally share the same object of belief. This too would be an example of a conception of objects of belief as not intrinsically representational. (One version of this is equivalent to the view I took of the objects of non-intentional psychology in Chapter 2.) By contrast, if we view the object of a state of believing that Caesar crossed the Rubicon as an ordered triple whose components are Caesar, the Rubicon, and the relation of crossing, that is an example of an intrinsically representational object of belief: it is intrinsically representational because its whole point is to represent the world as being such that Caesar crossed the Rubicon. Similarly, if we take as the object of the belief-state the set of possible worlds in which Caesar crossed the Rubicon (or, what comes to the same thing, the S-proposition that assigns 'true' to worlds end p.87
where Caesar crossed the Rubicon and 'false' to the others), that too is a conception of objects of belief as intrinsically representational. It seems to me that the most central philosophical question about intentionality is whether there are any good reasons to utilize intrinsically representational objects of belief (and if so, what those reasons are). It does not seem to me that one ought to assume without argument that we should utilize intrinsically representational objects. Two points need to be made in clarification of this claim. (1) To refuse to utilize intrinsically representational objects of belief is not to divorce belief-states and desire-states from the external world: for instance, a theorist of belief who shunned intrinsically representational objects of belief could still theorize about the causal influence of the external world on our belief-states and desire-states, and about the causal influence of our belief-states and desire-states on the external world. (Of course, such a theorist wouldn't have the intrinsically representing objects of states available to aid him in formulating the laws about how belief-states and desire-states influence and are influenced by the external world; but I think there is serious question as to whether intrinsically representational objects of mental states would ultimately be of much use in a serious effort to state such laws. Indeed it seems to me that the issue of whether we ought to use intrinsically representing objects of mental states turns largely on the answer to this question.) Moreover, (2) a theory of belief that shunned intrinsically representational objects of belief could still be used as the basis of 'believes that' and 'desires that' attributions: for instance, one might say that I believe that snow is white if I am in a belief-state that would typically lead to the production of the sentence 'Snow is white', and that someone else (not necessarily a language-user) believes that snow is white if he is in a state whose causal role is sufficiently similar to mine. That's very vague, of course, but working out a theory along these lines (as suggested by Stich 1982; 1983: Part I) would not require the resources required of a solution to the problem of intentionality with intrinsically representing objects: first, because the only relations that would need explanation in the theory of 'believes that' attributions would be state-state relations as opposed to state-world relations;3 and second, because almost anyone who took this view would hold that the state-state relations were highly vague and context-dependent, thereby inducing an asymmetry between the relative clarity of 'belief that' ascriptions to oneself and the relative unclarity of such ascriptions to others.
I suspect that a view of the sort just outlined (to which I now have considerable sympathy) underlies much of what Stalnaker had in mind as part of 'the linguistic picture' of intentionality; though it certainly does not fit all language of thought views. 4 Even the view outlined, however, does not reject the 3 Admittedly, the state-state relation in question would be based on the overall similarity of the states in their state-
world relations. (This overall similarity, incidentally, could take into account both similarity with respect to 'pragmatic' relations and similarity with respect to 'indication' relations; for a discussion of this distinction (Stalnaker's), see Sect. 3.) 4 For instance, I explicitly argued for the use of intrinsically representing objects to serve to describe (at least part of) the content of mental states in Sect. v of Ch. 2. (I no longer find the argument offered there conclusive. See fn. 16 below.) end p.88
pragmatic picture of beliefs and desires. For beliefs and desires are still invoked to explain behavior, and the serious explanation of behavior (that envisaged in (1) of the previous paragraph) works without appeal to language. The special appeal to language comes in only in the account of ordinary 'belief that' and 'desire that' attributions (discussed in (2) of the previous paragraph); and the point of such a theory is that these attributions do not themselves play a serious explanatory role, but serve only as devices for alluding to the more serious sort of explanation envisaged under (1). 5 In Stalnaker's discussion of the problem of intentionality, it is assumed that 'the objects of mental states' are to be intrinsically representational entities. It is illuminating to notice however that nearly all of the evidence that Stalnaker cites for S-propositions (or what is almost the same thing, sets of possible worlds) as objects of mental states could be obtained from structurally analogous entities that are not intrinsically representational. What is mostly relevant to sets of possible worlds as objects of mental states is their structural interrelations: the fact that they form a Boolean algebra. (On Stalnaker's view, a complete atomic Boolean algebra: see pages 54-6 of Inquiry.) Consequently, any other (complete atomic) Boolean algebra that was sufficiently large to make the psychological distinctions we need would do just as well for most psychological purposes; for most purposes at least, there is no need to think of the atomic elements of the algebra as possible worlds; they could be anything at all. For instance, they could just be numbers (possibly transfinite numbers, let's say, to insure that there are enough of them). Numbers, unlike possible worlds, do not purport to represent how the world is, so if we think of the atomic elements as numbers it will substantially alter how we conceive the problem of intentionality. Of course, numbers like possible worlds have their own special properties, and we would not want the problem of intentionality to be conceived in such a way that these special properties mattered. (For instance, we should take the project of explaining the difference between mental states that are assigned sets that include the number 17 and mental states that are assigned sets that don't include 17 as quite misconceived.) Rather, the point is that unless we can identify psychological laws that use features of sets of possible worlds that go beyond their structural features, then (if we buy Stalnaker's views on the identity conditions of the objects of mental states) we should conceive of the problem of intentionality as simply the problem of explaining the attribution of elements of a sufficiently large but otherwise arbitrary complete atomic Boolean algebra to our mental states. The project of explaining the difference between mental states that are assigned sets which include possible worlds according to which Caesar crossed the Rubicon and mental states that are assigned sets which do not include 5 The idea of separating the question of belief-that and desire-that attributions from the question of the contents of
belief (and of putting the main theoretical burden on the latter rather than on the former) is not unique to this approach. Indeed, Stalnaker himself argues for such a separation (and for such a view about where the burden of theory should go): see pp. 73-4. end p.89
such worlds would be as misconceived as the analogous project about numbers. 6 As I've said, Stalnaker conceives of the objects of mental states as intrinsically representational, which means that the differences between the set of possible worlds and other complete atomic Boolean algebras are taken to be somehow important (even though not important to much of psychological explanation). Even so, it is useful to divide the problem of intentionality into two parts: first, a part explaining the assignment of non-intrinsically-representing items (e.g., elements of an arbitrary complete atomic Boolean algebra) to our mental states; and second, a further part explaining the 'interpretation' of the non-intrinsically representing (e.g., Boolean) elements. (Compare this to the distinction in Fodor 1980, and Stich 1983, and in Chapter 2 above, between assigning uninterpreted sentences to mental states and assigning interpretations of those sentences. Here I'm simply making the same distinction, except freed from the presupposition of a language of thought.) Even the first stage of this project raises some interesting problems, especially if we assume that, to put it roughly, the important structural relations induced by the objects assigned to mental states must be explainable independently of
any talk of assigned objects.7 In Chapter 2 I interpreted an earlier paper of Stalnaker's (Stalnaker 1976) as denying this 'independence assumption' (and as using what I argued to be a misconstrual of the significance of functionalism to support that denial); but his remarks in the book (e.g., the remarks on the theory of measurement on page 10 and the remarks on functionalism in the first full paragraph on p. 15) suggest that this was a misinterpretation.8 (And I will give reasons in the next section why he ought to accept the independence assumption.) So let us ask how the Boolean relations among mental states that are induced by assigning S-propositions to those states are to be explained. This is not an issue that Stalnaker discusses, 6 For simplicity I have ignored here the fact that Stalnaker's S-propositions needn't be defined at all worlds, and
hence that they don't quite correspond to sets of worlds. The discussion would not need to be altered very much were this feature of Stalnaker's views taken into account. (In effect we would simply have to allow mention of the possibility of extending a given algebra of states to a larger algebra of which the original was a subalgebra.) 7 A less rough statement requires some ideas from the theory of measurement: see Sect. 2 of the postscript to Ch. 2, or the discussion of an earlier version of this material on p. 10 of Stalnaker's book. 8 My misinterpretation of Stalnaker has apparently led him to a serious misinterpretation of my argument in Ch. 2. Stalnaker points out that there is no difficulty in finding rather unproblematic naturalistic relations that connect objects to S-propositions, and he takes this to undermine my conclusion (Ch. 2, p. 49) that 'functionalism does not either solve or dissolve the problem of intentionality' (see Inquiry, 14-15). In so taking it, he misconstrues the import of the quoted remark on functionalism. (Indeed, his misconstrual is rather strange, since the goal of the first two sections of my paper (where I was presupposing for the sake of argument that we were to use sets of possible worlds as objects of mental states) was precisely to offer a proposal for how to naturalistically connect people to propositions in ways relevant to belief and desire.) The import of the quoted remark on functionalism was that on a naturalistic view, a functional relation between people and propositions is only possible if there are non-functional relations between people and propositions which ground each instantiation of the functional relation; so that if we are to solve the problem of intentionality, we need to find such non-functional relations. This is part of what I had been taking Stalnaker to be denying, but it is what he accepts in his first full paragraph on p. 15. end p.90
but it is highly relevant to the motivations that have led many to be sympathetic to the idea of a 'language of thought'. Let us focus on two particular mental states of a given agent: the first, s 1 , a state of idly wishing to trisect a 60 degree angle in a Euclidean plane with straight-edge and compass, and the second, s 2 , a state of having the conscious thought that Caesar crossed the Rubicon. (By a Euclidean plane I mean a plane that obeys the Euclidean axioms; I want a logically inconsistent wish.) The content assigned to s 1 is to be the empty set of possible worlds, or more generally, the zero element of a Boolean algebra; it is a subset of (or, bears the Boolean less than relation) to the content assigned s 2 , so we can say that the content of s 1 entails that of s 2 . Now, in virtue of what facts about s 1 and s 2 does the content of the first entail the content of the second? As part of the first stage of an answer to the problem of intentionality we must answer this question; if it proves unanswerable, the conclusion to draw is presumably that even the assignment of a Boolean structure to mental states is theoretically unjustifiable. I do not mean to suggest that the question is unanswerable. 9 A simpleminded attempt to answer it would be to suppose that the agent is disposed to logically reason by means of certain rules, and that s 1 entails s 2 10 because there is a possible chain of states beginning with s 1 and ending with s 2 each of which (besides the first) comes from previous members of the chain by one of the rules of logical reasoning. (This is obviously very simple-minded. To get a somewhat more realistic approach, one would have to deal with two sorts of rules (or prima facie rules) that are roughly on par: rules of direct implication as above, and rules for recognizing possibilities (to be used in showing failure of implication); and one would have to say something about how they interact, e.g., what happens when they come into conflict with each other). 11 Note however that to postulate anything like 'rules of logical reasoning' is to postulate something that goes far beyond the sort of Boolean structure that Stalnaker wants to build into the objects that he assigns mental states. And it is clear that one of the reasons that people have been attracted to language of thought theories over Boolean theories like Stalnaker's is precisely that language of thought theories assign mental states enough structure so that 'rules of logical reasoning' could be represented, thereby making the Boolean structure on the states less mysterious. I do not contend that any conceivable model of the Boolean structure that Stalnaker postulates requires the representation of anything like 'rules of logical reasoning': for instance, one might instead imagine that inside an agent's head is a huge array of boxes, one corresponding to each relevant possible 9 Kripke has forcefully stated (1982 : ch. 2) some serious doubts as to whether it is answerable; the simple-minded
suggestions that follow in this paragraph ignore the important challenge that Kripke poses. 10
Or more accurately, that the assignment of contents to s 1 and s 2 is such that the content of the first entails the content of the second. 11 The closest thing I've seen to a discussion of this is in Pollock (1974: ch. 10); the task Pollock is concerned with there is not one of providing psychological models, but it seems closely related. end p.91
world, and that what an agent believes is determined by which of these boxes has an 'X B ' in it. (The agent believes that Caesar crossed the Rubicon if all boxes corresponding to worlds in which Caesar did not cross the Rubicon contain an 'X B '.) It is clear however that Stalnaker would disavow any such model. (See Inquiry, p. 22.) And I think that any reasonably realistic model has got to give a place to something roughly on the order of 'rules of logical reasoning'; and that involves enlarging on the Boolean structure. Indeed, this extra structure is important not only in giving an idea of how the Boolean structure might be represented. It is also of direct importance in psychological explanations (indeed, more important than the Boolean notion of implying): think of the $1000 offer discussed in section 1. I do not claim that the enriched structure need be linguistic structure in any very obvious sense (and indeed, the bounds of what counts as linguistic structure and what doesn't are quite vague); my point is only that it is important to develop psychological models that contain that extra structure, and that 'language of thought' models at least have the virtue of doing this. Of course, Stalnaker recognizes that there is a distinction between a person whose beliefs we would describe with one sentence and a person whose beliefs we would describe with a logically equivalent (but unobviously equivalent) one: he suggests in one place (p. 23) that the objects of these two people's beliefs are the same, but that the forms by which the common object is represented in the two cases differs. Presumably, then, he could recognize 'rules of logical reasoning' (or some such thing): they would have to be stated in terms of the forms by which beliefs are represented, not in terms of the contents of those beliefs. I don't see that we can call this insistence on excluding form from the objects of belief wrong: he's free to use the term 'object of belief' as he likes. But Stalnaker seems to regard it as a substantive difference between his account and 'the linguistic picture' that the former recognizes the form/content distinction while the latter doesn't; he says that 'the conceptual separation between form and content is . . . the central feature which distinguishes the conception of thought implicit in the pragmatic picture from the one implicit in the linguistic picture' (p. 23). But the substance of this 'central issue' eludes me. As far as I can see the serious substantive issues are whether one needs to go beyond Boolean structure in giving psychological explanation, and if so what the nature of the extra structure postulated should be.
3. The Problem of Intentionality (2) Now let us turn to phase two of the problem of intentionality: the special problem that arises when one takes one's objects of belief to be 'intrinsically representing' entities. Half a chapter of Inquiry is devoted to a criticism of a strategy for solving this part of the problem of intentionality which I suggested in Chapters 1 and 2 above. I now have serious doubts about the adequacy of that strategy (based end p.92
largely on doubts about its presupposition that there is anything to be gained by using intrinsically representational objects of belief); but it will be useful to say a few words about Stalnaker's criticisms of it before presenting Stalnaker's own proposals. The strategy of my Chapter 2 was to assume that mental states of believing, desiring, and so forth could be given a roughly sentential structure, and then to impose on these states a componential semantics which specified truth conditions for sentences with analogous structures. That is, a person explicitly believes a proposition if he is in a state of believing that has that proposition as its content; where the association of contents with states of believing goes via a compositional semantics that utilizes the sentential structure of the states. (This was to hold for explicit belief— or core belief, as I called it there— rather than for belief generally. Belief generally was viewed roughly as a disposition to explicitly believe.) 12 The main ideas of this picture could be easily generalized so as to apply to lots of non-sententially structured states, e.g., states with pictorial structure. But the picture doesn't apply to states with only Boolean structure, and that of course is one of the reasons that Stalnaker is unsympathetic to the picture. Aside from that general reason, however, he suggests several specific criticisms of the strategy that I presented, which I will mention after describing the strategy in more detail. Underlying the strategy in question is a specific view of componential semantics: that the point of a componential semantics is to explain the semantic features of sentences in terms of the semantic features of their component parts. For instance, we might explain the truth conditions of 'It is not the case that Caesar crossed the Rubicon' in terms of
the fact that 'Caesar' stands for Caesar, 'the Rubicon' for the Rubicon, and 'crossed' for the relation of crossing (or if you prefer, the function assigning 'true' to the triples <w, x, y> such that x crossed y in world w), and the fact that 'it is not the case that' is a symbol of negation. Of course, a componential semantics says nothing about in virtue of what one of the non-logical components (such as 'Caesar' or 'crossed') stands for what it does; but I expressed hope that something vaguely like a causal theory of reference, applied to the structural components of a belief state (the 'morphemes of the language of thought') might serve to fill the lacuna. I skirted the issue of logical connectives (as Stalnaker points out— this is one of his criticisms), but it now seems to me pretty clear what I should have said about them: the facts in virtue of which 'it is not the case that' obeys the truth table for negation are facts about its conceptual role; more specifically, the standard truth tables for 'not', 'and', etc. are the only truth tables that make standard inferences involving these words truth-preserving. 13 12 Somewhat more accurately: to believe that p is to be disposed to realize that p follows from your explicit beliefs,
and to believe it for that reason, when you think of p or when p is suggested to you. For a good discussion of explicit belief and its relation to non-explicit belief, see Lycan 1986. (I have lifted the term 'explicit belief' from this paper, and the 'somewhat more accurate' formulation above is influenced by Lycan's discussion.) 13 A rather similar suggestion, but concentrating more on inductive inference than logical inference, is obtainable from Field 1977; the idea is developed explicitly in Wagner 1981. end p.93
Stalnaker seems to have two main objections to the sort of picture just sketched. The first centers on my mention of causal theories of reference. Stalnaker points out that believable causal theories of reference for words in public languages require intentional notions: for what a word means depends on the attitudes of the users of the language. The only conclusion to draw from this, however, is that the project of coming up with a non-intentional causal theory of reference has more prospect of success for the 'morphemes in the language of thought' than it has for words in public language. The prospects for a non-intentional theory for words in a public language seems irrelevant. The other main objection is deeper: Stalnaker says that the approach that I advocated is too atomistic in that it claims that the most basic kind of representation holds between words as opposed to sentences, or 'morphemes in the language of thought' as opposed to whole states. I have some feeling that there may be a legitimate complaint here, but I am unsure exactly what it is. Two points on this: 1. Stalnaker characterizes my view as holding that 'the name-object and predicate-property relations come first; the sentence-proposition relation is derivative' (p. 34, my italics.) Is this supposed to mean that first people invented names and predicates, and then some genius thought of putting them together to form sentences? Obviously I never held that view (nor the analogous view at the mental level). Rather, the view was that our goal is to explain the truth conditions of sentences and belief-states; name-object and predicate-property relations (and their analogs for components of belief states) are theoretical relations needed in characterizing those truth conditions; and an independent characterization of those theoretical relations is then needed (just as an independent characterization of valence in terms of chemical structure is needed, even though the goal of talk of valence is the holistic one of explaining chemical combination). 2. A theory which is formally atomistic in the sense required above may accommodate a great deal of interaction among the 'atoms'. Thus a theory of reference for names might well have the form 'reference is that relation R which is the first member of a pair
is the relation of indication, which Stalnaker takes to be particularly relevant to the problem of intentionality. He says Consider an object which has intrinsic states that tend, under normal or optimal conditions, to correlate with its environment in some systematic way, and where the object tends to be in the state it is in because the environment is the way it is. For example, the length of a column of mercury in a thermometer tends to vary systematically with the temperature of the surrounding air . . . (p. 12) Under these conditions we say that the length of the column of mercury indicates the temperature of the surrounding
air. The suggestion that will emerge later is that 'belief is a version of the propositional relation I called indication' (p. 18). I think there can be little doubt that if there is to be any hope of solving the problem of intentionality on the supposition of intrinsically representing objects of mental states, then something like indication must play a role in the account. The crucial question is, what exactly is the role it plays? In section v of both Chapter 1 and Chapter 2, I argued that the motivation for introducing intrinsically representing objects (or equivalently, a correspondence theory of truth) into semantics and philosophy of mind was to enable us to formulate such indication-relations as exist between beliefstates (and utterance-states) and the environment. Semantic notions such as 'truth conditions' were thus to be viewed as theoretical terms in a 'reliability theory', i.e. a theory of indication; and semantic notions such as 'refers' were viewed as still more theoretical notions of the theory, needed to formulate truth conditions which were in turn needed to formulate indication relations. An attractive feature of this way of viewing things is that many of the features which have been argued to be essential to a plausible theory of reference are features that clearly have to enter into any explanation of why my states or utterances are reliable indicators of the external world. For instance, to the extent that my utterances containing the phrase 'the Lyons silk-weavers strike' are reliable indicators of facts about the Lyons silkweavers strike, that reliability is obviously totally dependent upon the existence of 'experts' whose reliability I inherit; and the reliability of the experts (and hence, indirectly, of me) obviously depends on a causal network of some sort emanating from the silk weavers strike to their (and my) belief states and utterances. 14 14 My claim that we need a correspondence notion of truth if we are to formulate such indication relations as exist
between belief states (and utterance-states) and the environment was disputed by Grover, Camp, and Belnap (1975, 114-15). It is not entirely easy to evaluate their discussion— for one thing, they appear to smuggle what amounts to a correspondence notion of truth into their alternative account of indication relations, by taking the notions of believing a proposition and asserting a proposition for granted— but I think that their line of criticism may ultimately be correct. Indeed, I am now tempted by the position not only that a correspondence notion won't be needed to formulate specific indication relations, but also that such a notion won't be needed in theories that make generalizations about indication relations, and won't be automatically generated by theories involving indication either; so that a fortiori, notions like reference won't be automatically generated by theories involving indication relations. Consequently, I am tempted to view such notions as truth and reference in their correspondence senses (as opposed to their disquotational senses) as having no important value. But it still seems to me that we need some sort of explanation of whatever indication relations hold between mental states (or utterances) and the external world; and the parenthetical remark at the end of the paragraph indicates that much of the flavor of recent theories of reference would survive in this new guise. end p.95
That's one view of the relation between indication and belief. It would however be a bit misleading to sum up that view by saying that belief is a species of indication (since on that view there is no reason to suppose that belief states in general reliably indicate the states of affairs that we would regard as their contents); and in any case, since that view is atomistic, it seems clear that Stalnaker's view is different. But just what is Stalnaker's view? The obvious problem with making belief literally a species of indication is that most people are unreliable about a great many things. Of course, even tree rings and fuel gauges can err: that's why Stalnaker defines indication in such a way that for the rings of a tree to indicate the tree's age, the correlation between rings and age need hold only under 'normal or optimal conditions' (p. 12). Given the restriction to normal conditions, it does seem highly plausible that many of the most important belief states in people— including most of their perceptual belief states— do indicate the states of the world we take as their truth conditions. (Even in the case of perceptual belief states, though, there appear to be exceptions: that is the point of the thesis of the theory ladenness of observation, or at least, of the form of that thesis advocated by Feyerabend and plausibly illustrated by his 'tower argument'; see sections 6 and 7 of Feyerabend 1975.) When we widen our focus far beyond perceptual beliefs, though, it does not seem as if there is any very clear sense in which reliability is 'normal'. Are we to say that the unreliability of the average person's beliefs about science or about their own motivations or about politics is due to the fact that the average person is under 'abnormal conditions'? It is hard to see what content could be given to the notion of abnormality here that licensed this conclusion. (It should also be noted that belief states that do not reliably indicate the states of affairs that we take to be their truth conditions may well reliably indicate something other than their truth conditions: a member of a religious cult is likely to have lots of beliefs that reliably correlate (in both directions) not with the states of affairs that they represent, but rather with the assertions of the head guru of the cult.) It is not so odd, to be sure, to say that the average person is under less than optimal conditions with respect to such beliefs: the conditions couldn't be optimal, since so many of the person's beliefs are untrue! But if optimal conditions are to be defined as conditions under which a person's beliefs are mostly true, then it presupposes a notion of truth conditions; such a notion of optimality can not be used to explain a notion of indication which is used to explain the notion of truth conditions or representation. And it is very hard to see how to spell out a notion of optimality in nonintentional terms so as to make typical errors in non-perceptual matters due to non-optimal conditions. It seems initially
more attractive to try to loosen the connection between belief and indication, in such a way as to be compatible with the possibility end p.96
that even in quite normal and optimal conditions our beliefs are unreliable. The view discussed two paragraphs back was an attempt to do this, but it relied on a formally 'atomistic' conception of representation. Can we get the desired result without the 'atomism'? Perhaps the answer to this question will emerge from the second stage of Stalnaker's discussion (pp. 15-18 top). Here he suggests that there are more resources available than just the indication relation. He begins by saying: Belief and desire, the [pragmatic] strategy suggests, are correlative dispositional states of a potentially rational agent. To desire that P is to be disposed to act in ways that would tend to bring it about that P in a world in which one's beliefs, whatever they are, were true. To believe that P is to be disposed to act in ways that would tend to satisfy one's desires, whatever they are, in a world in which P (together with one's other beliefs) were true. (p. 15) That these facts hold of belief and desire is of course totally uncontroversial; the only question at issue is whether they can be used to give a non-circular explanation of the belief relation and the desire relation. The obvious way to try to use them for that purpose is as follows: The belief relation is the first component, and the desire relation the second component, of the unique pair
undetermined in a more extended discussion. 16 This is not to say that in order to give up on the idea of intrinsically representing objects of mental states, one would have to follow Stich (1983) in giving up on the prospects of a theory of indication or in excluding such a theory from psychology. It is merely to say that the results of a theory of indication should not be forced into the format of a theory involving intrinsically representing contents. (I think that some sort of theory of indication, not necessarily forced into the mold of a theory of content, is of considerable importance even within psychology, e.g., in giving an account of how psychological explanations are possible in the absence of detailed information about the agent's psychological state and the state of his environment. See Loar (1981: 194-5, and Schiffer 1981: 218-19.) end p.97
Stalnaker does not accept that conclusion; instead, he says, a pragmatic analysis gives only part of the picture (p. 18). The part of the picture that it gives is (i) that of determining which states are belief states and which states are desire states; and (ii) that of helping to determine the content of desire states once the contents of belief states have been independently determined. What determines the contents of belief states is the relation of indication. I quote the crucial passage: If belief is a dispositional state of the kind postulated by the pragmatic analysis, and also a kind of indication, then we have a fixed point to break into the circle that is responsible for the relativity of content [i.e. the fact that the attempted definition in stage two left the contents of beliefs and desires totally undetermined]. Beliefs have determinate content because of their presumed causal connections with the world. Beliefs are beliefs rather than some other representational state, because of their connection, through desire, with action. Desires have determinate content because of their dual connection with belief and action. (p. 19)
Here Stalnaker talks of 'causal connections with the world' instead of the indication relation, but he soon makes it clear (p. 21, 1st para.) that he does not have in mind incorporating such causal connections via an atomistic theory of representation. Rather, the sort of causal connection he has in mind is precisely that provided by the relation of indication. But if that is the solution, it is hard to see how there is any advance beyond stage one. 17 Our task in stage one was to provide an account of content. It was not part of the task to provide a solution to the question of which states were belief states and which desire states; everybody knew that we needed to use the causal role of the states to determine that. It is still the indication relation that Stalnaker uses to account for the content of belief states; and the problem I raised in my discussion of stage one is totally unresolved: how are we to bring in indication in our account of beliefcontent without flying in the face of the obvious fact that most people's beliefs about a great many things are extremely unreliable? Above I sketched one (admittedly quite programmatic) strategy for solving this problem, the strategy that I advocated in Chapters 1 and 2; but that is precisely the strategy to which Stalnaker is trying to find an alternative. It is quite unclear to me what the alternative strategy is to be. Indeed, there is strong reason to think that any remotely promising strategy for solving the problem of dealing with the widespread unreliability in human 17 Admittedly, Stalnaker's discussion does make explicit something that I slurred over in the discussion of stage one:
that the content of desires needn't be explained in terms of indication. Rather, we can use the indication relation solely to determine the contents of belief states, and to then use the pragmatic analysis in determining the contents of desire-states in terms of the contents of belief states. end p.98
beliefs must do something that Stalnaker doesn't want: it must invoke a finer structure in belief states than Boolean structure. For when we are interpreting the belief states of a person who is deceived about his own motivations or about what various politicians will do, why don't we maximize the agent's reliability by assigning nonstandard truth conditions that will make these belief states express truths? The reason is clear: those belief states are linked to others, via inferential links (how the agent is disposed to reason) and/or by composition from common constituents, and we want any assignment of truth conditions to the agent's states to be systematic in that it should respect such inferential and/or compositional links. So any reasonable approach to naturalizing the assignment of S-content (possible worlds content) to belief states must put a premium on assignments of content that are systematic in this way. But that means that even in giving a naturalistic account of the S-content of belief states, we must postulate the inferential and/or compositional links among belief states that the assignment of S-content is to represent; moreover, these links must be explainable independently of the assignment of S-content to the states, for otherwise the systematicity condition would be vacuous. (This point is argued in more detail in Field 1990.) The finer structure of belief states is needed not simply in explaining behavior, but in providing a naturalistic account of S-content: quite the reverse of what Stalnaker claimed in his critique of the use of epistemic possibility, in the passage quoted near the end of section 1. Stalnaker does make one suggestion about the naturalization of content that I have not discussed so far. He distinguishes backward-looking relations like indication, in which 'a state of an object is defined in terms of what tends to cause it', and forward-looking relations, in which 'a state of an object is defined in terms of what it tends to cause' (p. 14). In the main text, the only role that forward-looking considerations are given in solving the problem of intentionality is in determining what states are belief states and what states are desire states, and in determining the content of desire states by their role in action once the content of belief states has already been determined. So the main burden of determining content still rests upon the backward-looking (causal and indicational) considerations. Still, a natural suggestion, and one that Stalnaker makes in a footnote, would be to give the forward-looking considerations a more prominent role in determining content. This is an attractive idea. It is, of course, neutral to the issue of whether a theory of representation should be 'atomistic' in Stalnaker's sense, in that an atomistic theory of representation could just as well invoke forward looking considerations in its theory of reference as could a non-atomistic theory. But it suggests a kind of 'explicitly holist' approach to determining content, which Stalnaker might find attractive: a crude version of the holist approach would be that the S-content of a state is what is assigned to the state by that systematic function from states to sets of worlds which maximizes 'backward-looking' and 'forward-looking' constraints together. Nonetheless, the point of the last paragraph remains: (i) the systematicity condition is important, in that without it the account would never lead end p.99
to the ascription of familiar kinds of false beliefs, and (ii) the only way to explain the idea of systematicity is to postulate an internal structure in belief states of the kind that Stalnaker is adamantly opposed.
4. Failure of Logical Omniscience I now want to return to the difficulties that are apparently raised for Stalnaker's view by the fact that people's beliefs appear to be neither consistent nor closed under logical consequence. That there is a prima facie problem here is clear: Stalnaker wants to describe the beliefs of not only the 'ideally rational believer', but of ordinary believers as well, as relations to sets of worlds that are (at least)18 logically possible. How then can inconsistency and failure of closure under consequence be represented? I will confine my discussion here to the problem raised by inconsistency. Stalnaker has a discussion in chapter 5 which appears to me to plausibly handle many cases of inconsistency. Suppose that, in the course of theorizing about the philosophy of mind, I come to accept the existence of some sort of entity (perhaps propositions or properties or functions or sets) which I reject while doing ontology. If I notice the conflict, then presumably I will take an attitude of less than belief toward at least one of the conflicting existence claims. (There is a good discussion in this chapter of types of acceptance that do not involve full belief.) But if I don't notice the conflict, it is quite conceivable that I will believe in the entities in question in one context while disbelieving in them in another. Is this sort of inconsistency among beliefs a problem for one who wants to describe beliefs as relations to S-propositions or sets of possible worlds? Stalnaker points out that it is not. For the natural way to describe what is going on is that I have two belief states involving the existence of these entities— one belief state according to which the entities exist, the other according to which they don't. In one sort of context, one of these belief states enters into the explanation of my thought and action, in other 18 In fact, he assumes that the worlds are possible in some stronger sense, a sense in which not only denials of
logical truths are impossible, but so are denials of mathematical truths (pp. 73-7), denials of true a posteriori identity statements between names (pp. 85-6), and denials of certain sorts of true 'essentialist' claims (p. 75). The assumption that the worlds are possible in such a stronger sense worsens the difficulties that I will be discussing in Stalnaker's position. However, I have chosen not to stress that fact since I am unclear why Stalnaker wants to use such a strong notion of possibility anyway: none of the advantages he sees for the possible worlds conception seem to require it, and the fact that the boundaries of his strong notion of possibility are left unclarified serves only to make his notion of possible world more obscure than it would be had he stuck to logical possibility. Incidentally, on p. 58 Stalnaker attaches considerable weight to a rejection of the idea 'that there is one domain of all metaphysically possible worlds from which the restricted domains relevant to interpreting different kinds of possibility and necessity are drawn.' He does not seem to be merely saying that there are too many possible worlds to form a set; rather, it sounds from the context like a denial of any strongest sort of impossibility, even logical inconsistency. But it is clearly essential to much of Stalnaker's discussion that logical falsehoods (and perhaps mathematical falsehoods, false identity statements between names, etc.) are absolutely impossible: that is an essential premise to Stalnaker's conclusion that they can not be believed. So what he intends by the remark on p. 58 (which is part of an intriguing but obscure discussion of ontological commitment) is unclear. end p.100
contexts the other does. If I noticed the conflict, I would presumably try to integrate these states; but prior to my doing so, the description of me in terms of these two conflicting belief states seems intuitively correct. And there is no problem with describing the content of either of the two belief states in terms of S-propositions (or sets of logically possible worlds). I have no doubt that the above provides an attractive solution to the problem raised by this sort of example of inconsistency of belief. However, there is another sort of example, which Stalnaker does not discuss. Consider the belief states of Cantor while he was developing set theory. Cantor assumed a 'naive comprehension schema' for set theory; one instance of this proved logically inconsistent (Russell's paradox). Surely there is no plausible way to explain this along the lines above: rather, it would appear that one must attribute logical inconsistency to the content of a single belief state. Stalnaker can not accept the idea that a single belief state has inconsistent content. The reason is that on Stalnaker's view of content, two things hold. First, any belief state with inconsistent content has the same content as a state of believing that snow is both white and not white would have, and second, belief states with the same content have to enter into the explanation of behavior in essentially the same way. If these assumptions are both correct, then we have to find some way of describing Cantor's belief so that it has a consistent content. If we can not do this, that is good reason to reject at least one of the two assumptions. (Which one to give up is largely a matter of convention as to the use of the word 'content'.) Stalnaker discusses one other method which might be employed in an attempt to describe Cantor's belief state as having a consistent content: the method of going metalinguistic. That is, instead of ascribing to Cantor a belief state whose content is the infinite conjunction of the comprehension axioms of naive set theory (or some such thing), we ascribe to him a belief state whose content is that these sentences express truths. This maneuver strikes me as having
little to intuitively recommend it, but I will not press that point (except to note that it is an accident of the example that the prima facie subject matter of the belief is mathematical; the same sort of inconsistent characterization could arise in describing a subject matter with an uncontroversially non-linguistic content). A more basic doubt is whether the metalinguistic move will work. The move does succeed in giving Cantor's belief states consistent content, since it is consistent to hold that the sentences that in fact characterize sets characterize the decline of the Roman empire instead. The trouble is that it appears to be of little use to Stalnaker to ascribe to Cantor's belief state a content that is merely logically consistent; the content needs to be consistent with other facts that Cantor perfectly well knew. And among the facts that he knew, we may assume, were the fact that ' ' means set membership, that '&' means conjunction, and so forth, and hence the fact that the axioms of set theory mean just what they do in fact mean. If we now consider the metalinguistic belief that the naive comprehension axioms express truths and that they mean so and so (or, that the naive comprehension axioms express truths and that their meaning is determined end p.101
by such and such rules), then that metalinguistic belief is itself inconsistent. So the ascent from object-level propositions to metalinguistic propositions appears to have gained nothing. A number of people made essentially this objection to an earlier presentation of Stalnaker's (1976) views. In Inquiry (p. 76), Stalnaker considers the objection (in a more abstract setting, not tied to the Cantor example). His reply is to grant that even the metalinguistic beliefs ascribed to, say, Cantor form an inconsistent set; but then to introduce the discussion (which I've summarized in the second paragraph of this section) of how having an inconsistent set of beliefs is not ruled out by the possible worlds analysis, since the belief states that compose this inconsistent set may themselves be consistent. It seems to me however that this is plainly inadequate: Cantor's metalinguistic belief that the comprehension axioms express a truth and his other metalinguistic belief about the meaning of these axioms are not at all like a belief in properties while doing philosophy of mind and a disbelief in them while doing ontology; it isn't as if Cantor had two distinct belief states that could not both be operative at once. Rather, the two belief states were involved together in explaining his actions, and the discussion in Chapter 5 of 'compartmentalized' belief just seems irrelevant to the case at hand. 19 If, as I have been suggesting, we can not find a reasonable way to describe belief states such as Cantor's as having consistent content, then we must give up Stalnaker's fundamental idea (pp. 4-5, 85) that the explanation of behavior is to be done in terms of mental states described in terms (solely in terms) of the real possibilities that they represent. And if we give that up, there is little point in insisting on using the term 'content' in such a way that only the real possibilities that a state represents can be part of the state's content. It will help clarify the nature of the argument I have given in this section if I contrast it with another argument, the argument described in section 1 as expressing 'an almost universal first reaction' to a possible worlds view of belief and desire. The argument is that such a view of belief and desire can not be adequate, since people can clearly believe that P and disbelieve that Q even when P and Q are logically equivalent. I think that Stalnaker does a good deal in chapters 4 and 5 to undercut the force of that argument taken by itself (i.e., divorced from an argument against the model of explaining behavior that underlies it). For if we have good theoretical reasons for thinking that the explanation of behavior is to be given in terms of mental states described solely in terms of the real possibilities that they represent, then in many situations where a person sincerely assents to a sentence 'p' and sincerely dissents from a logically equivalent sentence 'q' it seems quite natural to conclude 19 Note added for this volume: In the final four paragraphs of his response to this paper (1986), Stalnaker points out
that compartmentalized belief has broader application than my 'philosophy of mind versus ontology' example suggests, and gives interesting examples that show how apparently 'inconsistent representation' in connectionist systems could be viewed as compartmentalized consistent representations. I'm skeptical that the model he suggests could handle the Cantor example, even in conjunction with the metalinguistic move, but I have to concede that the matter requires more discussion than I gave it here. end p.102
either that one of the sentences 'p' and 'not q' doesn't correspond to his beliefs (as would be determinable from his behavior) or that he is in a state of 'compartmentalized belief' not wholly unlike that described early in this section. Stalnaker also points out (pp. 72-4) that even the first alternative does not force one to deny that the person believes that p and disbelieves that q. For phrases of the form 'believes that p' needn't be analyzed straightforwardly in terms of content (e.g. as 'is in a state whose content is the set of possible worlds in which "p" is true'). Stalnaker does not provide an alternative analysis of phrases of the form 'believes that p', but it does not seem to me that invoking the prospects of such an alternative analysis is at all ad hoc. Still, two things must be noted. First, it is not at all clear that these points will help with all instances of 'the problem of equivalence': for instance, it is difficult to see how they help with the problem of the $1000 offer in section 1. That
problem arose not because of any assumptions about 'believes that' or 'desires that' locutions, but because the differences between a person trying to trisect a 60 degree angle and a person trying to find a map that needs more than four colors to color seem essential to explaining their non-linguistic behavior. It does not seem to me that one can plausibly explain this difference in behavior in terms of different attitudes to sentences like 'I will trisect a 60 degree angle'; and invoking 'compartmentalized belief' does not seem substantially more promising. Second, even if all instances of the problem of equivalence could be treated by going metalinguistic or by compartmentalized belief, that does not help in the least with the problem that has primarily occupied us in this section, namely the problem posed by the apparent existence of belief states with an inconsistent content. Stalnaker's denial that there can be such belief states does not stem simply from his decision to treat logically equivalent contents as identical: it would be possible to recognize inconsistent belief states compatibly with that decision, for that decision merely requires that inconsistent belief states all have the same content (the empty set of worlds). Rather, what forces Stalnaker to deny that there can be inconsistent belief states (or at least, to deny that such belief states can be used in the explanation of intelligent behavior) is his picture of action: the picture according to which the explanation of behavior is to be given in terms of mental states that are described solely in terms of the possibilities that they represent. It is that basic picture which I have argued in this section to be flawed. And my argument nowhere turned on the use of locutions such as 'believes that'; rather, the argument was that there is no plausible way to describe apparent cases of inconsistent belief states (such as in the Cantor example) so that they accord with Stalnaker's picture. Again, the basic difficulty with Stalnaker's views is not his use of the term 'content', but his picture of rational action. But once that picture of rational action is undercut (as I have tried to do in this section), then there is no longer any motivation for insisting that the word 'content' be used in such a way that only the real possibilities that a state represents can be part of its content. end p.103
4 Deflationist Views of Meaning and Content Abstract: Attempts to motivate and clarify a radically 'deflationist' view of the relations of 'meaning that' and 'having the content that', and to defend it against many of the arguments that are widely viewed as decisive against any such view. One of its claims is that standard work in the theory of reference (for instance, that of Kripke and Putnam) can be reconceived so as not to be primarily about reference at all. Includes a long postscript that, among other things, includes an account of how the degree to which behaviour is successful can be explained within a 'deflationist' framework. Keywords: belief, conceptual role, deflationism, interpersonal synonymy, Kripke, logical connectives, meaning, psychological explanation, Putnam, reference, representation, truth, truth-conditions
Hartry Field
1. Two Views of Meaning and Content I see the philosophy of language, and the part of philosophy of mind concerned with intentional states like believing and desiring and intending and the like, as pretty much bifurcated into two traditions. The traditions differ over the role that the notion of truth conditions plays in the theory of meaning and in the theory of the content of intentional states. One of the traditions, whose early advocates include Frege, Russell, the early Wittgenstein, and Ramsey, has it that truth conditions play an extremely central role in semantics and the theory of mind; a theory of meaning or content is, at least in large part, a theory of truth conditions. A strong prima-facie reason for the attractiveness of this position is that the way we standardly ascribe meanings and contents is via 'that' clauses, and the ascription of 'that' clauses is in effect the ascription of truth conditions: to describe an utterance as meaning that snow is white, or a belief state as a state of believing that snow is white, is in effect to say that the utterance or belief state has the truth conditions that snow is white. Since 'that' clauses and hence truth conditions play such a central role in our ascriptions of meaning and content, it would seem as if they ought to play a central role in the theory of meaning and content. However, it isn't easy to say precisely what this central role is that truth conditions allegedly play; that I think is a main motivation for the alternative tradition. As a crude paradigm of the other tradition, consider the verifiability theory of meaning. Here the main concept is not truth conditions but verification conditions. The verification conditions of a type of utterance might be given by the class of sensory stimulations that would or should lead to the acceptance of an utterance in that class. Notice that if the
verification conditions of a type of utterance are given in this way, then they are given without a 'that' clause; and indeed, there is no immediately clear way in which to obtain a 'that' clause from them. It is because of this that an advocate of the Frege-Russell-Ramsey-Tractatus tradition is likely to regard a verification theory of meaning as leaving out what is central to semantics: a theory of meaning according to which you can fully describe the 'meaning' of an utterance end p.104
without saying that it means that snow is white (or that something else) has left out the central element of meaning. That central element is truth conditions. Some advocates of the Frege-Russell tradition might prefer to describe what has been left out not as truth conditions but as propositional content. There is a way of understanding this that makes it unobjectionable, but it could be misleading. For a verificationist is likely to simply deny the charge that he has left out propositional content (or even, the charge that he has demoted it from a central place in his theory). A proposition, he can say, is simply a class of verification conditions; for an utterance or belief state to express the proposition is for it to have verification conditions in that class. So propositions in the verificationist's sense needn't be described by 'that' clauses: the most direct way to specify a proposition in the verificationist's sense is directly in terms of verification conditions. No one can deny that propositions so understood play a central role in the verificationist's theory. An advocate of the Frege-Russell tradition is likely to say that these 'propositions' aren't what she means by propositions, and that is doubtless so; that is because propositions as she conceives them must encapsulate truth conditions. (They may encapsulate more than truth conditions, but they encapsulate at least this much.) So it seems to me that to describe the difference between the two traditions in terms of truth conditions rather than in terms of propositional content brings to the forefront what is really central. I do not mean to suggest that a verificationist is precluded from speaking of truth or truth conditions; he can do so in either of two ways. One way, which seems to me a very bad idea, is to introduce some epistemic notion of truth conditions: that is, to define truth in terms of verification (e.g. 'would be verified in the long run'), so that truth conditions are derivatively defined in terms of verification conditions. I won't discuss this line here. The other way is to make use of what has been called a deflationary conception of truth. I take this to have several variants, but the variant I will primarily focus on is called pure disquotational truth. As a rough heuristic, we could say that for a person to call an utterance true in this pure disquotational sense is to say that it is true-as-he-understands-it. (This is not intended to provide a definition of pure disquotational truth in terms of some other notion of truth plus a notion of understanding; it is intended only as a heuristic, to motivate the features of pure disquotational truth I now proceed to enumerate.) As the heuristic suggests, a person can meaningfully apply 'true' in the pure disquotational sense only to utterances that he has some understanding of; and for such an utterance u, the claim that u is true (true-as-he-understands-it) is cognitively equivalent (for the person) to u itself (as he understands it). 1 A qualification is needed, since the claim that u is true involves an existential commitment to the utterance u, whereas u itself doesn't; this keeps the two from being fully cognitively equivalent. The qualified version is that the claim that u is true is cognitively equivalent to u relative to the existence of the utterance u; just as 1 Because of the paradoxes, exceptions must be made for certain utterances u that contain 'true'; I won't be
concerned here with just how the exceptions are to be carved out. end p.105
'Thatcher is such that she is self-identical and snow is white' is cognitively equivalent to 'Snow is white' relative to the existence of Thatcher. (To say that A is cognitively equivalent to B relative to C means that the conjunction of A and C is cognitively equivalent to the conjunction of B and C; so that as long as C is presupposed we can treat A and B as equivalent.) Having made this qualification, I will generally leave it tacit, and simply say that in the purely disquotational sense of 'true', the claim that u is true (where u is an utterance I understand) is cognitively equivalent for me to u itself (as I understand it). 2 There are both oddities and attractive features to this as a reading of 'true'; I will discuss these later on, starting in section 5. But odd or not, the cognitive equivalence of the claim that u is disquotationally true to u itself provides a way to understand disquotational truth independent of any nondisquotational concept of truth or truth conditions (and independent of any concept of proposition). That is: if I understand 'Snow is white', and if I also understand a notion of disquotational truth as explained above, then I will understand ' "Snow is white" is true', since it will be equivalent to 'Snow is white'. This will hold on any view of understanding, even a crude verificationist view according to which understanding a sentence is simply a matter of having verification conditions for it: for the verification conditions of ' "Snow is white" is true' will be precisely those of 'Snow is white'. Certainly there is no need to presuppose that understanding an utterance involves correlating a proposition or truth condition with it, as propositions and truth conditions are understood in the Frege-Russell tradition. Because of this, there can be little doubt that the notion of pure disquotational truth is a notion which a verificationist is entitled to— and which other opponents of the Frege-
Russell tradition who have a more sophisticated alternative than verificationism are entitled to as well. (Adherents of the Frege-Russell tradition are entitled to it too, though not all want it.) And using this notion of truth, even a crude verificationist can grant the legitimacy of talk of truth conditions of his own utterances. For the cognitive equivalence of ' "Snow is white" is true' and 'Snow is white' will lead to the (more or less indefeasible) acceptance of the biconditional ' "Snow is white" is true iff snow is white'; and a natural way to put this (more or less indefeasible) acceptance is to say ' "Snow is white" has the truth conditions that snow is 2 There is more than one acceptable way to understand 'cognitively equivalent' here. My own preferred reading, for
what it's worth, is that to call two sentences that a person understands 'cognitively equivalent' for that person is to say that the person's inferential procedures license a fairly direct inference from any sentence containing an occurrence of one to the corresponding sentence with an occurrence of the other substituted for it; with the stipulation, of course, that the occurrence to be substituted for is not within the context of quotation marks or an intentional attitude construction. (Cognitive equivalence relative to some other assumption is the same, except that that assumption is allowed to be used in the inferences.) I would also take the claim of cognitive equivalence to imply that the inferences are more or less indefeasible. (More specifically, that they are empirically indefeasible, and close to indefeasible on conceptual grounds as well, and that the person is not in possession of defeaters for them. These stipulations are motivated by the assumption that we should describe someone who doesn't know the semantic paradoxes as a person for whom ' "p" is true' is cognitively equivalent to 'p' across the board; as the person comes to terms with the paradoxes he revises his standards of cognitive equivalence on conceptual grounds.) end p.106
white'. A pure disquotational notion of truth gives rise to a purely disquotational way of talking about truth conditions. An opponent of the Frege-Russell tradition can try to use this purely disquotational way of talking about truth conditions, or some variant or extension of it, to allow the legitimacy of talk of the truth conditions of utterances and mental states without giving truth conditions the central role that they play in the Frege-Russell tradition. For instance, if one of our mental states can be described as an attitude of believing or accepting the sentence 'Snow is white', then it can be described both as a state of believing that snow is white and as a state with the truth conditions that snow is white. The connection of 'that' clauses to truth conditions noted in the second paragraph thus remains, but the initial impression that it leads inevitably to the Frege-Russell view is at least lessened. It may not be obvious that all legitimate talk of the truth conditions of mental states can be handled in this way or by an extension of it, but that is what the opponent of the Frege-Russell tradition contends. (This may involve disallowing as illegitimate certain uses of truth conditional talk that an advocate of the Frege-Russell view would allow.) 3 Such a view might naturally be called a deflationist view of meaning and content; or more accurately, a deflationist view of meaning that and having the content that, or a deflationist view about the role of truth conditions in meaning and content. The first of the three labels (which I have used in the title for brevity) could be misleading, since some versions of the view are in a sense quite undeflationist about meanings: for instance, the crude verificationism I described is not in the least deflationist about verification conditions; and it identifies them with meanings. What it is deflationist about is truth conditions; or, the locutions 'means that' and 'has the content that'. From now on I'll simply use the label 'deflationist'. So that my labels not be thought as prejudging the merits of the views, I'll call the Frege-Russell tradition 'inflationist'. The division between the inflationist and deflationist positions is in some ways the most fundamental division within the theory of content and meaning (though as with many fundamental divisions in philosophy, the line between the two views is not absolutely sharp). I myself strongly feel the attractions of each position, though I have come to favor deflationism. In this paper I will formulate a fairly radical version of deflationism,4 say a few things to try to motivate it, and try to defend it against some obvious objections. I will also mention some deeper lines of objection that I cannot deal with here; these seem to me the places where the battle between deflationism and inflationism must ultimately be fought. 3 Certain uses of truth conditional talk in explanations seem to me the most likely target for excision. 4 More radical, I think, than the one advanced in Horwich 1990. While I like much that Horwich says I have substantial
disagreements: see Field 1992b. end p.107
2. More on the Deflationist/Inflationist Distinction The version of deflationism I outlined in my opening remarks was built around a crude verificationism. It hardly needs arguing that such a version of deflationism isn't satisfactory. But the main idea behind deflationism doesn't require verificationism, it requires only that what plays a central role in meaning and content not include truth conditions (or relations to propositions, where propositions are conceived as encapsulating truth conditions).
I must admit at the start that the question of whether truth conditions (or propositions encapsulating truth conditions) play a central role in meaning and content is not a very clear one, for three reasons. First, the notions of meaning and content aren't clear as they stand. Some ways of clarifying them involve legislating truth conditions into or out of meaning and content, which would make the issue of deflationism totally uninteresting. My way of resolving this unclarity will be to interpret 'meaning' and 'content' as broadly as possible short of explicitly legislating truth conditions into meaning or content. Second, the idea of a central role isn't initially clear either. My goal is to clarify this as I go along, by making explicit the limited role that I think the deflationist can give to truth conditions, and identifying kinds of role that deflationism cannot allow truth conditions to have. Unfortunately I cannot complete the job in this paper: to keep the paper to a manageable size I have had to leave to another occasion discussion of the crucial subject of the role of truth conditions in psychological explanation. [But see section 7 of the postscript.] There is also a third source of unclarity to the question, one which I think makes a complete sharpening impossible. If deflationism is to be at all interesting, it must claim not merely that what plays a central role in meaning and content not include truth conditions under that description, but that it not include anything that could plausibly constitute a reduction of truth conditions to other more physicalistic terms. And it is well known that there is no sharp line between reduction and elimination, so this crucial tenet of deflationism is not an altogether precise one. In other words, a theory of content and meaning might end up not employing the notion of truth conditions directly (in a central role), but employing (in that role) certain physicalistic relations that could be regarded as reducing the relation 'S has the truth conditions p', but could only be so regarded on a rather loose conception of reduction. Such a theory of content and meaning would occupy the borderline between a deflationist and inflationist theory: it would be rather a matter of taste which way we describe it. I'll have a bit more to say about this possibility later. I remarked earlier that a deflationist need not be a verificationist. What elements instead of, or in addition to, verification conditions does he have available for inclusion in meaning or content? One element that can certainly be included in content is conceptual or computational role: role in a (perhaps idealized) computational psychology that describes how the agent's beliefs, desires, etc. evolve over time (partly in response to sensory end p.108
stimulations). 5 The conceptual role of a belief state includes its verification conditions, but includes much more besides. (It includes a wider variety of evidential relations;6 it includes the conceptual consequences of having the belief; etc.) But conceptual role isn't enough: it is both 'internalist' and 'individualist', and a plausible deflationism is going to have to give to content both 'externalist' and 'social' aspects. In describing these further aspects of content, it will help if I make what I hope is a harmless assumption. The assumption is that we can speak of a language-user as believing and desiring sentences of his or her own language— or at least, as believing and desiring internal analogs of them in which some or all of the ambiguities may have been removed— and that this relation can be made sense of without a prior notion of content for the belief or desire states or of meaning for the sentences. In other words, this relation is one that a deflationist can perfectly well appeal to. There is a good bit that needs to be said about just what this assumption comes to, but let me simply say that I think that when it is unpacked it is pretty unobjectionable.7 Now let's get back to the aspects of content that go beyond conceptual role. The most obvious 'externalist' element that a deflationist can put into content is indication relations. It is a fact about me that I am a pretty good barometer of whether there is rain falling on my head at that moment: when there is rain falling on my head, I tend to believe 'There is rain falling on my head'; conversely, when I do believe this sentence, usually there is rain falling on my head. This is simply a correlation, there to be observed; and a deflationist is as free to take note of it as is anyone else, and as free as anyone else to deem it an ingredient of what he calls content. The correlation of people's belief states to the world outside presumably extends past the directly observed; the beliefs of astronomers in sentences of form 'The location of Halley's comet at time t will be x' generally correlate pretty well with the location of Halley's comet at time t, and in certain circumstances the beliefs of physicists in sentences of form 'A particle has recently tunnelled through the potential barrier' will correlate with whether a particle has indeed done so. 5 Of course, conceptual or computational role can be described at different levels of abstraction, so there is more than
one notion of conceptual role that meets this description. I think the notions of meaning and content are too indeterminate for there to be a right answer about which notion of conceptual role to include; one could even include more than one. 6 If the probability of S given E is higher than the probability of S, then (especially if this fact is rather 'robust', that is, if except for very special sentences F, P(S|E&F) remains higher than P(S|F)) it is natural to include this fact in the conceptual role of S, even if E is in no sense 'observational' and could never be established with enough certainty or whatever to count as evidence for anything. It was central to verificationism to exclude such facts from verification conditions when E was not observational.
7 Part of the idea behind this assumption is that when you read or hear a given unambiguous sentence, there is a
type of thought state that you typically undergo in processing it, where this type is identifiable in purely computational terms; and that this type of thought state also typically occurs when you utter such a sentence. This coupled with some other plausible assumptions about the computational processes of the language-user make it possible to define in computational terms a correspondence between many of the person's states of believing and desiring and their unambiguous sentences. A natural computational story about the processing of ambiguous sentences allows us to extend this to such sentences too. I will not attempt to give the details of any of this here. end p.109
These observations might make one think that a deflationist is bound to recognize (a non-disquotational version of) the relation 'S has truth conditions p', in fact if not in name, since these indication relations constitute the truth-conditions relation. But this overlooks the fact that the project of giving anything close to a believable reduction of talk of truth conditions to talk of indication relations is at best a gleam in the eye of some theorists. A way to see the point is to notice that there are plenty of examples where the indication relations don't reflect what we would intuitively regard as truth conditions. Consider the ancient Greek whose judgements (that we would translate as) 'Zeus is throwing thunderbolts' are reliably correlated with the presence of lightning in the vicinity. That reliable correlation is a key fact about the use of the sentence, and is part of its 'content' in the deflationist's sense, but it is not a matter of the truth conditions of his utterance as normally understood. (It would be possible to maintain that the Greek's judgements had as their truth conditions that lightning is present; I wouldn't want to say that that is exactly wrong, since I doubt that there is a clear matter of right and wrong here, but it certainly fits ill with the fact that there were systematic inferential connections between this sentence and others containing 'Zeus', such as 'Let's build an idol so Zeus will stop throwing thunderbolts'.) In many other cases, the facts about indication relations are clear but a decision about truth conditions seems rather arbitrary. The judgement 'The sun is rising' today may have as its truth conditions that the angle from the horizon to the sun is small, positive and increasing, but was this so at earlier times when people believed that its motion relative to the earth was absolute motion? I have no idea how to decide, but from a deflationist point of view it doesn't matter: the indication relations are there anyway, and they are what is important. I have given examples from earlier theories since we obviously can't recognize any such serious errors that may infect our own observational claims, but it is not beyond imagination that even some of our own observational reports reliably indicate something at variance with their own truth conditions. When we move beyond observation reports to more theoretical sentences, this is still more likely. There are of course less systematic indication relations than the ones I have considered. Maybe I systematically exaggerate, so that my believing a sentence of the form 'It is n feet high' is strongly correlated with the object before me being f(n) feet high, where f(x) starts dropping off rapidly from x after about 6 feet or so. Or worse, maybe my beliefs of the form 'In Bosnia, p' don't stand in any interesting correlations with the actual facts about what's happening in Bosnia, but just reflect what appears in the newspapers I read. The deflationist can recognize these facts about indication too, and attribute explanatory importance to them. The deflationist can also use the disquotation schema to formulate the distinction between these cases and cases like my reliability about whether rain is falling on my head: in the latter case my belief state reliably indicates its own truth conditions (as given by the disquotation schema), end p.110
in the former it indicates something else. 8 What he can't do, it seems to me, is say that this distinction is of much explanatory importance: for that would give truth conditions (rather than just indication relations in general) a central role in the theory of mind; and the claim that truth conditions have a central role in the theory of mind is the defining characteristic of inflationism.9 But that isn't to say that he can't differentiate the exaggeration and Bosnia cases from the rain falling on my head case on other grounds, e.g. of how systematic the indication relations are and how well they survive under the addition of evidence. And here it is worth noting that on these scores the Zeus example and the rising sun example (and analogous examples of as-yet-undetected errors in our own observation reports) come out more like the rain falling on my head example than like the exaggeration or Bosnia examples: they are rather systematic indication relations not dislodged by any small amount of additional evidence. It is these factors, rather than the factor of whether what is indicated is truth conditions, that are explanatorily important. Anyway, the deflationist can include indication relations in content; this is enough to make content externalist, but not enough to make it social. However, since we are regarding many of our most important beliefs as in effect attitudes toward sentences in our language, the means to make content social is at hand. That is, it seems reasonable to decree that the content of these belief states is to be influenced by the meaning these sentences have for us; that meaning is of course partly influenced by the content of our intentional states, but since language is social it will be influenced by the contents of the states of other members of our community as well. In particular— and removing the appearance of circularity— we can understand 'content' in such a way that the conceptual roles and indication relations of other people's states of believing a certain sentence are counted as relevant to the content of my state of believing 10
that sentence; such a way of understanding 'content' is not only possible, it is natural when one thinks of the ways that which sentences I believe influence and are influenced by which sentences others believe. Again, construing content as social in this way is quite compatible with deflationism, that is, 8 The deflationist can also formulate the distinction for other peoples' belief states, by relying on translation, though to
the extent that that is indeterminate the distinction there can obviously be given on objective significance. 9 I have heard the objection that a deflationist is free to give truth conditions an explanatory role as long as it is only
'deflationary truth conditions' and not 'heavy duty truth conditions' that are given that role. This seems to me a confusion. I think it is quite misleading to speak of two kinds of truth conditions, the heavy duty ones and the disquotational ones: this suggests that the advocate of 'heavy duty' truth conditions thinks that we should ascribe to 'Snow is white' some truth conditions other than that snow is white, and that of course is absurd. It would be better to say that there are two kinds of relation between truth conditions on the one hand and utterances and mental states on the other, the disquotational relation and the heavy duty relation. But the only ways I can see to make sense of the distinction between these relations directly or indirectly preclude giving the disquotational relation certain kinds of explanatory role. (In my view, the preclusion is indirect, and stems from the deflationist's account of how truth conditions attach to sentences and mental states.) 10 This last way of putting things gives a way to make content social without relying on an independent notion of meaning. It thus leaves room for explaining meaning in terms of content if so desired. end p.111
with keeping truth conditions (and hence 'that' clauses) out of the fundamental characterization of content.
3. The Semantics of Logical Operators 'Deflationism' and 'inflationism' are broad labels, each encompassing many different views. I think it would help to make the contrast between the deflationist and inflationist traditions clearer if I focussed on several problems in the theory of content, and showed how they looked from each of the two broad perspectives. The first problem I will discuss is the problem of saying what determines the 'referent' of a logical operator like 'or' or 'not' or 'all'; or to put it less Fregeanly, the problem of saying what determines the contributions that such a logical operator makes to the truth conditions of sentences that contain it. I think it is clear in broad outline what an inflationist's best strategy is for answering this question. The answer will have two stages. The first stage involves spelling out the 'conceptual role' that 'or' has (for a given person, or for the linguistic community generally): in this case, it would largely be a matter of its role in deductive inference and perhaps inductive and practical inference as well. For instance, among the relevant facts about my use of the word 'or' is that I tend to accept inferences to a disjunction 'A or B' from either disjunct, and to the negation of the disjunction from the negations of both disjuncts, but not to the disjuncts from the disjunction; and when I do occasionally slip and infer according to the latter rule, I can be brought to correct myself. Also relevant perhaps is that I tend to assign to a disjunction a degree of belief higher than the degrees of belief of each disjunct but no higher than the sum of the degrees of belief of the disjuncts. (Indeed my degree of belief in a disjunction 'A or B' tends to be about equal to the sum of my degrees of belief in the disjuncts minus my degree of belief in A times my conditional degree of belief in B given A.) And so on. But what does the word's having this conceptual role have to do with its standing for the usual truth function (that is, with its contributing to truth conditions according to the usual truth table)? Perhaps the conceptual role of 'or' determines that it stands for that truth function in this sense: any word in any linguistic community that had that conceptual role would stand for that truth function. But that doesn't seem very helpful, for it doesn't tell you why that conceptual role should be associated with that truth function. There is, though, a rather natural story to try to tell. The idea is that if we assign truth conditions to our sentences (and hence our belief states) according to this rule of truth for 'or' (and the standard rules of truth for 'not'), it will make our deductive inferences involving 'or' truth-preserving, and presumably will make our inductive inferences involving 'or' highly reliable; 11 11 I mention 'not' because the deductive inference rules I listed include the rule that we can infer 'not (A or B)' from
'not A' and 'not B' together. It is possible to replace this by a rule involving 'if . . . then' instead of 'not'; or (as in natural deduction systems and sequent calculi) by a rule involving a notion of implication. end p.112
whereas if we used any other truth table, the deductive and inductive inferences would come out totally unreliable.
Reliability considerations seem to give just the sort of link between conceptual role and reference that we want in this case. There are of course some holes here. One hole is that though it is true that no alternative truth function makes our inferences involving 'or' at all reliable, this just shows that reliability considerations make the usual truth table for 'or' more satisfactory than alternative truth functions; but maybe there is an alternative to a truth functional account that would make us equally reliable? A second hole is that in explaining the contribution of 'or' to truth conditions, we can't legitimately assume the contribution that 'not' makes to truth conditions unless that can be independently established; so unless it can be independently established (which is doubtful), or unless we can avoid mentioning it in our account of what determines the truth table for 'or', then we really need an account of how reliability considerations give us the rules of truth for 'or' and 'not' together. (Using another operator like 'if . . . then' or 'implies' instead of 'not' would give rise to similar problems.) This deepens the first hole: for now we need to compare the package containing the usual truth functional accounts of 'or' and 'not' to alternative packages in which both truth rules are varied, and where there is no constraint that either truth rule be truth functional. Even if the second hole can be avoided, the problem with the first hole is greater for more complicated operators, like quantifiers: for instance, in the case of 'some' our usual deductive inferences turn out truth preserving if we suppose that its contribution to truth conditions is that of the unrestricted objectual existential quantifier, but making it a restricted quantifier or substitutional quantifier of a certain sort would also make the inferences truth preserving, and (depending on the details of the restriction) might make us at least equally reliable— even with the truth rules for the truthfunctional connectives and 'implies' unchanged. So what determines that our word 'or' or 'not' or 'some' makes the contribution to truth conditions that we assume it makes? An inflationist has three choices. Either she can find some sort of naturalistic facts that could be cited in answer to this question (facts involving reliability considerations might work in the end, despite the questions above); or she can say that there is simply no fact of the matter, that we have a surprising example of referential indeterminacy here; or she can say that it is a non-naturalistic fact about us that our word 'some' contributes to truth conditions according to the rules of the unrestricted objectual quantifier and that 'not' and 'or' contribute according to the usual truth tables. But this third option strikes me as rather repellent; the first may be hard to carry out; and the second may not be altogether free of philosophical difficulties. end p.113
The same sort of problem arises in the case of the predicate 'is identical to' (and the numerical quantifiers such as 'there are exactly two' that are definable from it). The axioms that we accept as governing this predicate determine (given a reliability assumption) that it stands for a congruence relation, that is, an equivalence relation for which substitutivity holds; but this is not enough to determine that it stands for genuine identity, that is, for the relation that everything bears to itself and to nothing else. It is especially not enough if the extensions of the other predicates in the language are up for grabs: as Quine has famously pointed out, it isn't so easy to say what facts about us make 'identical to' stand for identity and 'rabbit' for rabbits, rather than 'rabbit' for rabbit stages and 'is identical to' for the relation of being stages of the same object. And even if we can find facts about usage that would rule out this example as violating a plausible reliability requirement, it is a tall order to find facts about usage which would rule out all choices of nonstandard congruence relations. Again, an inflationist is apparently faced with the three options of finding such facts, or accepting it as simply a brute fact that our word stands for genuine identity, or accepting a surprising level of referential indeterminacy in our basic logical vocabulary. This is a choice that not everyone would be happy to have to make. A main motivation for deflationism is that it apparently avoids having to make this sort of choice. 12 According to the simplest version of deflationism— that relying on the pure disquotational truth predicate mentioned earlier— it is an entirely trivial matter to explain why one's own word 'or' obeys the truth table it does: this follows from truth functional logic together with the logic of the disquotational truth predicate, with no mention of any facts at all about our usage of 'or'. Recall that if 'true' is used purely disquotationally, then ' "p" is true' is cognitively equivalent to 'p', for any sentence 'p' that we understand. As a result, we get that each instance of the 'disquotation schema' (T) 'p' is true if and only if p holds of conceptual necessity, that is, by virtue of the cognitive equivalence of the left and right hand sides. But for any sentences q and r, instances of the following schemas are also instances of (T), and hence conceptually necessary: 'q or r' is true if and only if q or r 'q' is true if and only if q 'r' is true if and only if r.
And now you can use the latter two to substitute into the right hand side of the first, so as to get the conceptual necessity of each instance of the following: (TF) 'q or r' is true if and only if 'q' is true or 'r' is true. 12 It may not avoid the threat of indeterminacy in the end: I discuss this in Ch. 8. Even so, the account to follow
offers an attractive alternative to the inflationist's approach to the determination of the way that our logical symbols contribute to truth conditions; and I will generalize it in Sect. 6. end p.114
The conceptual necessity of the instances of (TF) really isn't enough to justify the claim that 'or' obeys the usual truth table: we need the conceptual necessity of the generalization (TFG) For all sentences S 1 and S 2 of our language, S 1 or S 2 is true iff S 1 is true or S 2 is true. But it is easy to get that if we start not from (T) but from some generalized form of (T). One such generalized form of (T) employs a universal substitutional quantifier: (TG) Πp['p' is true if and only if p]. 13 (I assume a theory of substitutional quantification that avoids the semantic paradoxes). There are also ways to get the generalization without substitutional quantifiers. One alternative is to incorporate schematic letters for sentences into the language, reasoning with them as with variables; and then to employ two rules of inference governing them: (i) a rule that allows replacement of all instances of a schematic letter by a sentence; (ii) a rule that allows inference of x(Sentence(x) A(x)) from the schema A('p'), where A('p') is a schema in which all occurrences of the schematic letter p are surrounded by quotes. Such a formalism corresponds to a very weak fragment of a substitutional quantifier language, and is probably preferable to using the full substitutional quantifier. In such a formalism, (T) and (TF) themselves are part of the language, rather than merely having instances that are part of the language; and from (T) we can derive (TF) and thence (by rule (ii) and a bit of syntax) (TFG). Note that on any of these versions of the pure deflationary account, facts about the meaning of our word 'or'— e.g., its conceptual role— need not be explicitly referred to in explaining why our word 'or' obeys the usual truth table. Of course, we do need to use logical reasoning, and in particular, the deductive inferential rules that are in fact associated with the word 'or'. But we needn't mention in the explanation that these rules are associated with 'or'; in that sense, the meaning of 'or' plays no role in the explanation. Does this mean that a deflationist can't make sense of the idea that 'or' obeys the truth table it does because of its conceptual role? Not really: the fact that the deductive inferential rules for 'or' are used in the explanation of 'or' having the truth table it has is enough to give one quite natural sense to the claim that it obeys that truth table because it obeys those deductive inferential rules. However, the sense of 'because' here does not support counterfactuals. That is, because the fact that these rules are associated with 'or' plays no role in the explanation of 'or' having this truth table, we don't get the conclusion that if 13 The argument from (T) to (TF) generalizes as an argument from (TG) to
ΠpΠq['p or q' is true if and only if 'p' is true or 'q' is true]. By use of the additional axiom x[x is a sentence of our language Σp(x = 'p')], we get (TFG). end p.115
different rules had been associated with 'or', its contributions to truth conditions would have been different. (At least, we don't get this conclusion on the most straightforward reading of the counterfactual.) Indeed, it is clear that this counterfactual (on its straightforward construal anyway) is unacceptable if 'truth conditions' is understood in a purely disquotational sense. (This may seem an objection to the purely disquotational notion of truth, but later on I will argue that it is not. I will also consider whether a deflationist can make sense of a modified notion of truth conditions according to which such a counterfactual would be straightforwardly true; if so, the conceptual role of 'or' might have to be explicitly referred to in explaining why 'or' 's truth conditions in this modified sense of truth conditions are those of the usual truth table.) What I've said here about the explanation of the truth-theoretic properties of the logical connectives goes for other words as well. Consider 'rabbit': an inflationist presumably thinks that the set or property that my term 'rabbit' stands for is determined from the facts about this word's conceptual role for me, together with its conceptual role for other members of my community, together with the facts about what my believing various 'rabbit' sentences tends to be
correlated with, together with the same sort of facts for other members of my community, and so on. This raises the question of precisely how it is determined; and it seems to me that if inflationism is to be believable then the inflationist needs to have some story to tell here. Of course, Quine's well-known difficulty about rabbits v. rabbit-stages would be solved if we solved the problem for identity: reliability with respect to sentences like 'For all x and y, if x and y are nearby rabbits then x = y' ties the extension of 'rabbit' to that of 'identical' in such a way as to handle such individuation problems. But there are further problems: for instance, what makes 'rabbit' stand for the rabbits rather than the rabbitsor-realistic-imitations? I don't say that the inflationist can't tell a reasonable story about this, only that there is a story to be told, and perhaps there is room for skepticism about the possibility of telling it adequately. If so, that provides a motivation for deflationism. For the deflationist view is that there is nothing here to explain: it is simply part of the logic of 'refers' (or 'is true of') that 'rabbit' refers to (is true of) rabbits and to nothing else. The deflationary view of truth and reference as I've presented it here applies only to words and sentences that we understand. This may well seem worrisome, and it is important to ask both whether it should seem worrisome and whether it could be avoided. I'll return to these matters in later sections, but first I want to further explore the contrast between the deflationist and inflationist viewpoints. end p.116
4. Inverting the Theory of Reference One qualm that one might reasonably have about the deflationist perspective is that a lot of work that has gone into the theory of reference in recent years seems to be onto something, and it seems at first hard to explain just what it could be onto if truth conditions play no central role in the theory of meaning. After all, if truth conditions play no central role, reference can hardly play a central role: whatever importance reference has surely derives from its contribution to truth conditions. Let's consider first Kripke's observation (Kripke 1972) that description theories of the reference of our proper names 14 are incorrect. If truth conditions play no central role in meaning, and truth is fully explained by the disquotation schema (and of value only as a logical device, in a manner soon to be explained), then the same is true of reference: for the reference of singular terms, the relevant schema is (R) If b exists then 'b' refers to b and nothing else; if b doesn't exist then 'b' doesn't refer to anything. But if this tells us everything we need to know about the reference for our own words, what could Kripke's critique of description theories be telling us? We need a concrete example before us. Consider Kripke's case against a version of the description theory according to which the referent of our term 'Gödel' is determined by the associated description 'prover of the incompleteness theorem': the referent of the name (if it has one) is to be whatever uniquely fits the description. Kripke's case against this view is a thought experiment, in which we discover the following fact: (F) The incompleteness theorem was proved by a man baptized 'Schmidt' and who never called himself anything other than 'Schmidt'; a certain person who called himself 'Gödel' and got a job under that name at the Institute for Advanced Study stole the proof from him. In this situation, Kripke asks, who is it natural to say we have been referring to when we used the name 'Gödel', the guy who called himself 'Schmidt' and proved the incompleteness theorem, or the guy who called himself 'Gödel'? Nearly everyone says the latter, so the description theory (or this version of it) is wrong. Surely there is something important in this critique, and if the deflationist can't make sense of it then something is wrong with deflationism. I think that the deflationist can make sense of Kripke's observations. On the deflationist viewpoint, though, the observations aren't at the most basic level about reference but about our inferential practice. That is, what Kripke's example really shows is that we would regard the claim (F) as grounds for inferring 'Gödel didn't prove the incompleteness theorem' rather than as grounds for inferring 'Gödel was baptized as "Schmidt" and never called himself "Gödel" '. 14 Or at the very least, description theories that use only non-metalinguistic descriptions. end p.117
Reference is just disquotational. It does comes in indirectly: from (F) I can indirectly infer 'Gödel' doesn't refer to the guy that proved the incompleteness theorem. But that isn't because of a causal theory of reference over a description theory, but only because I can infer Gödel isn't the guy that proved the incompleteness theorem,
and then 'semantically ascend'. This does seem to me a fairly plausible account of what the Kripke point shows. We see that at least in this case, the deflationist picture leads to a reasonably plausible inversion of standard views, and tends to demote the importance of theories of reference and truth conditions from accounts of language use and cognitive functioning. But how about the positive view of reference that emerges from the writings of Kripke, Putnam, etc.? The positive view is that a typical name of mine refers to what it does because of a causal network of beliefs and utterances involving that name: in the simplest cases, some people acquired beliefs involving a name as a result of direct causal interaction with the thing named; these beliefs led to their using the name in utterances, which led other people to have beliefs involving the name; these in turn passed beliefs on to others, and so forth. And it is because my uses of (say) 'Hume' stand in a causal network of roughly this sort, a network whose dominant causal source is Hume, that my uses of the name refer to Hume. The previous discussion shows how a deflationist can partially capture the force of this: he can say that it is just part of our inferential procedure to regard claims of roughly the form 'The dominant causal source of our beliefs involving 'b' is b' as pretty much indefeasible. But this doesn't seem to fully capture the importance of the causal theory of reference. It has seemed to many that the causal network emanating from Hume to my uses of the name 'Hume' explains what is otherwise mysterious, namely, how my name could be about Hume. Obviously a deflationist can't say this: the whole point of deflationism is that the only explanation we need of why my word is about Hume is given by the disquotation schema. Nonetheless, the deflationist can agree that this causal network story is explanatory of something: what it explains is the otherwise mysterious correlation between a knowledgeable person's beliefs involving the name 'Hume' and the facts about Hume. You probably believe quite a few sentences that involve the name 'Hume', and a large proportion of them are probably disquotationally true: that is, the conditional probability that Hume was Φ given that you believe 'Hume was Φ' is quite high. Surely this correlation between your 'Hume' beliefs and the Hume-facts cries out for explanation. 15 And the general lines of the explanation are clearly suggested in the Kripke-Putnam 15 This incidentally is why the Benacerraf problem about mathematical knowledge (Benacerraf 1973) isn't dissolved by
a disquotational theory of truth. [See the reformulation of the problem in sec. 5 of ch. 11.] end p.118
account: you acquired your 'Hume' beliefs largely through interactions with others, who in turn acquired theirs from others, and so on until we reach believers with a fairly immediate causal access to Hume or his writings or whatever. Moreover, the causal network has multiple independent chains, and contains historical experts who have investigated these independent chains systematically, so the chance of large errors surviving isn't that high. The role of 'experts' that figures heavily in Putnam's account of reference thus also has its analog when focus is put instead on explaining our reliability— indeed, when the focus is put on reliability it is obvious why the reliance on 'experts' should have such a central place. Earlier I pointed out that the reliability of my beliefs under the disquotation schema is simply an objective fact about me, stateable without semantic terms, which a deflationist can hardly be debarred from taking note of in his account of meaning. Here I am expanding the point, to observe that he can hardly be debarred from wanting an explanation of it; and the explanation is bound to involve some of the ingredients that inflationists tend to put into their theories of reference. This of course reraises the possibility mentioned earlier, that by the time the deflationist is finished explaining this and similar facts, he will have reconstructed the inflationist's relation 'S has the truth conditions p', in fact if not in name. My guess is that this will turn out not to be the case, but to assert that conclusion with confidence requires a more thorough investigation than I will be able to undertake in this paper. All I really hope to motivate here is that we should be 'methodological deflationists': that is, we should start out assuming deflationism as a working hypothesis; we should adhere to it unless and until we find ourselves reconstructing what amounts to the inflationist's relation 'S has the truth conditions p'. So methodological deflationism is simply a methodological policy, which if pursued could lead to the discovery that deflationism in the original sense ('metaphysical deflationism') is workable or could lead to the discovery that inflationism is inevitable. It could also turn out that we end up constructing something that might or might not be regarded as the inflationist's relation 'S has the truth conditions p'; in that case, the line between inflationism and metaphysical deflationism will turn out to have blurred. 16
5. More on Disquotational Truth It is time now to be a bit more precise about what exactly deflationism involves. In order to do this, it will help to ask what a deflationist should say about why we need a truth predicate. If truth conditions aren't central to meaning, why
not drop talk of truth altogether? As is well known, the 16 Methodological deflationism doesn't even preclude that we might in the end come to accept an unreduced
inflationist relation S has the truth conditions that p: we might conceivably see the need for introducing non-physical relations as explanatory. I think it does put the burden of proof against such a position, but I think that doing so is appropriate. end p.119
deflationist's answer is that the word 'true' has an important logical role: it allows us to formulate certain infinite conjunctions and disjunctions that can't be formulated otherwise. 17 There are some very mundane examples of this, for instance, where we remember that someone said something false yesterday but can't remember what it was. What we are remembering is equivalent to the infinite disjunction of all sentences of form 'She said "p", but not-p'. A more important example of this which has been widely noted arises in discussions of realism. 'Realism' has been used to mean many things, but one version of it is the view that there might be (and almost certainly are) sentences of our language that are true that we will never have reason to believe, in contrast to 'anti-realist' doctrines that identify truth with long-run justifiability or whatever. To assert realism in this sense one needs a notion of truth. But the reason for this is purely logical: that is, if only finitely many sentences could be formulated in our language, we could put the realist doctrine without use of a predicate of truth: we could say: It might be the case that either the number of brontosauruses that ever lived is precisely 75,278 but we will never have reason to believe that; or the amount that Michael Jackson spent on underwear in his lifetime is exactly $1,078,852.72 but we will never have reason to believe that; or . . . where in place of the ' . . .' goes similar clauses for every sentence of the language. It is because we can't complete the disjunction that we need a notion of truth. Or perhaps I should say, because we can't otherwise complete the disjunction: for the claim that it might be the case that some sentence of our language is true but that we will never have reason to believe it can be viewed as simply a way of formulating the disjunction. 18 Another example of 'true' as a device of infinite conjunction and disjunction is the desire to utter only true sentences or to have only true beliefs: what we desire is the infinite conjunction of all claims of the form 'I utter "p" only if p' or 'I believe "p" only if p'. It is sometimes claimed that a deflationist cannot grant that there is any 'substantial norm' of assertion beyond warranted 17 This needs two small qualifications. The first is that the generalizations can be formulated otherwise if the language
contains certain devices of infinite conjunction such as substitutional quantifiers. However, this qualification is not a terribly severe one, since the presence of such substitutional quantifiers would allow you to define 'true': S is true iff Πp(S = 'p' p). (Again, I assume a theory of substitutional quantification that avoids the paradoxes.) Note that the use of schematic letters as an alternative to substitutional quantification, mentioned in Sect. 3, does not allow us to formulate infinite disjunctions without 'true', and therefore does not allow us to explicitly define 'true'. The second qualification is that the view I've given (in either the substitutional quantifier version or the free schematic variable version) allows statements that are really a bit stronger than infinite conjunctions, in the same way that first order quantifications are stronger than the totality of their instances even when every object has a name: see the discussion in Sect. 3 of how to get the generalization (TFG). 18 For present purposes I am regarding sentence types as 'necessary existents', so that the fact that the disjuncts don't all entail the existence of sentence types doesn't debar a sentence that does entail this from counting as their 'infinite disjunction'. end p.120
assertibility.19 This seems to me a serious mistake: any sane deflationist will hold that truth and warranted assertibility (even long-run warranted assertibility) can and do diverge: as the previous paragraph should make clear, their divergence is a consequence of the truth schema together with quite uncontroversial facts. Consequently, a norm of asserting the truth is a norm that goes beyond warranted assertibility. But there is no difficulty in desiring that all one's beliefs be disquotationally true; and not only can each of us desire such things, there can be a general practice of badgering others into having such desires. Isn't this enough for there being a 'norm' of asserting and believing the truth? Admittedly, this account of norms in terms of badgering is a bit crude, but I see no reason to think that on a more sophisticated account of what a norm is, norms of striving for the truth won't be just as available to the deflationist as they are on the crude account. I'll give a fourth example of the utility of a device of infinite conjunction and disjunction, since it clearly brings out some points I want to stress. Consider some theory about the physical world, formulated with finitely many separate axioms
and finitely many axiom schemas (each schema having infinitely many axioms as instances). For an example of such a theory, one can take a typical first order version of the Euclidean theory of space (which is not finitely axiomatizable). Suppose that one rejects this theory without knowing which specific part of it to reject; or alternatively, suppose that one accepts it but regards it as contingent. In the first case one will put the rejection by saying 'Not every axiom of this theory is true'; in the second case, by saying 'It might have been the case that not every axiom of the theory was true.' But the intended purpose in the first case is of course to deny the infinite conjunction of the axioms, and in the second case to assert the possibility of the negation of this infinite conjunction. My reason for focussing on this example is that it shows clearly the importance of what I earlier called pure disquotational truth. Pure disquotational truth involves two important features that certain other truth concepts lack. The first is that I can understand 'Utterance u is true' only to the extent that I can understand utterance u; the second is that for me, the claim that utterance u is true in the pure disquotational sense is cognitively equivalent to u itself as I understand it. (The first pretty much follows from the second: if I can't understand u, and if an attribution of truth to u is cognitively equivalent to u, then I can't understand an attribution of truth to u.) 20 The second feature of 19 Both Hilary Putnam (1983: xiv-xv, 279-80) and Crispin Wright (1992: 12-24) object to deflationism on this ground.
However, their discussions are directed at a version of deflationism which says that truth is not a property (Putnam) or not a 'substantial property' (Wright), and I'm not clear enough as to what that is supposed to mean to know whether the authors would have intended to be arguing against deflationism as I have defined it. (In a long footnote, Wright does say that he thinks his objection applies to the version of deflationism in Horwich 1990, despite the fact that Horwich disavows the 'no property' claim.) 20 Opponents of deflationism sometimes try to pin on it the claim that when we come to understand new sentences, or when new words and hence new sentences are added to our language, our concept of truth changes. But a deflationist can agree that on the most natural ways to individuate concepts over time, adding new sentences to the language (or to our domain of understanding) leaves the concept of truth unchanged: after all, once we have come to understand the new sentences, the truth schema dictates how the word 'true' is to be applied to them. This doesn't conflict with what is said in the text: the point there was that we don't now understand attributions of disquotational truth to sentences containing future words. (Also, there is no reasonable sense in which such attributions now have disquotational truth conditions, since what truth conditions they may come to have will depend on how the future words will be used.) end p.121
the pure disquotational notion of truth means that this notion is of a use-independent property: to call 'Snow is white' disquotationally true is simply to call snow white; hence it is not to attribute it a property that it wouldn't have had if I and other English speakers had used words differently. In ordinary English we seem to use a truth predicate that does not share these two features. It is not entirely obvious, though, that what we say can't be understood indirectly in terms of disquotational truth. (More on this in Sections 8 and 9.) In any case, these two features of pure disquotational truth make it ideally suited to serve the logical need for a device of infinite conjunction and disjunction illustrated in the examples above, particularly the Euclidean geometry example. In the first place, the only sentences we ever literally conjoin or disjoin are sentences we understand, so it is clear that a notion that is inapplicable to utterances we don't understand will serve our needs for truth as a device of conjunction and disjunction. Second and more important, the use-independence of disquotational truth is required for the purposes just reviewed. For if 'All sentences of type Q are true' is to serve as an infinite conjunction of all sentences of type Q, then we want it to entail each such sentence, and be entailed by all of them together. This would fail to be so unless 'S is true' entailed and was entailed by S. But the only way that can be so is if 'true' doesn't ascribe a use-dependent feature to S. Suppose for instance that Euclidean geometry is true, and that we try to express its contingency by saying that the axioms together might have been false. Surely what we wanted to say wasn't simply that speakers might have used their words in such a way that the axioms weren't true, it is that space itself might have differed so as to make the axioms as we understand them not true. A use-independent notion of truth is precisely what we require. Of course, there is another use-independent notion of truth besides disquotational truth that would be usable for these purposes: truth as applied to propositions, where propositions are construed as language-independent (rather than, say, as equivalence classes of utterances). But I think that pure disquotational truth is better for these purposes, for two reasons. The first and less important is ontological: it is better to avoid postulating strange entities unnecessarily. The more important reason is that unless one is very careful to limit the use made of the notion of expressing a proposition, the introduction of such a notion of proposition would beg the question in favor of inflationism.21 There may be good arguments for inflationism, but the need to express 21 We can introduce a purely disquotational notion of what it is to express a proposition, with the property that claims
of the form ' "p" expresses the proposition that p' are necessary truths. We can also introduce certain impurely
disquotational notions of what it is to express a proposition; these are analogous to the impure disquotational views of truth that we will consider later. Deflationists about meaning-that and believing-that think that such disquotational notions of expressing are the only legitimate ones; inflationists disagree. This distinction between inflationist and deflationist notions of expressing a proposition obviously needs to be clarified, if one is going to talk in terms of propositions; I think that some of the rest of the paper suggests a way to clarify it, but I will obviate the need to discuss this by avoiding talk of propositions. end p.122
infinite conjunctions and disjunctions can be handled with the more minimal apparatus of purely disquotational truth, and it is best to use only the more minimal apparatus as long as the need of the more powerful apparatus is unargued. I have been arguing that inflationists as well as deflationists need some use-independent notion of truth as a device of infinite conjunction and disjunction, and that such a notion is needed only for sentences that one understands. Pure disquotational truth serves the need admirably; and I think that a deflationist should take pure disquotational truth as the fundamental truth concept. This is not to say that it is the only notion of truth available to the deflationist: in Section 9 I will consider the possibility of introducing a modified kind of disquotational truth, which both applies to sentences in languages we can't understand and is not use-independent. But I have argued that a deflationist needs a notion of pure disquotational truth, and I think that he should regard it as the basic notion of truth in terms of which others are to be explained.
6. Disquotational Truth versus Tarskian Truth In this section I consider the relation between Tarskian truth definitions and disquotational truth. To facilitate the discussion I will pretend that English is a language without indexicals or demonstratives or ambiguous sentences, so that we can apply talk of truth to sentence types. (I will consider what the deflationist should say about indexicality, ambiguity, and the like in Section 10.) As I have explained disquotational truth, axiomatic status is given to some generalized version of the truth schema (T) 'p' is true if and only if p. The generalized version might be the result of prefixing the schema with a universal substitutional quantifier; alternatively, we might prefer the weaker approach involving schematic variables, mentioned earlier. (Again, I assume sufficient restrictions on the schema to avoid semantic paradoxes.) Actually, if our language contains a modal operator we need something a bit stronger, as the Euclidean geometry example of the previous section indicates: the schema (T) should be replaced by: (NT) □['p' is true if and only if p]. But even when the language doesn't have a modal operator, the left hand side of (T) is to be understood as cognitively equivalent to the right hand side. (Any end p.123
view, no matter how inflationist, accepts the instances of (T) as material biconditionals.) Tarskian approaches are somewhat different. First I should remind the reader that though Tarski was interested in explicitly defined truth predicates, he also proved a severe limitation of such explicitly defined predicates: they are available only for fragments of one's language. If we explicitly define a truth predicate in a language, not only must we exclude its application to some or all sentences of that language that contain semantic terms, we must also exclude its application to a substantial body of other sentences of the language without semantic terms but with unrestricted quantifiers. And this of course limits the use of the truth predicate as a device of infinite conjunction and disjunction: for instance, we can't use it to form infinite conjunctions and disjunctions of sentences with unrestricted quantifiers, at least if arbitrarily large numbers of alternations of those quantifiers appear among the conjuncts or disjuncts. If we want to avoid such limitations on formulating infinite conjunctions and disjunctions, we should give up on the idea that our truth predicate must be explicitly defined. Even if a Tarskian were to give up the insistence on defining truth, there would still be an important difference between the Tarskian approach and the full-fledged deflationary approach that takes a generalized version of the truth schema as an axiom. For central to what is usually thought of as the Tarskian approach (though perhaps it owes more to Davidson than to Tarski himself) is that truth 22 is characterized (inductively if not explicitly) in terms of compositional structure. 23 This gives compositional principles of truth a much more central role than they have on the full-fledged deflationary account. Recall the deflationist story I gave about why the word 'or' obeys the usual truth table: the principle
If S 1 and S 2 are sentences of our language then (S 1 or S 2 ) is true iff S 1 is true or S 2 is true is built directly into the inductive characterization of truth on the Tarskian approach, whereas on the deflationary approach it is simply a consequence of the generalized truth schema (together with the principle ' S(S is a sentence Σp(S = 'p')', on the substitutional quantifier version of the generalized schema). Something similar holds for other compositional principles, such as 22 Or a more basic notion like satisfaction from which truth is defined. 23 Actually even the demand of a compositional inductive definition must be relaxed, if we want to both get a notion
of truth that applies to sufficiently many sentences that contain semantic terms and simultaneously avoid limiting the truth predicate to sentences containing no unrestricted quantifiers: the usual methods like Kripke's for defining a notion of truth that applies to sufficiently many sentences that contain 'true' require a restriction of quantifiers even in the inductive definition (not just in turning it into an explicit definition); that is, what is being defined isn't truth but only 'comes out true under such and such a restriction of the quantifiers'. I think that a lesson of the paradoxes is that we must go to an axiomatized truth theory rather than defining 'true', even inductively. But this doesn't undermine the point intended in the text: for one might well demand (more or less in the spirit of Tarski or at least Davidson) that an axiomatized truth theory be built around explicit compositional axioms like 'A disjunction is true if and only if at least one disjunct is true'. A more full-fledged deflationist view would be that such compositionality has no intrinsic importance. end p.124
that a sentence consisting of a 1-place predicate and a referring name is true iff the predicate is true of what the name denotes: on a deflationist account this is simply a consequence of generalized disquotational schemas for 'true', 'true of', and 'refers' (together with principles like the one just given parenthetically, together with some obvious syntactic principles and laws of concatenation and quotation marks). 24 Why does it matter whether such principles are built into an inductive characterization of truth or viewed as consequences of generalized disquotational schemas? One reason why it matters is that there is no guarantee that for all regions of discourse there will be such compositional principles that follow from the generalized disquotation schemas. The compositional principles for 'or' and other truth functional operators simply fall out of the logical rules of inference that govern those operators (together with the disquotational principles for 'true'); the same is basically true of other standard compositional principles (except that we sometimes need subsidiary semantic notions such as satisfaction, and disquotational principles governing them). But there is no obvious reason to think that in the case of all operators, e.g. non-extensional ones, there will be subsidiary semantic notions that yield compositional principles. This shouldn't upset the full-fledged deflationist: for the full-fledged deflationist, such compositional principles have no particular interest in their own right; we can explain why we have them when we do, since they follow from the basic disquotational schemas whenever a substitutivity principle is part of the logic, but they are not of foundational importance. For the Tarskian they are of much more fundamental importance: they are needed in order to have a satisfactory notion of truth. This is one important way in which a Tarskian approach isn't fully deflationist. A Tarskian approach may or may not be partially deflationist: whether it is depends on the status that is accorded the Tarskian 'truth definitions'. In particular, what (if anything) makes such a definition correct for a given population? For a view to count as in any serious sense deflationist, it must say either that there is no sense at all to be made of speaking of one Tarski-predicate for a population as 'correct', or that the only sense that can be made of it is based on stipulating that the homophonic truth predicate counts as correct for us. To elaborate: it is easy to use Tarski-predicates in a theory that is in no way deflationist. One view of the significance of a Tarskian truth definition (whether inductive or explicit) is as partially characterizing an abstract 24 Let P be a one-place predicate and N a name. By the assumptions indicated, ΣF(P = 'F') and Σa(N = 'a'). By laws
of concatenation and quotation marks, it follows that ΣFΣa(P N = 'Fa'). By syntax, ΠFΠa['Fa' is a sentence], so by the truth schema, ΠFΠa['Fa' is true iff Fa]. By first order logic, we can rewrite this as ΠFΠa['Fa' is true iff x(Fx and x = a)]. But then the appropriate schemas for truth-of and reference will allow us to rewrite this as ΠFΠa['Fa' is true iff x('F' is true of x and 'a' refers to x)]. Combining with the previous paragraph, we get that P N is true iff x(P is true of x and N refers to x). The step labelled 'by first order logic' implicitly assumes that N denotes, that is, that Σa[N = 'a' and x(x = a)]. If this supposition is abandoned, some form of free logic must be used; the details of the free logic affect the details of the compositional principle that is derived. end p.125
language, that is, a language viewed as an abstract entity that exists whether or not anyone actually speaks it: the language is defined by the rules governing it, including Tarskian truth rules. (See Lewis 1975; Soames 1984; Stalnaker
1984.) So in addition to English there is an abstract language English*, with the same rules of grammar but different truth rules: e.g. in English* 'or' might obey the truth table for conjunction and 'rabbit' might be true of all and only the dinosaurs. Then questions about the truth conditions of a person's sentences are relocated as questions about which abstract language she speaks. An inflationist thinks that there are facts about a person's employment of her sentences by virtue of which it is one abstract language rather than another that she is speaking; such facts determine that Aristotle spoke in abstract Greek, and in the same way determine that we are speaking abstract English. The problem is to say what these facts are, and how they do the determining. But a deflationist thinks that a homophony condition guarantees that we are speaking English rather than English*, without the need of any such facts about our employment of the language. 25 There is of course a strong suggestion of the deflationist viewpoint in Tarski himself, stemming from his famous adequacy condition on truth definitions, 'Convention T'. (I should actually say, a strong suggestion of the partial deflationist view, because of the points earlier in this section.) My point is only that if you deemphasize these and focus on the truth definitions themselves, and think it is a straightforwardly factual question which truth definition is correct for a given population, then there is nothing in the least deflationist about the resulting view.
7. Objections to Deflationism There are many different sorts of worries that one might have about deflationism. Some seem to me to be based on misunderstandings. Under this heading I would include the commonly voiced worry that deflationism cuts language off from the world. The worry is that if I simply accept 'T-sentences' like 'There are gravitational waves' is true iff there are gravitational waves, then unless the sentence on the right hand side already has truth conditions of a more-than-disquotational sort, it has nothing to do with gravitational waves, and so the T-sentence doesn't really supply truth conditions to 'There are gravitational waves'. The response is that we do indeed need to establish a connection between the use of 'There are gravitational waves' and gravitational waves, independent of the truth schema. But deflationism allows for this (at least in the version I have sketched): it admits that among the important facts about the use of 'There are gravitational waves' are facts, stateable 25 At least, this is so for what I've called a 'pure deflationist'. The idea of 'impure deflationism' will be considered in
Sect. 9. end p.126
quite independent of disquotational truth, that relate the sentence to gravitational waves. For instance, the laws of physics are such that gravitational waves, if they exist, will cause pulses in a quadrupole antenna, and such pulses are one of the things that would increase our confidence in the sentence 'There are gravitational waves'. The objection tacitly assumed that the only kind of connection between gravitational waves and the use of 'There are gravitational waves' (other than the derivative connection that arises simply from the use of 'true' in accordance with the disquotation schema) comes in the non-disquotational assignment of truth conditions to that sentence; but this is simply false. In contrast, many of the worries that one might have about deflationism raise deep issues. In my view the most serious worry about deflationism is that it can't make sense of the explanatory role of truth conditions: e.g. their role in explaining behavior, or their role in explaining the extent to which behavior is successful. Unfortunately it is a big job even to state the worry clearly, and a bigger job to answer it; I must save this for another occasion.26 Another kind of worry about deflationism that is too big to discuss here is that it can't make the distinction between vague and nonvague discourse, or between discourse that is thoroughly factual and discourse that falls short of this in various ways. I discuss this worry in Chapter 8. But there remain some less serious objections that I can discuss here; discussing them will help clarify what I am taking deflationism to involve. The objections to be discussed are 1. That deflationism can't handle the attribution of truth to utterances in other languages. 2. That deflationism gives the wrong modal properties to 'true'. 3. That deflationism can't handle ambiguous utterances, indexicals, demonstratives, and the like. 4. That it can't make sense of how we learn from others. I think all of these can be easily answered, but it is worth spelling out how this is to be done, for my treatment will separate my version of deflationism from certain other less radical versions. I will treat these objections in order in the final four sections.
8. Applying 'True' To Sentences in Other Languages I have emphasized that a person can meaningfully apply a pure disquotational truth predicate only to utterances that he understands. This raises the question of what we are doing when we talk of the truth or falsity of sentences in other languages. 26 I made a rather confusing attempt to state the problem in Field 1986, Sections iv and v . I now think there is a way
around the problem there raised. (Stephen Leeds helped set me straight on this: see Leeds 1995, a draft of which I received just as this paper was being completed.) [For further discussion, see sec. 7 of the postscript.] end p.127
There are at least three different approaches that a deflationist might take toward talk of truth for sentences in other languages. Perhaps the most obvious— but to my mind the least satisfactory— would be to use a notion of interlinguistic synonymy: regard 'S is true' (where 'S' stands for a foreign sentence) as equivalent to 'S is synonymous with a sentence of ours that is true in the purely disquotational sense'. This option brings the semantic notion of synonymy into the characterization of truth for foreign sentences. I will call truth as so defined 'extended disquotational truth'. Some may object that extended disquotational truth is not a notion that a deflationist can allow himself, since it involves interlinguistic synonymy, which must be understood as involving sameness of truth conditions. I hesitate to say anything so strong; I say only that a deflationist who wants to employ a notion of interlinguistic synonymy faces a serious challenge, the challenge of showing how to make sense of it without relying on a prior notion of truth conditions. (Recall that the deflationist/inflationist contrast was explained in terms of whether truth conditions play a role in semantics and the theory of content.) But perhaps this challenge can be met. If for instance one were to adopt a crude verificationism, synonymy would be just interpersonal sameness of verification conditions— assuming of course that that can be spelled out. On a more reasonable deflationist view, meaning will involve a variety of components: as discussed earlier, the meaning of a sentence of mine will depend on its inferential role for me, its inferential role for my fellow speakers, and its indication relations for me and my fellow speakers. So whether one of my sentences is synonymous with a sentence used by someone else (e.g., a foreign speaker) will depend on a variety of disparate facts. In that case there is no obvious way to get a useful and well-behaved synonymy relation. 27 Still, it isn't idiotic to suppose that there is an unobvious way to do it; so it isn't initially clear that a deflationist can't admit a useful and wellbehaved notion of extended disquotational truth. But extended disquotational truth isn't the only option the deflationist has available for talking about truth conditions for foreign sentences. A second option is more flexible: simply define what it is for a foreign sentence to be true relative to a correlation of it to one of our sentences: a sentence is true relative to the correlation if the sentence of ours correlated with it is true in the purely disquotational sense. This approach differs from the previous in that no claim is made that the correlated sentences are synonymous. Of course, if we accept the notion of interpersonal synonymy, we can use it to get the effect of the first option: we can say that a foreign sentence S is true in the extended disquotational sense iff there is a correlation of S to a synonymous English sentence such that S is true relative to this correlation. But the second option can 27 I'm inclined to think that the problem isn't just that interpersonal synonymy ('is a good translation of') is a highly
vague notion, but that it isn't a fully factual notion at all; rather, it is an evaluative notion. This is compatible with many claims of interpersonal synonymy being factual relative to the goals of the translator; but not all interpersonal synonymy claims need be factual even relative to the goals. (For more on the idea of evaluative claims as 'not fully factual', see Chs. 8 and 13.) end p.128
be applied by those like Quine who reject a notion of interpersonal synonymy, or by those who accept it but want to consider truth relative to correlations that violate it, or by those who want to keep track of where notions like synonymy are being used and where they aren't. I should note that even those like Quine who reject a notion of interpersonal synonymy grant that there are standards of translation: it is beyond controversy that some translations of a foreign sentence are better than others, at least for certain purposes. If someone hears an utterance of 'Comment ça va?' and translates it as 'You have just passed your French exam', then that grossly violates any reasonable standard of good translation. The skeptic about interpersonal synonymy holds that though we of course have standards of translation, they are a matter of being better or worse, not of right and wrong, and also are highly context-dependent (since the purposes for which they are better or worse vary from one context to the next); and that these features together keep them from generating any useful notion of synonymy. I don't want to discuss whether this rejection of interpersonal synonymy is a reasonable position. But note that even if it is, standards of better and worse translation can be employed to partly capture the import of claims about
extended disquotational truth: what we are doing when we conjecture whether some utterance we don't understand is true is conjecturing whether a good translation of the utterance will map it into a disquotationally true sentence we do understand. If 'good translation' is a highly context-sensitive and interest-relative notion that doesn't reflect an objective synonymy relation, then there is a certain non-objectivity to the question of whether the utterance we don't understand is true, but that consequence is not obviously undesirable. (Even a deflationist who thinks there is an objective notion of synonymy for good translations to reflect will have a similar consequence to swallow. For presumably if there is an objective synonymy relation, then there could be utterances that we know don't bear this relation to any utterances we understand, and yet presumably it could still sometimes make a certain amount of sense to discuss whether such an utterance was true. An inflationist can take such a discussion at face value; but as far as I can see, the only way for a deflationist to deal with this, even if he accepts a notion of 'objective synonymy', is to appeal to context-sensitive and interest-relative standards of translation that do not reflect objective synonymy.) In addition to extended disquotational truth and truth relative to a correlation, there is a third option for a deflationist who wants to apply 'true' to foreign utterances, and that is simply to use the concept of pure disquotational truth as originally defined in connection with the foreign utterance, without relativization. This may seem odd: didn't I say that my concept of pure disquotational truth applied only to utterances in my own language? No: I said that it was defined only for utterances that I understand. I understand the sentence 'Der Schnee ist weiss'. Following Quine (1953b: 135), I hold that I can apply schema (T) to it, getting end p.129
'Der Schnee ist weiss' is true iff der Schnee ist weiss, which I can perfectly well understand despite its not conforming to the grammatical rules of any standard language. (If it is objected that there would be a problem had I picked a sentence with a different meaning in German than in English, my response is that in that case 'Der Schnee ist weiss' would be ambiguous for me, and there would be no problems in this case that don't arise equally for ordinary ambiguity within a language. I will discuss whether ambiguity creates a problem for deflationism in Section 10.) I don't see any reason to choose between the second and third options: the deflationist can use whichever one suits his purposes of the moment. For some purposes the third option is better than the second. For instance, if I understand a foreign sentence that has no exact equivalent in English, the third option can be used to give more adequate disquotational truth conditions than the second. The third option is also more useful in certain explanatory contexts, when we want to explain a foreigner's behavior without bringing the English language into the explanation. But the second option is often more useful than the third, in that it can in principle be used even for sentences we don't understand. Sometimes that broader application can come in handy: for instance, in trying to come to understand a foreign sentence we may try out several different correlations to our own sentences to see which makes most sense. And in some contexts we want to use the second option in connection with semantic or quasi-semantic notions like synonymy or acceptable translation: for instance, in a perjury trial we might want to know whether there is any acceptable translation of an utterance that makes it come out true. Finally, there are contexts in which the second option is useful even in connection with our own utterances: if we are considering a counterfactual situation in which my use of words is very different, I may be interested in the truth conditions of my sentences relative to a mapping of my language into itself that is not the identity mapping. So a deflationist should not dispense with the second option. In many contexts though I think that the third option is the most natural. It is the one that accords with my original gloss on disquotational truth: a sentence is disquotationally true iff it is true-as-I-understand-it. I think then that the notion of extended disquotational truth should be avoided if possible: it is not obvious that a deflationist can make sense of it, it is less obvious that he needs it, and it seems better to keep the logical aspects of truth cleanly separated from any semantic and quasi-semantic elements.
9. The Modal Properties of 'True' A further point about extended disquotational truth as I have defined it is that it is a concept of little theoretical utility. That is because it is a curiously hybrid notion: whether a foreign utterance is true in this sense depends on semantic facts, but whether one of our own sentences is true in this sense doesn't (since end p.130
as applied to our own sentences, extended disquotational truth is equivalent to pure disquotational truth, which is useindependent). Recall that certain logical uses of the pure disquotational truth predicate 'true pd ' (for instance, the
modal version of the Euclidean geometry example) require that □ ('Snow is white' is true pd iff snow is white). So even if 'snow is white' had been used in English in pretty much the way that 'Grass is red' is actually used, 'snow is white' would still have been true in the purely disquotational sense, and hence in the extended disquotational sense as well. Moreover, if in addition grass and the German language had been as they actually are, then 'Das Gras ist rot' would have been synonymous with 'Snow is white', and hence would have been true in the extended disquotational sense too. The counterfactual supposition of a drastic change in the use of English sentences leaves their extended disquotational truth conditions the same but alters the extended disquotational truth conditions of foreign sentences! This, surely, is not a theoretically useful way to make truth conditions use-dependent. Obviously what we would need to do, if we wanted a more useful notion of truth conditions that has semantic features built into it, is to build in such features for our own sentences as well as for the sentences of others. And of course inflationary theories of truth conditions do precisely this. The question is, can one do this on a theory of truth conditions with any claims to being deflationary? There is an obvious modification of the extended disquotational approach: I don't know if a view of this sort should be called deflationary or inflationary, so I'll call it quasi-deflationary. It requires that we have not only a notion of synonymy, but a prior notion of meaning such that two sentences are synonymous if they have the same meaning; but meaning is to be defined independently of truth conditions. (The crude illustration would be that meanings are just verification conditions; slightly less crudely, perhaps meanings are some sort of equivalence classes of pairs of conceptual roles and indication relations; the point of using equivalence classes being of course to allow that small differences of conceptual role don't make for a difference of meaning.) Then we define quasi-disquotational truth (truth qd ) and quasi-disquotational truth conditions by: □ (S is true qd iff Σp[ m (m is the meaning of S and @(m is the meaning of 'p')) and p]), and analogously, □ [S has the quasi-disquotational truth conditions that p iff m(m is the meaning of S and @(m is the meaning of 'p'))]; where '@' is an 'actually operator' which 'temporarily undoes the effects of' the modal operator. 28 This seems to have the modal properties we desire if we 28 See e.g. Crossley and Humberstone 1977. Quantification over meanings seems necessary in this definition:
certainly you can't replace the right hand side by 'Σp[@(S is synonymous with "p")) and p]', since this makes only the actual meaning of S relevant to whether S would be true in a non-actual context, contrary to intentions. And you can't simply take meanings to be equivalence classes of expressions under the (actual) synonymy relation: for then the proposed definition would collapse into the inadequate one just mentioned. In these remarks I'm assuming that you don't quantify over possible worlds: if you do, then you can use the relation expression e as used in world w is synonymous with e' as used in w', and take meanings to be equivalence classes of pairs of expressions and possible worlds under this equivalence relation. end p.131
want to mimic an inflationary truth predicate in making the truth of sentences use-dependent. Indeed, it mimics it well enough so that it is natural to wonder whether it shouldn't count as an inflationary notion of truth and truth conditions. For note that according to it, a sufficient condition for having the truth conditions that snow is white is having a certain 'meaning' m: perhaps a certain combination of such things as conceptual role, indication relations, and the like. Why then shouldn't this count as an explication of truth conditions in terms of such things as conceptual role and indication relations? It does nonetheless seem to me that there may be something to be said for counting a view like this as somewhat deflationary, if the idea of a meaning is explained in a sufficiently 'un-truth-theoretic' way. After all, it is compatible with this view that there be no very systematic connection between 'meanings' and truth conditions: for instance, it is compatible with this way of defining truth conditions, albeit not very plausible, that the 'meaning' of 'Snow is white or grass is green' have few systematic links to the 'meanings' of 'Snow is white', 'Grass is green', and 'or'. Note further that this view assigns truth conditions only to sentences that are synonymous with English sentences (or anyway, sentences that we understand). This points up the fact that there need be no 'natural connection' between the 'meanings' it assigns and truth conditions: the connection between the meanings it invokes and truth is supplied entirely by the disquotation schema for sentences we understand. So the view is somewhat deflationary, though just how deflationary it is will depend very much on the details of what it says about 'meanings'. One kind of worry about views of this sort concerns whether any acceptable notion of 'a meaning' and of 'having a meaning' will do the job required. Putting aside purely ontological concerns, the main worry concerns the notion of synonymy (having the same meaning) that would be generated by any such view of meaning and the 'having' of
meanings. Long ago Quine observed that it is an open question whether interpersonal synonymy and intrapersonal synonymy can be viewed as two aspects of the same general equivalence relation. (Quine 1953a, 56; see also Quine 1960, Sections 9 and 11.) Obviously we employ different criteria in the two cases— for instance, in the intrapersonal case we use a speaker's willingness to substitute one expression for the other, and this criterion is unavailable in the interpersonal case— and reflection on the criteria we do use in the two cases can lead to skepticism as to how well they cohere. I won't try to elaborate this skepticism here, I simply want to note that such skepticism is possible, and that it would make it impossible to regard intrapersonal synonymy and interpersonal synonymy as both cases of the general relation of 'having the same meaning'. (In addition, end p.132
the explanation of synonymy as having the same meaning requires synonymy to be an equivalence relation; this is another assumption about synonymy that has been questioned.) I'm not sure how seriously to take these worries, but as I said in connection with extended disquotational truth, I think it advantageous to cleanly separate the semantic assumptions one is employing from the logical aspects of truth: failing to do so can easily lead to obscuring the role to which one is putting one's semantic assumptions. (I think in fact there is a serious danger of this in the case of quasi-disquotational truth conditions: it is easy to fall into trying to give them a kind of explanatory role that in fact is precluded given the peculiar way they are defined. But this and other issues about explanation are beyond the scope of this paper.) In any case, I think a full-fledged deflationist should make no commitment to a notion of interpersonal synonymy, let alone to interpersonally ascribable meanings. Let us return now to counterfactuals like (1) If we had used 'Snow is white' in certain very different ways it would have had the truth conditions that grass is red. Presumably the average person would call such a sentence true. Doesn't this show that the average person is clearly not using the word 'true' in its purely disquotational sense? And doesn't that in turn show that a version of deflationism that puts purely disquotational truth at the center of things, and which refrains from endorsing intersubjective meanings and hence quasidisquotational truth, is gratuitously departing from common sense? I don't think so. We could after all say that the 'cash value' of (1) is (1*) in considering counterfactual circumstances under which we used 'Snow is white' in certain very different ways, it is reasonable to translate it in such a way that its disquotational truth conditions relative to the translation are that grass is red; and that this is all the ordinary person really means to assert. However, I am suspicious of this and all other claims about what the ordinary person means in making assertions involving 'true': I rather doubt that there is a consistent way to make sense of all ordinary uses of this notion. I am rather inclined to think that many ordinary uses of 'true' do fit the purely disquotational mold, though I regard the question of whether this is so as of only sociological interest. If, as may well be true, there is an ordinary meaning of (1) that (1*) fails to capture, then that would show that ordinary speakers are committed to a notion of truth that goes beyond the purely disquotational. But if we can lessen those commitments in a way that is adequate to all practical and theoretical purposes, and if in doing so we can still capture the 'cash value' of ordinary utterances as well as (1*) captures the cash value of (1), then the charge that we are 'gratuitously departing from common sense' is quite unfounded. end p.133
10. Ambiguity and Indexicals So far I have been a bit cavalier about whether a disquotational notion of truth applies to sentence-types or to utterances. I want it to apply to utterances, so that if an ambiguous sentence or a sentence containing an indexical or demonstrative is uttered on different occasions, we can regard some utterances of it as true and others as false. 29 One substantial worry about deflationism is whether it can accommodate this. In dealing with this worry it is important to bear clearly in mind that talk of truth conditions as I am primarily construing it is thoroughly non-semantic. 'True' in the purely disquotational sense means 'true as I understand it': it doesn't mean 'true on the correct understanding of it', because the idea of a 'correct understanding' of a sentence or utterance is a semantic notion that has no place when we are discussing purely disquotational truth. If on my understanding of 'Der Schnee ist weiss' it is equivalent to 'E = mc 2 ', then for me this sentence is disquotationally true iff E = mc 2 .30 Anyone in the grip of the Frege-Russell tradition will think that this shows that we need a notion of truth conditions very different from the disquotational one; but I don't think it at all obvious that they are right. As I observed in Section 8, a deflationist can grant that there are standards of adequacy of translation that do not presuppose truth conditions, and can grant the legitimacy of saying that relative to a translation manual that meets those standards the sentence will
have different truth conditions. So a notion like 'true as I ought to understand it (given my goals of translation together with the facts about how the sentence is used)' certainly make sense; it just needs to be distinguished from the purely disquotational notion that I am taking to be primary. Can the purely disquotational notion be applied to ambiguous utterances and indexical utterances? Let's start with ambiguity. I believe that it poses little problem for the deflationist. There is a rather commonsensical story of how we process ambiguous sentences, which I assume the deflationist can help himself to: the story has it that when we come upon a two-way ambiguous sentence like 'Visiting relatives can be boring' or 'I met my lover at the bank', there are two alternative ways of processing it and using it in inferences, one corresponding to each of the interpretations; indeed, interpreting it one way rather than the other just is processing it in one way rather than the other (for instance, storing it in memory in such a way that it will be used in inferences of one kind rather than the other). When we are undecided as to how to interpret an ambiguous sentence, what we are undecided about is how to process and store it and which of the two sorts of inference to make with it. The fact 29 Incidentally, I remind the reader that I am counting it as an example of ambiguity if someone understands the same
sentence as meaning one thing in German and something else in English: this was required by my adopting the 'third option' for dealing with sentences in other languages, discussed in Sect. 8. 30 Since it is disquotationally true iff der Schnee ist weiss, and hence, by the equivalence hypothesis, iff E = mc 2 . end p.134
that we can process and store a sentence in two different ways can be put metaphorically by saying that we attach 'inner subscripts' to lexically ambiguous elements and 'inner syntactic markers' to mark different ways of resolving the syntactic ambiguities. This metaphor allows us to talk, if we like, as if each person thinks in an ambiguity-free language. On this way of talking, to say that an utterance is disquotationally true is to say in effect that the sentence in my ambiguity-free inner language which I associate with the utterance is disquotationally true. Of course, if I am undecided how to interpret the utterance, there is no unique ambiguity-free inner sentence to use and hence no unique disquotational truth conditions will be generated. We can regard that as a case where we don't understand the utterance, so that the notion of disquotational truth doesn't apply— though of course we can also ascribe it truth conditions relative to each of the ways of understanding it. The metaphor of the ambiguity-free language is not really playing an essential role here: I could equally well have said that I accept all instances of 'That utterance of "Visiting relatives can be boring" is true iff visiting relatives can be boring', but may process the right hand side in either of two ways on different occasions. (I disallow inference from the right hand side processed one way to the right hand side processed the other way.) Now let's turn to sentences containing indexicals and demonstratives. Here it is natural to divide up the account of truth into two stages. The first stage concerns sentence types: here of course there is no hope of defining an unrelativized truth predicate, since sentence-types like 'I don't like her' don't have a truth value, but we can associate a truth value to this type relative to a pair of objects
interpret-it', the obvious thing to say is that an utterance of 'p(i 1 ,. . . ,i n )' is disquotationally true (for me, that is, as I understand it) iff the sentence is true relative to the values of a 1 ,. . . ,a n I regard as appropriate to associate with the indexicals.31 When I say that I 'associate values' with an indexical, of course, what I do is associate a mental occurrence of one of my own expressions (possibly itself indexical) with it. If I can't associate a term with an indexical in a sentence, then I can't attach disquotational truth conditions to the sentence. This is just an extension of a point made earlier for non-indexical utterances: sentences that I don't understand have no disquotational truth conditions for me.
I've said that the expression I associate with an indexical may itself be indexical: I may assign disquotational truth conditions to an utterance of 'She is here now' by thinking 'She is here now'. As with ambiguity, the deflationist needs to say something about how thoughts involving indexicals are processed, but the story is very similar to the story for ambiguity. When I think a thought involving 'she' to myself on a given occasion, that thought will typically hook up causally to a certain 'internal file drawer' of thoughts involving other singular terms, perhaps 'Sheila' and/or 'that woman I saw at the beach last Friday'. (It is possible that all the singular terms in the 'file drawer' are indexical: you can use an indexical to 'open a new file drawer'. Even when there are other terms in the 'file drawer', nothing need pick out any one person uniquely: the function of these associated terms isn't to explain how we can be referring to a certain person, if indeed we are; the function is simply to tell an obvious story of what it is for me to regard a given occurrence of 'she' as 'about Sheila' or 'about that woman I saw at the beach'.) Again, you could if you like say that the internal occurrences of indexicals are 'subscripted' so as to remove their indexicality; I take this to be just a dispensable manner of speaking for something better stated in terms of the internal processing. We see that there is no special difficulty in explaining disquotational truth conditions for indexical tokens. If indexicals raise any special difficulty for deflationism, it is that for indexicals it is less believable that we don't need a more inflationist notion of truth conditions. Surely, it may be said, there will typically be a correct answer to the question of who another person was referring to with a particular application of 'she'; and the deflationist seems unable to accommodate this. It seems to me though that the internal processing story does a lot to accommodate it. In a typical case where we misinterpret a token of 'she'— where we incorrectly interpret the speaker as meaning Mary when in fact she meant Sheila— we do so because of false beliefs about the speaker's internal processing: we think that her token was connected up to an internal file drawer of thoughts involving terms like 'Mary' when in fact it was connected up to a file 31 Typically these values will be the ones I take the producer of the utterance to have intended, but not always: e.g.,
'now' on the sign intended to be read by various readers at different times. And since we're talking disquotational truth for a hearer, the actual intentions of the producer don't matter, what matters is the reading the hearer gives. end p.136
drawer involving terms like 'Sheila'. Standards of appropriate translation are going to rule that in that case, 'Mary' is a bad translation of 'she'. The sort of facts that this brings into the standards of good translation are not in themselves semantic, so it seems hard to argue that the deflationist isn't entitled to appeal to them. Of course, it may be possible to argue that when we describe the standards of acceptable translation for indexicals in detail we will have to bring in machinery that is powerful enough to provide a reduction of the semantic notion of reference to non-semantic terms; if this is so, then the would-be deflationist is in fact turning himself into a reductionist inflationist. As I remarked earlier, the distinction between deflationism and reductionist inflationism may not in the end be altogether clear. At any rate, it certainly isn't obvious without argument that the standards of acceptable translation for indexicals give the machinery necessary to reduce the semantic notion of reference to non-semantic terms.
11. Learning from Others The final objection that I will consider is that you need an inflationist view of truth conditions to make sense of how we learn from others. Much of our information about the world is acquired from others— in particular, from their utterances, and from the beliefs we infer them to have. The notion of truth figures in this process: typically, what we do is assume that certain things the other person tells us are probably true— or that certain things we can see that she believes are probably true. It might be thought that an understanding of this process requires a non-deflationary notion of truth; or at least, that it requires more than purely disquotational truth. There are actually two distinct lines of argument that might be given for this conclusion. The first is that since we learn from other people, including people who speak different languages, translation must somehow be involved, so that some notion of interpersonal synonymy is presupposed in the inference. If this were right it would show that pure disquotational truth, without a synonymy relation, was inadequate for these purposes: you'd need at least extended disquotational or quasi-disquotational truth, and maybe a more thoroughgoing inflationary notion, for the purpose. I'll get back to this after discussing the other line of argument. The second line of argument purports to show that even pure disquotational truth plus synonymy isn't enough. Let's slur over the possible need for a synonymy relation, by imagining that I am learning from Charley, a speaker of my idiolect. Charley says: 'You know, on one day in 1936 there was over a foot of snow on the ground in Mobile Alabama.' Do I infer that one day in 1936 there was over a foot of snow in Alabama? For most Charleys and most circumstances, surely not. But suppose that in the past Charley has asserted each of the following: 'You know, there are parts of Virginia that are north of parts of New Jersey'; 'You know, in 1928 the Soviet government secretly supplied
end p.137
arms to Chiang Kai-Shek to be used against Mao'; 'You know, Leon Russell was one of the studio musicians for the Fleetwoods.' Each time I've found the claim almost unbelievable, but on checking it has turned out true. If he has never said 'You know, p' for a p that has turned out false, then I think I might well believe his claim about Mobile. A crude way to formalize my inference is as an enumerative induction: all the sentences of a certain sort that Charley has uttered under certain circumstances that have been independently checked have been true; 'There was a foot of snow in Mobile' is a sentence of that sort uttered under those circumstances; so (in the absence of defeating evidence) probably that sentence is true. In addition to the inductive inference, we need an instance of the disquotation schema, to get from the claim about truth to the object level claim. Obviously the disquotationalist has no trouble with the last step. But the inductive part of the inference might seem more questionable. For suppose we were to try to formalize the inductive inference without use of the word 'true'. What we would get is this: 1. There are parts of Virginia that are north of parts of New Jersey. . . . n. In 1928 the Soviet government secretly supplied arms to Chiang Kai-Shek to be used against Mao. n+1. Charley said 'You know, there are parts of Virginia that are north of parts of New Jersey.' . . . 2n. Charley said 'You know, in 1928 the Soviet government secretly supplied arms to Chiang Kai-Shek to be used against Mao.' 2n+1. Charley said 'You know, on one day in 1936 there was over a foot of snow in Mobile Alabama.' Therefore (probably): On one day in 1936 there was over a foot of snow in Mobile Alabama. And a possible reaction to this inference so formulated is that it has no inductive force whatever: none of the premises have anything to do with snow or Mobile; the first n have to do with diverse topics, and the last n+1 have to do only with noises that Charley uttered. So, it might be argued, when we reformulate the premises and conclusion of the inference in terms of truth, this can't be the innocent reformulation that the disquotationalist says it is: for the displayed inference has no inductive force, but the reformulation in terms of truth does have inductive force. So the reformulated inference works by attributing a more substantive property than disquotational truth to Charley's utterances, a substantive property on which it is legitimate to enumeratively induce. end p.138
I have tried to set out this line of argument persuasively, but in my view it rests on a very naive theory of induction. (I'm not talking about its use of enumerative induction: I take the assumption that learning from Charley is based on an enumerative induction to be a simplifying assumption on which nothing important is likely to turn.) I grant that to someone who had no concept of truth whatever, the displayed inference would probably have no inductive force. But that doesn't imply that we who do have a concept of disquotational truth should attach no inductive force to the very same inference (or equivalently, to the inference as reformulated in terms of disquotational truth). To be sure, our attaching force to this inference does not inevitably follow from our having a disquotational truth predicate; in principle, someone could perfectly well have a disquotational truth predicate and regard it as 'unprojectable'. But the fact that we aren't like that, that we do regard it as 'projectable', doesn't seem to me to have much to do with whether the predicate is 'substantive' in the sense employed in the objection: it doesn't show that 'true' isn't just a logical predicate defined by its role as a device of infinite conjunction. (It's worth noting that someone could equally well introduce a more theoretically loaded truth predicate and find it 'unprojectable': 'grue' after all is a 'substantive' predicate in the sense in question here, that is, it is not a mere logical device in the way that disquotational truth is.) Our acceptance of enumerative inductions on 'true' is presumably due to having discovered that we get useful results by doing so, 32 and such a discovery does not require that truth be anything more than what the disquotation schemas tell us. If this is right then the second argument that learning from others is incompatible with deflationism has no force. How about the first argument, that learning from others requires an interpersonal notion of synonymy, so that pure disquotational truth by itself is inadequate to the job? This too seems wrong. Suppose Charley speaks another language, which I think I understand: anyway, I have a translation manual that I use for it. Suppose I have found out in the past that when Charley utters sentences, they usually turn out to be true as I understand them— that is, their
translations turn out true, on the translation manual I use. Then we have an inductive argument that the next one will be true, on the manual I use: in this way, we can inductively argue to the snow in Mobile, as before. As far as I can see it doesn't matter in the least whether this manual accords with the genuine interpersonal synonymy relation between his sentences and ours, assuming that such talk of the correct interpersonal synonymy relation even makes sense: truth-relative-to-the-manual is the only notion we need in learning from others. 32 We may well also have an innate predisposition to believe other people, barring evidence to the contrary; this does
not affect my point. end p.139
12. Conclusion I repeat that there are many other arguments to be considered for the need of a more-than-disquotational notion of truth or truth conditions, some much more complicated to deal with than any given here. What I have tried to do here is simply to motivate a fairly strong version of deflationism— strong in that it does not rely even on a semantic notion of synonymy— and to sketch how such a version of deflationism can overcome various objections. I think what I have done here makes some case for what I've called 'methodological deflationism', the idea that we should assume fullfledged deflationism as a working hypothesis. That way, if full-fledged deflationism turns out to be inadequate, we will at least have a clearer sense than we have now of just where it is that inflationist assumptions about truth conditions are needed.33 33 I've had helpful conversations with many people about these issues in recent years. Among those whose
comments have influenced my presentation are Marian David, Michael Devitt, Paul Horwich, Barry Loewer, and Stephen Schiffer. I have also incorporated some useful suggestions by the editors. end p.140
Postscript I'll start with two quasi-technical issues, then move on to issues of broader philosophical interest.
1. Proving Generalizations About Truth Tarski pointed out long ago that we do not get an adequate theory of truth by simply taking as axioms all instances of the truth schema: this doesn't give the generalizations we need (such as that whenever modus ponens is applied to true premises, the conclusion is true), but only gives their instances. In section 3 I made a proposal for how to get around the problem, but what I said could have used more explanation. What I suggested takes off from an idea of Feferman (1991) and others, of building schematic variables into the language and subjecting them to a rule of substitution. This allows their scope to be automatically extended as the language expands. Feferman used the idea in connection with schematic theories like number theory and set theory. It is a bit more complicated to apply the idea in connection with the theory of truth, since the schematic letters appear both inside and outside quotes. My proposal was to handle this (in a language with quotation mark names) by using not only a substitution rule, but also a rule that allows you to pass from a schematic theorem in which all occurrences of a certain schematic letter appear inside quotes to a generalization about all sentences. One thing I should have said is that the syntactic theory needs to be done with schematic variables as well, for we need a schematic version of syntax in the suggested approach to truth theory. Consider the passage from the schema (TF) 'q or r' is true if and only if 'q' is true or 'r' is true (derived from three applications of the truth schema, using a substitutivity principle) to the generalized version
(TFG) For all sentences S 1 and S 2 of our language, S 1 or The required derivation uses schematic syntax to go from (TF) to (TF*) 'q'
'or'
'r' is true if and only if 'q' is true or 'r' is true,
is true iff S 1 is true or S 2 is true.
which by rule (ii) of Section 3 yields end p.141
(TFG*) For all sentences S 1 and S 2 of our language, S 1 'or'
S 2 is true iff S 1 is true or S 2 is true;
(TFG) is simply a rewrite of this using the corner-quote notation.
How does this schematic variable approach compare with Tarski's? (Or rather, with the variant of Tarski's approach that gives up the idea of defining truth, but takes over the compositional clauses that Tarski built into his definition as axioms. My reasons for favoring the axiomatic version of Tarski over the definitional are given in Section 6 of the Chapter.) I'm not sure that the difference between the schematic variable approach and the axiomatic variant of Tarski is all that significant, but I do see a few advantages to the schematic variable approach. First, we have the same advantage that Feferman stresses for schematic number theory and schematic set theory: the theory of truth doesn't need to be revised as the language is expanded. Second, the use of ' "p" is true if and only if p' (understood now as a schematic formula that is part of the language) in the axiomatization seems to more directly capture the core of the notion of truth. (That's very vague, and I wouldn't put a great deal of weight on it.) Third and most significant, the theory doesn't depend on our having a compositional account of the functioning of the other devices in the language. An axiomatic Tarskian truth theory for certain constructions, for instance, belief sentences, is notoriously hard to provide; the schematic variable approach works without such an account. This last reason may well be thought to backfire: surely the compositional structure built into Tarski's axiomatization is important, and a theory of truth that leaves it out would be deficient. But the schematic variable approach doesn't leave it out, for I just sketched how a typical compositional axiom (the disjunction rule) can be derived. The derivation was based on a prior schematic derivation of (TF): 'p or q' is true iff p or q 'p' is true iff p 'q' is true iff q So 'p or q' is true iff 'p' is true or 'q' is true. The last line of this prior derivation follows from the earlier three only by a substitutivity principle that has no analog for problematic constructions like belief sentences, explaining why compositional principles are not generally available. The fact that truth is compositional in some cases but not others isn't fundamentally a fact about truth, but rather about the underlying logic of the domains to which it is applied. In summary, it is this last point that provides the main advantage of the schematic variable approach over the axiomatic Tarskian approach.1 But in most respects, the axiomatic Tarskian approach is itself quite 'deflationist'; for 1 The schematic variable approach can also be used if what we take as basic isn't the Tarski schema, but the
equivalence between A[True('p')] and A[p] where the context A[. . . ] isn't quotational or intentional or anything like that. Strictly speaking, it is that rather than the Tarski schema that I recommend taking as basic in sect. 1 of the chapter. end p.142
instance, the deflationistic 'inversion' of the theory of reference discussed in Section 4 would work just as well on a Tarskian approach as on a schematic variables approach.2
2. Semantic Paradoxes The paper was slightly cavalier in its treatment of the paradoxes, mostly because I have vacillated as to how best to avoid them. I think that our ordinary concept of truth involves principles that in classical logic lead to contradiction; any way of avoiding the contradiction within classical logic can only be a recommendation for how to improve our concept.
(Generally I find the distinction between revising our concept and revising our views while keeping the concept fixed to be a murky one at best, but this is one case that clearly seems to fall under the former heading.) Actually I should be more cautious: any reasonable candidate for our ordinary concept of truth involves principles that in classical logic lead to contradiction. The reason for the retreat to the more cautious statement is that it is not entirely implausible that 'our ordinary concept' bifurcates over whether certain instances of the schema 'if p then "p" is true' are correct (e.g., instances where 'p' crucially contains vague terms, or evaluatives: see the distinction between weak and strong truth in Chapter 8). Even so, it seems almost beyond doubt that on our ordinary concept, the converse schema holds, and that when one has produced a solid argument for 'p' one is entitled to conclude ' "p" is true'. These are all that are needed to derive a contradiction involving Liar sentences in classical logic. We can view the disquotational theory as put forth in this paper either as an account of our ordinary concept, or as an account of a concept that improves on our ordinary concept in certain ways. (As I've said, I don't think the distinction between these alternatives to be entirely clear.) But because of the paradoxes, further improvements in the concept will be needed if classical logic is assumed to apply to it. (For a very useful survey of some of the revised theories of truth that are available in classical logic, see Friedman and Sheard 1987.) Will the revisions needed for a consistent theory of truth in classical logic undermine the 'deflationary' nature of the resulting conception? Certainly not, if we count a pure Tarskian theory (the kind of axiomatic theory discussed in Section 6) as 'deflationary'. For all the revisions would do is complicate the axioms; they wouldn't, for instance, undermine the deflationistic 'inversion' of the theory of reference discussed in Section 4. On the other hand, to the extent that one builds into deflationism a special role for the schemas, deflationism would be somewhat undermined. I'm not entirely content to leave the matter there, because the revisions in the theory of truth required to restore classical consistency also tend to undermine 2 Shapiro (1998) and Ketland (1999) argue that the schematic variable approach undermines deflationism, at least if
one assumes that logic is first order. For a reply, see Field (1999). end p.143
central ways in which we use the notion of truth. (That's why the restoration of consistency seems such a clear case of alteration of the concept.) For this reason, I am tempted by the idea that we ought to keep the standard truth schemas, but weaken the underlying logic so that these schemas will no longer lead to contradiction. It is not clear that this requires a general weakening of classical logic: I'm tempted to think of the weakened logic as simply a special system of conventions designed to restore coherence to truth and related concepts. At any rate, if we can keep the schemas by weakening the logic then we don't need to qualify the deflationary nature of the truth theory in the least. As a result of Kripke's influential work (Kripke 1975), the first thing one thinks of under the heading of a revision in classical logic to deal with the paradoxes is the use of a logic based on the strong Kleene truth tables: more particularly, the logic obtained by taking the valid inferences to be those that are guaranteed to preserve 'correct assertion', and taking an assertion to be correct if and only if it has the 'highest' of the three semantic values. Unfortunately, this 'Kleene logic' as it stands does not fully support deflationism as we've been understanding it, because the Tarski biconditionals do not all come out correct. The inference from 'p' to ' "p" is true' and its converse can be maintained (as it can in classical logic); more generally (and unlike the situation with classical logic), one can maintain that 'p' and ' "p" is true' are everywhere intersubstitutable in extensional contexts. Maybe the latter preserves enough of deflationism, but it would certainly be desirable to be able to assert the Tarski biconditionals. 3 (Moreover, the Kleene logic seems unsatisfyingly weak, in that 'if A then A' is not generally assertable.) One initially attractive response is to add a new kind of conditional A→B, not equivalent to ¬AVB, together with the corresponding biconditional.4 This may require that in the semantics, the underlying space of semantic values be enlarged, in a way that doesn't alter the valid principles for the other connectives. An especially appealing proposal along these lines is to use the Lukasiewicz continuum-valued semantics; the principle that every continuous function from the unit interval into itself has a fixed point yields an easy argument that any ordinary paradoxical sentence can consistently be assigned a semantic value in this semantics, and in a very appealing way. ('Ordinary' paradoxical sentences include all those in which the only role of the quantifiers is to achieve self-reference. In particular, any Curry paradox sentence 'If this sentence is true then p' gets a truth value, even when 'p' has a value other than 0 or 1: its value is always (p+1)/2.) Unfortunately, certain non-ordinary paradoxical constructions give rise to a serious problem: though the truth 3 However, a proponent of Adams' (1975) thesis about conditionals might argue that what really ought to be involved
in a strong deflationism has nothing to do with the acceptance of sentences involving a biconditional connective. Rather, what ought to be involved is conditional degrees of belief: Cr(p | 'p' is true) and Cr('p' is true | p) should both be 1. I don't know if the details of this position could be satisfactorily worked out, but if so it would be another way of
defending a deflationism somewhat stronger than one that relies on compositional axioms. It would also remove the parenthetical difficulty that follows in the text. 4 Typically such a conditional will not have the monotonicity property that the Kleene connectives have, in which case Kripke's fixed-point constructions will obviously be inapplicable. end p.144
schemas don't produce outright inconsistency when applied to them, they do produce an ω-inconsistency in number theory and hence in protosyntax: see Restall 1992.5 I should also note that the inferential role of '→' that is validated by the semantics is quite unnatural: for instance, the inference from A→(A→B) to A→B is disallowed. Probably a still further weakening of the logic would allow us to escape the ω-inconsistency problem, but such a logic is likely to be even less natural. More attractive, I'm now inclined to think, is an idea that has been popularized by Graham Priest (e.g., Priest 1998): on Priest's view, the Liar sentence and its negation are each correctly assertable (as is their conjunction), and the rules of logic are weakened so that this does not allow the assertion of just anything. The particular version of this that I think most appealing (Priest's LP) is also based on the strong Kleene 3-valued truth tables, and again takes the valid inferences to be those that are guaranteed to preserve 'correct assertion'; but it takes an assertion to be correct if and only if it has one of the two highest of the three semantic values. The truth schema can be consistently added, as can the intersubstitutivity of 'p' and ' "p" is true' even in the interior of sentences.6 (The Curry paradox is also blocked, since the only conditional in the language is the material conditional, i.e. the one defined in the usual way from ¬ and V, and this conditional does not fully validate modus ponens in this logic.)7 The distinction between the Kleene logic and the Priest variant is commonly described as being that Kleene's takes 'deviant' sentences (e.g. the Liar) to have truth value gaps, i.e., to be neither true nor false, but takes no sentences to have truth value gluts, i.e. to be both true and false; whereas Priest's takes deviant sentences to have gluts and takes no sentences to have gaps. But for anyone who wants to maintain an equivalence between ' "p" is true' and 'p' (even in the interior of sentences), this is wholly misleading. For to assert a truth value gap in a sentence 'A' would be to assert '¬[True ("A") V True ("¬ A")]', which should be equivalent to '¬(A V ¬ A)'; but no sentence of that form can ever be a legitimate assertion to an advocate of Kleene logic, 5 Consider a sentence P that we can informally write as 'There are natural numbers n for which my n-fold jump is not
true'; where the jump of a sentence A is ¬(A → ¬ A) (which is a strengthening of A in this logic), and where the n-fold jump is the result of iterating the jump n times. (In effect P says 'I am not super-true', where to be super-true is to have all of one's n-fold jumps true, which on any reasonable way of relating truth to the semantic values is to have semantic value 1.) Even if we enlarge the value range from [0,1] in the reals to [0,1] in a non-Archimedean field and generalize the Lukasiewicz semantics accordingly, it is clear that the only value that can be assigned to P consistently with the truth schema in the semantics is within an infinitesimal of 1. But then for any standard natural number n, the n-fold jump of P also has values within an infinitesimal of 1; so the truth schema demands that for standard n, 'the nfold jump of P isn't true' has a value within an infinitesimal of 0. Clearly the only consistent resolution involves taking the 'there are natural numbers' in P to have nonstandard numbers in its range; in other words, the only consistent resolution involves a protosyntax with sentences (in particular, jumps of P) that are not genuinely finite in length. 6 Restricting to transparent contexts, of course. 7 There is something counterintuitive about the absence of a stronger conditional, but the counterintuitiveness can be somewhat ameliorated by using conditional probability judgements as a surrogate for conditionals, as in fn. 3. Alternatively, one could add stronger conditionals, as long as they fail to obey either modus ponens or the deduction theorem (or contraction): there are natural ways to do this by explaining the conditional modally, or in terms of derivability in an appropriate formal system. end p.145
whereas when 'A' is deviant this is a correct assertion according to Priest's logic! Similarly, to assert the absence of a truth-value glut in sentence 'A' would be to assert '¬[(True ("A") & True ("¬ A")]', which should be equivalent to '¬(A & ¬ A)'; and an advocate of Kleene logic can not assert this absence for deviant sentences, whereas an advocate of Priest's logic can. If one wants to state the distinction between the logics in terms of a deflationary notion of truth, the right way to do so is this: (i) on the Kleene logic one can't assert of a 'deviant' sentence either that it has a truth-value gap or that it doesn't, nor can one assert either that it has a glut or that it doesn't; (ii) in Priest's logic, one can assert both that it has a truth-value gap and that it doesn't, and one can also assert both that it has a glut and that it doesn't. 8 There is no distinction between gaps and gluts in either logic! What differentiates the logics is simply the 'threshold of assertability': deviant sentences are always assertable in Priest's, never in Kleene's. Since sentences attributing truth or non-truth or falsity or non-falsity to deviant sentences are themselves deviant, this means that the advocate of Kleene logic can say nothing whatever about the truth value of deviant sentences, whereas the advocate of Priest logic can say anything she
pleases about them. 9 My own view is that Priest's logic LP is quite appealing as a logic for dealing with semantic concepts, now that we know that they engender classical inconsistencies: it is probably the simplest way of coping with classically inconsistent concepts. (It does not, I think, have much appeal elsewhere, e.g. in dealing with vague and indeterminate concepts.) To summarize: (1) The paradoxes do not undermine deflationism as a reconstruction of our ordinary truth-theoretic concepts; (2) they certainly don't undermine deflationism even for our improved concepts, if 'deflationism' is used in the weak sense on which a Tarskian theory is deflationist; and (3) there are several ways of maintaining even a strong form of deflationism for improved truth-theoretic concepts.
3. The Generalizing Role of 'Refers' I stressed in the paper (following Quine and Leeds) that the term 'true' increases the expressive power of our language, by serving as a device for expressing infinite conjunctions and disjunctions (which could have been expressed by more obviously logical means like substitutional quantifiers). The idea obviously works equally well for 'true of'. But one might doubt that the same holds for 'refers' as applied to singular terms: we already have quantifiers 8 In Priest's logic one can also assert of any deviant sentence that it is 'solely true' (true and not false), and that it is
'solely false' (false and not true). 9 Priest often expresses his view as the claim that contradictions can be true. It might be better to say that sentences
of form 'p and not-p' aren't, or aren't always, genuinely contradictory. In any case, the claim that some such sentences are true is a bit misleading as a statement of the view, since it tends to make one overlook the fact that on his theory these true 'contradictions' are also not true! end p.146
that play a generalizing role in the name position, so it might seem that no expressive power is gained. I think this argument incorrect: ordinary quantifiers don't allow for generalizations of names that appear both inside and outside quotation marks, but that is what we would need to express what is said by 'Every name that came up in Department X's discussion of who to hire referred to a male' without use of 'refers'.
4. Untranslatable Utterances (1) I now think that my restriction of an agent's disquotational truth predicate to the sentences the agent currently understands wasn't quite right, and is in considerable tension with my remarks on our commitment to extending the truth schema as we expand the language. Those latter remarks suggest that once we come to regard an expression S as an acceptable declarative sentence, then even if we have no understanding of it (or virtually none) still we ought to accept the corresponding instance of the truth schema. Admittedly, the cognitive equivalence thesis implies that to the extent that we don't yet understand S, we don't understand the attributions of truth to it; this may seem to make the extension of the truth schema to it idle. But it is not completely idle, for we tie any future improved understanding of 'true' as applied to S to the future understanding of S. One way to think of the matter is in terms of indeterminacy: a sentence I don't understand at all is maximally indeterminate in content for me, and so is an attribution of truth to it; but the indeterminacies are 'tied together' (see the discussion of 'correlative indeterminacy' in Chapter 7; and Chapter 9, sects. 1 and 2) in such a way that the truth schema holds. The main point of modifying my view in this way is to better accommodate talk of truth for untranslatable utterances, utterances that are in no sense equivalent to anything in our current language. When I wrote the paper I was tempted to bite the bullet and say that we simply can't apply the notion of truth to such utterances; I wasn't happy with this, but could see no good alternative. Stewart Shapiro (unpublished) presented a nice problem case for this position. In Shapiro's example, we have a guru who makes pronouncements about set theory; a disciple who thinks that everything the guru says is true, but who doesn't understand set theory; and a logician who distrusts the guru's set-theoretic pronouncements, but likes to draw their number-theoretic consequences, which the disciple does understand. Shapiro argues that if in addition the disciple trusts the logician's acumen about logic (though not about set theory), then the disciple ought to be able to reason that since everything the guru says is true, and consequence preserves truth, and the logician truly says that the number-theoretic sentence 'p' is a consequence of what the guru says, then 'p' must be true, so p. But though the disciple understands this conclusion, the reasoning is blocked if we can't meaningfully apply 'true' to the guru's utterances. end p.147
There are ways of handling this example without modifying the view of the Chapter, but they aren't entirely natural, and I'm not sure that they would extend to all alterations of the example.10 But the modified view suggested two paragraphs back handles the example very neatly: the disciple regards the guru's sentences as in a potential expansion of the disciple's language, so that there is no difficulty in his carrying out the reasoning. It may seem mysterious that the reasoning can be carried out in this framework, since according to this framework the guru's utterances and the notion of truth for these utterances have quite an indeterminate content for the disciple. But the joint acceptance of the claims (i) that the guru's utterances are true, (ii) that they logically imply the number-theoretic claim 'p', (iii) that logical consequence preserves truth, and (iv) the truth schema, give just enough determinacy to enable the reasoning to be carried out.11 (The reasoning could be carried out even if the guru's language were disjoint from the disciple's, rather than an expansion of it; though in that case some might prefer to appeal to expansions of the disciple's language that translate the guru's utterances rather than that include them.) Incidentally, the guru problem is a prima-facie problem not only for the version of deflationism in the paper, but for versions of deflationism that take truth to be attributed primarily to propositions. The latter, if they are to have any claims to the title of deflationism, must explain what it is for a sentence to express the proposition that p in terms of equivalence in content to our sentence 'p'; but this explanation is unavailable for the guru's sentences. For those views as for mine, the idea that the attributions have a quite indeterminate content, constrained only by such beliefs as that they have specific number-theoretic implications, provides the way out.
5. Untranslatable Utterances (2) The incorporation model for dealing with foreign utterances has wider application than my remarks in Section 8 of the Chapter suggest. Indeed, in the next to last paragraph of Section 8 I explicitly ruled out using it for sentences that we don't understand. As the discussion of Shapiro's example should make clear, I recant. 10 For instance, in the example as it stands, where the guru and the disciple share an underlying logic, the disciple
has a notion of consequence that applies to the guru's language; one might argue that the disciple's faith in the guru is inadequately represented by the claim that everything the guru says is true, it should be represented by the claim that all consequences of what the guru says (indeed, all consequences of those together with other truths) are true. In that case, we would get the desired conclusion without the need to reason as in the previous paragraph. 11 It isn't essential to suppose that the disciple accepts the application of the truth schema to the guru's sentence prior to his knowing that an extraordinarily reliable guru uttered it and prior to knowing that it has the number-theoretic consequence. These pieces of information do give the disciple some very minimal understanding of the guru's sentence, so it isn't really necessary to hold that we can apply 'true' to sentences of which we have no understanding at all. end p.148
Consider our reaction when on a certain occasion Z we hear another speaker, Mary, use a proper name such as 'George', and we take her to be 'talking about someone we have never heard of before' (as we colloquially put it). If we are interested in what she says, we may pick up the name 'George' from her and start speculating with it ourself. ('I wonder if George lives in California.') When I say that we pick it up from her, I don't just mean that we start using the sound 'George': we probably used that sound already, e.g. in connection with various U.S. presidents. But we pick it up with a new use, not equivalent to these prior uses, and a use that is in some way deferential to Mary's: we base our own beliefs on hers (more accurately, on what we take her beliefs to be). 12 If you like, you can put this by saying that we mentally introduce a new term of form 'George j ' where 'j' is an 'inner subscript' that we haven't previously used in connection with 'George', and we use 'George j ' to translate those of Mary's assertions that we regard Mary as having 'mentally subscripted' by the same subscript that she mentally attaches on occasion Z. I do not see any reason to think that understanding this incorporation of Mary's term requires an antecedent grasp of the notion of reference. Of course, once we have a disquotational notion of reference, we can use it in this connection: I can say ' "George" (as I'm using it now) refers to George', using the term 'George' to incorporate Mary's use of it on the occasion in question. Or on the 'hidden subscript' picture: ' "George j " refers to George j (and to nothing else)'. Since we translate Mary's use of 'George' on occasion Z with 'George j ', we can also say that Mary's use of 'George' on occasion Z refers to George j (and to nothing else). In other words, I can conclude: George j is identical to the referent of Mary's term 'George' on occasion Z. This allows me to use the phrase 'what Mary was referring to by "George" on occasion Z' instead of 'George j '. Since
the inner subscripts are silent (and other people don't know my inner subscripts even were I to vocalize them!), doing so is sometimes useful: it is a very good way to make clear which way I'm using 'George' on this occasion. Of course, I could have introduced other means for doing the disambiguating: I could have just said 'I'm going to introduce the new term 'George Mary, Z ' (or 'Shrdlu') as an equivalent of Mary's use of 'George' on occasion Z; but since we have a notion of disquotational reference already, there is no need for such new terms. The approach carries over straightforwardly to demonstratives. Suppose that on occasion Z Mary uses the term 'that', and suppose that I have no idea how to pick out in English (either demonstratively or descriptively) what I take her to be talking about; alternatively, suppose I have little or no idea what to take her as talking about. Here it might be misleading to incorporate her use of 'that' into my language with my own use of 'that': normal conventions about 12 This is so even if we know Mary to be unreliable. For instance, if we know her to have exaggerated beliefs, we
might merely accept 'George is over 6′4″ ' when Mary accepts 'George is over 6′7″ '. Or if Mary gushes over George, we might accept 'Mary is infatuated with George' instead of 'George is a wonderful guy'. end p.149
the use of 'that' might suggest that my term stands for something quite different. But I could always introduce some special term, say 'Oscar' or 'THAT Mary, Z ', as a device for incorporating her use of 'that' on this occasion. But if I did so, I could argue just as above that in a disquotational sense of 'refers' (extended by translation), Oscar is identical to the referent of Mary's term 'that' on occasion Z; so given that we have a notion of disquotational reference already, there is no need for these special terms. The approach just suggested is quite similar to, and perhaps identical to, the proposal of Brandom 1984. What I've said is that we should view the phrase 'the referent of Mary's "that" on occasion Z' as functioning as a device for incorporating her use of 'that' on occasion Z into our language. What Brandom says is that it functions as a device of cross-speaker anaphora. I don't entirely like Brandom's way of putting things, because taken by itself it seems to suggest that there is a non-deflationary notion of reference applicable to Mary's pronoun, and that our anaphorically dependent phrase gets its referent from it. However, Brandom says things incompatible with that understanding of anaphora, and in support of some sort of deflationary story, and I think the incorporation model is at least very close to what he intended. The point that I've been making for reference can be made for truth as well. There are many reasons why a direct incorporation of another person's utterance into my language could be inconvenient (e.g. if I can't pronounce it) or misleading (e.g. if it contains indexicals that would shift their reference were I to use them). In such cases, we can use 'Her utterance is true' as a means of incorporating her utterance. This indeed is what Grover, Camp, and Belnap (1975) see as the main function of 'true'. I prefer to view the main function of 'true' as the disquotational function. But as with reference, I can argue from the disquotational use of 'true' to the incorporation use. For if I were to introduce a new sentence, say 'UTT Guru, Z ', to incorporate the guru's unpronounceable utterance on occasion Z, then the disquotational properties of truth give an equivalence between (1) 'UTT Guru, Z ' is true and (2) UTT Guru, Z ; but since 'UTT Guru, Z ' is my translation of the guru's utterance, (1) is equivalent to (1′) The guru's utterance is true, and so (1′) must be equivalent to (2). That is, (1′) is legitimized as a way of incorporating the guru's utterance, so we don't need the special sentence 'UTT Guru, Z ' any more. The discussion in this section (and in Brandom, and in Grover, Camp, and Belnap) shows another way, besides as a device of infinite conjunction and disjunction, in which 'true' and 'refers' can increase expressive power: or at least, end p.150
can increase the range of what can be expressed easily and in a way not subject to confusions of ambiguity. Again,
this expressive function could have been achieved by other means ('George Mary, Z ' or 'Shrdlu' in the case of 'refers'); but 'refers' and 'true of' are a convenient way to do it, and I have argued that if we did use another device for doing it we would have no need for the other device once we had the disquotational use of 'refers' and 'true'.
6. 'Pure Disquotational' and 'Quasi-Disquotational' Truth The central feature of 'pure disquotational truth' is that ' "p" is true' is cognitively equivalent to 'p' (at least, modulo the existence of the sentence). I say in the chapter that 'true' in the pure disquotational sense is to be understood as something like 'true-as-I-understand-it'; what I really meant was 'true-as-I-actually-understand-it'. As I emphasized, this is just a heuristic, and I claim that the notion does not require that we take 'ways of understanding' to be understood in terms of propositional content. I have a bit more to say about this in the next chapter. The paper recommends using this notion of pure disquotational truth where possible, but it of course recognizes that we use the notion of truth in connection with other people's utterances and in connection with our own utterances in counterfactual circumstances, and it suggests ways of accommodating this fact. One way is to use what I call 'quasidisquotational truth'; I down-grade it because it assumes a stand on Quinean issues about interpersonal synonymy that I prefer to remain neutral about. But the general idea behind quasi-disquotational truth could have been put in a way that is more neutral about interpersonal synonymy: (*) S is true qd (at possible world v) ↔ S is to be translated by a sentence of mine (in the actual world) that is purely disquotationally true (at v). My method of 'explaining away' modal intuitions about the truth of our sentences at the end of Section 9 really amounts to suggesting that it is true qd in the sense of (*) rather than pure disquotational truth that we are operating with in many contexts. I have come to think that it is unnecessary to use two distinct truth predicates, the purely disquotational and the quasidisquotational. We can use a single truth predicate as long as we take the entities in its extension not as orthographic types but as computational types: equivalence classes of (potential) tokens under the relation of computational equivalence. I explain what I mean by computational equivalence in the next chapter, but an important feature of it is that it is defined only within an individual X in a given possible world u: it doesn't make sense to ask if one of my tokens is computationally equivalent to a token of yours, or to a token of a counterpart of me in another possible world. So we can rewrite (*) with the individual and the world made explicit, as follows: end p.151
( ** ) S X,u is true qd (at possible world v) ↔ S X,u is to be translated by a sentence of mine (in the actual world) that is purely disquotationally true (at v). But then 'pure disquotational truth' can be viewed as just the special case where the individual is me and the world is the actual world (by assuming that in that case the 'translation' is just the identity function); ( ** ) is merely a way to generalize from the special case of me in the actual world to others and to my counterparts. (Of course the heuristic 'true-as-I-actually-understand-it' is appropriate only for the special case where translation drops out. It is worth emphasizing, though, that this special case plays a central role in the account: the other cases result from the special case by translation.) If sentences are typed orthographically, the same sentence can be used in distinct ways in distinct worlds and we need to distinguish between a truth predicate that 'holds meaning constant' and one which doesn't; but if sentences only exist within a single world then no such distinction is required. If ordinary or quasi-disquotational truth is defined either as in the text or by (*) or ( ** ), it has the curious feature that utterances not translatable into our language are not true. It seems more natural to suppose it indeterminate (or maybe just undetermined) whether they are true. The matter may be somewhat academic: if the points made in the two preceding sections of this post-script are correct, the notion of an untranslatable utterance does not have entirely clear application. Still, I would now prefer to do things in the more natural way. The way to do so is clear: instead of defining ordinary or quasi-disquotational truth in terms of purely disquotational truth, we explain it by a schema: ( ***) If S X,u is translatable as 'p' then □(S X,u is true iff p). [And if S X,u is translatable as 'p' then □(S X,u has the truth conditions that p).] In the paper I attached some importance to the idea that we ought to define quasi-disquotational truth, in terms of the more-or-less logical notion of purely disquotational truth together with translation. I must have thought that the motivation for demanding a substantive theory of truth and truth conditions and reference, like that advocated in Chapters 1 and 2, could only be undercut if these notions are explicitly defined in terms of more-or-less logical notions plus translation. But this seems a mistake: the argument for a substantive theory of truth conditions and reference
depends on taking truth conditions and reference as having a certain kind of 'causal explanatory' role, as noted in the postscript to Chapter 1; and introducing a notion of truth conditions by means of the schema ( ***) does nothing to make the notion 'causal explanatory'. Of course, using ( ***) to introduce the notion of truth conditions doesn't guarantee against putting the notion to the relevant sort of 'causal explanatory' use, but as long as one doesn't do this— and also doesn't make unwarranted determinacy claims for these notions (see footnote to postscript to Chapter 1)— then one can maintain a deflationist position. end p.152
7. Bigger Issues There are two big areas in which much more needed to be said than I said in this Chapter. First, there is little discussion of the aspects of meaning that go beyond truth, reference, and related notions. Don't we need a deflationary account of those aspects of meaning too? The answer is that we do need a deflationary account of meaning generally, but that there is no special difficulty for getting one once we have a deflationary account of truth and reference; I hinted as much at the beginning of the chapter, and say more about it in the next chapter. The other area in which a lot more needed to be said was the explanatory role of the assignment of truth conditions to mental states, both in the explanation of behavior and in the explanation of the extent to which behavior is successful in achieving various results. In a paper written some years earlier (Field 1986), I had flirted with deflationism but ended up tentatively arguing against it, on the ground that such explanations required a role for the assignment of truth conditions to mental states that the deflationist could not legitimize. The argument was so abstract and convoluted it couldn't possibly have convinced anyone; I was skeptical myself, but couldn't pinpoint what was wrong with it. I now think it made a number of mistakes that collectively deprived the argument of all force. Or as Jimmy Carter said of his attempt to rescue the American hostages in Iran (an attempt that had to be aborted with eight deaths before the rescue helicopters even reached Iran): it was 'an incomplete success'. It isn't worth going through the argument in detail here, but I'll sketch the sort of considerations it involved and what I now think a deflationist should say about them. One key point (which was recognized in the 1986 paper, but not made sufficiently salient) will be that there is nothing in deflationism that prevents the use of 'true' in explanations as long as its only role there is as a device of generalization. It is perfectly obvious that in explaining how a pilot manages to land a plane safely with some regularity, one will appeal to the fact that she has a good many true beliefs: beliefs about her airspeed at any moment, about whether she is above or below the glideslope, about her altitude with respect to the ground, about which runway is in use, and so forth. (None of these beliefs need be based very directly on observation; she might be flying in bad weather with some of her instruments not working, so that she must rely on complicated cues.) The deflationist obviously needs to grant this explanatory role for the truth of her beliefs, and will have to say that it is somehow licensed by the generalizing role of 'true'. Can this deflationist strategy be maintained? A detailed explanation of how the pilot functions would involve the existence of some class C of internal representations, intuitively representing airspeed, such that when she believes a representation in C that represents too high an airspeed she slows the plane and when she believes one that represents too low a speed she speeds it up; end p.153
and the representations in C that she believes tend to be true. The last part can of course be rephrased as a generalization: she tends to believe a representation in C that represents too low a speed when the plane is in fact too slow, etc. But is this restatement of the claim that she has true beliefs about airspeed 'innocent' from the deflationist viewpoint? What may well seem a problem is that it is put in terms of what the pilot's internal representations 'represent'. The basic idea behind the tortuous argument of the 1986 paper was that the deflationist is forced to understand the sense in which the representations 'represent' airspeed in terms of their translation into the explainer's language, but that this is inappropriate since this should be an objective explanation in which the explainer plays no causal role. 13 An obvious strategy for responding to this argument is to say that talk of representation is serving a merely heuristic role in the explanation of the pilot's ability; put without the heuristic, the explanation involves the existence of some class C of internal representations in the pilot and two subclasses C 1 and C 2 of C such that (i) when she believes a representation in C 1 she slows the plane and when she believes one in C 2 she speeds it up, (ii) there is a 1-1 function f from C to a certain set of real numbers such that (a) C 1 is that subclass of C that is mapped into numbers above a certain threshold and C 2 is that subclass of C that is mapped into numbers below a certain (slightly lower) threshold, and (b) she tends to believe a representation r in C when the airspeed in knots is approximately f(r). This makes clear
that we have a perfectly objective explanation, expressible without translating the pilot's representations or talking of their truth conditions or of what they represent. Of course part (ii)(b) of the explanation uses an indication relation, but no deflationist could object to that. (See the postscript to Chapter 2.) The function mapping internal representations into airspeeds needn't even give the intuitive truth conditions of those representations in all cases: one could tell a story in which the pilot's beliefs about what she was doing were so weird that it would be natural to assign quite different truth conditions to her representations. (Perhaps she believes she isn't in an airplane at all, but is using the controls to direct US ground forces on a foreign mission.) The representations would be reliably correlated with airspeed but would represent something quite different that they were not reliably correlated with. It is also important to realize that in normal cases (as opposed to cases where the pilot has very weird beliefs about what she is doing) we can explain the 13 As noted in Leeds (1995: 28-9), arguing in such a manner seems prima facie to be misguided: of course it is the
correlation between the agent's states and external conditions that is of primary explanatory importance, but why should the fact that we use a correlation between an agent's states and our sentences as a means of setting up the correlation be thought to undermine this? Why should we think that there must be some means of specifying the relevant correlation that doesn't go via translation into our language but is given by a substantive theory of truth conditions? I would rather not press this line, however, because doing so leads quickly to complicated issues. (Indeed, the most confusing parts of the 1986 paper concern precisely this.) Instead of arguing that the basic line of argument against the deflationist is misguided in very conception, I will argue merely that it cannot be carried out in detail. I believe that this way of proceeding has the additional advantage of making clearer the role that talk of representation plays in the explanation. end p.154
pilot's competence in less detail, by simply saying that she tends to believe truths about the airspeed. (We would have to rest content with this if we had no idea that the pilot needed to maintain airspeed within a certain range to safely land, but knew only that she needed to base her actions somehow on her airspeed.) The objective core of such an explanation is clear: it is that a functional correspondence between internal representations and airspeeds like that mentioned above is playing some sort of role that we may not be in a position to specify. But when we are not in a position to specify that role with enough precision, it is inevitable that we use the notion of truth in one way or another. One way to do so would be to just say: some explanation roughly like the one in the previous paragraph is true; here it is transparent that truth is playing simply a role in generalizing, and so clearly this way of proceeding is available to the deflationist. But this way of proceeding requires that we have a sample objective explanation, however crude, to use as a model, and it is not always convenient to provide one. But the alternative way of proceeding mentioned at the start of this paragraph does not require this: if we just say 'She tends to believe truths about the airspeed, and this somehow enters into her landing successfully', we can be vague on the details without even sketching a sample precise model. Is this latter way of proceeding available to the deflationist? 14 When we give this sort of explanation, we say in effect (i) that she has certain internal representations that are similar enough in their role to our representations of form 'The airspeed is n' to count as 'saying the same thing', (ii) that they tend to be disquotationally true under the translation scheme just alluded to, and (iii) that the fact that their disquotational truth conditions under this mapping tend to be satisfied plays an important role in explaining the safe landing. This is a 'projectivist' or 'second class' explanation, in that we make use of a similarity between the agent and ourselves to avoid the need to fill in explanatorily crucial details; but it is a projectivist explanation of a broader sort than those considered in Section 4 of the postscript to Chapter 2. The projectivist explanations discussed there are ones that in effect assert the existence of fuller explanations in the agent's computational psychology, explanations which they leave only vaguely specified. Here, on the other hand, the projectivist explanations in effect assert the existence of fuller explanations in the agent's extended psychology which they leave only vaguely specified; where the extended psychology is the computational psychology together with the 14 In probably the most crucial passage in my 1986 paper (p. 97 middle to 99 top), I assumed not, without any clear
argument. (In the immediately preceding pages I had discussed avoiding mention of truth conditions in explanations, along the lines mentioned two paragraphs above. I then raised the question about what to do when we didn't know the full explanatory details. The discussion is obscure: it recognizes that it is legitimate for a deflationist to use a disquotational truth predicate as a way of giving an explanation sketch in absence of full knowledge of the details (and that even an inflationist would need to resort to this), but concludes for reasons that it doesn't make at all clear that we would need something more 'inflationist' in addition.) Stephen Leeds rightly focuses much of his critique of my paper on this crucial passage (Leeds 1995: 17-20), and is understandably baffled as to what my argument could have been. For what it's worth, my own diagnosis is given in the next footnote. end p.155
assumptions about correlations between the agent's internal states and the external world. But the central point is the
same: assignment of truth-conditional content is something we need only when we don't know how to fill in the details of the explanation. 15 The same basic point holds in cases where the agent is less reliable in his beliefs about the external world. Suppose I explain Kennedy's having avoided nuclear war over Cuba in terms of a lucky guess as to what Khrushchev would do. Again, the objective core of the explanation is that Kennedy's action tends not to lead to war if Khrushchev acts as he did but to lead to war otherwise, and that the action was crucially based on his having believed a certain representation; if we knew a lot of detail about Kennedy and his decision-making process, we could say a great deal about this representation and the role in his psychology, but typically we don't know enough to do this. So here we specify the representation by its truth conditions; or what amounts to the same thing, by a translation into our own idiolect. That is, our explanation is that Kennedy believed (or anyway, assumed as a basis for action) some representation which had for him roughly the role that 'Khrushchev will blink' has for us, and his believing or assuming this was a salient cause of his actions, and that those actions didn't lead to war only because Khrushchev blinked. Again, it is a 'second class' explanation, in that a comparison to our psychology is used in place of specifying the relevant details of the agent's psychology. As such, it is nothing that causes any problem for a deflationist. 15 I think the key problem in the 1986 paper was an ambiguity about 'projectivist explanation'. In some passages I
used it in the more inclusive sense, as when I argued that the deflationist could appeal to the truth conditions of another agent's states only in projectivist explanations. But in other passages I used it in the narrower sense, as when I argued that the role to which we put the assignment of truth conditions in explanations of 'success phenomena' like landing the plane with regularity can't be projectivist. The confusing structure of the paper disguised the ambiguity. end p.156
5 Attributions of Meaning and Content Abstract: Is concerned with the special epistemological status possessed by attributions of meaning to sentences we understand: such attributions seem a priori, in a strong sense that includes empirical indefeasibility. What explains this special status? One explanation involves the idea that attributions of meaning (and of belief, etc.) express relations between an expression (or an agent) and a linguistic item in one's own language; in the special case of a meaning attribution to one's own language, the attribution is trivial. Argues that this linguistic view of attributions can be defended against well-known objections. Also argues that alternatives in terms of propositions are perfectly acceptable, if suitably understood; they can even be developed in a way that accommodates Quinean doubts about interpersonal comparison, by viewing the propositions as 'local entities'. But (whether or not one accepts the Quinean doubts), it is essential to view the assignment of propositions in a somewhat deflationary spirit, not altogether removed from a linguistic view of them, if the special epistemological status is to be explained. Keywords: belief, interpersonal synonymy, meaning, propositions, Quine, truth, truth-conditions
Hartry Field
I Consider Schemas like (M s ) 'p' means that p, where 'p' is to be replaced by any meaningful sentence of our language; or more generally, (M) 'e' means < e > , where 'e' is to be replaced by any meaningful expression of our language. The bracket quotes are an invented notation, but 'Hund' means < dog>
is intended to be simply a formal representation of our usual devices of meaning-attribution for subsentential expressions; it is intended to be neutral to the question of what sort of entity if any the bracket-term refers to. In the case of sentences we don't need special bracket terms, because we have 'that' clauses, but we can regard phrases of form ' < p > ' as simply an alternative notation for the corresponding phrases of form 'that p', so that (M S ) is simply a special case of (M). I should say something about the use of the schematic letters 'p' and 'e' in the schemas above. My focus will be on the individual thinker. An individual thinker has an idiolect— a set of expressions that he or she understands (in a way that may or may not accord with how others understand it)— which will typically not coincide with the boundaries of any language, or of the union of any languages. I will take the appropriate substitution instances for the schematic letters in the above schemas for a given person (at a given time) to be the expressions in that person's idiolect (at that time) that are of the appropriate syntactic category. (Understanding comes in degrees, so the boundaries of a person's idiolect are somewhat vague, and so there is some vagueness as to which substitution instances are appropriate. I don't think that this fact will matter much in what follows.) Of course, if there are ambiguous expressions in the person's idiolect, the above schemas need to be adjusted for this. One way to do this would be to rewrite them as end p.157
'p' (as I'm now understanding it) means that p and 'e' (as I'm now understanding it) means < e > . Some would want to explain 'ways of understanding' sentences in terms of propositions (and uses of subsentential expressions in terms of 'parts of propositions'— concepts, in one sense of that term). I don't refuse to talk in terms of propositions in all contexts, but I think that such talk often obfuscates the issues, and that it would do so here. When a sentence in a speaker's idiolect is ambiguous, the speaker processes different tokens of it in systematically different ways: for instance, some tokens of 'Visiting relatives can be boring' have a direct inferential connection to 'My cousin is usually boring when he visits me' and others don't. We don't need to invoke relations to distinct propositions to make the distinction between the two 'readings' of this or any other ambiguous sentence. (The point can be extended to sentences that contain indexicals: two of my tokens of 'She is beautiful' said of obviously different women have different 'readings' in that they are 'linked up with distinct inner file drawers'. I'm using 'means' in such a way that they differ in what they mean.) Instead of invoking 'ways of understanding', it might be clearer to stick to the original versions of (M S ) and (M), but stipulate that the quotation mark name refers not to an orthographic type but to a computational type: it refers to a class of expression tokens (or potential tokens) in the meaning-ascriber's actual idiolect that are treated computationally as equivalent. This is somewhat reminiscent of the dot quotes employed in Sellars 1962. But those were supposed to refer to interpersonally ascribable conceptual roles, and thus give rise to Quinean problems about which beliefs affect conceptual role and which don't. The notion of sameness of computational role I'm employing should be regarded as meaningful only for expressions in the same idiolect. Clearly the schema (M) and its special case (M S ) have a special epistemological status: they are not open to serious doubt on empirical grounds. Prima facie, this is somewhat puzzling: it certainly isn't beyond doubt or empirical discovery what a word in another language means; how then can it be beyond doubt or empirical discovery what a word in my language means? (Of course, there is an obvious sense in which what it means in my language isn't beyond empirical discovery: I might learn that 'metacompact' doesn't mean quite what I suppose it does. But the point is that I can't learn that it doesn't mean < metacompact > .) What explains the special epistemological status of the schema?
II One possible answer is the linguistic view of meaning attributions. Putting aside doubts about the clarity of interpersonal synonymy for now, here's an initial way to put the view: to say that a word means < rabbit > is simply to
say that it end p.158
means the same as 'rabbit' as I actually understand that word. (Or as I'm actually understanding it now, if the word is ambiguous.) The 'actually' is there to handle counterfactual contexts: in a world like the actual world except that the English usage of 'rabbit' and 'cockroach' are interchanged, 'cockroach' and the French 'lapin' mean the same as 'rabbit' actually means for me, so they mean < rabbit > . The key idea is that although a meaning attribution may appear to relate a word to a mind-independent entity (a meaning), it really just relates the word to one of the meaning-attributor's words (computationally typed). Or perhaps, to one of the meaning-attributor's 'mental symbols' that is causally correlated to his use of the word; I'll count that as not essentially different. Of course, the relation of the foreign word to my word (or my 'mental symbol') is based on characteristics of each word separately, e.g. on the inferences that govern certain kinds of sentences containing the words and on the wordly conditions that typically lead to the assent to other kinds of sentences involving the words. Let's call such characteristics of a word (on which translation of the word or synonymy judgements about it are based) its meaningcharacteristics. To make this explicit, we can restate the linguistic view roughly as follows: to say that a word means < rabbit > is just to say that it has the same meaning characteristics as the meaning characteristics of my actual use of 'rabbit'. Here the meaning-characteristics of a word do not include relations to intentional entities (meanings or concepts); indeed, I take it to be possible to describe the meaning-characteristics of a word without using any semantic terms at all, including terms like 'refers'. This may need a small refinement: the requirement of exact sameness of meaning characteristics is obviously appropriate only if you understand 'meaning characteristics' in a suitably coarse-grained sense. So as not to require this special coarse-grained notion, we can restate the idea in terms of a special equivalence relation M on the set of possible meaning-characteristics, finely described: (LV) To say that a word means < rabbit > is just to say that it has meaning characteristics that are M-equivalent to the actual meaning characteristics of my term 'rabbit'. Of course, interpersonal sameness of meaning is at least somewhat indeterminate, which means that if we do things this way, we should recognize that it is indeterminate precisely which equivalence relation 'M-equivalence' should be taken to stand for. Indeed, on a more Quinean view the problem with interpersonal synonymy goes beyond mere indeterminacy; on such a view, we should avoid mentioning equivalence relations at all, and just say something like: (QLV) To say that a word means < rabbit > is to say that its meaning characteristics make it appropriate to translate the word into my actual language as 'rabbit'. end p.159
I will not decide between these two variants of the linguistic view. If the linguistic view (in either form) is acceptable, then it is obvious why the meaning schema should have special epistemic status: its instances in effect have the form (*) p if and only if actually p, whose special epistemic status is generally recognized and understood. (More accurately, they have the form of generalizations of sentences of form (*): they say that for any meaning characteristic C, my word has C if and only if it actually has C.) It seems clear that any considerations favoring the linguistic view of meaning attributions would equally favor a linguistic view of saying-that attributions, belief-attributions, desire-attributions, knowledge-attributions, and so forth.
III There are a number of possible objections to the linguistic view of meaning-attributions. One is the Church translation argument (Church 1950): it is alleged that if ' "lapin" means < rabbit > ' says that 'lapin' means the same as 'rabbit', then its German translation ought to be ' "lapin" bedeutet < rabbit > ' rather than ' "lapin" bedeutet < Kaninchen > '; which goes against both the practices and purposes of normal translation. (Recall that the brackets are just my notation for ordinary meaning-attributions; if sentences were in question, I could just use 'that' clauses.) Schiffer (1987) poses the
point slightly more sharply: simplifying his formulation a bit, let's imagine that Karl and Fritz are monolingual German speakers, and Karl is told that Pierre's utterance means the same as 'Der Schnee ist weiss' while Fritz is told that it means the same as 'Snow is white'. Schiffer argues in effect that if we were to interpret 'S said that snow is white' linguistically, as 'S said something equivalent to "Snow is white" ', this would have the absurd implication that it is Fritz rather than Karl who believes that Pierre said that snow is white. In my view, the Church-Schiffer argument against the linguistic view shows only that the view needs to be formulated a bit more carefully. Before saying what the argument does and doesn't show, let me tell a little story. Imagine an English novelist whose books are translated into French. He is dismayed to discover that in the French translation of the first novel, the translator (a philosopher of language) left the quotations untranslated, on the ground that when the novel said that the character uttered a certain English sentence it would be a mistranslation to have the character uttering a French sentence. To guard against such bloody-mindedness, the novelist in his second novel used special quotation marks, #-quotes, when he wanted the dialog translated. (He decided that the need to resort to this device had unexpected benefits, because occasionally he wanted English quotations left untranslated, and this allowed the translator to know which was which.) Of course, he had underestimated just end p.160
how bloody-minded philosophers of language can be: his translator still left the quotations untranslated, saying that whatever the author wanted, the #-quotes still referred to English sentences and so it would be a mistranslation to translate them otherwise. So for his third novel, the author told his publisher not to hire a translator, hire instead a quasi-translator who will translate within #-quotes. The advocate of the linguistic view as described above is like the novelist: he is undone by bloody-minded philosophers of language. The idea behind the linguistic view was that the referent of an English 'that'-clause is the sentence contained in it, and more generally the referent of a bracket-quotation is the same as an ordinary quotation; semantic claims like (1) 'Lapin' means < rabbit > and (2) 'La neige et blanc' means that snow is white, which appear to relate French expressions to intentional entities, really just relate them to English expressions. (Similarly, the sententialist will hold that when we say that a person says that snow is white or believes that snow is white, we are expressing a relation between the person and the sentence 'Snow is white'.) So stated, I think the idea is quite defensible; but what needs to be added is that the bracket-quotes and 'that'-constructions don't behave quite like ordinary quotation marks (at least as those are standardly construed by philosophers of language), they work instead like the novelist's #-quotes. This means of course that we want them to be quasi-translated rather than 'literally translated' (on the Church-Schiffer understanding of what literal translation involves). But this point about the kinds of translation we want should be taken as reflecting a view of what features of language we care about. We don't care much about 'literal meaning', if that is what is preserved in 'literal translation' as conceived by Church and Schiffer; what we care about, rather, is what is preserved in quasi-translation. We might in deference to Church and Schiffer call this quasi-meaning, though I think it is what most would simply call meaning. The rationale for our interest in 'quasimeaning' rather than (what I reluctantly call) 'literal meaning' is that what each of us must do in understanding foreign utterances is to compare them with utterances we already understand, and such comparisons are what is preserved in quasi-translation but not in 'literal translation'. To summarize, one key part of sententialism is that our interest is in 'quasi-translation' and 'quasi-meaning' rather than 'literal translation' and 'literal meaning'. The other key part is that the way to understand (2) is as: (2′) 'La neige et blanc' means the same as # Snow is white # ; or better, as (2″) 'La neige et blanc' quasi-means the same as # Snow is white # ,
where again, quasi-meaning is what is preserved in quasi-translation. The use of #-quotes rather than ordinary quotes around 'Snow is white' guarantees that the quasi-translation of (2′) or (2″) into German involves ' # Der Schnee ist weiss # '. end p.161
The use of 'quasi-means' instead of 'literally means' in (2″) is unimportant when discussing 'La neige et blanc', since 'literal meaning' and quasi-meaning coincide for that sentence, but it becomes important when a sentence with #quotes is under discussion. (This includes a sentence with a 'that' clause or bracket-quotes, given the linguistic construal of these.) The fact that it is quasi-meaning rather than meaning that is important to the sententialist is one good reason for preferring (2″) to (2′) (or rather, to (2′) with 'means' interpreted as 'literally means'); another good reason is implicit in the next paragraph. I claim that this version of the sententialist doctrine is immune from the arguments of Church and Schiffer.1 The point with regard to Church is too obvious to require comment. The Schiffer argument is a bit more interesting, in turning on a double intentional context: believing that a person says that p. Clearly any sententialist who elevates quasi-meaning over meaning will hold that to believe that q is to accept a sentence that quasi-means the same as # q # . So to believe that Pierre said that snow is white is to accept a sentence that quasi-means the same as (3) Pierre said that snow is white; on our sententialist doctrine, that will be a sentence that quasi-means the same as (4) Pierre said something that quasi-means the same as # snow is white # . Since clearly it is Karl rather than Fritz who accepts such a sentence, the Schiffer argument has no force against this version of the linguistic view.
IV A rather more interesting set of issues to raise about the linguistic view is its implications about explanation: in particular, the fact that if we explain what a person does in terms of her belief that p, the linguistic construal has it that 1 Stephen Leeds (1979) gives a superficially quite different response to the Church argument, which may not
ultimately be as different as it initially seems. A key part of his response is that the German word 'bedeuten' doesn't literally mean the same as 'means', or even have the same extension: the first relates 'dog' and 'Hund' to 'Hund' but not 'dog', the second relates them to 'dog' but not 'Hund'. But 'bedeutet' and 'means' do nonetheless stand in an important homology relation, similar to that between the extensionally different predicates 'the temperature-inFahrenheit of x is r' and 'the temperature-in-Centigrade of x is r', which makes it reasonable to translate 'bedeutet "Hund" ' as 'means "dog" '. Leeds, like me, insists that capturing the literal content is simply not important in translation of 'means' sentences (in his case, it's because we have no way to translate that preserves literal content). Indeed, I see no reason to think that Leeds and I differ as to the 'quasi-content' that should be preserved. The accounts of quasi-content in terms of literal content differ, but that needn't reflect a difference about quasi-content, but merely the difference in the views about literal content: I'm assuming the Church-Schiffer view, according to which 'means' and 'bedeutet' literally have the same content, while he is assuming an alternative. Other comparisons: The fact that translators merely 'quasi-translate' was also emphasized by Dummett (1973: 372); he too says that this undermines the Church argument. It is mostly because of the added twist in Schiffer's argument that I saw fit to give more detail. Sellars (1962) used his dot quotes (mentioned above) for attribution of meanings and attitudes; if I'm right in saying that his dot quotes were to name intersubjectively ascribable inferential roles, this isn't a linguistic view (nor is it an adequate view, in light of standard externalist objections to sameness of inferential role as a criterion of translation). end p.162
we are in a sense bringing ourselves into the explanation: we are saying that the person accepts a sentence (or mental representation) which has a role in her psychology like the role that the sentence 'p' (or a mental representation that we associate with it) plays in ours. I don't think this consequence of the linguistic view is terribly upsetting: in explaining the behavior of things of whose detailed workings we are ignorant, we often bring in other things we assume to work in roughly analogous ways. ('I bet your car is acting up like that because it has the same engine problem that my car had
last week.') In fact, I think it a virtue of the linguistic view that it brings out this feature of intentional explanation; but I can not discuss these issues here. (See the postscripts to Chapter 2 and Chapter 4.) Instead, I will simply note an initial reason for thinking that it is not propositions, but rather sentences we understand, that do the work in explanations of what people do. For in order to use Rebecca's beliefs in any normal explanation of her behavior you must be able to supply a sentence you understand (whether in public language or Mentalese) that expresses her belief: you must be able to say 'she believes that p', where 'p' is a sentence you understand. Being able to designate the proposition Rebecca believes in some other way (e.g. via a sentence you don't understand, or a list of possible worlds, or a name like Horace) would be of no use to us in our normal explanatory practice. (Objection: 'That's because these wouldn't tell me which proposition it is.' Response: Maybe, but if so, that's because our standard for knowing which proposition it is that the belief expresses is being able to supply a sentence we understand that expresses the proposition. That should give the game away: it is the sentence we understand that does the explanatory work.)
V One might also object that on this view of meaning, to say that 'Der Schnee ist weiss' means that snow is white will not imply, without auxiliary premises, that it is true if and only if snow is white. (It will of course imply that it is true if and only if our sentence 'Snow is white' is true; but the claim that our own sentence 'Snow is white' means that snow is white will be trivial on this view, and won't imply that our own sentence is true if and only if snow is white. So saying that 'Der Schnee ist weiss' is true if and only if 'Snow is white' is true won't imply, without further information, that it is true if and only if snow is white.) Because of this, it might be objected that on this account, attributions of meaning to sentences leave out a key ingredient of meaning, viz. truth conditions. Similarly, it might be objected that on this account, the description of the meaning of a singular term or predicate has no implications for the conditions under which it refers to or is true of a given object, again leaving out a key ingredient of meaning. I don't really see why an attribution of meaning to 'Der Schnee ist weiss' must imply that it is true if and only if snow is white all by itself, i.e. without end p.163
use of the uncontroversial background fact that 'Snow is white' is true if and only if snow is white. But even if one accepts that it must, an advocate of the linguistic theory has an easy response if he adopts a disquotational theory of truth, reference, and the like. According to this disquotational view, the claim that 'Snow is white' is true if and only if snow is white, and the claim that 'the teacher of Alexander' refers to something if and only if that person was the teacher of Alexander, are not contingent empirical claims about our language, they are analytic truths. So when I say that a foreign term is synonymous with 'the teacher of Alexander', I imply in a quite literal sense that the foreign term refers to an object iff that object is the teacher of Alexander, and so the objection that the view divorces meaning from reference has no weight. Indeed, it is worth noting that an advocate of the linguistic view can adopt a version of disquotationalism that is often regarded as more plausible than standard versions. Standard versions of disquotationalism take as central such schemas as (T) 'p' is true if and only if p and (R) For any x, 'b' refers to x if and only if b = x. But an advocate of the linguistic view can instead take as central such schemas as (T*) It is true that p if and only if p and (R*) For any x, < b > refers to x if and only if b = x;
for if we adopt a linguistic view of 'that' clauses and other bracket terms, these amount to the same thing. The only advantage of (T) and (R) over (T*) and (R*) is in making unambiguously clear that it is our own sentences and expressions that we are talking about, rather than special intentional entities. The equivalence of (T) and (T*) on a linguistic view implies that they have the same modal status. In my favored version, they are both necessary. The view of quotation mark names as naming computational types can be used to motivate this. For just as sameness of computational type seems ill-defined across speakers, it seems ill-defined across possible worlds, or at the very least, across possible worlds where our use of language is importantly different. To say that at another possible world w a computationally typed sentence is true can only mean that the sentence as understood at our world is true-at-w. So to say that □('Snow is white' is true iff snow is white), i.e. that at all worlds w, 'Snow is white' is true iff snow is white, should be understood as meaning that for all worlds w, 'Snow is white' as understood at our world is true at w iff at w snow is white. And this is correct. The linguistic view also allows us to put the deflationary view in a more general way, applicable to arbitrary languages (and to our own language in other possible worlds): the generalized versions are end p.164
(T G ) For any sentence S (of any linguistic community C, in any possible world u), if S means that p (for C in world u) then S (as used by C in world u) is true (at world w) if and only if p (at w). and (R G ) For any term t (of any linguistic community C, in any possible world u), if t means < b > (for C in world u), then for any x, t (as used by C in world u) refers to x (at world w) if and only if x is b (at w). These together with the appropriate instances of (M) (namely, 'p' means that p and 'b' means < b > ) give rise to (T) and (R). This more general formulation of disquotationalism has advantages; its disadvantage is that the unwary might fail to see that it contains a hidden bias toward our own language, which is here built into the notion of meaning that p or meaning < b > . (The generalization is in effect the same as the 'quasi-disquotational' truth discussed in the previous chapter, except that here I have been neutral as to whether the notion of sameness of meaning on which the meaning attributions are based is purely factual. See also Section 6 of the postscript to that chapter.)
VI So far I have simply defended the possibility of a linguistic theory of meaning attributions, and said that it offers a neat answer to the question of why the instances of the meaning schemas don't seem open to doubt on empirical grounds. I don't claim that the linguistic theory is inevitable. One rather natural alternative to the linguistic theory is that 'that' clauses and other bracket-terms are not denoting terms: 'believes that' and 'means that' and 'means <>' are to be viewed as operators that convert sentences and other expressions into predicates, with the component expression more or less semantically inert. Stephen Schiffer has pointed out to me that there are inferences in which 'that' clauses seem to figure as singular terms: e.g. the inference from 'Sue believes that E = mc 2 ' and 'Einstein's theory is that E = mc 2 ' to 'Sue believes Einstein's theory'. But one could also validate the inference by supposing that the first premise was short for 'Sue believes the theory that E = mc 2 ' and the second premise was short for 'Einstein's theory is the theory that E = mc 2 '. This does save the view of 'that' clauses as non-denoting, but of course it requires that belief sentences can take singular terms as well as 'that'clauses as complements, and so we still must decide whether those singular end p.165
terms denote linguistic expressions or intentional entities. The non-denoting operator view just defers the question.
The other alternative we need to consider is that 'that' clauses, and bracket terms more generally, refer to some sort of intentional entities: propositions, concepts, and the like, taking concepts not to be components of mental states, but to be intentional entities assigned to such components. (I will occasionally use the term 'meaning' as a catchall for properties, concepts, and the like; as remarked earlier, this is somewhat idiosyncratic usage in the case of sentences with indexicals, for it involves taking two tokens of 'She is beautiful' said of obviously different people as being assigned different 'meanings'. Frege's term 'sense' or Kaplan's term 'content' might be better, but I trust that no confusion will ensue.) I am not strongly opposed to this intensionalist proposal, as long as the intentional entities are construed in a sufficiently bland way. For instance, it is certainly possible to introduce, for each expression e of our language (or rather, each use of a such an expression) a corresponding intentional entity i e i ('the intentional entity corresponding to "e" ' ) with no particular properties other than those induced from the properties of e; more or less in the way that Frege introduced directions on the basis of lines. 2 (As Stephen Schiffer once nicely put it: propositions should be viewed simply as shadows of sentences, and concepts simply as shadows of certain subsentential expressions.) The theory that introduces these entities can be set up so that it conservatively extends the underlying theory, in which case it is hard to see how it can be regarded as seriously problematic.3 The danger of doing so, however, is that we will start talking about propositions, concepts, etc. in ways not licensed by this method of introducing them. Notice for instance that this method of introducing propositions would seem compatible with viewing them as not importantly different from either (i) classes of synonymous sentence-tokens (on the conception of synonymy introduced before, on which tokens of 'She is beautiful' said of obviously different people are non-synonymous); or (ii) equivalence classes of meaning characteristics (as defined in Section 2). Either of these would make the propositional view basically just a notational variant of the linguistic view (though the second involves a commitment to the 'not radically Quinean' linguistic view (LV), as opposed to (QLV)). So to assume that propositions are not to be viewed as simply classes of synonymous sentences or as classes of meaning characteristics is to assume something that goes beyond what is licensed by the Fregean introduction technique, and the intensionalist would seem to need an argument for the further assumption. But having made this point, I will not press it: from here on I will concede to the intensionalist that propositions are not just classes of linguistic entities. 2 I use these new i -quotes instead of the earlier < -quotes because my use of the latter is neutral to what if anything the <>-terms refer to. i -quotes and # -quotes are competing accounts of < -quotes. 3 I should add that a large part of the reason that conservative extensions of a theory are unproblematic is that it is
possible to construe the added entities as fictitious, or convention-dependent; we can talk in terms of such entities without really being Platonist about them. But I will not press this point. end p.166
My main qualm about talking in terms of propositions and the like isn't that doing so is incorrect, but that it tends to obscure the fact that the explanatory role of propositions derives from that of the sentences that they 'shadow'. I discussed this in Section 4. Moreover (as Davidson 1967 observed) it seems impossible to develop any nontrivial theory of these intentional entities. For instance, how do they combine? The view of meaning-combination that seems most in line with the ordinary notion of meaning is that sameness of meaning requires being built up in the same way from synonymous parts. In that case, the approach requires the introduction of a concatenation operation on meanings, analogous to that on expressions. Given that, we could of course give a simple inductive clause for correlating expressions with meanings: for any expressions E 1 and E 2 , if E 1 means m 1 and E 2 means m 2 then E 1 E 2 means m 1 m 2 . How exciting!4 We could extend this 'theory', by connecting sameness of meanings with sameness of meaning characteristics (in the non-intentional sense introduced in Section 2): two expressions have the same meaning iff they have equivalent meaning-characteristics. But even with such an extension, the theory as just developed makes no connection between these meanings on the one hand and truth-conditions and reference-conditions on the other: that needs to be added. Nothing tells us, for instance, that the meaning associated with 'Plato' picks out Plato. We must add some sort of theory of reference (perhaps a disquotational theory) to get this. (It won't help to somehow build into the notion of i Plato i that no intentional entity can be i Plato i unless it picks out Plato. For if we did so, it would no longer be trivial that the meaning associated with 'Plato' is i Platoi : we would need an explanation of why the meaning of 'Plato' is i Plato i , and it isn't obvious why the claim that the meaning of 'Plato' is i Plato i should be beyond reasonable doubt.) And we must add similar 'theories of reference' for other expressions, e.g. predicates and logical symbols; for instance, a theory that tells us that any expression whose meaning is i or i picks out the appropriate truth-function. We also must add a theory of truth, showing how the meanings of complex expressions contribute to truth conditions; though presumably this will reduce to the usual inductive account of truth in terms of reference, plus the account of reference. And isn't all the interesting work now being done by the theories of reference and truth, plus the account of which equivalence relation on the set of meaning characteristics underlies translation? The intentional meanings seem to be
completely idle. Note for instance that the inductive specification of the meanings has no real base clause other than the base clause for reference. You can't specify a nontrivial base clause by conjoining all sentences of form ' "e" means i e i ' for atomic e, because we have given no content 4 We could also include definitions of the types of meanings, which would entail e.g. that if m1 is a 1-place predicate
concept and m2 an individual concept then m1 m2 is a proposition. This doesn't add much to the excitement that the theory induces. end p.167
to the phrase ' i e i ' other than 'the actual meaning of "e" in our own language'; the contemplated base clause adds nothing. The only substantial base clause available is the conjunction of sentences like 'an expression that means i Plato i refers to Plato', 'an expression that means i or i obeys the truth-table disjunction ', and so forth. So the compositional theory can't say anything about how the meanings associated with coreferential atomic expressions differ (unless at least one of them is synonymous with another expression), it can only say that they differ somehow. The danger that the introduction of intentional entities will appear more explanatory than it really is becomes especially acute if we write our theories of truth and reference in the form (T G ) and (R G ) used at the end of the previous section. I remarked there that on a linguistic view of meaning attribution, these are just extensions of (T) and (R) to other languages via sameness of meaning; the central role of our own language is still easily seen, because of its role in the meaning attributions used in (T G ) and (R G ). But if one introduces propositions and other intentional entities, and takes 'that' clauses and other bracket terms to refer to them, then this central role of our own language is easily blurred. Indeed there can then be a real unclarity as to whether (T G ) and (R G ) express a 'deflationary' view of truth not essentially different from disquotationalism, or a thoroughly 'inflationary' view of truth, or something in between. The view is deflationary as long as 'the proposition that Plato taught Aristotle' is interpreted as simply meaning (A) the proposition actually meant by our sentence 'Plato taught Aristotle'; for then the theory of truth (T G ) builds in the fact that 'Plato taught Aristotle' as we actually use it is true if and only if Plato taught Aristotle. The presence of that in the theory of truth (as opposed to, in a result that we hope to arrive at from the theory of truth together with the facts about the use of our language) is disquotationalism. If on the other hand (i) 'the proposition that Plato taught Aristotle' is interpreted as meaning something like (B) the set of possible worlds in which Plato taught Aristotle or (B') the ordered triple
terms like 'proposition' and 'concept', and that real care is required to avoid doing so. But if one keeps in mind the explanatory idleness of meanings (propositions, concepts, etc.), and is clear enough in the use of phrases like 'the proposition that Plato taught Aristotle' to avoid inviting confusion, then I think introducing such talk is harmless. And if we indulge in such talk, we can understand attributions of meaning, belief, etc. in terms of propositions and the like. One might suppose that Quinean considerations count against this. Quine noted in some of his early writings that the notion of synonymy is much clearer when restricted to words or sentences in the same idiolect than it is in the interidiolect case. If someone accepts 'The Son of Sam was a serial killer' but not 'David Berkowitz was a serial killer', or
even if he merely can imagine evidence that would lead him to accept one but not the other, that's enough to show that 'The Son of Sam' and 'David Berkowitz' don't mean the same thing for that person. This suggests that intra-idiolect sameness of meaning ought to be explainable by some sort of substitutivity criterion: without trying to be very precise, the idea is that two terms of the same idiolect mean the same if substitution of one for another inside a sentence (except in certain special contexts like quotation marks or sentential attitude constructions) doesn't affect the speaker's epistemic attitudes toward the sentence; for instance, it doesn't affect what observations or what other sentences count as supporting the sentence. Terms that are synonymous within one idiolect of a language tend to be synonymous within other idiolects of the language as well, so it does little harm to speak of intra-linguistic synonymy. But substitutivity criteria obviously can't work in the case of inter-linguistic synonymy. 5 (Such criteria don't really work even for inter-idiolect synonymy, though there we can stipulate that smooth communication under the identity translation should suffice for synonymy.) And Quine thought that nothing else would work in the inter-linguistic case either: the distinction between differences of meaning and differences of belief can be given no principled basis. We can still invoke translations from one language to another, but shouldn't view them as attempts to reflect a pre-existing synonymy relation. It isn't entirely clear that the considerations Quine adduced show more than the indeterminacy of inter-linguistic synonymy. If that is all it shows, then of course there is no threat to talk of propositions and the like, provided that we recognize an indeterminacy in that notion. But Quinean considerations are often thought to cut deeper than this: to show that 'there is no synonymy relation that translation is an attempt to reflect'. (I don't say that this wears its meaning on its face, but I think it probably can be made sense of. One possible approach to doing so was suggested in Chapter 4, footnote 27 . And one reason why someone might want to maintain such a view will be mentioned in 5 This is certainly obvious when there are no bilingual speakers. And the presence of bilinguals doesn't change
anything: the first bilingual's choice of a mapping from one language to the other is partly arbitrary; if other bilinguals were to make a choice independent of what the first bilingual has done, there is no reason to think they would make the same choice. 27 I'm inclined to think that the problem isn't just that interpersonal synonymy ('is a good translation of') is a highly vague notion, but that it isn't a fully factual notion at all; rather, it is an evaluative notion. This is compatible with many claims of interpersonal synonymy being factual relative to the goals of the translator; but not all interpersonal synonymy claims need be factual even relative to the goals. (For more on the idea of evaluative claims as 'not fully factual', see Chs. 8 and 13.) end p.169
Chapter 10, footnote 2 .) But even this is compatible with the innocent introduction of intentional entities, as long as one construes those entities as local entities. That is, we take each language to have its own supply of meanings, not objectively comparable to the meanings of any other language. (If you like, you could think of the meanings of the expressions of language L as just equivalence classes of meaningful L-sentences under the equivalence relation of intra-linguistic synonymy; though this construal is no more essential to the idea than the analogous construal when we were using a more general synonymy relation.) We could take it to be an objective matter which of the L 1 -meanings an L 1 -sentence means, and which of the L 2 -meanings an L 2 -sentence means, but L 1 -meanings and L 2 meanings are simply distinct entities; a mapping of L 1 -meanings to L 2 -meanings can be suitable or unsuitable relative to certain goals, but talk of objective correctness makes no sense here. This would allow us whatever convenience meanings as entities might be thought to have (e.g. in the semantics of belief sentences), without any commitment to views about interpersonal comparison that might be deemed suspect. Indeed, the Fregean analogy with directions has an illuminating extension to this case. Let's pretend that space as a whole is highly non-Euclidean, but that it has a number of large Euclidean regions within it. Within each Euclidean region, geometry is as you're used to. In particular, within each region the notion of parallelism of line segments makes clear sense, and is an equivalence relation. So within each region we can assign to a line segment a direction: a direction is simply what all parallel line segments in the region have in common. We can do this within each of the flat regions. Now, a naive philosopher might say, ' ok , within region 1 we can assign directions to line segments, and sensibly ask whether two line segments have the same direction; and within region 2 we can also assign directions to line segments, and sensibly ask whether two line segments have the same direction. So it ought to make sense to ask whether a line segment in region 1 has the same direction as a line segment in region 2.' But in fact it doesn't make sense: if you take a line segment in region 1 and transport it to region 2, keeping it as straight as possible along the journey, how it ends up aligned in region 2 will depend on the path of transport. (You can see this in two dimensions by taking the regions to be small flat regions on the surface of the earth, one on the equator and one at one of the poles.) Directions are perfectly well-behaved local entities, in that within each flat region we can assign local directions in a completely consistent manner; but the local directions for one region aren't
straightforwardly comparable to those for another, they are comparable only relative to a method of transport. On a strong understanding of Quine's indeterminacy thesis, meanings can be viewed as just like that: they make pretty good sense locally (i.e. within a language, or maybe just within a speaker), but they aren't straightforwardly comparable inter-linguistically, but rather, are comparable only relative to a method of translation.6 2 A more moderate view would be that we can have an unrelativized notion of reference extending beyond our terms,
it will just be highly indeterminate. But the more moderate view is not available to one who wants to deny indeterminacy in our own terms, for it postulates an indeterminacy in our term 'refers'; that is why I have tentatively ascribed to Leeds the view that 'refers' is inapplicable to other terms except by relativization. Note that this less moderate view requires that we have no notion of inter-theoretic synonymy that could be used to break the relativization: inter-theoretic synonymy is not merely indeterminate, it makes no sense. (Indeterminacy of translation would be allowed, but it would have to be viewed as indeterminacy of how to translate, not as indeterminacy of 'the synonymy relation'.) 6 Note that in the geometric analogy, it is natural to say not merely that it is indeterminate whether line segments in different regions are parallel, but that the question of their parallelism makes no sense except relative to a method of transport. I am inclined to say the same in the synonymy case, though in both cases the matter deserves more discussion. end p.170
Whether this is a good picture is not something I want to argue here. My only claims are (i) that as long as one is careful enough in talking of propositions, we can accommodate the Quinean picture; and its flip side (ii) that we need to be careful not to smuggle in the assumption that the Quinean picture is wrong by uncritically assuming that the propositions one has innocently introduced in analogy with directions are globally defined. This fits with the general picture that propositions and other intentional entities are things that we can take or leave as we see fit: we don't need them for attributions of meaning or belief or the like, but as long as this is clearly recognized, and we avoid certain confusions of ambiguity, they do no harm.
VII Early in the paper I noted that the meaning schema (M) seems to have a special epistemological status: its instances are not open to doubt on empirical grounds. And I said that the linguistic view of meaning attributions gives one possible explanation of this. Are there other explanations? In this final section I'll consider several such. It might be thought that there is a completely obvious explanation of the special status of (M). It involves the idea that to understand an expression is to know its meaning. I should note in passing that this is ambiguous between two interpretations. One is that to understand an expression is to know what it means for me; the other is that to understand an expression is to know its meaning in the public language. These two interpretations should not be viewed as giving conflicting theses about understanding, they should be viewed as showing an ambiguity in the notion of understanding. In one sense I can understand a term even if my understanding doesn't coincide with other people's; in another I can't. Given the focus in this paper on the idiolect of an agent, it is the former sense that is the more relevant. So: for me to understand an expression is for me to know what it means for me. But then isn't the special status that the schema (M) has for me quite trivial? After all, the instances of the schema that I can be said to believe are just those where the term 'e' in question is one that I understand. If I have no understanding of 'grug', then I won't accept 'grug' means < grug > . (And even if I did it wouldn't count as a belief.) But for me to understand 'e' is just for me to know what it means, so of course I am going to know the meaning of all the terms I understand! Where's the mystery, or the need of the linguistic theory of meaning attributions? end p.171
I think this is mistaken. It's worth noting that while the claim that to understand an expression is to know its meaning is in a way unexceptionable, it may also be misleading. What is unexceptionable is that for an ordinary person (one who has the concept of meaning), understanding a term goes along with knowing what it means; one understands 'metacompact' if and only if one knows that it means < metacompact > . What would be misleading is to take for granted that this knowledge that 'metacompact' means < metacompact > explains one's understanding of 'metacompact'. In my opinion, that would be getting things backwards. Understanding a term (in the idiolect sense of 'understanding')
involves, roughly, having a sufficiently rich conceptual role for it. 7 It isn't a matter of knowing that p, for any p whatsoever let alone a p about the meaning of the word. It is the understanding of 'metacompact', together with one's acceptance of all instances of the schema (M), which explains the fact that one accepts 'metacompact' means < metacompact > , (and the fact that accepting it suffices for believing it). And in conjunction with the impropriety of revising (M) on empirical grounds, this yields the conclusion that the instances of (M) are known and empirically unrevisable. But of course this presupposes the special epistemological status of (M): it presupposes that the instances of (M) should be accepted whatever the empirical evidence. And it is that special status that was supposed to be explained. So at best, the proposed explanation of the special status of (M) is one that works only on a very contentious (not to say ridiculous) version of the connection between meaning and understanding: a version on which propositional knowledge about meaning plays an explanatory role in understanding. But even on such a view of the connection between meaning and understanding, the explanation of the special status of the instances of (M) is faulty. For all it does is 'explain' the generalization For any expression, if we understand it then we know what it means (by reducing this generalization to a vacuity). That together with the fact that 'metacompact' means < metacompact > gives us a trivial explanation of If we understand 'metacompact' then we know that it means < metacompact > . But what was asked for wasn't this, but an explanation of the consequent. Well, we'd have that if we had an explanation of the antecedent. But on the view of understanding in question, we could only explain the antecedent by explaining our knowledge of what 'metacompact' means: viz., that it means < metacompact > ! So even on this contentious view of understanding, the 'explanation' of the status of (M) is completely circular. I turn now to a second attempt to explain the special status of (M). The idea of the approach is to note that the instances of (M) are built into the notion 7 Understanding it in the more social sense involves, roughly, having a sufficiently rich conceptual role for it that
accords in the appropriate way with the conceptual role that others have given it. end p.172
of meaning, in the following sense: it is part of learning to use the word 'means' that we come to accept all instances of (M), and our acquiring this body of beliefs is central to the notion of meaning serving the purposes it serves. Its being part of the meaning in this sense is all we need to legitimate the practice of giving default status to all instances of (M); only an absurdly foundationalist epistemology would require more. And nothing in this explanation commits one to the linguistic view of meaning attributions. As an explanation of the default status of the instances of (M), I think this approach is unexceptionable. But what I asked for was more: an explanation of why we should regard the instances of (M) as empirically indefeasible, that is, of why we should think there can be no empirical evidence against the instances of (M). To regard (M) as empirically indefeasible is much stronger than merely to give it default status. There are many examples to illustrate this: consider for instance 'temperature', or (better for my purposes) the comparative predicate 'has higher temperature than'. I take it that a few hundred years ago, part of the ordinary mastery of that predicate was the acceptance of such beliefs as that if one body felt substantially warmer than another to a normal observer in normal conditions then its temperature was higher. Such beliefs would have been taken to be central to the explanatory purposes of the notion of having a higher temperature. These beliefs were in a perfectly good sense built into the meaning of 'has a higher temperature', and had a default status in that no one was required to argue for them on empirical grounds. But these beliefs were not empirically unrevisable: once people started exploring the physical underpinnings of temperature and of feelings of warmth, it was discovered that the beliefs were false, in that things other than temperature-differences (for instance, differences in thermal conductivity) play a substantial role in determining feelings of comparative warmth. In the case of 'means' however this cannot happen: if one insists on construing the claims that 'dog' means < dog> and that 'metacompact' means < metacompact > as claims about 'dog' bearing the meaning relation to i dogi and 'metacompact' bearing the meaning relation to i metacompact i , then these claims about the extension of the meaning relation seem empirically unrevisable. So our having learned to accept the instances of (M) as central to the meaning of 'means' is not by itself enough to explain their immunity to empirical revision.
I think it is fairly clear that explaining the empirical unrevisability of (M) requires a somewhat 'deflationary' attitude toward meaning attributions to our own terms. The reason that our views about the relation of having higher temperature than can be empirically revised even when 'built into' the meaning of 'temperature' has to do with the fact that 'temperature' is an explanatory concept. If the attribution of intentional entities (meanings) to our own terms isn't taken to be explanatory, there is no mystery as to how it can be reasonable to hold onto such attributions whatever empirical discoveries we might make. Of course there is no doubt that attributing what I've called meaning characteristics (conceptual roles, indication relations, etc.) to expressions is end p.173
explanatory: that is explanatory in a straightforward causal sense. And attributions of intentional entities to expressions can be used in explanations, in so far as the attribution of an intentional entity is used simply as a way of (perhaps vaguely) picking out the relevant meaning characteristics. But the way in which the ascription of an intentional meaning picks out meaning characteristics makes use of the fact that intentional meanings are mere shadows of expressions. That is, if we explain a person's uttering (or silently believing) a certain German sentence on the basis of its meaning i there have been many British kings i , the intentional entity we've picked out in the explanation has been picked out via our expression 'there have been many British kings'; our explanation amounts to the claim that the person's utterance (or inner state) has meaning characteristics sufficiently like those actually possessed by our sentence 'there have been many British kings' for the purposes at hand. Nothing about the relation between expressions and meanings, other than the fact that sameness of meaning goes by sameness in type of meaning characteristic, plays any role in this explanation; or in any other legitimate explanation, so far as I can see. It is no wonder that if meanings are viewed as simply shadows of our expressions, and the shadowing relation is not put to explanatory work, then there can be no empirical reason to revise the meaning schema. But if intentional meanings are only shadows of expressions, then they would seem to be entities that we can take or leave as we see fit. And if intentional meanings have a more thoroughly explanatory role than this, then it is hard for me to see why the special epistemological status of (M) isn't a mystery. 8 8 Stephen Schiffer's skeptical reaction to the previous chapter provided the main impetus for writing this, and he
made many useful comments along the way; Paul Horwich and Brain Loar gave me comments on a previous draft that led to a major change in emphasis. Ken Akiba, David Barnett, Jared Blank, Ray Buchanan, David Enoch, Joshua Schechter, Brad Skow, and James Woodbridge all made comments that significantly influenced the final version of this chapter and/or the related parts of the postscript to the previous chapter. end p.174
Part 2 Indeterminacy and Factual Defectiveness end p.175 end p.176
6 Theory Change and the Indeterminacy of Reference Abstract: There is a natural argument for the incommensurability of scientific theories, based on examples where there is more than one equally good candidate for what an earlier term referred to. The chapter argues that such examples actually do not support incommensurability in any very serious sense. They do show a difficulty in applying standard semantic vocabulary in a determinate way to such earlier theories, but the chapter argues for a generalization of semantic vocabulary that largely avoids the difficulties. Includes a new postscript. Keywords: incommensurability, indeterminacy, reference, supervaluation, theory change, truth
Hartry Field In this Chapter I will argue that considerations about scientific revolutions show that many scientific terms are referentially indeterminate—there is no fact of the matter as to what they denote (if they are singular terms) or as to what their extension is (if they are general terms). In the opening section I will try to establish this general point of view by considering a particular scientific revolution, that in which Newtonian mechanics was replaced by the special theory of relativity. I will argue that this revolution reveals that the word 'mass' as used before relativity theory was discovered had no determinate denotation. Later on in the chapter I will suggest that this indeterminacy in the terms of earlier scientists is evidence for an indeterminacy in many of our own scientific terms. In addition to establishing the existence of indeterminacy, I want to show its import for semantic theory. There is a long tradition in semantics according to which the truth value of what someone says is determined by certain semantic features of the words (or word-tokens) he uses in saying it. Thus the truth value of an utterance of 'John loves Mary
and Mary loves Bob' is determined by the denotations of the utterer's tokens of 'John', 'Mary', and 'Bob' (i.e., by the people the utterer referred to in using these names) and by the extension of his token of 'loved': the utterance is true if and only if the extension of 'loved' contains both the ordered pair whose first member is the denotation of (the token of) 'John' and whose second member is the denotation of 'Mary', and the ordered pair whose first member is the denotation of 'Mary' and whose second member is the denotation of 'Bob'.1 It has seemed to many philosophers that it is only because there are linguistic rules for determining the truth value of a sentence from the denotations of its component names, extensions of its component predicates, etc., that the notions of truth and falsity even make sense. 1 It would be possible to formulate referential semantics in terms of properties rather than sets: we could say that the
utterance 'John loves Mary' is true if and only if the person denoted by 'John' bears the relation (2-place property) denoted by 'loves' to the person denoted by 'Mary'. I believe that if properties are carefully distinguished from meanings, there are certain advantages to proceeding in this manner; but for our present concerns it is irrelevant whether we choose sets or properties, so I have stuck to the more common policy of relying on sets. end p.177
This point of view (I'll call it referential semantics) was adhered to by Frege, Carnap, and Tarski,2 and has become increasingly common in recent years. But referential indeterminacy creates a serious problem for referential semantics. For we'll see that there are sentences with perfectly determinate truth values which contain referentially indeterminate names and predicates, so that it makes perfectly good sense to ask whether the sentence is true or false even though it doesn't make sense to ask what the name really denotes or what the real extension of the predicate is. Clearly, then, the fact that it makes sense to speak of the truth and falsity of such sentences cannot be based on the existence of linguistic rules that determine truth value from denotation and extension. So if my argument for referential indeterminacy works, it follows that we have to revise the program of referential semantics to some extent. The second part of the paper will be concerned with the question of what sort of revision is required in order to handle the sort of indeterminacy whose existence is revealed by scientific revolutions. It turns out that the required revision is not a very drastic one.
I Before Einstein's special theory of relativity, physicists accepted a great many assertions involving the term 'mass' that are no longer accepted today. For instance, Newton and his successors accepted the following claims: (1) The mass of a particle is equal to twice its kinetic energy divided by the square of its velocity. and (2) Mass is conserved in all interactions. Part of the novelty of Einstein's theory was that (1) and (2) and most of Newton's other assertions involving 'mass' were given up. This fact has led some philosophers (e.g., Thomas Kuhn 3 ) to claim that before relativity theory 2 This is more true of what Tarski ought to have said (given his logical discoveries) than of what he did say: see Ch.
1. 3 Kuhn 1962: 100-1. A similar conclusion is suggested by a passage from Quine (1960: 16), about the term 'neutrino'.
Judging from this passage we might expect that Quine would argue as follows: Since Newton's and Einstein's theories differed so much, they cannot have been referring to the same quantity. The fact that both physicists used the word 'mass' is not significant. To discern two phases here, the first an agreement as to what physical quantity is involved (viz., mass) and the second a disagreement as to what it's like (what laws it obeys), is absurd. But this would be an ignoratio elenchi: the issue is not whether Newton 'knew what mass is' or whether he and Einstein 'agreed as to what mass is,' but whether he referred to mass and whether he and Einstein referred to the same thing. Even if one is not inclined to say that Newton 'knew what mass is,' it is still possible to say that he referred to mass, and saying so might be important in enabling you to characterize the truth value of his assertions in terms of referents of the component parts. end p.178
was discovered the term 'mass' did not denote (or refer to) 4 the same physical quantity 5 it denotes (or refers to) today. These philosophers, in other words, deny the following claim: (3) Newton's term 'mass' denoted mass; i.e., Newton was referring to the same physical quantity that we refer to when we use the word 'mass'.
It seems to me very unlikely that the falsity of (3) can be established solely on the sorts of grounds that Kuhn utilizes, viz., on the fact that Newton had many beliefs involving 'mass' that are no longer accepted: there is nothing at all incoherent in the position that Newton was referring to mass even though he had a great many false beliefs about it (cf. note 3 ). Moreover, we should note that the false beliefs about mass that Newton must have had for (3) to be right are all approximately true—e.g., we can approximate (1) by the weaker claim (1*) At low velocities, the mass of a particle is almost precisely equal to twice its kinetic energy divided by the square of its velocity. and this approximating claim is true according to relativity theory. And there is certainly nothing incoherent in the position that Newton was referring to mass even though he had a great many false-but-approximately-true beliefs about it; indeed, the fact that most of Newton's beliefs involving 'mass' come out approximately-true-though-strictlyfalse if we regard them as being about mass looks like evidence for (3) rather than against it. Nevertheless, there are other considerations about Newtonian physics and its relation to special relativity which do call (3) into question: these considerations show, I contend, that it simply doesn't make sense to ask what physical quantity Newton and other pre-relativity physicists referred to when they used the term 'mass'. If one opens a textbook on special relativity, one may find assertions such as the following: (R) The mass of a particle is equal to the total energy of the particle divided by the square of the speed of light. Since the total energy of a particle with respect to one frame of reference differs from the total energy with respect to other frames of reference, but the speed of light is the same in all frames of reference, it follows that the mass of the particle has different values in different frames. That's what one will find in some textbooks, but in others one will find something quite different: (P) The mass of a particle is equal to the nonkinetic energy of the particle divided by the square of the speed of light. Since the nonkinetic energy of a particle is the same in all frames of reference, and a similar claim 4 I use these terms more or less interchangeably. (Usually I say that a person refers, and that the word-token he uses in referring to x denotes x.) 5 A physical quantity can be viewed as a function that assigns a value to each physical object (or measurable set of spatial points). A quantity-term like 'mass' can take the role of a function symbol, as in (1) above, or it can take the role of a singular term, as in the abbreviated form of (1): 'Mass equals twice kinetic energy divided by velocity'. In either case I say that it denotes the physical quantity. 3 Kuhn 1962: 100-1. A similar conclusion is suggested by a passage from Quine (1960: 16), about the term 'neutrino'. Judging from this passage we might expect that Quine would argue as follows:
Since Newton's and Einstein's theories differed so much, they cannot have been referring to the same quantity. The fact that both physicists used the word 'mass' is not significant. To discern two phases here, the first an agreement as to what physical quantity is involved (viz., mass) and the second a disagreement as to what it's like (what laws it obeys), is absurd. But this would be an ignoratio elenchi: the issue is not whether Newton 'knew what mass is' or whether he and Einstein 'agreed as to what mass is,' but whether he referred to mass and whether he and Einstein referred to the same thing. Even if one is not inclined to say that Newton 'knew what mass is,' it is still possible to say that he referred to mass, and saying so might be important in enabling you to characterize the truth value of his assertions in terms of referents of the component parts. end p.179
holds for the speed of light, it follows that the mass of the particle has the same value in all frames. (R) and (P) appear to conflict with each other; but is this conflict genuine? A closer examination of the two textbooks will reveal that both agree that the total energy divided by c 2 (where c is the speed of light) has a different value in different frames, and that the nonkinetic energy divided by c 2 has the same value in all frames: perhaps, then, the 'disagreement' is simply over which of these quantities is to be called 'mass'. This hypothesis is confirmed by the fact that the textbooks agree about the ratio of these quantities: they agree that
(where ν is the speed of the particle). In fact it turns out that if you take any formula of Textbook P (the textbook from which (P) was extracted) and replace 'mass' everywhere by 'mass times
', you get a formula that is
acceptable according to Textbook R, and if you take any formula of Textbook R and replace 'mass' everywhere by 'mass divided by ', you get a formula that is acceptable according to Textbook P. It seems reasonable to say, then, that there are two physical quantities involved here: Textbook R uses the term 'mass' for a quantity that can be called relativistic mass, which is equal to total energy/c 2 ; and Textbook P uses the term 'mass' for a different quantity that can be called proper mass, which is equal to nonkinetic energy/c 2 . The appearance of conflict between the two textbooks arises solely from the fact that one uses the word 'mass' for one quantity and the other uses the same word for a different quantity. With these facts in mind, let us return to the question of what physical quantity Newton was referring to when he used the term 'mass'. It is natural to imagine Physicist R making the following claim: (3R) Newton's term 'mass' denoted mass; i.e., it denoted total energy/c 2 . But it is equally natural to imagine Physicist P claiming (3P) Newton's term 'mass' denoted mass; i.e., it denoted nonkinetic energy/c 2 . Putting their claims into unambiguous terminology, we get (HR) Newton's word 'mass' denoted relativistic mass and (HP) Newton's word 'mass' denoted proper mass. How are we to decide between these claims? My claim is that we can't. Before relativity theory was discovered (I will argue), the word 'mass' was referentially indeterminate: it did not lack denotation, in any straightforward sense; on the contrary, there are two physical end p.180
quantities that each satisfy the normal criteria for being the denotation of the term. To put the point more precisely, let a positive analytical hypothesis for Newton's use of the term 'mass' be a sentence of the form Newton's word 'mass' denoted X. where 'X' is the name of a physical quantity, and let the negative analytical hypothesis be the sentence (HΛ) Newton's word 'mass' denoted nothing whatsoever. I claim that the negative analytical hypothesis must be rejected, and so must all positive hypotheses except for (HP) and (HR). I also claim that (HP) and (HR) are each extremely plausible (or more properly, that each would be extremely plausible were it not for the existence of the other), and that there is no basis for choosing between them. It isn't merely that we don't know whether Newton was referring to proper mass or to relativistic mass; I claim that there is no fact of the matter as to which of these quantities he was referring to. The first point I will argue is that there is no basis for choosing between (HR) and (HP), i.e., for accepting one of these hypotheses and rejecting the other. In order to appreciate the difficulty of choosing between them, it is necessary to know two central tenets of Newtonian mechanics: (4R) Momentum = (mass) · ν
and (5P) For any two frames of reference, mass with respect to frame 2 = mass with respect to frame 1. The first of these was formulated quite explicitly in every presentation of Newton's theory, often as the 'definition' of momentum; the second was considered so obvious that it never occurred to anyone to state it, but certainly an advocate of the theory would have assented to it had anyone thought to ask him, and its truth was presupposed in all of the theory and practice of Newtonian mechanics. Now, were these two tenets of Newtonian mechanics correct? They can't both have been correct; for, according to relativity theory, the momentum of a particle divided by its velocity has different values in different frames of reference, whereas (4R) and (5P) together entail that the momentum divided by the velocity has the same value in different frames. Relativity theory, then, shows that the conjunction of the Newtonian tenets (4R) and (5P) was false; and so it would be natural to expect that it would also show that a particular one of these tenets was false. Unfortunately, this natural expectation seems to be wrong. For the fact is that momentum does equal relativistic mass times velocity, and does not equal proper mass times velocity; whereas (as we've seen) relativistic mass does not have the same value in all frames, whereas proper mass does. In other words, some physicists today use the word 'mass' in such a way that their tokens of (4R) are true and their tokens of (5P) false, and if Newton was using the word in the way that they do (i.e., for end p.181
relativistic mass), then his tenet (4R) was true and his tenet (5P) was false. But other physicists use the word 'mass' in such a way that their tokens of (4R) are false and their tokens of (5P) are true, and if Newton used the word 'mass' in the way that they use the word (i.e., for proper mass), then his tenet (4R) was false and his tenet (5P) was true. I contend that there is no basis whatever for deciding between these two possibilities. The conjunction of Newton's tenets (4R) and (5P) was objectively false, but there is no fact of the matter as to which of the conjuncts was true and which false, and hence no fact of the matter as to whether the word 'mass' as it occurred in them denoted relativistic mass or proper mass. The difficulty of choosing between (HR) and (HP) arises, then, from the fact that (4R) and (5P) were both extremely central to Newton's theorizing and to his scientific practice; and I can see no basis for asserting that a particular one of them was more central than the other. Perhaps, however, we could decide between (HR) and (HP) on the basis of other tenets of Newtonian physics? I don't believe so. Most of the theoretical claims of Newtonian physics, like (1) and (2), come out false (though approximately true) whether we take Newton as referring to relativistic mass or to proper mass; and most of Newton's experimental claims, like (6) The mass of Object A is between 1.21 and 1.22 kilograms [said after putting Object A onto a pan balance and accurately weighing it].6 come out true either way. Another way we might try to choose between (HR) and (HP) is to look, not at Newtonian mechanics, but at special relativity; for if one were to find that one of the quantities played a much more important role in special relativity than the other one played, some 'principle of charity' might provide some basis for asserting that the more important quantity was the one that Newton was really talking about. Unfortunately, however, proper mass and relativistic mass play about equally important roles in special relativity theory. Some laws of physics come out looking simpler in terms of relativistic mass than in terms of proper mass: thus if we let 'mass' stand for relativistic mass we get (4R), but if we let 'mass' stand for proper mass we get the more complicated formula:
(4P) But other laws of physics come out simpler in terms of proper mass than in terms of relativistic mass: when 'mass' stands for proper mass we get the simple transformation law (5P), but when 'mass' stands for relativistic mass we get instead 6 I use here the fact that Newton made experimental assertions only about objects that were moving slowly (in
comparison with the speed of light), and for a slowly moving object the proper mass and relativistic mass are virtually identical. (It might be asked 'Slowly moving with respect to what reference frame? No specific frame is mentioned in (6).' But no specific frame is mentioned in assertions like 'Object A is traveling at 10 ft/sec' either, and yet we often
adjudicate the truth or falsity of such assertions. We do so by supposing that one frame of reference is intended by the speaker, and evaluating the truth or falsity of the utterance with respect to the intended frame. So the answer to the question is: slow-moving with respect to the intended frame.) end p.182
(5R) (where ν 1 and ν 2 are the speeds of the particle in frame 1 and frame 2, respectively). Taken together, the laws of special relativity come out looking just about as simple when formulated in terms of one of these quantities as they look when formulated in terms of the other quantity. This explains why the division between those physicists who find it more aesthetically pleasing to formulate the theory in terms of relativistic mass and those who prefer to formulate it in terms of proper mass is just about 50/50. I think that my last four paragraphs make it highly plausible that there is no basis for choosing between the hypotheses (HP) and (HR). But this contention is compatible with a number of positions which I now want to argue against. The first position is (HN) Newton wasn't referring either to proper mass or to relativistic mass; instead he was referring to a quantity called 'Newtonian mass' which had some of the properties of each. (More precisely, Newtonian mass is equal to momentum divided by velocity—in this it is unlike proper mass but like relativistic mass. But, unlike relativistic mass, Newtonian mass has the same value in all reference frames. Newtonian mass also has some properties that distinguish it both from proper mass and from relativistic mass: e.g., it is equal to twice the kinetic energy divided by ν 2 , and it is conserved in all interactions.) The view that what Newton was referring to was 'Newtonian mass', as just characterized, seems to be advocated by Kuhn. In discussing Einstein's concepts of space, time, and mass, Kuhn writes: The physical referents of these Einsteinian concepts are by no means identical with those of the Newtonian concepts that bear the same name. (Newtonian mass is conserved; Einsteinian is convertible with energy . . . ) (1962: 101, emphasis mine). 7 But this view seems wholly unreasonable, for what Einstein showed is that there is no such quantity as 'Newtonian mass'; and unless one holds that the world used to obey Newton's laws but started obeying Einstein's laws one day, it is clear that there was no 'Newtonian mass' in Newton's time either. I'm not denying that in Newton's time the word 'mass' meant something different than the word 'mass' (or its counterparts 'relativistic mass' and 'proper mass') means today; on certain construals of the term 'meaning', this seems to be perfectly true. What I'm denying is not a claim about meaning, but a claim about reference or denotation: I'm denying that there is or ever was such a quantity 7 It is odd that Kuhn adopts this view. Only a few pages earlier (97) he had correctly asserted 'Einstein's theory can
be accepted only with the recognition that Newton's was wrong'; but if Newton was referring to 'Newtonian mass' and Einstein to 'Einsteinian mass', both theories would be right. end p.183
as 'Newtonian mass', and hence I'm denying that Newton could have ever referred to 'Newtonian mass' when he used the word 'mass'. Once we've rejected all appeals to 'Newtonian mass', it becomes clear that there are no positive analytical hypotheses that are more plausible than (HR) and (HP). But now we must examine the negative analytical hypothesis (HΛ), that Newton's word 'mass' was simply denotationless, in the way that 'Santa Claus' is denotationless. I claim that, if we regard Newton's tokens of 'mass' as simply denotationless, we are forced to assign the wrong truth values to Newton's sentence tokens. In order to make the argument good, we have to know how a denotationless term like 'Santa Claus' affects the truth value of sentences that contain it. One view, sometimes attributed to Frege, is that all sentences containing denotationless terms lack truth value; another view is that sentences containing denotationless terms are all false. If either of these views were true, my argument would be simple: I would merely point out that Newton made many utterances with the term 'mass' that all current physicists would agree to, whichever way they used the term 'mass'. For instance, he doubtless made experimental assertions like (6), and he doubtless made theoretical assertions like (7) To accelerate a body uniformly between any pair of different velocities, more force is required if the mass of
the body is greater. When Newton made such assertions as these, he would certainly seem to have been saying something true; and, since his tokens of (6) and (7) were true, the tokens of 'mass' that occur in them can't simply have lacked denotation in the way that 'Santa Claus' lacks denotation. As it stands, the above argument will not do; for it depends on the view that a sentence containing a term like 'Santa Claus' is always truth-valueless or false, and I regard this view as incorrect. However something like that view is correct, and once we see what it is we can see how the above argument can be patched up so that it really does show that Newton's tokens of 'mass' were not denotationless. Consider the following three sentences which contain the word 'Santa Claus': (8) Santa Claus doesn't exist. (9) Johnny saw Santa Claus today. (10) Santa Claus had a wart on his left shoulder. I would regard the first of these as true, the second as false, and the third as truth-valueless. Others may have different attitudes toward (9) or (10), but one thing will, I think, be generally agreed: if you take any other nondenoting singular term and substitute it for 'Santa Claus' in (8), (9), and (10), the resulting sentences will have the same truth value as the original, or have no truth value if the original lacked truth value. For instance, let's assume that 'Moses' doesn't denote anyone; then certainly the sentence end p.184
(8′) Moses doesn't exist. is true. Moreover, anyone who regards (9) as false will also regard (9′) Johnny saw Moses today. as false, and anyone who regards (9) as truth-valueless will regard (9′) as truth-valueless. Similarly, (10′) Moses had a wart on his left shoulder. will appear truth-valueless to those who regard (10) as truth-valueless, and false to those who regard (10) as false. These examples suggest that Leibniz's well-known principle (LL) If two terms each denote the same object, then substitution of one term for the other (in nonquotational, nonintentional, etc., contexts) always preserves truth value. has a counterpart for non-denoting names: (LL*) If two terms each denote nothing whatsoever, then substitution of one term for the other (in nonquotational, nonintentional, etc., contexts) always preserves truth value (or lack of truth value). I think that (LL*) accords with nearly everyone's pretheoretic views as to the truth values of sentences containing terms like 'Santa Claus' and 'Moses'.8 It is fortunate that this is so, for (LL*) is just as important a principle for the semantics of nondenoting terms as (LL) is for the semantics of denoting terms. If two names each have no denotation whatsoever, they are completely alike from a denotational point of view; so how could substitution of one for the other affect truth value? The answer that will be given by any advocate of referential semantics is that it couldn't: if substitution of one term for another affects truth value, these terms must be different from each other, denotationally speaking, and that means that they are not both simply denotationless. Using this principle, we can show that the term 'mass' (as used by Newton) was not denotationless. For suppose we replace the word 'mass' in (6) and (7) by the denotationless term 'phlogiston', getting (6′) The phlogiston in Object A is between 1.21 and 1.22 kilograms. and (7′) To accelerate a body uniformly between any pair of different velocities, more force is required if the phlogiston in the body is greater. Clearly no one will regard (6′) or (7′) as true; so if 'mass' were denotationless, then (6) and (7) wouldn't be true either. We're talking, of course, about (6) and 8 It might be objected that 'Santa Claus flies reindeer' is true, whereas 'Moses flies reindeer' is not. But it seems to
me that 'Santa Claus flies reindeer' is true only when it is elliptical for 'The story says that Santa Claus flies reindeer'; in that sentence, 'Santa Claus' occurs in an intentional context and so is immune to (LL*). end p.185
(7) as used by Newton: the analytical hypothesis (HΛ) does not commit us to the claim that when (7) is used today it is untrue, but only to the claim that when (7) was used by Newton it was untrue. But even that seems bad enough— since all relativity theorists (whether they use 'mass' for proper mass or for relativistic mass) agree to the truth of (7), it seems grossly uncharitable to deny that Newton can have said something true by uttering (7). Similarly, it seems uncharitable to deny that when Newton referred to a slow-moving object whose proper mass and relativistic mass were nearly identical and were between 1.21 and 1.22 kilograms, he could have said something true by uttering (6). By calling Newton's utterances of (6) and (7) false, (HΛ) conflicts with the principle of charity; in this respect it is vastly 9 inferior both to (HR) and to (HP). 10 We now have all the components of the argument for the referential indeterminacy of Newton's word 'mass'. We've seen that, of all the positive and negative analytical hypotheses, the only two with any plausibility are (HP) and (HR). Each of these two hypotheses would be plausible, were it not for the existence of the other, but we have found no basis whatever for choosing between the two. It could be contended, I suppose, that our inability to choose between (HP) and (HR) is due simply to ignorance—that one of (HP) and (HR) is correct and the other incorrect, and the only trouble is that we don't know which. But the only way to give this contention any plausibility is to state what sort of information is likely to decide between the two possible denotations; and the only prima facie likely bases of choice which I can think of have been ruled out earlier in the paper. It seems implausible, then, that one of (HP) and (HR) is correct and the other incorrect: the situation is not that we don't know what Newton's word denoted, but that Newton's word was referentially indeterminate. But can this conclusion be tolerated? Surely (HP) and (HR) can't both be correct, and so if it is not the case that one is correct and the other incorrect, it follows that neither (HP) nor (HR) is correct. Yet we have also rejected all the obvious alternatives to (HP) and (HR) (i.e., the other positive analytical hypotheses, and the negative one), and this seems to show that there is simply no coherent way of using the term 'refers' in connection with Newton's word 'mass'. In spite of this, there are many of Newton's utterances containing the word 'mass' that we want to regard as true—(7) was one example—and there are many also that 9 Note that (HΛ) is incompatible even with the view that (6) and (7) are approximately true; for any approximating claim would also come out untrue if evaluated according to (HΛ). Similarly, (1) and (2) come out approximately true according to (HP) and (HR), but not according to (HΛ). (These remarks hold not only on the conception of approximate truth introduced earlier, but on any other conception in which the approximate truth of a sentence depends on the denotations and extensions of its parts.) 10 I would prefer not to have to rest my argument on the appeal to charity: I would prefer it if I could show that the purpose for which we want the notion of truth would not be satisfied if we declared (6) and (7) false in these circumstances. I don't know in detail how to show this, but I suspect that by further developing the remarks on the purpose of the notion of truth in the last section of Chapter 1 (and also arguing that we have good reason to suspect that many of our own words will someday be in the position that Newton's word 'mass' is in today), we could get the desired proof. end p.186
we want to regard as false, e.g., the conjunction of (4R) and (5P). It follows, then, that the truth and falsity of these utterances simply cannot be explained on the basis of what Newton was referring to when he used the word 'mass', for there is no coherent way of explaining what he was referring to. In other words, indeterminacy rules out the possibility of referential semantics.
II According to Quine If there were . . . an unknown ['ideal theory' θ (say a limit that would be attained by applying scientific method forever) which we could regard as completely true, still this would not settle the truth-value of] actual single sentences. We could not say . . . that any single sentence S is true if it or a translation belongs to θ, for there is in general no sense in equating a sentence of a theory θ with a sentence S given apart from θ. Unless pretty firmly and directly conditioned to sensory stimulation, a sentence S is meaningless except relative to its own theory; meaningless intertheoretically. (1960: 23-4) It is inessential to Quine's claim that θ be taken as a 'limit theory'; we could equally well take θ to be current scientific theory, which is treated in most contexts as if it were completely true. Quine's point then, in the first part of the passage, is that there are sentences S from theories not now accepted such that (i) we can not equate them with any sentences of current scientific theory and (ii) we can not find any objective grounds for deciding whether they are true or false. The preceding section provides a confirmation of this part of Quine's claim: both (4R) and (5P) are examples
of such an S. In other words, the observations of the preceding section show not only a failure of the concept of reference to do what it was supposed to do (viz., explain truth), but also a failure of the concept of truth at least as applied to certain single sentences of a theory: Newtonian physics as a whole is objectively false, but there is no fact of the matter as to how the falsity of the theory as a whole is to be distributed among the individual sentences of the theory. This much I agree,11 but still there is part of Quine's claim that I do not accept (or even fully understand): the part that concludes that S is 'meaningless intertheoretically'. In the rest of this paper I will sketch a way to handle referential indeterminacy (of the sort we've been discussing) which does not accord with this last Quinean claim. My approach will be to try to preserve as much of referential semantics, and of the claim (3), as can be preserved in the face of indeterminacy. The argument of the last paragraph of the preceding section shows, I believe, that there is at present no coherent way of using the term 'denotes' in connection with Newton's word 'mass'. I think that in this situation we have to develop a new semantic terminology that is capable of handling referentially indeterminate 11 Except that the last sentence holds only if we restrict our consideration to methods of distribution that accord with
classical 2-valued semantics. This will become clearer shortly. end p.187
expressions. The terminology I propose is 'partially denotes': I want to say that Newton's word 'mass' partially denoted proper mass and partially denoted relativistic mass; since it partially denoted each of them, it didn't fully (or determinately) denote either. The novel feature of partial denotation is that a singular term can partially denote more than one thing, and a quantity term like 'mass' can partially denote more than one quantity. A term that partially denotes more than one quantity is called referentially indeterminate;12 a term that partially denotes exactly one quantity is said to fully (or determinately) denote that quantity; and a term that doesn't partially denote anything is said to be denotationless. I do not try to define 'denotes' in terms of 'partially denotes', for I don't think that this can be done in any acceptable way. (Probably the best such definition is to take 'denotes' as equivalent to 'fully denotes'; but this would have the unattractive feature that indeterminate terms would not denote anything and yet would not be denotationless.) I find it more natural simply to abandon the term 'denotes', except when dealing with terms that are perfectly determinate (or can be assumed, in a given context, to be perfectly determinate). 13 If the term 'denotes' is abandoned, we can no longer express (HP), (HR), and (HΛ). But of course we can express close analogs of them, viz., (HP*) Newton's word 'mass' fully denoted proper mass; i.e., it partially denoted proper mass and nothing else. (HR*) Newton's word 'mass' fully denoted relativistic mass; i.e., it partially denoted relativistic mass and nothing else. (HΛ*) Newton's word 'mass' was denotationless; i.e., it didn't partially denote anything at all. But none of these analytical hypotheses are acceptable, for precisely the same reasons that none of (HP), (HR), and (HΛ) were acceptable. The problem with (HP*) and (HR*) is that each of them discriminates between proper mass and relativistic mass, and there appears to be no basis for such discrimination. In this respect, (HΛ*) is better than (HP*) and (HR*), but only at a high cost: according to (HΛ*), Newton's word 'mass' bears no semantic relation to anything at all, and as a consequence 'mass' turns out to be just like 'phlogiston', 12 This definition and that which follows are intended to apply to term tokens, not to term types. This is necessary in
order to distinguish indeterminacy from ambiguity. A term is ambiguous if different tokens of it have different semantic features; thus 'mass' as used today is ambiguous, since physicist R's tokens denote relativistic mass while physicist P's tokens denote proper mass. There is thus a sense in which the term type 'mass' as used today denotes (and hence partially denotes) more than one thing; but this does not demonstrate the existence of indeterminacy or show the failure of referential semantics; it merely shows that referential semantics has to be applied to tokens instead of to types. For a case of indeterminacy one needs to look to pre-relativity physics: here each token of 'mass' partially denoted two different quantities. 13 I would defend this last approach on the grounds that the word 'denote' is itself referentially indeterminate: it partially signifies the relation of full denotation and partially signifies the relation of partial denotation. For referentially determinate terms, full denotation and partial denotation coincide, and so we can safely apply the word 'denotes' to these terms without having to take the indeterminacy of 'denotes' into account. But if a term t partially denotes an object x without fully denoting it, the statement that t denotes x will lack truth value; and so to avoid the complication of truth-valueless metalinguistic sentences, it is best to drop the indeterminate term 'denotes'. end p.188
denotationally speaking. These objections to (HP*), (HR*), and (HΛ*) are not shared by the new analytical hypothesis (HI) Newton's word 'mass' partially denoted proper mass and partially denoted relativistic mass and didn't partially denote anything else. In particular, (HI) rules that 'mass' and 'phlogiston' are not alike, denotationally speaking: for clearly 'phlogiston' either does not partially denote anything, or else partially denotes a whole bunch of things, none of which is either form of mass. If we accept the idea that Newton was partially referring to proper mass and partially referring to relativistic mass when he used the term 'mass', which of these partial denotations are we to say affected the truth values of Newton's utterances? The answer of course is 'both'. When Newton uttered the sentence (5P*) Mass is independent of the frame of reference. did he say something true? The relevant facts, you'll recall, are that relativistic mass is not independent of the frame of reference, but proper mass is; so the answer is that when Newton uttered this sentence what he said was sort of true and sort of false—it was imprecise, and the imprecision mattered. There are also cases where the imprecision does not matter: (7) was such an example, since both the modern analogs of (7) are true. In order to turn these observations into an adequate truth definition for sentences containing indeterminate expressions, we have to extend our analysis of indeterminacy beyond singular terms and quantity terms, to other parts of speech such as general terms, for it is plausible that indeterminacy can arise in them also. An example, I suspect, is the word 'gene' as used by all biologists until a very few years ago: one analytical hypothesis governing the use of this term by earlier biologists is that it had the set of cistrons as its extension, a second is that it had the set of mutons as its extension, and a third is that it had the set of recons as its extension. 14 There is also a negative analytical hypothesis, that the extension of 'gene' was the empty set.15 But I think it could be argued, as was argued before with the term 'mass', that the negative hypothesis is inadequate, and that all the positive hypotheses are inadequate too, since they all ascribe to the word 'gene' (as used by earlier biologists) a determinate extension when the word was in fact indeterminate. To handle such cases of indeterminacy, I introduce the term partial extension; this allows us to express analytical hypotheses such as 14 See Schaffner 1967: 142-3; and references cited there. 15 For predicates actually, there are two possible 'negative analytical hypotheses': (a) that the predicate has the empty
set as its extension, and (b) that it doesn't have an extension at all. (b) sounds a bit more reasonable than (a), but only because (b) is compatible with the hypothesis of indeterminacy set out later in this paragraph; (b) is not, in other words, a viable alternative to the hypothesis of indeterminacy, any more than (a) is, for taken by itself (i.e., without use of the concept of partial extension) (b) would obliterate the semantic differences between any two referentially indeterminate predicates. end p.189
The word 'gene' had the set of cistrons as one partial extension, the set of mutons as another, and the set of recons as a third. It is convenient also to introduce the words signify and partially signify, by saying that a determinate predicate signifies its extension and that an indeterminate predicate partially signifies each of its partial extensions. In order to give a semantics for indeterminate expressions, let's introduce the term 'structure'; a structure for a sentence is a function that maps each name or quantity term of the sentence into some object or quantity, and maps each predicate into some set. The structure m corresponds to the sentence if each name or quantity term of the sentence partially denotes the thing that m assigns to it, and each predicate partially signifies the set that m assigns to it. Now, for each structure m, we can apply the standard referential (Tarski-type) semantics to determine whether the sentence is m-true or m-false, i.e., true or false relative to m. (To say that the sentence is m-true is to say that it would be true if the denotations and extensions of its terms were as specified by m.) We can then say that a sentence is true (false) if it is m-true (m-false) for every structure m that corresponds to it. Putting all these definitions together, we get definitions of truth and falsity in terms of partial denotation and partial signification. A few examples will illustrate how the truth definition works and show the plausibility of it. Consider first the sentence (5P*), and for the sake of illustration regard everything after the 'mass' as a primitive predicate. Then there are two structures corresponding to this sentence: both assign to the predicate 'is-independent-of-the-frame-of-reference' the same set (viz., the set of frame-independent quantities), but one (m 1 ) assigns proper mass to 'mass', whereas the other (m 2 ) assigns relativistic mass to 'mass'. Relative to m 1 the sentence is true, and relative to m 2 it is false, and, since these relative truth values differ, the sentence is not determinately true or false. A similar analysis will show that (4R) had no determinate truth value for Newton, but that (6) and (7) were determinately true.
Now let's consider sentences in which the same indeterminate term occurs twice. An interesting example is the conjunction of (4R) and (5P). Again there are two structures corresponding to this sentence. Relative to the one that assigns proper mass to 'mass', the first conjunct is false and the second true, so that the whole sentence is false; relative to the other structure the first conjunct is true and the second false, so that again their conjunction is false. The truth definition tells us, then, that the conjunction of (4R) and (5P) was false—even though each conjunct lacked truth value (so that there is no fact of the matter as to which conjunct was false). This is precisely the conclusion that seemed desirable earlier. Finally, we have to consider the very important case where there are two different indeterminate expressions in the same sentence. One such example of double indeterminacy is (11) The mass of a gene is less than a microgram. end p.190
(as used by pre-1950 genticists). Here the semantics says, plausibly enough, that the utterance was true, since for cistrons, recons, and mutons, both the proper mass and the relativistic mass are less than a microgram. But it would be well to consider also a case where a sentence contains two indeterminate terms from the same theory. In order to get such an example, let us suppose not only that was Newton's word 'mass' indeterminate, but that his word 'velocity' was also indeterminate—it was indeterminate between rate of change of distance with respect to normal time t, and rate of change of distance with respect to proper time T. (Perhaps this is implausible, but it illustrates the semantics admirably.) Now let us reconsider sentence (4R) on this supposition. We find that, since momentum momentum momentum momentum
,
equals (relativistic mass) · dx/dt does not equal (relativistic mass) · dx/dT does not equal (proper mass) · dx/dt equals (proper mass) · dx/dT.
As a result, (4R) comes out m-true for two of the structures m that correspond to it and m-false for the other two; therefore it comes out truth-valueless, just as it did before. It seems to me that this is just what we would intuitively want to say if we believed that 'mass' and 'velocity' were both indeterminate in the way I've supposed.16 I think these example make it plausible that the truth definition I've given, together with the appropriate hypotheses about what things the earlier theorist was partially referring to or partially signifying, leads to the sort of results we want. If so, then I have established the utility of the concept of partial denotation in dealing with theory change, and shown that scientific revolution can affect the semantic features of scientific terms in more complicated ways than many philosophers have allowed. No longer are we confined to the three alternatives: (i) term t has switched denotations during revolution r [as (HN) held]; (ii) t has acquired a denotation during r [as (HΛ) held]; and (iii) t has kept the same denotation through r. Now there are other possibilities, most notably (iv) t has undergone a denotational refinement during r; i.e., the set of things that it partially denoted after r is a proper subset of the things it partially denoted before. This added possibility is of considerable importance for debates about 'incommensurability' in science: it shows that we can accept the claim that we can't always equate a term from one theory with a term from 16 This may perhaps be controversial: some might want to say that Newton's utterances of (4R) were true, and that
only the first and last of the four facts listed above ought to be considered relevant to the truth value of these utterances. In other words, it might be held that only certain combinations of the partial denotations and partial extensions of the terms ought to affect the truth value of the whole sentences and that therefore my semantics needs to be complicated by imposing some restrictions on which combinations are considered. I don't object to this contention in principle, but I also don't see any very good reason for supposing it to be true (for the kind of indeterminacy under consideration in this paper). There may be other sorts of indeterminacy for which the more complicated semantics is necessary, but I will defer consideration of that more complicated kind of indeterminacy (correlative indeterminacy, I call it) to a later paper (Ch. 7). end p.191
a later theory, and still deny the incommensurability thesis, i.e., the thesis that the earlier and later terms cannot objectively be compared with respect to referential properties. It is worth making two final points about denotational refinement. First, the 'mass' example is not an example of denotational refinement pure and simple: it is an example of double refinement, since some physicists have refined 'mass' into a word for relativistic mass while others have refined it into a word for proper mass. I hope that by concentrating on this example I have not obscured the fact that simple refinement (where the term is refined in only
one direction) is also a possibility. That this really is a possibility should be obvious: 'mass' would have undergone a simple refinement rather than a double refinement if everyone had followed Einstein's example in adhering to (4R) and rejecting (5P). Simple refinement will doubtless be less obtrusive than multiple refinement (since there will be no linguistic disagreement resulting from the revolution like the linguistic disagreement between Physicists R and P); but it seems plausible that it does sometimes occur. 17 Second point: if I am right in thinking that denotational refinement is a fairly common feature of scientific revolutions, that suggests that future scientists may very well refine many of our current scientific terms, and hence that many of our current scientific terms are referentially indeterminate. (In fact, induction from the indeterminacy of terms in earlier theories may even suggest that science will never reach the stage where all of its terms are perfectly determinate.) I don't think that this conclusion conflicts with anything that is reasonably called 'scientific realism'; but an adequate discussion of this matter is beyond the scope of the present paper. I will conclude the paper by returning briefly to the comparison with Quine's views begun earlier in this section.
III By modifying the program of referential semantics in the way I have suggested, we come to rather different conclusions about indeterminacy from those reached by Quine. Quine thinks the existence of indeterminacy shows that scientific terms are 'meaningless [and denotationless] except relative to [their] own theory; meaningless [and denotationless] intertheoretically.' This claim is then used to buttress his tenet that 'truth . . . is immanent [i.e., relative] 17 One might even be tempted to speculate that all scientific revolutions involve denotational or significational
refinement, but I think that that is unlikely. Consider the sequence of revolutions leading from Thomson's theory of the atom to our own. In all these revolutions, it is hard to see any significational refinement in the term 'electron': there is nothing that Thomson can plausibly be said to have partially signified, beyond the set of electrons. Moreover, the kind of argument used against (HN) and (HΛ) rules out the possibility that Thomson's word signified 'Thomsonian electrons' (things that satisfied all of Thomson's beliefs involving 'electron'), or that it signified nothing at all; the only alternative seems to be (iii), that the word kept the same denotation despite the radical change of theory. (Actually the matter becomes a good deal more complicated when one reflects that 'electron' might be referentially indeterminate today (cf. next paragraph); but even so, the above considerations do not completely lose their force in showing that scientific revolution is possible without denotational refinement.) end p.192
to the conceptual scheme':18 for the natural way to get an objective (nonrelative) notion of truth is to try to explicate that notion in terms of denotation and signification, but if those semantic notions were also relative to the conceptual scheme then the desired objectivity would not be achieved. Now, what I contest in this argument is the assumption that the semantic relations of denotation and signification are in any interesting sense 'relative to the conceptual scheme'; on my view they are perfectly objective relations which hold between terms on the one hand and extralinguistic objects or sets of objects on the other. The existence of referential indeterminacy shows only that the relations of denotation and signification are not well-defined in certain situations, and that if we want to apply semantics to those situations we have to invoke the more general relations of partial denotation and partial signification. But these more general relations (like the less general ones, in the situations in which those less general relations are well-defined) are perfectly objective relations between words and extralinguistic objects (or sets of objects), and are not in any interesting sense 'relative to the conceptual scheme.' This account of the differences between Quine's view of indeterminacy and my own view is meant only to be suggestive; I will provide a more detailed investigation of the differences, and an argument for my view as opposed to Quine's, in the next chapter.19 There I will also extend the above semantics to certain other sorts of indeterminacy which could not be discussed in this chapter.20 18 Quine (1984b: 304). This view is set out more fully in §6 of Quine 1960. 19 But sections iii and v of Ch. 1 are also of considerable relevance to our disagreement. 20 I am grateful to Michael Friedman, Clark Glymour, and David Hills, for helpful suggestions about this paper. My
work on the paper was supported in its final stages by the National Endowment for the Humanities. end p.193
Postscript 1. This paper was simultaneously an attempt to call attention to the subject of semantic indeterminacy and to motivate a non-Quinean approach to it.
Quine took translation to be the central semantic notion, and so when he argued for semantic indeterminacy, he took this to be at bottom a matter of indeterminacy of translation. In contrast, when I wrote this paper I thought that the primary semantic notions were single-language notions like reference, and so the central kind of indeterminacy should be not indeterminacy of translation but indeterminacy of reference. And, I argued, indeterminacy of reference didn't call for a very radical revision of the kind of (nondisquotational) referential semantics advocated in Chapter 1 of this volume. My plan was to use the view of referential indeterminacy advocated in this paper to undermine Quine's attack on the ordinary (nondisquotational) notion of reference. Quine had argued that since alternative translations that don't preserve reference can be equally good, there is something wrong with the notion of reference as normally understood. My view (argued in Chapter 7 of this volume and Field 1975) was that the Quinean examples of indeterminacy (if one took them seriously) could be handled by the same sort of devices used in the case of indeterminacy discussed here. My assertion of this at the end of the paper puzzled quite a few readers, but I think that the later papers bore out my claim. This may not be the best way to deal with indeterminacy—indeed, Quine's view seems to me now more attractive than it did when I wrote this paper, and I advocate a view that differs from both in Chapter 10—but its possibility shows the limits of using indeterminacy as an argument against the basic idea of referential semantics. 2. The main example was prompted by reflections about the extent to which historical accident is involved in how we extend terms in prior theories.1 In the early days of relativity theory, and in the relativity textbooks I read as an undergraduate, the unqualified term 'mass' was primarily used for relativistic mass. I had been persuaded by Hilary Putnam that reference is usually preserved through scientific revolutions, so thought there was an initial inclination to suppose that Newton was referring to relativistic mass. But it also seemed a historical accident that the unqualified term 'mass' wasn't primarily used for rest mass in relativity theory. (Indeed, more recent textbooks were tending to use it in that way.) If it had been, different judgements about what Newton was referring to would have been natural. Of course, how we use the term 'mass' is really secondary: we do, after all, think that some terms shift 1 The opening example in Wilson 1982 is another good illustration of how historical accident can affect the
naturalness of claims about what people with radically different theories from ours are referring to. end p.194
their reference. The key question is, what candidates are there for what Newton's term referred to; and there are two natural choices. (See footnote 18 to Chapter 8 for a response to arguments by Earman and Fine that rest mass is the only correct choice.) I should say more about my contention that we shouldn't adopt the 'analytical hypothesis' that Newton's term straightforwardly lacked denotation. I defended this by pointing out that on any reasonable semantics for denotationless expressions (e.g. Scott 1967), the term's lacking denotation would have the consequence that if Newton said things like 'The mass of the earth is less than the mass of the Sun' what he said would not be literally true. 2 This consequence seemed unattractive, but it is probably not in itself beyond defense: we could regard the sentence as 'conveying information' in the same way that Priestley's assertions about loss of phlogiston convey information about gain in oxygen. But if we take that route in the case of Newton, then unless we adopt a view that gives privilege to our own language, we must say something analogous in our own case: we must say that if there are two quantities, mass 1 and mass 2 , such that our own beliefs that we would express using 'mass' are about equally true of each, then we are saying nothing true when we say 'The mass of the Earth is less than that of the Sun'. That conclusion about our own case would be hard to swallow, and this was part of the motivation for not adopting the analogous conclusion about the Newtonian case. (This was hinted at, not very clearly, in footnote 10 of the paper.) Of course this way of arguing would lose some of its force if we adopted a translational approach to semantics that gave a special role to our own language. 3. My main qualm about this paper today is that it may take the notion of reference too seriously. On the most radical version of Quine's views, when we ask what Newton was referring to there is nothing to be right or wrong about, there is only the question of whether translating his term one way or the other would meet our goal-driven standards, standards which may vary from one context of translation to the next. In some contexts the translation 'relativistic mass' may be preferred; in other contexts the translation 'rest mass' may be preferred; in other contexts we may prefer to translate it by a made up term 'Newtonian mass', which we recognize to be true of nothing. (Relative to this latter translation we will not adopt the Kuhnian claim that Newton's term referred to Newtonian mass, we will adopt the claim that it didn't refer at all.) The idea isn't that one of these translations is correct for some of Newton's uses of 'mass', a second for others, and the third for still others;3 rather, it is 18 This last point seems to be missed by Earman and Fine (1977). On other points they raise: (1) Fine makes a big
point of the fact that Einstein in his later writings decided that it was better in formulating relativity to use 'mass' for rest mass than for relativistic mass (in contrast, I might add, both to his original usage and to the usage of many textbooks). I personally agree that this later policy makes for a more aesthetically pleasing presentation of special
relativity; but this seems at best loosely related to the question of what we take Newton as having been referring to. (2) Earman's remark 'If . . . "relativistic mass" denotes a new kind of mass, then for every distinct kind of noninertial coordinate system there will be yet a new kind of mass' (p. 537) seems to me highly misleading. We don't say that there is a new kind of energy for each kind of coordinate system, but that energy is a frame-dependent quantity: in effect, a function from objects and coordinate systems to numbers. Similarly for relativistic mass. 2 Once indeterminacy is recognized, we can of course also consider the 'analytical hypothesis' that Newton's term partially referred to rest mass, partially referred to relativistic mass, and partially lacked denotation: given any standard semantics for denotationless terms, the approach adopted in this paper carries over, as does the obvious quantitative extension where partial denotation and therefore truth come in degrees. Newton's term being 'partially denotationless' would make the Newtonian sentence about the mass of the Earth and the Sun fail to be completely true; but it could come out 'close to' true if the 'degree to which the term lacked denotation' was low, which would not be so unattractive. 10 I would prefer not to have to rest my argument on the appeal to charity: I would prefer it if I could show that the purpose for which we want the notion of truth would not be satisfied if we declared (6) and (7) false in these circumstances. I don't know in detail how to show this, but I suspect that by further developing the remarks on the purpose of the notion of truth in the last section of Chapter 1 (and also arguing that we have good reason to suspect that many of our own words will someday be in the position that Newton's word 'mass' is in today), we could get the desired proof. 3 That is a determinacy hypothesis, with little to recommend it I think, since it makes many apparently valid Newtonian inferences suffer from fallacies of ambiguity. end p.195
that even when translating a given use of 'mass' there are multiple acceptable standards of translation, depending on the translator's needs of the moment, with talk of 'correctness' not straightforwardly applicable. From this viewpoint, it is unobvious that recognition of any indeterminacy in our own terms makes sense: it seems inevitable that 'mass' (as we use it) determinately refers to mass (assuming there is such a thing) and that's that; no discovery that future scientists might make can undercut this. I'm inclined to separate (i) the radical Quinean view of translation from (ii) the suggestion in the last sentence that no sense can be made of indeterminacy in our own language. I think that (ii) does require (i): a less radical view of indeterminacy of translation, according to which indeterminacy of translation merely shows indeterminacy in the notion of synonymy we employ, is incompatible with (ii) since it requires that our own language contains the indeterminate term 'synonymous'. But I think that not only does (i) not require (ii), there are considerable difficulties in maintaining (ii) given the perspective on translation required to maintain (i). These matters are discussed in Chapter 10, where (as in Chapter 8) I advocate freeing the notion of indeterminacy from any reliance on the theory of reference. On the view of indeterminacy discussed in these later chapters, the notion of partial reference does not bear the explanatory weight that I suggested for it in this chapter. Indeed, we could do without it. Alternatively, we might want to introduce the notion of partial denotation for one's own language via a schema: say x ['b' partially refers to x iff ¬ D ¬ (b = x)], where 'D' means 'determinately'. This involves quantifying into the determinately operator; the interpretation of this would require some discussion, especially when we have no way of determinately referring to the objects in the range of the quantifier. 4. The views developed in this chapter have an application to Rudolph Carnap's and David Lewis's views on the semantics of theoretical terms (Carnap 1956, ch. 26; Lewis 1970b and 1972); an application that I alluded to without explanation in Chapter 2. Let's review the background: if a new theoretical term T is introduced into our language by a theory Θ(T), Carnap proposed that the content of the theory introduced is really just the content of the Ramsey sentence xΘ(x). In the special case where Θ(x) is uniquely realized, i.e., where !xΘ(x), we can say that T denotes the unique thing that satisfies Θ(x). But when Θ(x) is multiply realized, what does T denote? It seems that it must denote something, since otherwise we can't explain why Θ(T) is true (as it must be, given Carnap's thesis that it has the content of xΘ(x)). Carnap proposed that when Θ(x) is multiply realized, T denotes an arbitrary one of the things satisfying Θ(x); but Lewis very plausibly objected to this, on the grounds that it is hard to see by what mechanism the user of T manages to single out one as opposed to the others. Lewis thus felt compelled, reluctantly, to suppose that the content of the end p.196
term introducing theory Θ(T) is not xΘ(x) but the modified Ramsey sentence !xΘ(x): in other words, the theory is false if Θ(x) is multiply realized, so that T can be taken as lacking denotation and the problem of how it picks out its
denotation doesn't arise. As he noted in 1970b, this does have a high intuitive cost, because it is not in the least evident that the person who asserts Θ(T) means to commit himself to the claim that nothing other than T satisfies Θ(x). Its cost became greater in Lewis 1972 where he applied his approach to theoretical terms to functionalism, because the whole point of functionalism is to allow for multiply realized states. Actually, because of the way that Lewis's approach to theoretical terms is applied to the case of functionalism (which involves a hidden relativity in Θ(x) to an organism and a time), there is no problem allowing different realizations of a functional state in different organisms, or in the same organism at different times; and this is enough to accommodate an important part of functionalism. But many functionalists would want to also allow multiple realizations in the same organism at the same time, and this seems ruled out by the Lewis approach to the semantics of functional predicates. That is where the present paper comes in: Lewis didn't need to revise Carnap's view that the content of Θ(T) is just xΘ(x), he could have just said that when Θ(x) is multiply realized, T partially denotes each of the realizers.4 In particular, a functionally defined psychological term, when applied to a given organism at a given time, partially denotes each of the physical states that satisfy the functional property in that organism at that time. Multiple realization of a functional property even within the same organism at the same time becomes unproblematic. In the above discussion I considered only theories that introduce a single basic new theoretical term, from which all other new theoretical terms can be defined (allowing the old vocabulary as well to appear in the definitions). For the more typical case of a theoretical term introduced in a package with other equally basic theoretical terms, the point is unchanged as long as one uses the idea of correlative indeterminacy mentioned in this chapter and treated in detail in the next. (In the language of the next chapter, one takes the n-tuple of theoretical terms that are introduced together as an 'atom'.) This is important in functional theories, where (e.g.) terms like 'belief' and 'desire' are bound to be introduced together. If belief and desire are multiply realized in a given organism, say belief by the relations B 1 and B 2 (relating organisms to inner representations) and desire by the relations D 1 and D 2 , then whereas the pairs and must realize the term-introducing theory, the pairs 4 This may seem to give Lewis less than he wanted: he wanted to define theoretical terms, not merely specify their
denotations, and how are we to do that? The answer is that we can do it in just the way that Lewis himself did (viz., T = df ιxΘ(x)), as long as we adopt a slightly nonstandard theory of descriptions, according to which a description 'ιxFx' partially denotes each thing that satisfies 'Fx' (thereby making a significant difference between cases where 'Fx' is multiply satisfied and cases where it isn't satisfied at all). (When I presented the idea of partial denotation in a colloquium at Princeton in 1971, Lewis himself noted the possibility of such a nonstandard theory of descriptions. I don't know if he had in mind the application to defining theoretical terms; it didn't occur to me until a few years later.) end p.197
and need not.5 That is the reason, obscurely alluded to in footnote 24 of Chapter 2, why one can't allow for multiple realization in a single organism simply by using Lewis's formulation except with the uniqueness requirement dropped. The proper remedy, also obscurely alluded to there, is to take the terms introduced together as an 'atom' that partially denotes as a whole; this allows the needed association of the realizers of the believing relation with the realizers of the desiring relation in determining the truth values of belief-desire sentences. 5 For simplicity I ignore the fact that other notions besides belief and desire—including the basic syntactic relations
among inner representations—will surely need to be taken as primitives in the basic functional theory. 24 The uniqueness requirement is Lewis'. I think that it must be taken with a grain of salt, but I do not have the space
here to explain my reservations or to develop the machinery needed to avoid it. The word 'unique' in (2) cannot merely be dropped. [Note added for this volume: These remarks are explained in the Postscript to Chapter 6.] end p.198
7 Quine and the Correspondence Theory Abstract: A correspondence theory of truth explains truth in terms of various correspondence relations (e.g. reference) between words and the world. Quine's doctrine of indeterminacy is often supposed to undermine correspondence theories of truth, and Quine himself argued this in Ontological Relativity with his doctrine of 'relative reference'. This chapter argues that the relativization of reference that Quine proposed makes no sense, and that we can accommodate indeterminacy by generalizing the kind of correspondence relations we appeal to; this will save the correspondence theory. Includes an appendix on vagueness and indeterminacy in the metalanguage, and a new postscript. Keywords: correspondence theory of truth, higher-order vagueness, identity, indeterminacy, ontological relativity, Quine, reference, supervaluation, vagueness
Hartry Field
Quine (1968a) has stated the following doctrine of 'ontological relativity': What makes sense is to say not what the objects of a theory are, absolutely speaking, but how one theory of objects is interpretable or reinterpretable in another. (p. 50) It seems clear from the context that the phrase 'objects of a theory' is intended to apply to the objects denoted by the singular terms of the theory and the objects in the extension of the general terms of the theory. So if we say that a predicate 'signifies' its extension, we can rewrite the above quotation as: (1) What makes sense is to say not what the terms of a theory denote or signify, absolutely speaking, but how one theory is interpretable or reinterpretable in another. This is a very radical contention, for it seems to preclude the possibility of a correspondence theory of truth. By a correspondence theory of truth, I mean a theory that says that the notion of truth can be explained by appealing to the relation between words on the one hand and the objects that they are about on the other. The objects that words are about are (by and large) extralinguistic objects; so the central feature of a correspondence theory is that it explains truth in terms of some correspondence relations between words and the extralinguistic world. But (1) denies the possibility of such a correspondence theory: it says that the only interesting correspondence you can get is a correspondence between the words of one theory and the words of another. Quine's only argument for (1) is based on his thesis that semantics is radically indeterminate. But I will argue that even if semantics is as indeterminate as Quine says it is, we ought to believe in a correspondence theory of truth and reject (1). If I am right about this and about several of the other points I shall make about indeterminacy, I think it will follow that Quine's radical indeterminacy thesis, while certainly not without philosophical interest, is less devastating to traditional semantic theory than is usually supposed. Quine's main argument for the claim that semantics is radically indeterminate is physicalistic. At the beginning of 1968a he writes: end p.199
[K]nowledge, mind, and meaning are part of the same world that they have to do with, and . . . are to be studied in the same empirical spirit that animates natural science. There is no place for a priori philosophy. (p. 26) He then goes on to suggest that once this position is taken seriously, one is bound to recognize the existence of indeterminacy. Suppose, for instance, that we are interested in determining the extension of the foreign term 'gavagai.' If we look at the matter physicalistically, we see that there is no sense in saying that 'gavagai' has the set of rabbits as its extension as opposed to the set of undetached rabbit parts, unless we can find physical facts— facts about the speaker's behavioral dispositions, his causal relations to rabbits, and so on 1 — which determine that it is the set of rabbits rather than the set of undetached rabbit parts that is the real extension of the term. And Quine thinks it is obvious that there are no physical facts underlying the use of the term that could allow us to say that the term signifies one of these sets rather than the other. To set out the matter in a bit more detail, let us suppose that 'gavagai' and 'potrzebie' are foreign terms that are most naturally translated as 'rabbit' and 'dinosaur,' and that 'glub' is a term of the same language that is most naturally translated as 'is identical to.' Then according to Quine, there is no fact of the matter as to whether (i) 'gavagai' signifies the set of rabbits, 'potrzebie' the set of dinosaurs, and 'glub' the identity relation; or (ii) 'gavagai' signifies the set of undetached rabbit parts, 'potrzebie' the set of undetached dinosaur parts, and 'glub' the paridentity relation (that is, the relation of being undetached parts of the same object). To see why this seems plausible to Quine, suppose that there were a fact of the matter; suppose, for instance, that (i) were really true and (ii) really false. Then it ought to be possible to state facts about the way that 'gavagai' is used which make this word a word for rabbits rather than for undetached rabbit parts. One place we might look for such facts is in the causal links between the rabbits on the one hand and the uses of 'gavagai' on the other. But this does not seem to work: any causal links between rabbits and uses of 'gavagai' are also causal links between undetached rabbit parts and uses of 'gavagai', so it appears that causal connections by themselves will not do the job. What then are we to supplement them with? Perhaps the foreigners' dispositions to assent and dissent? Suppose that when we place a foreigner in an environment containing exactly one visible rabbit, he tells us that for every pair of nearby gavagais x and y, x glub y. This fact about the foreign speakers determines that if 'glub' is a word for identity then 'gavagai' is a word for rabbits, and that if 'glub' is a word for paridentity then 'gavagai' is a word for undetached rabbit parts. But what facts determine whether 'glub' is a word for identity or paridentity? There is no 1 Quine thinks that it is sufficient to consider behavioral dispositions alone; but to a large extent his discussion can be
freed from this dubious assumption (and from his equally dubious verificationist assumptions), as the next paragraph illustrates. end p.200
obvious answer to this question that does not assume an answer to the question of whether 'gavagai' or some similar word is a word for 'whole objects' or for their undetached parts. Again the attempt to find physical facts which decide between (i) and (ii) fails, and it is hard to see where else such physical facts are to be found. Someone might perhaps claim that either (i) is right and (ii) is wrong, or vice versa, even though no physical facts determine which; but this is the position that Quine calls 'the myth of the museum' and which he rejects on physicalistic grounds. I think that Quine is correct in holding that 'the myth of the museum' is a totally unreasonable position: if indeterminacy is to be rejected, it must be rejected by finding physical facts which do in some sense decide between (i) and (ii). I believe that there are physical facts which (in the relevant sense) decide between (i) and (ii), but I shall say nothing about that in this paper. Instead, I shall pretend to believe that Quine is right about this example of indeterminacy, and consider the consequences that such indeterminacy would have for the correspondence theory of truth. The first thing I want to do is note that if such indeterminacies exist, then at least half of (1) is correct: it makes no sense to ask what the terms of a language or theory refer to (denote or signify). Actually, what I have just said goes beyond (1) is some respects, for what Quine claims in (1) is that it does not make sense to ask what the terms of a language or theory refer to 'absolutely speaking', while my remark in the previous sentence suggests that (if the indeterminacy thesis is right) it does not make sense to ask what terms refer to either in the absolute sense or in Quine's relativized sense. For the moment, however, let us ignore the notion of relativized reference, and use the terms 'refer,' 'denote,' and 'signify' in the ordinary, absolute way. In that case, since there is no fact of the matter as to what a term like 'gavagai' signifies (has for its extension), it seems pretty obvious that there is no sense in speaking of 'the extension' of the term. This conclusion may seem to rule out the possibility of a correspondence theory of truth, for the most obvious form of the correspondence theory of truth is one which explains truth in terms of such 'correspondence relations' as denotation and signification. 2 What I claim, however, is that a correspondence theory is still possible: all that is needed is the introduction of certain more general correspondence relations between words and extralinguistic objects (or sets of objects). For instance, the difficulty with the relation of signification was that we had to choose between saying that 'gavagai' signified the set of rabbits and saying that 'gavagai' signified the set of undetached rabbit parts, and that according to the indeterminacy thesis there is no physical basis for such a choice. So to avoid having to make such a choice, why not 2 Such a correspondence theory would say, for instance, that for 'Caesar crossed the Rubicon' to be true there must
be objects x and y and a relation (in extension) R such that 'Caesar' denotes x, 'the Rubicon' denotes y, 'crossed' signifies R, and x bears R to y. This illustrates how such a correspondence theory would work for relatively simple sentences; and by employing Tarski's work on truth we can easily extend the treatment to more complex sentences. (See Ch. 1.) end p.201
introduce a new correspondence relation— call it 'partial signification'— and say that the term 'gavagai' bears this correspondence relation both to the set of rabbits and to the set of undetached rabbit parts? (Each of these sets will then be called partial extensions of the term 'gavagai': so even though the term has no extension, it has a number of different partial extensions.) Of course, the introduction of new correspondence relations like partial signification is not too interesting unless it proves possible to explain truth in terms of them; this raises problems which I will turn to shortly. But let us ignore these problems of detail for the moment, and say more about the general idea of handling indeterminacy by introducing new correspondence relations. To see the import of this idea, and how the idea differs from Quine's ideas, let us shift our attention from foreign languages to our own language— or, better yet, to my own language as used by me right now. It is clear that the indeterminacy argument outlined before applies just as cogently to our own language as it does to the foreign language. In other words, we can argue that there is no fact of the matter as to whether (i*) 'rabbit' (as I use it now) signifies the set of rabbits, 'dinosaur' the set of dinosaurs, and 'is identical to' the identity relation; or (ii*) 'rabbit' (as I use it now) signifies the set of undetached rabbit parts, 'dinosaur' the set of undetached dinosaur parts, and 'is identical to' the paridentity relation, on the grounds that there are no physical facts that could decide between (i*) and (ii*). To say, in the face of this argument, that (i*) is really true and (ii*) is really false is to fall victim to the myth of the museum. 3 So I think that if we believe in this example of indeterminacy, we must give up speaking of the extension of 'rabbit,' and say instead that 'rabbit' (as we use it right now) has the set of rabbits as one partial extension and the set of undetached rabbit parts as
another.4 3 Note that this argument shows that the 'disquotation schema'
'—' signifies the set of —'s and nothing else cannot be accepted by anyone who believes in indeterminacy. The schema does provide a partial axiomatization of the concept of signification; but to axiomatize a concept is not to show that the concept is physicalistically acceptable. (For more on physicalistic acceptability, see Secs. iii-v of Ch. 1.) It does not follow that the believer in indeterminacy is deprived of all disquotation schemas: he may be able to adhere to unusual schemas like '—' partially signifies the set of —'s and partially signifies the set of undetached — parts. Such schemas might axiomatize, and clarify, the concept of partial signification to precisely the same degree that the more usual schemas axiomatize, and clarify, the more usual semantic concepts. 4 The concept of partial signification is not really as unfamiliar as it may sound, for we implicitly employ it in giving the semantics of vague expressions. Suppose we were asked what the extension of the English phrase 'tall man' is— is it the set of men taller than 6′0″, or the set of men taller than 6′1/2″, or what? Clearly there is no fact of the matter as to which of these sets is 'the real extension' of the English phrase 'tall man'; for clearly 'tall man' does not simply signify a particular set, but partially signifies various different sets— viz., those sets of form {x|x is a person whose height is greater than h} where h takes on values in some region centering around six feet and extending a few inches in either direction. (A still better account of vagueness could be given by quantifying the notion of partial signification: thus— if we pretend that the set of possible heights is discrete— we could say that 'tall man' signifies various different sets to various different degrees.) That the notions of partial denotation and partial signification (or quantified forms of these notions) are of use for dealing with vague expressions is by no means a novel point; e.g., it is suggested in the appendix to Lewis 1970a. These cases are unexciting because they are very unsurprising, and because we can easily do without such vague terms whenever our conversation turns to exact and serious purposes. But there is at least an abstract possibility that something similar to vagueness occurs even where we least expect it: it is possible that there are much more deep and pervasive ways in which our terms have indeterminate application to the world. And what Quine's argument about 'gavagai' seems to me to suggest is that this abstract possibility is in fact realized. end p.202
The point of view I have just outlined is very different from Quine's. On Quine's view, there is no need to give up the ordinary semantic notions of denotation and signification (or extension); instead, we can relativize them. Consider first the case where we are giving a semantics for a foreign language. On Quine's view there is no need to abandon all talk of what a foreign term like 'gavagai' signifies: what we must do, however, is say that relative to the obvious translation manual it signifies the set of rabbits, and that relative to an unobvious but nonetheless acceptable manual it signifies the set of undetached rabbit parts. The central role that translation manuals play in Quine's semantics reflects the doctrine of ontological relativity: the view that it makes no sense to speak of 'absolute' correspondence relations between words and extralinguistic objects, and that what does make sense is to say how one language or theory is translatable and retranslatable into another. There is, however, a serious difficulty with Quine's view: the notion of a general term signifying a set relative to a given translation manual (or of a singular term denoting something relative to a given manual) does not seem to make any sense. (1) seems to suggest that we can explain the idea of signification relative to a manual as follows: (2) To say that a term T used in one language signifies the set of rabbits, relative to a translation manual M, is in effect just to say that M translates T as 'rabbit.' But this is not a satisfactory explanation. The difficulty becomes clear when we try to define explicitly the notion of relative signification on the model that (2) suggests. The obvious first attempt is (3) For every predicate T, set {x | Fx}, and manual M, T signifies {x | Fx} relative to M if and only if M maps T into 'F.' But this clearly involves a use-mention confusion since we are trying to quantify over a variable 'F' that appears both inside and outside quotation marks.5 Can (3) be modified so as to avoid this defect? Yes: we can say 5 The point would perhaps be a bit clearer for singular terms than for general terms like 'gavagai,' since certain irrelevant issues about the use of predicate letters as variables would not then arise. For singular terms, the unacceptable 'definition' analogous to (3) is
(3′) For every name T, object x, and manual M, T denotes x relative to M if and only if M translates T as 'x'. end p.203
(4) For every predicate T, set y [or {x | Fx}], and manual M, T signifies y [or {x | Fx}] relative to M if and only if M maps T into some term which signifies y [or {x | Fx}]. But (4) defines Quine's relativized notion of signification only in terms of an unrelativized notion of signification applied to our own language (the language into which we translate). So it seems that we have to understand this unrelativized notion before we can understand the relativized notion employed in (2). The difficulty is obvious: the whole point of relativizing the notions of denotation and signification to a translation manual was that due to the indeterminacy of reference (or 'inscrutability of reference,' to use Quine's phrase) the unrelativized notions of denotation and signification are not physicalistically acceptable. But the foregoing remarks show that once this indeterminacy is taken seriously and applied to our own current language as well as to other languages, the manual-relative notions of denotation and signification are not acceptable, either. By employing them, Quine himself has become a victim of 'the myth of the museum.' It is clear from these remarks that what Quine needs for his notion of reference-relative-to-a-translation-manual to make sense is some link between the word 'rabbit' of our own language and the actual rabbits. But the problem is that the indeterminacy thesis denies the existence of any such connection: it denies the existence of any connection between 'rabbit' and rabbits that does not also hold between 'rabbit' and undetached rabbit parts. One possible response to this would be to simply give up the notion of denotation and extension, and make do with (i) the notion of a singular term in another language being codenotational with (denoting the same thing as) a singular term of our language (relative to a translation manual M), and (ii) the notion of a general term in another language being coextensive with a general term of ours (relative to a manual M). Quine does not take this course, and for good reason: the notions of codenotationality and coextensiveness are not powerful enough semantic notions to be of much utility— e.g., they are of no use whatever in a theory of truth, for it is impossible to define truth (or even material equivalence) in terms of them. Quine's response in 1968a is different: he proposes to introduce a new sort of relativity— relativity to a background language. He admits that we cannot say that 'rabbit' (as we use it) refers absolutely to the rabbits; but, he suggests, surely no one can complain if we say that 'rabbit' refers to rabbits relative to our own language? Thus he writes: It is meaningless to ask whether, in general, our terms 'rabbit,' 'rabbit part,' 'number,' etc., really refer respectively to rabbits, rabbit parts, numbers, etc., rather than to some ingeniously permuted denotations. It is meaningless to ask this absolutely; we can meaningfully ask it only relative to some background language. . . . Querying reference in any more absolute way would be like asking absolute position, or absolute velocity, rather than position or velocity relative to a given frame of reference. (1968a: 48-9) end p.204
Unfortunately, Quine's suggestion will not work: if the indeterminacy thesis precludes us from making sense of any connection between 'rabbit' and the rabbits that is not equally much a connection between 'rabbit' and the undetached rabbit parts, it is clear that merely by relativizing to our language (that is, to our word 'rabbit' and to a whole bunch of other expressions) there is no hope of improving the situation. Quine tries to evade this conclusion by invoking his well-known analogy between the doctrine of ontological relativity and Leibniz' relational doctrine of space. Leibniz held— in analogy to (1)— that it makes no sense to speak of relations between physical objects and absolute space, and that what does make sense is to speak of spatial relations between physical objects. But Leibniz held that locutions which appear to state relations between physical objects and absolute space (e.g., 'object x has position p or velocity v') do not need to be given up; we can reinterpret them so that they do not state such relations, by relativizing them to a coordinate system composed of physical objects. Similarly, Quine suggests, we can adhere to (1) without giving up the predicates of denotation and signification, as long as we relativize them to a 'coordinate system' of words. But there is a crucial disanalogy here: on Leibniz' theory, we can understand relativized claims about the relations of physical objects to places only because places are understood as constituted by the relations of physical objects; whereas no one holds that physical objects are constituted by the relations of words. This difference leads to a further difference of more direct relevance: whereas the relativized predicate 'x has velocity v relative to y' is definable in terms of the spatial relations between x and y (viz., as the time derivative of the distance), the relativized predicate 'T 1 denotes x relative to T 2 ' is not definable in terms of the linguistic relations between T 1 and T 2 . In fact (and this was my argument earlier), there is no hope of defining this last predicate at all unless we can establish a relation between
either T 1 or T 2 on the one hand, and x (and no object other than x) on the other. But that is just what the indeterminacy thesis applied to our own language precludes us from doing. Given this crucial disanalogy, it is clear that Quine's indeterminacy thesis forces us to give up not only the absolute notions of denotation and signification, but even the relativized notions which Quine has proposed as surrogates for them. I have urged, however, that there is nothing in the indeterminacy thesis which forces us to give up various generalizations of the notions of denotation and signification— for example, the notions of partial denotation and partial signification, which are just like the notions of denotation and signification except that a single term is allowed to partially denote more than one object or to partially signify more than one set. Still, the existence of such correspondence relations as these is not a cause for much satisfaction unless we can use them in an explanation of truth and falsity. What I want to do is investigate the question of how this can be done. In doing this, it will be convenient to start with a simpler example of indeterminacy. (The example is one which I employed in the previous chapter; and end p.205
I believe that it is a genuine example of indeterminacy, though it is an indeterminacy of a much more limited variety than the indeterminacy which Quine's examples purport to establish.) I claim that we can translate certain outdated physical theories into current theory in a number of ways; no one translation is best. Suppose, for instance, that we want to translate Newtonian mechanics into special relativity. Then there are two natural ways to translate the word 'mass': we could translate it as 'relativistic mass,' or we could translate it as 'rest mass.' If we translate it in the first way, Newton's tenet (5) Momentum is mass times velocity comes out strictly true, but his tenet (6) Mass is invariant (that is, independent of the frame of reference) comes out false (though approximately true at low velocities). If, on the other hand, we translate it in the second way, (6) comes out strictly true and (5) comes out false (but approximately true at low velocities). I claim that there is no fact of the matter as to which of these translations is 'the correct one,' and hence no fact of the matter as to which of Newton's sentences (5) and (6) was strictly true. 6 I hope that my earlier remarks have made clear the sort of way I want to treat such examples of indeterminacy: on my view, what the example shows is that Newton's word 'mass' partially denoted both relativistic mass and rest mass; since it partially denoted both these quantities, it did not fully (or determinately) denote either. Perhaps I should explicitly mention that the notion of partial denotation is meant to apply to terms that have determinate application as well as to terms that do not: a term with determinate application is a term that partially denotes exactly one thing. So when we give a semantics for sentences that include both determinate and indeterminate expressions, there is no need to employ the concept of denotation in addition to the notion of partial denotation: the latter concept is just a generalization of the former. Let us now provisionally sketch such a semantics. The first step is to introduce the model-theoretic notion of a structure. For our purposes, we can say that a structure for a language L is a function that maps all the names of L into extralinguistic objects and all the predicates of L into sets of extralinguistic objects. Note that a language will have many structures: for instance, there are structures which assign Ted Williams to Newton's term 'mass,' and there are other structures which assign the set of aardvarks to 'is invariant.' But if we ignore the existence of indeterminacy for a moment, there is one structure for the language L that is worth singling out— namely, the one that assigns to each name of L the object that name denotes, and that assigns to each predicate 6 The 'mass' example illustrates Quine's contention that two acceptable manuals for translating a foreign language or
theory might 'dictate, in countless cases, utterly disparate translations; not merely mutual paraphrases, but translations each of which would be excluded by the other system of translation. Two such translations might even be patently contrary in truth value, provided there is no stimulation that would encourage assent to either' (Quine 1960: 73-4; italics mine). end p.206
of L the set of objects that that predicate signifies. Let us say that this structure is the one that accords with the semantics of L. Once we remember the existence of indeterminacy, however, we lose the ability to single out a unique structure in this way. But we can do the next best thing: we can introduce a class of structures, each of which partially accords with the semantics of L. We can provisionally define this notion of 'partially according' as follows: (7) A structure m partially accords with the semantics of L if and only if each term of L partially denotes or partially
signifies the entity which m assigns to it. Partial accordance is of course just a generalization of accordance; when all the terms of L are determinate, only one structure partially accords with the semantics of L— namely, the one that fully accords with the semantics of L. Now that we have the notion of a structure, it is easy to explain the notion of truth-in-a-structure: to say that a sentence is true in the structure m is in effect to say that it would be true if all the terms in the sentence were determinate and if they denoted or signified just those entities which m assigns to them. Of course, this definition of truth-in-a-structure is only a vague and intuitive one; but anyone familiar with Tarski's work on truth will know that it can be made perfectly clear and precise. For present purposes, it will be enough to illustrate the definition by the following example: relative to a structure that assigns Ted Williams to the English word 'mass' and the set of aardvarks to 'is invariant,' the English sentence 'Mass is invariant' comes out untrue (since Ted Williams is not an aardvark). This example makes it clear that the notion of truth-in-a-structure is not of much interest for every structure: the fact that Newton's sentence (6) comes out untrue-in-the-structure-just-considered has no bearing whatever on whether it was true in any ordinary sense of 'true.' It is obvious, then, that we must somehow restrict our attention to those structures which are in close accord with the semantics of the language. When all the terms of L are determinate, so that there is a unique structure which perfectly accords with L, it is clear enough how this should be done: we should say that a sentence is true if and only if it is true in the unique structure that accords with L. But what do we do when there are indeterminate terms and hence no uniquely privileged structure? Obviously, we want somehow to restrict our attention to those manuals which partially accord with the semantics of L. But there are two very different ways in which this 'restriction of attention' can be accomplished. The most natural way to do it is to define 'true' in terms of 'true-in-m' and 'partially accords,' as follows: (8) A sentence of L is true if and only if it is true-relative-to-m for every structure of m that partially accords with the semantics of L. This would have the consequence that when Newton uttered the disjunction of (5) and (6), what he said was true, and that when he uttered their end p.207
conjunction what he said was false (where to say that a sentence is false is to say that its negation is true). It would also have the consequence that when Newton uttered (5) by itself, or (6) by itself, what he was saying was untrue (and also unfalse, if 'false' is defined as above). It seems to me that this is precisely the conclusion we want. For it would be unreasonable to say that (5) was true and (6) was untrue: this would suggest that there was a fact of the matter, that Newton's term was really a word for relativistic mass. It would be equally unreasonable to say that (6) was true and that (5) was untrue: again that would suggest that there was a fact of the matter as to which conjunct was true. So unless we are to say that both (5) and (6) were true, which is clearly unreasonable, 7 the only alternative is to say that neither was true. (8) lets us say just this, without forcing us to give up the view that the disjunction of (5) and (6) was true and the conjunction false. In spite of what I have said in defense of (8), philosophers of a more Quinean persuasion may prefer to handle the matter differently. The alternative they may propose is to make an arbitrary choice among all of the structures which partially accord with the semantics of L, and to let 'true' stand for 'true relative to m o ' where 'm o ' denotes the structure arbitrarily chosen. This procedure would have the consequence that the pair consisting of (5) and (6) contains one true sentence and one false sentence; though which one is true and which one false would depend on which structure is arbitrarily chosen. This proposal seems to me to be completely unreasonable. A semantics according to which (5) was true and (6) false seems to me to be a semantics committed to the idea that 'mass' denoted relativistic mass, and a semantics according to which (6) was true and (5) false seems to me to be a semantics committed to the idea that 'mass' denoted rest mass. To make an arbitrary choice between these theories is to make an arbitrary choice between two theories each of which is inadequate (given the existence of indeterminacy). For this reason I think we should reject such arbitrary choices and adhere to the definition (8). Perhaps it is worth mentioning another respect in which the 'arbitrary choice' approach is inferior to (8). The difficulty arises from the fact that determining which structures partially accord with a particular language L and which ones do not is a matter of empirical linguistics, and it is easy to make errors in the process (especially in the case of foreign languages). This creates a difficulty for the 'arbitrary choice' approach, because the only way for a person to make a choice is for him to choose among the structures he thinks to be in partial accordance with the language, so that in choosing he may end up with a structure m o that does not accord with the language at all. Suppose this happens. If in this situation he adheres to the 'arbitrary choice' approach by letting 'true' be an abbreviation of 'true relative to m o ,' then a sentence might be 'true' in his sense even though it is false-relative-to-m for every manual m that partially accords with the language. This result seems extremely unpalatable,
7 A set of true sentences ought to have only true consequences, and yet (5) and (6) together have the false
consequence that momentum divided by velocity is invariant. end p.208
and I know of no way to avoid it other than to forgo arbitrary choices and to adhere to definition (8). Let us now look at what we have accomplished. If we combine (7) and (8) with Tarski's definition of truth-in-astructure, the result is a definition of truth in terms of partial denotation and partial signification which I will call the restricted truth-definition. The restricted truth-definition is a generalization of the usual variety of truth-definition, which is highly restricted: the highly restricted truth-definition defines truth in terms of denotation and signification, and hence works only for languages containing no indeterminate terms (if there are any such languages); while the restricted truthdefinition can handle not only those languages but also languages with expressions that are indeterminate in the way that Newton's term 'mass' was indeterminate. But even the restricted truth-definition cannot adequately handle a phenomenon that I call correlative indeterminacy. Correlative indeterminacy is an all-pervasive phenomenon, if Quine is right, for the 'gavagai' example is an example of correlative indeterminacy if it is an example of indeterminacy at all. To see what I mean by correlative indeterminacy, let us consider a foreign sentence that contains both 'gavagai' and 'glub,' together with various logical connectives which we will assume to have a determinate semantics. Actually there is no need to focus on foreign words, for since we have already rejected the translation-theoretic approach to semantics, we can just as well give the semantics directly for English. Consider then the sentence (9) x y(x and y are nearby rabbits x is identical to y), which we imagine to be uttered in the environment of exactly one rabbit. Obviously, (9) ought to come out true in this environment, but if we try to apply the restricted truth-definition to it we do not get this desired result. The reason is that, by our earlier suppositions, 'rabbit' partially signifies both the set of rabbits and the set of undetached rabbit parts, and 'is identical to' partially signifies both identity and paridentity. But then (7) has the consequence that not only are the following two structures relevant: (a) 'rabbit' → {rabbits}, 'is identical to' → identity (b) 'rabbit' → {undetached rabbit parts}, 'is identical to' → paridentity (where a relevant structure is one which partially accords with the semantics of L); it also has the consequence that the following two 'undesirable' structures are relevant: (c) 'rabbit' → {rabbits}, 'is identical to' → paridentity (d) 'rabbit' → {undetached rabbit parts}, 'is identical to' → identity. It is (d) that causes particular problems for (9): since not all the undetached end p.209
rabbit parts are identical to each other, (9) comes out false in some relevant structures, and therefore by (8) it cannot be true. It is clear that no advantage is to be gained by abandoning (8) and going back to the 'arbitrary choice' approach to indeterminacy: obviously what needs revision is (7). But it is equally obvious that there is no possibility of making the required modification of (7), unless we introduce new correspondence relations besides partial denotation and partial signification. (I emphasize that these must be correspondence relations as that term was explained in my introductory remarks: that is, they must be relations which hold between words and extralinguistic entities rather than between the words of one language and the words of another.) What we need to do, then, is introduce slightly more complicated correspondence relations than the ones we have so far, and then define relevance in terms of them. The new definition, when combined with (8) and with Tarski's definition of truth-in-a-structure, will give us an unrestricted truthdefinition that explains truth in terms of our new correspondence relations. In order to see what new correspondence relations we need, let us return to a consideration of the details of the 'rabbit' example, as they were set out at the beginning of this paper. Let us focus first not on the predicate 'rabbit,' but on the predicate 'is identical with.' A philosopher who did not believe in the existence of indeterminacy would presumably want
to say that this predicate signified identity— that is, that it signified the set of those ordered pairs whose first member and last member were the same. The problem that Quine has raised for such a philosopher, however, is to say in virtue of what this predicate signifies that relation. The most obvious facts to cite are that people use the predicate 'is identical with' in such a way that it obeys certain laws: the laws governing equivalence relations, and the substitutivity schema, ' x y(x y Fx Fy)' (restricted to those contexts we call extensional). But the fact that people use the predicate 'is identical with' in accord with these laws is not sufficient to rule out the hypothesis that it signifies paridentity— at least, not without the assumption that the language contains predicates true of one object and false of a paridentical object, and it seems that we can always refuse to grant this assumption (for example, by saying that 'white' is true of the undetached parts of white things, rather than being true of the white things themselves). So there is certainly a prima-facie difficulty here in assuming that 'is identical with' is a word for identity rather than paridentity, and unless this difficulty can be resolved there is little to say except that it partially signifies both. Now let us turn to the words 'rabbit' and 'dinosaur.' It seems fairly clear that there are facts about the causal relations (for example, perceptual relations) between rabbits and their undetached parts on the one hand and our uses of 'rabbit' on the other by which we could hope to explain that 'rabbit' was a word for some kind of rabbitish entity (whether the rabbit itself, or its undetached parts, or whatever); and it seems equally clear that there are facts about the causal relations between dinosaurs and ourselves (for example, our perceptual end p.210
relations to dinosaur skeletons) with which we could hope to explain that 'dinosaur' was a word for some dinosaurish kind of entity. The only problem is to figure out precisely which kind of entity. We get a certain amount of help here from the circumstances under which people are disposed to assent to (9): this tells us that if 'is identical with' were a word for identity, then 'rabbit' would signify the set of rabbits, and that if 'is identical with' were a word for paridentity, then 'rabbit' would signify the set of undetached rabbit parts. But the indeterminacy thesis forces us to assume that 'is identical with' is not a word for either identity or paridentity alone: it is a word that partially signifies both. What then are we to say about 'rabbit'? One thing we could say is that it partially signifies the set of rabbits and partially signifies the set of undetached rabbit parts. But we can also say something a bit more informative: we can say that relative to a correlation of identity with 'is identical to,' 'rabbit' signifies the set of rabbits; and that relative to a correlation of paridentity with 'is identical to,' 'rabbit' signifies the set of undetached rabbit parts. What I am suggesting, then, is that if we take Quine's radical indeterminacy thesis seriously we should take 'rabbit,' 'dinosaur,' and so forth as being dependent predicates, predicates whose extension is a function of the extension of another predicate, 'identical' (which I will call the basis of the dependent predicates). Then if this basis predicate turns out to be indeterminate, the other predicates which functionally depend on them will turn out indeterminate too (that is, they too will partially signify more than one set); but the functional dependence of these other predicates on the basis predicate 'identical' will allow us to correlate the partial extensions of one predicate with the partial extensions of others. This means that in working out the key step of our unrestricted truth-definition— namely, the analogue of (7)— we will be able to toss out those structures which assign uncorrelated partial extensions to our predicates. The above paragraph should make clear how the new definition of 'partially according' is going to work. A central concept we must employ is that of one term t 1 being the basis of another term t 2 (which I will write as 't 1 = b(t 2 )'). The notion of a name or predicate being dependent is of course definable from this: a name or predicate is dependent if it has a basis. In the Quinean example lately considered, the dependent terms 'rabbit' and 'dinosaur' had as their basis the independent term 'identical.' We do not have to require in general that the basis always be independent, but we do have to require that if a term has a dependent term as its basis, then either the basis of the basis is independent, or the basis of that is independent, or . . . Call this the grounding requirement. We can now define relevance as follows: (7*) A structure m partially accords with the semantics of L if and only if (a) each independent term t of L partially denotes or partially signifies m(t); (b) each dependent term t of L denotes or signifies m(t) relative to the correlation of m(b(t)) with b(t). end p.211
If the grounding requirement is met, (7*) is a formally acceptable recursive definition, and combined with (8) and with Tarski's definition of truth-in-a-structure it gives us the definition of truth that we wanted. 8 The treatment of indeterminacy that I have been advocating raises a number of epistemological questions: for example, how do we know which things a term partially refers to; and how do we know which terms have bases and what their bases are? I cannot adequately discuss these epistemological questions in this paper, but perhaps a few observations are in order.
One important factor to take into consideration in deciding what partially denotes what, what is the basis of what, and so on, is the consequences that such decisions have for the truth and falsity of sentences. The main reason for inventing the notions of denotation, partial denotation, basis, and so forth, was to give us a reasonable theory of truth, and if our decisions as to what partially denotes what (and so forth) lead us to unreasonable conclusions about truth then those decisions are inadequate. But this requirement (which I label 'Requirement 1') is not a sufficient one: we also want the semantic notions we employ to make physicalistic sense (call this 'Requirement 2'). Many views about what partially refers to what that are compatible with Requirement 1 fail to meet Requirement 2. For instance, suppose someone were to hold that 'gavagai' determinately signifies the set of rabbits (that is, partially signifies that set and no other), and that 'glub' determinately signifies identity. This hypothesis would lead to reasonable conclusions about the truth and falsity of sentences; but if the Quinean argument sketched earlier in the paper is correct (that is, if we cannot make physicalistic sense of any semantic connection between 'gavagai' and the set of rabbits that does not hold equally between 'gavagai' and the set of undetached rabbit parts), then the hypothesis would be unsatisfactory since it would rule out the possibility of a physicalistic explanation of partial signification. A similar point can be made for the notion of basis. Thus we could get reasonable results about truth and falsity by singling out one of the natural-kind terms of the language— for example, 'gavagai'— and letting this term serve as the basis of all of the other natural-kind terms (and for the identity term). But this would be unreasonable since there seem to be no facts about the use of 'gavagai' which give us objective grounds for singling it out for special treatment 8 Added generality could be obtained by replacing 'denotes' and 'signifies' in clause (b) of (7*) by 'partially denotes' and 'partially signifies'; and it is easily seen that this added generality is necessary if we are to handle 'mass'-type indeterminacy and 'gavagai'-type indeterminacy together. (Consider especially 'mass'-type indeterminacy for predicates.) A second way to generalize (7*) is to allow dependent terms to have more than one basis; a third way is suggested in the ensuing paragraphs on Benacerraf's problem. Putting the three generalizations together, we get (7 ** ) A structure m partially accords with the semantics of L if and only if (a) each independent atom t of L partially denotes or partially signifies m(t);
(b) each dependent atom t of L partially denotes or partially signifies m(t), relative to the correlation of m(b 1 (t)), . . . , m(b n t)) with b 1 (t), . . . , b n (t), respectively. I think that this definition is general enough to handle every plausible example of indeterminacy. There are some apparent exceptions to this claim— e.g., Quine's Japanese classifier example (1968a: 35-8); but in the appendix I will suggest a device by which such apparent exceptions can be handled. end p.212
over all of the other natural-kind terms of the language. It seems then that if we are to avoid physically undetermined choices, we have to treat all natural-kind terms on a par; and the only way I know to do this and still get reasonable results about truth and falsity is to let all the natural-kind terms be dependent terms whose basis is 'glub.' 9 I am suggesting, then, that we should decide what partially denotes what, what is the basis of what, and so forth, largely by the interaction of two considerations: considerations 'from above' about what consequences those decisions have for truth and falsity, and considerations 'from below' about what consequences the decisions have for the prospects of some day understanding semantics physicalistically. In other words, we should decide on the application of the semantic notions I have introduced in pretty much the same way that people have in the past decided upon the application of semantic notions like denotation and signification. The reason for replacing the notions of denotation and signification by the more general notions of partial denotation, partial signification, and so forth was that unless we made this generalization there would be situations in which it was impossible jointly to satisfy Requirement 1 and Requirement 2. (Thus before making the generalization we could satisfy only Requirement 1 by saying either that 'gavagai' signified the set of rabbits or that it signified the set of undetached rabbit parts; while physicalistic considerations seemed to preclude 'gavagai' from bearing any semantic relation to one of these sets without bearing it to the other set as well.) But I believe that if we adhere to the semantics developed in this paper (or the slight generalization of it suggested in note 8), there is no reason to doubt that Requirements 1 and 2 can always be jointly satisfied. The semantic theory I have been developing is of some interest not only for Quine's radical indeterminacy thesis, but also for the philosophy of mathematics. Consider the following two ontological views: (i) there are infinitely many physical objects, but no abstract objects; (ii) there are sets in addition to physical objects, but no abstract objects other than sets. On neither view are there any entities which are obviously numbers; therefore on both views there is a prima-facie problem of accounting for the truth and falsity of sentences expressed in number-theoretic terminology. But since both
views admit the existence of infinitely many objects, the idea suggests itself of correlating an object with each numeral in such a way that no two numerals have the same correlated object. We could then say that each numeral denotes 9 Actually, there is a more positive reason for regarding the identity predicate as the basis for the other predicates
(assuming we believe Quine's example). The reason is that the causal link alone is not sufficient to explain how 'rabbit' could partially signify the set of rabbits and partially signify the set of undetached parts: perhaps the causal link could be used to explain how 'rabbit' partially signified one or more sets whose union was {rabbits} {undetached rabbit parts}, but it is hard to see how the causal link could divide this union up into the two sets that are required. This problem does not arise if we view 'rabbit' as acquiring its partial referents through identity considerations in the manner outlined on pp. 210-11. end p.213
the object correlated with it, that a predicate such as 'prime' signifies the set of objects correlated with '2,' '3,' '5,' and so forth; and in this way we could get a definition of truth that did not require the existence of any objects except those allowed by (i) or by (ii). Benacerraf 1965 raises a difficulty for this reductivist proposal. The difficulty is that many correlations of the sort just discussed are possible, and it is clearly impossible to specify one correlation that gives 'the real referents' of the number-theoretic words. The reductivist might try to escape this difficulty by saying that it is not important to his purposes to hold that number-theoretic terms referred to these correlated objects all along; it is sufficient (he might say) that we be able to replace number-theoretic talk by talk of the correlated objects. (Quine has suggested this line of escape in 1960, sections 53 and 54.) But this will not do, for it suggests that earlier number theorists such as Euler and Gauss were not referring to anything when they used numerals, and (barring a novel account of mathematical truth) this would imply the unpalatable conclusion that virtually nothing that they said in number theory was true. Benacerraf's observation, then, seems to rule out the possibility of the reductivist proposal. In fact, however, the reductivist has another answer to Benacerraf. 10 All that Benacerraf's observation shows is that if numerals are related to physical objects, they are not related to them in a determinate (that is, one-one) way. What the reductivist needs is an explication of number-theoretic truth in terms of correspondence relations between numerals on the one hand and physical objects and/or sets on the other; and a slight generalization of the semantics that works for Quine's 'rabbit' example will work here as well. The generalization is this: we need to be able to apply semantic locutions like 'partially denotes' not just to individual terms, but to certain sequences of terms that can be naturally thought of as working together in a unit. Such sequences can be called atomic sequences, and we can stipulate that no term is contained in more than one atomic sequence. Then if we let the atoms consist of the atomic sequences and the terms that are not members of any atomic sequences, we can let the variable 't' in the (a) and (b) clauses of (7*) range over atoms instead of over terms. The idea will become intuitively clear from the number-theory example. The first thing we want to say is that the ωsequence of numerals as a whole is semantically related to certain ω-sequences of objects— namely, to those ωsequences of which no two members are the same. Let us say, then, that the ω-sequence of numerals is an atomic sequence that partially denotes precisely the ω-sequences of objects just mentioned. Now what are we to say about number-theoretic predicates like 'odd' and 'prime'? The obvious answer is that they are dependent predicates whose basis is the sequence of numerals. Relative to any ω-sequence σ that is correlated with 10 Or more accurately, to philosophers who think that Benacerraf's observation shows that there are abstract objects
distinct from sets— viz., numbers. In the final section of his paper Benacerraf himself expresses a reluctance to adopt such a platonistic position; so perhaps he would approve of the reductivist position suggested below. end p.214
the numerals, the word 'prime' signifies the set of objects in the prime positions of σ, and the word 'odd' signifies the set of objects in the odd positions of σ. These seem to be rather natural stipulations, and they have the consequence that precisely the right number-theoretic statements come out true. Also, because we have adopted (8) instead of making arbitrary choices, the stipulations have the consequence that bizarre statements like 'The number two is Julius Caesar' come out neither true nor false. So the semantics of correlative indeterminacy shows that Benacerraf's observation is not ultimately of much relevance to the possibility of adhering to austere ontologies like (i) and (ii) and yet retaining the notion of mathematical truth. But my main concern in this paper is not with the value of the unrestricted truth-definition for the philosophy of mathematics, but with its value in showing that much of what Quine says about the significance of indeterminacy is wrong. My view about the significance of indeterminacy differs from Quine's view in two central respects. The first respect was stressed in the early pages of this paper: I view indeterminacy as showing not that a correspondence theory of truth is hopeless, but that a correspondence theory of truth must be a bit more complicated than we might have once thought; and I view indeterminacy as lending no support whatever to Quine's thesis of ontological relativity
(which I have argued to be untenable). The second important respect in which my view differs from Quine's is that I do not see indeterminacy as showing any arbitrariness in semantic theory.11 According to Quine, whenever we give a semantic theory for a foreign language, there is another semantic theory equally as good as the first, but incompatible with it. The reason Quine believes this is that on his view we can give a semantic theory for a language only after making an arbitrary choice among all the acceptable translation manuals mapping that language into ours; and if we had chosen a different manual, we would have gotten a different (and incompatible) semantic theory for the language. On my view, however, what we should do in this situation is to transcend the individual manuals: if many translation manuals are acceptable (or better, if many structures are relevant), then any semantic theory for the language that looks at only one of them is inadequate, and an adequate theory has to look at all of them. This has the consequence that now there are no alternatives to our semantic theory that are just as acceptable as it, but incompatible with it; therefore the objectivity of our semantic theory is restored. 11 I am excluding from consideration here the uninteresting kind of indeterminacy mentioned in the last paragraph of
the appendix, the kind of indeterminacy which rests solely on the vagueness of current semantic terms; for it is true, but trivial, that any science whose terms are currently vague can be developed in more than one way equally well. The arbitrariness that Quine believes to exist is more radical, since he believes that arbitrary choices are necessary even apart from any vagueness that our semantic terms might possess. (See last sentence of appendix.) end p.215
Appendix: Indeterminacy in the Metalanguage In footnote 4 I suggested that on my view, indeterminacy is a lot like vagueness: the semantics of partial denotation and partial signification provides an adequate treatment of both (except that for vagueness, it is necessary to quantify the semantics by talking of degrees of denotation and of signification, and hence degrees of truth). It might be thought, however, that it is possible to treat vague predicates in the way I (following David Lewis) have suggested only if there are some predicates in our language which are not vague— namely, the predicates we use in specifying the partial extensions of the vague predicates. Analogously, it might be thought that it is possible to treat indeterminacy in our own language by the methods I have advocated only if there are determinate predicates in our own language to use in specifying the partial extensions of the indeterminate predicates. If this were so, then my treatment of Quine's example would be inadequate, for I tried to describe the partial extensions of the (allegedly) indeterminate term 'rabbit' by using the (allegedly) indeterminate expressions 'rabbit' and 'undetached rabbit part.' The reason for thinking that we can use only determinate terms in describing the partial extensions of indeterminate terms is, presumably, that it is not clear how assertions like (10) 'rabbit' partially signifies the set of rabbits and partially signifies the set of undetached rabbit parts could be understood by anyone who regarded the last two tokens of 'rabbit' as indeterminate. But I will argue that though it is psychologically easier to understand assertions like (10) if we pretend that these assertions are phrased in a determinate meta-English, still they are just as intelligible, and have precisely the same truth conditions, if we remember that our meta-English is indeterminate. To argue for this claim, it is necessary to know what a believer in indeterminacy would say about the expression 'undetached rabbit parts.' Relative to identity, of course, it signifies the set of undetached rabbit parts; but what does it signify relative to paridentity? Let us suppose that the answer to this question is that it signifies the set of undetached parts of undetached rabbit parts; and let us assume that undetached parts of undetached rabbit parts are not undetached parts of rabbits (or of anything other than undetached rabbit parts).12 Let us now turn to a more important question: what should a believer in indeterminacy say about the expression 'partially signify'? Relative to identity, of course, it signifies the relation R 1 ŷ (x partially signifies y). But what does it signify relative to paridentity? To answer, note that 'partially signify' is a two-place relation symbol whose left-hand argument ranges over abstract entities (namely, expression types) and whose right-hand argument ranges over sets of physical objects. For such expressions, we should expect the believer in indeterminacy to accept the schema 4 The concept of partial signification is not really as unfamiliar as it may sound, for we implicitly employ it in giving the
semantics of vague expressions. Suppose we were asked what the extension of the English phrase 'tall man' is— is it the set of men taller than 6′0″, or the set of men taller than 6′1/2″, or what? Clearly there is no fact of the matter as to which of these sets is 'the real extension' of the English phrase 'tall man'; for clearly 'tall man' does not simply signify a particular set, but partially signifies various different sets— viz., those sets of form
{x|x is a person whose height is greater than h} where h takes on values in some region centering around six feet and extending a few inches in either direction. (A still better account of vagueness could be given by quantifying the notion of partial signification: thus— if we pretend that the set of possible heights is discrete— we could say that 'tall man' signifies various different sets to various different degrees.) That the notions of partial denotation and partial signification (or quantified forms of these notions) are of use for dealing with vague expressions is by no means a novel point; e.g., it is suggested in the appendix to Lewis 1970a. These cases are unexciting because they are very unsurprising, and because we can easily do without such vague terms whenever our conversation turns to exact and serious purposes. But there is at least an abstract possibility that something similar to vagueness occurs even where we least expect it: it is possible that there are much more deep and pervasive ways in which our terms have indeterminate application to the world. And what Quine's argument about 'gavagai' seems to me to suggest is that this abstract possibility is in fact realized. 12 Some such assumption is necessary if Quine's 'gavagai' example is to work for a language that contains a predicate which is naturally translated as 'is an undetached part of.' We can imagine the assumption true by stipulating that for something to be an undetached part of an x, it must be precisely half the size of an x. end p.216
Relative to paridentity, '——' signifies the relation y).
ŷ (x——'s the set of objects whose undetached parts are in
In particular, he will hold that relative to paridentity, 'partially signifies' signifies the relation R 2 ŷ (x partially signifies the set of objects whose undetached parts are in y). It is clear that if we evaluate (10) according to the semantics of correlative indeterminacy outlined in this paper, we will get that (10) is true if and only if (i) 'rabbit' partially signifies the set of rabbits and partially signifies the set of undetached rabbit parts, and (ii) 'rabbit' partially signifies the set of objects whose undetached parts are in the set of undetached parts of rabbits, and partially signifies the set of objects whose undetached parts are in the set of undetached parts of undetached rabbit parts. ([i] comes from the structure in which 'identical' is correlated with identity and R 1 is correlated with 'partially signifies'; [ii] comes from the structure in which 'identical' is correlated with paridentity and R 2 is correlated with 'partially signifies.') But (ii) is equivalent to (i), given the assumptions about the undetached-part-of relation made above: the weird treatment of 'rabbit' and 'undetached rabbit part' is canceled by the weird treatment of 'partially signifies.' We see then that (10) has the same truth conditions whether one pretends that meta-English is perfectly determinate or one recognizes that it too is indeterminate. I think it is clear that the same is true for the other sentences which I used in developing the semantics of indeterminate expressions, and for this reason I think that the objection to my approach fails. I now want to consider a different objection to the semantics I have been advocating. This second objection, if it were valid, would also tell against Lewis-type treatments of vagueness; in fact, the objection is most easily formulated as an objection against such treatments of vagueness, so that is how I will formulate it here. Consider sentences of the form (S h ) All men of height greater than h are tall. On a Lewis-type semantic theory there are two critical heights h 1 and h 2 (where h 1 < h 2 ), such that S h is absolutely false if and only if h ≤ h 1 , and absolutely true if and only if h≥h 2 . The advantage of a Lewis-type treatment of vagueness over a standard semantic theory (one which does not recognize partial signification) is that Lewis's treatment accounts for the obvious fact that S h does not jump suddenly from absolute falsehood to absolute truth as h increases; rather, there is an interval (between h 1 and h 2 ) during which S h takes on successively higher degrees of truth between absolute falsehood and absolute truth. But it might be argued that although Lewis's treatment goes part way toward accounting for vagueness, it does not go far enough. For according to it, there is a precise point h 2 such that is absolutely true while no sentences of the form are absolutely true (though some of them have extremely high degrees of truth). Suppose now that another Lewis-type theorist held that the critical point was really not h 2 , it was h 2 −0.001 inch; could we reasonably suppose that there was a fact of the matter as to which Lewis-type theorist was correct? Obviously not, and so it seems that Lewis-type theorists have merely put the problem of vagueness one
step back: they explain why end p.217
there is no fact of the matter as to whether is absolutely true or absolutely false, by saying that it is neither, is absolutely true. but they do not explain why there is no fact of the matter as to whether If this objection to Lewis-type treatments of vagueness were valid, it seems likely that an analogous objection could be made to my treatment of indeterminacy (though the examples are considerably more complicated to describe). In fact, however, the objection is invalid. What the objection overlooks is that a Lewis-type theorist need not hold that 'tall' is the only vague term that is relevant to the discussion; he can hold that the semantic terms 'true' and 'partially signifies' are also somewhat vague. The vagueness of 'tall' serves various practical purposes; but the vagueness of our semantic terms serves no useful purposes, and as semantic theory develops we can expect these terms to become more will be absolutely true; on other ways of making them precise, precise. On some ways of making them precise, it will not be absolutely true but will have an extremely high degree of truth. (On no way of making them precise will be either absolutely true or absolutely false.) The fact that there is no fact of the matter as to whether is absolutely true, as we currently use 'absolutely true,' is simply due to the fact that our current use of 'absolutely true' is imprecise. This answers the objection to Lewis's semantics, and the analogous objections which might be raised against my own semantics. I should also mention that the idea utilized in answering the objection— the idea of positing vagueness in our semantic terms— is essential for dealing with some of Quine's less interesting examples of indeterminacy— for example, the Japanese classifier example. But there is no hope of treating the 'gavagai' example as merely illustrating the vagueness of our semantic terms: for the difficulty that the 'gavagai' example raised was that it seemed impossible to find any hypothesis about what 'gavagai' signifies that satisfies both Requirement 1 and Requirement 2; and it is obvious that this difficulty (if it really exists) cannot be alleviated by making the term 'signifies' more precise. 13 13 Adam Morton, Gilbert Harman, Paul Benacerraf, and the editors of the Philosophical Review, made helpful
comments on earlier versions of this paper. My work was supported by the National Endowment for the Humanities. end p.218
Postscript 1. Quine's own discussions of indeterminacy had been in terms of indeterminacy of translation, and discussions of Quine at the time this paper was written were put in those terms too. In this paper I tried to reorient the discussion to indeterminacy of the semantics of a given language, and took the semantics of a given language to have an autonomous semantics (as opposed to a translation into a previously understood language). Perhaps I should have made this more explicit, and made more explicit that as a consequence, some of Quine's examples of indeterminacy were not examples for me: for instance, his Japanese classifier example (Quine 1968a), which creates an indeterminacy of translation because of a structural difference between Japanese and English. There is, to be sure, an indeterminacy in the application of the semantic terms that we have developed for English to the Japanese classifiers, but the example doesn't show a semantic indeterminacy in the classifiers themselves: it is unlike the 'gavagai' case, where the indeterminacy arises in the object-level terms. This was mentioned at the end of the Appendix, and discussed in more detail in Field 1975, esp. sections iii and iv . Similarly, since I assumed that English has an autonomous semantics of which a causal theory of reference is a part, I thought that the problem of 'bad' reference schemes that would clearly be ruled out by a causal theory of reference was not terribly serious. In effect, then, I made a distinction between 'Quinean indeterminacy', involving cases like the 'gavagai' case where nothing like a causal theory of reference seems to rule out the alternative proposed reference schemes, and 'Wallace-Putnam indeterminacy' where this requirement is not imposed. (I name this after the indeterminacy arguments in Wallace 1977 and Putnam 1981, ch. 2. Even though Quine was not sympathetic to a causal theory of reference, he did often grant that the stimulus meanings of terms could be used to rule out alternative reference schemes; this rules out examples like those of Wallace and Putnam, so I think that my restrictive use of the phrase 'Quinean indeterminacy' is not wildly inappropriate.) The present paper attempts to deal only with Quinean indeterminacy. Wallace-Putnam indeterminacy was discussed in Field 1975: that paper argued that if these are genuine cases of indeterminacy at all, then (as with the Japanese classifiers) the indeterminacy would reside only in our semantic terms; it also argued that if such an indeterminacy exists at all then it can be removed by conventionally adopting something like a causal requirement on reference. (The distinction between the two types of indeterminacy may not be so important if one accepts the 'deflationist' view that no autonomous semantics is needed.) 2. As I say early in the chapter, and as has been explained in chapter 3 of Fodor 1995, the 'gavagai' example doesn't really work. It would work for a very
end p.219
simple language in which, in the standard interpretation, there are no overlapping objects: e.g., if the standard interpretation talks of rabbits it can't talk of rabbit toes or rabbit feet, or mereological sums of rabbits and cows. This no-overlap requirement on the standard interpretation would ensure that each undetached part quantified over in the nonstandard interpretation has a unique counterpart in the standard interpretation; this uniqueness is needed in setting up the nonstandard interpretation in the way the text suggests. (It may be possible to substantially weaken or even entirely avoid the 'no over-lap' requirement on the standard interpretation, by restricting the domain of the nonstandard interpretation to some special kind of undetached parts in such a way as to ensure a unique correspondence of a 'standard object' to each undetached part; but it is up to the proponent of a believable example to suggest how this might be done. Doing this becomes especially problematic when one tries to run the example for a language that contains the notion 'undetached part of', as I observed in footnote 12 .) But it seemed to me that despite the problems, Quine's discussion is suggestive. As I note on pp. 210-11, the fundamental question is the determinacy of our notion of identity, and from a non-deflationist point of view, this becomes the question of whether the facts about our use of '=' determine that it stand for identity as opposed to some other relation. Our assent to the usual axioms of identity seems sufficient to ensure that '=' stands for an equivalence relation, and indeed for a congruence relation with respect to the other predicates in our language. But even without questioning the normal interpretation of those predicates, it seems doubtful that this suffices to rule out all the equivalence relations other than identity. We do of course have a commitment to continuing to accept the substitutivity of identity as we add new predicates to the language; this means that we have a commitment to gradually reducing the indeterminacy of '=' in a certain direction as we expand the language, but it is hard to see how it makes the predicate determinate now, before such expansions have taken place. The preceding remarks did not question the normal interpretation of the other predicates in the language: predicates like 'rabbit'. But Quine's discussion makes clear that it would be question-begging to simply assume the normal interpretation of the other predicates, since the kinds of facts that we would have to appeal to in explaining why 'rabbit' is true of the rabbits will very prominently include our identity claims (and related claims like numerical quantifications). I suspect that proper attention to this fact would increase the degree of indeterminacy in '=' that inevitably results from 'inflationist' assumptions (and will thereby show a kind of indeterminacy in terms like 'rabbit' that one would not have suspected prior to Quine). Again, the degree of indeterminacy would presumably be reduced as we expand the language. Even if paridentity is a 'partial referent' of '=' in a primitive language that has no predicate 'is an undetached part of', it is likely that adding that predicate to the language (and using it in the appropriate way) would remove this candidate for the referent of identity (and thereby cut down the indeterminacy in terms like 'rabbit'). But other candidates would remain. 12 Some such assumption is necessary if Quine's 'gavagai' example is to work for a language that contains a
predicate which is naturally translated as 'is an undetached part of.' We can imagine the assumption true by stipulating that for something to be an undetached part of an x, it must be precisely half the size of an x. end p.220
Again, these remarks all presuppose the 'inflationist' semantic framework of an autonomous semantic theory for each language. I take it to be an advantage of the kind of 'deflationist' view advocated in Chapters 4 and 5 that it lessens the pressure for indeterminacy. 3. I now reject the claim made in footnote 3 of the chapter, that recognition of indeterminacy in nonsemantic terms requires a revision of the disquotation schemas for 'primitive reference'. This seems wrong even on inflationist assumptions: as I note in Chapter 9, we can keep the disquotation schemas by supposing that any indeterminacy in nonsemantic terms generates a corresponding indeterminacy in semantic terms. (It would be possible to suppose that the only indeterminacy resides in the semantic terms, rather than in the nonsemantic terms like 'rabbit'. That is in fact the position that I take in Field 1975 for other of Quine's examples, e.g. the Japanese classifiers. But as a general view it seems unmotivated (if semantic terms can be indeterminate, why can't nonsemantic terms be indeterminate?), and in Field 1975 I give considerations (presupposing inflationism) against treating the 'gavagai' example in this way.) For related reasons, the supervaluationist approach to the truth of sentences with referentially indeterminate terms is not as inevitable as I tried to make it seem; this too is discussed in later chapters. 3 Note that this argument shows that the 'disquotation schema'
'—' signifies the set of —'s and nothing else cannot be accepted by anyone who believes in indeterminacy. The schema does provide a partial axiomatization of the concept of signification; but to axiomatize a concept is not to show that the concept is physicalistically acceptable. (For more on physicalistic acceptability, see Secs. iii-v of Ch. 1.)
It does not follow that the believer in indeterminacy is deprived of all disquotation schemas: he may be able to adhere to unusual schemas like '—' partially signifies the set of —'s and partially signifies the set of undetached — parts. Such schemas might axiomatize, and clarify, the concept of partial signification to precisely the same degree that the more usual schemas axiomatize, and clarify, the more usual semantic concepts. end p.221
8 Disquotational Truth and Factually Defective Discourse Abstract: Some discourse (e.g. involving vague or indeterminate terms, normative language, or conditionals) can seem 'factually defective': it generates issues about which there seems to be 'no fact of the matter'. But there seems to be a difficulty in making sense of factually defective discourse, and in explaining the division between it and other discourse, if one holds that the basic notion of truth is a fairly minimal one. The chapter argues that we can overcome the difficulty: we can recognize and account for several different kinds of factual defectiveness while adhering to a very minimal notion of truth. Keywords: conditionals, disquotation, indeterminacy, interpersonal synonymy, non-factuality, normative discourse, reference, supervaluation, truth, vagueness
Hartry Field 'Deflationism' is the view that truth is at bottom disquotational. I take this to mean that in its primary ('purely disquotational') use, (1) 'true' as understood by a given person applies only to utterances that that person understands, and (2) for any utterance u that a person X understands, the claim that u is true is cognitively equivalent for X to u itself. 1 If X is an English speaker, and u is an utterance of an unambiguous sentence that he understands with no indexicals or demonstratives in it, such as 'Snow is white', we can put (2) by saying that for X, ' "Snow is white" is true' is cognitively equivalent to 'Snow is white'; which means that the sentence 'Snow is white' is true iff snow is white is more or less 'analytic' or 'logically true' for X, by virtue of the cognitive equivalence of the left and right hand sides. In general then, the instances of the 'disquotation schema' (DS) 'p' is true if and only if p where we substitute for both occurrences of 'p' a sentence (not containing ambiguities, indexicals, etc.) that X understands, will be more or less analytic for X, if X understands the word 'true' in its purely disquotational sense. 2 1 I take cognitive equivalence to be a matter of conceptual or computational role: for one sentence to be cognitively
equivalent to another for a given person is for that person's inferential rules to license (or, license fairly directly) the inference from either one to the other. (The inferential rules strictly apply not to the spoken or written sentences, but to internal analogs of them, which we might call 'sentence-readings'; so we really have an account of cognitive equivalence for sentence-readings. The distinction between sentences and sentence-readings can be ignored when we are ignoring ambiguity. But for sentences that are ambiguous for a speaker, that is, sentences that the speaker can connect up with more than one reading, it is obviously crucial that we regard cognitive equivalence as holding among the readings rather than the sentences. But it is worth stressing that readings as here defined are identified in computational rather than semantic terms, and cognitive equivalence between them is likewise explained computationally.) 2 (DS) must be qualified in light of the semantic paradoxes. There is actually no need to qualify (1) and (2): the paradoxes point up the fact that the move from cognitive equivalence as defined in the preceding note to validity of the biconditional isn't really valid in general, though it surely holds with some restrictions. (But in fn. 2 of Ch. 4 I used a stronger definition of cognitive equivalence, which does imply validity of the biconditionals.) For present purposes it will do no harm to ignore the paradoxes. end p.222
(There is an analogous purely disquotational notion of falsehood, in which ' "p" is false' is cognitively equivalent to 'not p'.) The intelligibility of such a disquotational notion of truth should not be in doubt: you could think of it as an indexical concept, meaning in effect 'true on my understanding of the terms involved' (as opposed to, e.g., 'true as meant by the
producer of the sentence'). What is controversial in deflationism is the claim that any legitimate concept of truth is based on the disquotational concept. More specifically, the deflationist allows that there may be certain extensions of the purely disquotational truth predicate that don't have features (1) and (2); but he requires that any other truth predicate be explainable in terms of the purely disquotational one, using fairly limited additional resources. Of course, there is a certain vagueness here, about what additional resources count as 'fairly limited'; obviously, deflationism threatens to become uninteresting unless this is interpreted rather narrowly. I intend to leave this vague, to allow for 'deflationisms' of differing strength; but I should say that I think the most interesting versions of deflationism are the radical ones that do not regard interpersonal synonymy as one of the 'limited' resources to which we can appeal.3 In addition to these claims about truth, a deflationist will make analogous claims about related semantic notions such as reference and satisfaction. For instance, the sentence 'Object b satisfies formula "F(v)" ' can be understood purely disquotationally (by someone who understands 'b' and 'F(v)' as well as a disquotational satisfaction predicate): so understood, it is equivalent to 'F(b)'. Deflationism has it that the only legitimate notions of satisfaction are this disquotational notion and other notions explained in terms of it. Even the radical versions of deflationism that disallow talk of interpersonal synonymy don't keep us from applying 'true' (and 'satisfies', etc.) to foreign utterances: if on my understanding of 'Der Schnee ist weiss' it is cognitively equivalent to 'Snow is white', then of course ' "Der Schnee ist weiss" is true' will be cognitively equivalent for me to both of them. (And if on my understanding of 'Der Schnee ist weiss' it is cognitively equivalent to 'Coal is black', then ' "Der Schnee ist weiss" is disquotationally true' will be cognitively 3 Non-radical versions can still be quite non-trivial: for instance, if they hold that interpersonal synonymy is to be
explained in terms of similarity in computational role. However, the chance of explaining a reasonable notion of interpersonal synonymy on this basis strikes me as meager; and as you expand the basis or hold the interpersonal synonymy relation to be irreducible, you come closer to triviality. On the radical versions of deflationism that I prefer, it doesn't make sense for X to ask whether one of Y's sentences is true as Y really intended it, but only whether it is true as X understands it. This isn't to deny that translations can be better and worse for various purposes (and that empirical discoveries about the production of an utterance are relevant to deciding how good a given translation of it is); but it is to deny that talk of correctness of translation makes good sense. I take radical deflationism to involve not just the claim that 'synonymous' (applied interpersonally) is vague; the claim rather is that it should be understood as not a straightforwardly factual predicate at all, but an evaluative predicate (less misleadingly rendered as 'is a good translation of'). Of course, to oppose the evaluative to the straightforwardly factual is controversial, and might even be thought contrary to deflationism; but I will try to undermine this last view later in the paper. end p.223
equivalent for me to 'Coal is black'.) Nor does ambiguity pose a problem: 'Visiting relatives can be boring' is ambiguous for me, because I process different tokens of it in different ways and subject them to very different sorts of inference. Typically if I say (3) u is (disquotationally) true if and only if visiting relatives can be boring, I am processing the right hand side in some definite way that resolves the ambiguity. What I say will then be correct as long as u is an utterance that I interpret as equivalent to 'Visiting relatives can be boring' as used on the right hand side of (3); this will be true of some but not all of my utterances of 'Visiting relatives can be boring'. One might object to deflationism on the grounds that we sometimes conjecture whether some utterance we don't understand is true, showing that our use of 'true' isn't purely disquotational (because of (1)). But I think a deflationist can say that what we are doing is conjecturing whether a good translation of the utterance will map it into a disquotationally true sentence we do understand. (A radical deflationist adds that 'good translation' should be taken to be a highly context-sensitive and interest-relative notion; it should not be understood as meaning 'correct translation', for talk of correctness of interpersonal translation makes no sense.)4 Doubtless this response raises some issues that need looking into, but I think it is enough to show that the proffered objection to deflationism is not obviously decisive. Another objection to deflationism might be that we regard it as a contingent fact that we use 'Snow is white' in such a way that it is true if and only if snow is white, so again our use of 'true' isn't purely disquotational (this time because of (2)). But the deflationist can of course agree that we might have used 'Snow is white' very differently, e.g. with the verification conditions that we in fact attach to 'Grass is purple'; and he can agree that we should use a nonhomophonic translation when we consider utterances that we would have produced under those conditions. It is arguable that this is enough to accommodate our intuitions of contingency. Deflationism is at odds with much current theorizing in the philosophy of language and of mind. That theorizing takes the relation between an utterance or mental state and its 'truth conditions' or 'propositional content' as a theoretically important relation, a relation that applies as directly to other people's utterances and mental states as to our own, and typically at least a relation for which we would expect some sort of rough naturalistic account. It is not hard to see that
all of this is precluded on a deflationist view (though I will not argue the point here). Basically, accepting deflationism requires dethroning truth conditions from the central place in the theory of meaning and the theory of 4 The non-radical deflationist thinks there is an objective notion of synonymy for good translations to reflect, so at first
blush he seems better off. However, presumably there could be sentences that we know don't bear this alleged objective synonymy relation to any sentences we understand, and presumably we could conjecture whether they are true. I think that the only way for even a non-radical deflationist to deal with this is to appeal to context-sensitive and interest-relative standards of translation. end p.224
intentionality that Frege and Russell and many others have given to them. My current view is that this is probably a good thing. Much more could be said about this 'deflationist' view of truth (and truth conditions, and satisfaction, etc.), both by way of elaborating it and explaining its significance and by way of responding to natural objections that many think decisive. But my concern in the rest of this paper is simply with one class of objections to the view; and these objections are relatively insensitive to the details of what one takes deflationism to involve. (For a more general discussion of deflationism, see Chapter 4.) The class of objections I propose to discuss here have to do with what I will call 'factually defective discourse'. Under this label I include discourse that involves vague expressions like 'bald' or 'heap'; discourse that involves referentially indeterminate expressions, that is, expressions which seem to refer but for which there seems to be no clear fact as to which of several things they refer to; and discourse for which some sort of nonfactualist account seems plausible, as for instance evaluative discourse and discourse involving indicative and/or subjunctive conditionals. Such discourse can give rise to two worries. The first worry is that deflationism yields the wrong account of truth for such discourse; the second (and I think more important) worry is that a deflationist cannot explain the distinction between such discourse and discourse which is not 'factually defective'. Of course, it isn't obvious that the problems raised by one kind of factually defective discourse are the same as those raised by other kinds; indeed, I think that they are not quite the same, but that they are related. So I will consider factually defective discourse of several different kinds. I will do my best to argue that the deflationist can deal with the worries about factually defective discourse; nonetheless, I think that (especially in the cases of vagueness and indeterminacy) such discourse raises questions about deflationism that need clearer answers than I will be able to give.
1. Vagueness The first kind of factually defective discourse I will consider is discourse containing vague terms. There are two ways one might base an objection to deflationism on such discourse. First, one could argue that a deflationist view is incompatible with the proper use of 'true' in connection with vague sentences; second, one could argue that a deflationist cannot explain the distinction between vague and non-vague language. I begin with an initial explanation of the second worry, which I think is the deeper one. What does it mean to say that 'bald' is vague? Presumably it means that the predicate admits borderline cases; 5 it means in other words that there are or 5 Or if you like, that it stands for a property that admits of borderline cases; but I take the formulation in terms of
predicates to be the more basic. (If we do want to talk of properties, it seems to me a matter of convention whether to say that 'bald' stands for a property that admits of borderline cases or to say that it 'indefinitely stands' for a precise property or for each of a range of precise properties.) end p.225
might be things that are neither definitely bald nor definitely not bald. But what does it mean to say that Jones is neither definitely (determinately) bald nor definitely (determinately) not bald? It is natural to answer this question by taking 'definitely p' as short for 'p is true', or 'c is definitely F' as short for 'c satisfies F'. If so, then to say that Jones is a borderline case amounts to saying (4) Neither 'Jones is bald' nor 'Jones is not bald' is true; or (virtually equivalently) (5) Jones satisfies neither 'bald' nor 'not bald'. But then accepting Jones as a borderline case requires either a non-disquotational notion of truth or satisfaction or a rather substantial revision of classical logic: if 'true' and 'satisfies' are used disquotationally, (4) and (5) each imply that
Jones is neither bald nor not bald. This violates the law of excluded middle (LEM). Indeed, it is not merely a refusal to accept all instances of LEM, as with intuitionism (better called 'Dutch LEM disease'). Rather, it is an acceptance of the negation of an instance of LEM, something one doesn't get with intuitionism. Such a substantial revision of classical logic is not altogether plausible, since unless other plausible rules are given up, one could get from ¬(A V ¬A) to ¬A & ¬¬A and from that to A & ¬A and to anything else. Putting aside such substantial revisions of logic, we see that the proposed analysis of vagueness requires either the claim that no predicates are vague or a non-disquotational notion of truth; and thus it may seem hard to see how a deflationist can accept the existence of vagueness. That would of course be a decisive objection to deflationism. I think that the only reasonable course, for the deflationist who is unwilling to revise classical logic in the radical way just mentioned, is to grant that we understand an operator 'definitely' (or 'determinately'), where 'definitely p' is a strengthening of 'p'; but to reject the explanation of it in terms of truth or satisfaction. One attempt to explain 'definitely' independent of truth-theoretic concepts is suggested by Paul Horwich (1990: 82-4): Horwich takes the claim that it isn't definite whether Jones is bald to mean that no physical information could verify whether he is bald. But such 'ultimate epistemological indefiniteness' doesn't sound to me like vagueness: it sounds rather as if Horwich is allowing 'bald' to stand for a very precise property whose instantiation unfortunately isn't verifiable by knowledge of the instantiation of ordinary physical properties. (One could avoid this objection by modifying Horwich's proposal to say that 'it isn't definite whether Jones is bald' means that the definite facts aren't enough to determine whether he is bald; but that definition would obviously be circular.) I think it unlikely that there is any way to capture in epistemological terms what we are saying when we say of someone that he is neither definitely bald nor definitely not bald. Nor can I think of any other approach to defining 'definitely' end p.226
that is promising, if we put aside the definition in terms of 'true'. Does this mean that we must understand 'definitely' in terms of 'true'? Not necessarily: the alternative is to say that it is not a notion we can hope to define; rather, 'definitely' is a primitive operator that we come to understand in the same way we come to understand such operators as negation and disjunction and universal quantification: by learning to use it in accordance with certain rules.6 Paramount among these rules are the logical laws governing the operator. What are these logical laws? Assuming that one adheres to classical logic for the ordinary connectives (the connectives other than 'definitely'), the natural thing to say about the connective 'definitely' is that it obeys the same laws as 'necessarily' in the modal system T (so that 'definitely' distributes over conjunction but not disjunction or negation). On this view, either Jones is bald or Jones is not bald, by excluded middle; indeed, it is definitely the case that he is either bald or not bald. But it may still be the case that he is neither definitely bald nor definitely not bald (since distributivity over disjunction fails); when he isn't, he is a borderline case. Some may think it more attractive to give up excluded middle for vague utterances. On the most plausible versions of this view (and the only versions I will consider until the very end of this section), de Morgan's laws and double negation elimination are to be preserved and explicit contradictions never asserted: so negations of instances of excluded middle can never be asserted either, as noted earlier. The views have it then that just as we can neither assert 'Jones is bald' nor assert 'Jones is not bald', when Jones is a borderline case, so too we can assert neither 'Jones is bald or Jones is not bald' nor 'It is not the case that either Jones is bald or Jones is not bald'. But with a 'definitely' operator, we can assert 'It is not the case that either Jones is definitely bald or Jones is definitely not bald', when Jones is a paradigm borderline case. 7 And without excluded middle, we presumably also want to assert 'It is not definitely the case that either Jones is bald or Jones is not bald'; this means that there is now no obvious obstacle as there was in the classical case to holding that 'definitely' distributes over disjunction. It might be thought that a deflationist who gives up excluded middle doesn't need a definitely operator. Admittedly, he can't convey that Jones is a borderline case of baldness by saying that neither 'Jones is bald' nor its negation is true; for on a disquotational view of truth that amounts to denying an 6 It is probably more natural to take 'It is indefinite whether p' rather than 'Definitely p' as the primitive locution, since
in most contexts, asserting 'p' already carries the suggestion that you believe 'Definitely p': it is the belief that it is indefinite whether p that requires the most explicit marking in conversation. But the two locutions are interdefinable, and the logical laws are simpler when put in terms of 'definitely'. 7 That is, a clear case of being a borderline case, as opposed to a borderline case of being a borderline case. I should note that vague predicates are typically used according to contextually varying standards— someone who counts as bald in the context of a search for an actor on an Alberto VO5 commercial may count as not bald in the context of a search for a Yul Brynner look alike. So when I speak of a paradigm borderline case, I mean a borderline
case given the contextually determined standards. end p.227
instance of excluded middle, which we are assuming for the moment not to be acceptable. (Nor can we say that 'Jones is bald or Jones is not bald' is not true, if our notion of truth is disquotational, for the same reason.) But perhaps we can convey that we regard Jones as a borderline case of baldness not by saying anything but by withholding assertions? For instance, by refusing to assert either 'Jones is bald' or 'Jones is not bald'? One problem with this is that we might withhold our assertion not because we think Jones is a borderline case but because we are completely ignorant as to his hirsuteness, or because we have information that he is either a clear case of baldness or a clear case of non-baldness but aren't sure which. (An analog of this problem arises for the suggestion that we convey that we regard Jones as a borderline case by refusing to assert 'Either Jones is bald or Jones is not bald': this conveys only lack of certainty that he is not a borderline case.) A second problem is that it doesn't seem enough to convey that we regard a given individual as a borderline case of vagueness; we would like to be able to make general assertions about borderline cases. (It will not do to try to define 'borderline case' as something we would withhold assertion about given complete information about the physical world, for reasons discussed in connection with Horwich.) So even if we abandon excluded middle in the way contemplated, a deflationist needs a 'definitely' operator. Before returning to the issue of whether the deflationist can give an adequate account of our understanding of 'definitely', let's turn to the less serious of the objections with which I began, namely, the objection that deflationism is incompatible with the proper use of 'true' in connection with vague sentences. I claim that this objection is misconceived: there is no uniquely proper use of 'true' for vague sentences; and a deflationist can account for each of the two natural ways of using it. One natural way to use the word 'true' (I'll call it the weak use) is the one given by the disquotation schema. If we adhere to classical logic (as is my preference), this involves the claim that either 'Jones is bald' or 'Jones is not bald' is true. This may sound like it is denying that Jones is a borderline case, but it isn't. The idea rather is that just as 'bald' is vague, so is 'true', and that indeed 'true' inherits all the vagueness of 'bald' in that if it is indeterminate whether Jones is bald then it is equally indeterminate whether 'Jones is bald' is true. Since (on this approach) we are adhering to excluded middle, we are retaining the view that either Jones is bald or Jones is not bald, even when Jones is neither definitely bald nor definitely not bald; so why not also retain the view that either 'Jones is bald' is true or 'Jones is not bald' is true, even when neither 'Jones is bald' nor 'Jones is not bald' is definitely true? In short, on the weak notion of truth (and classical logic), both 'Jones is bald' and 'Jones is not bald' are borderline cases of truths. The other natural use of 'true' (the strong use) has it that if Jones is a borderline case of baldness then neither 'Jones is bald' nor its negation is true. If we adhere to classical logic, we'll want the disjunction 'Jones is bald or Jones is not bald' to come out true even on the strong reading, but we'll say that end p.228
though the disjunction is true, neither disjunct is true. (If on the other hand we give up excluded middle, then when Jones is a paradigm borderline case we will deny the truth of the disjunction 'Jones is bald or Jones is not bald', on the strong reading of 'true'; whereas on the weak reading we will merely refrain from asserting its truth.)8 On the strong reading then, 'true' is much less vague than it is on the weak reading: it doesn't inherit all the vagueness of 'groundlevel' terms like 'bald'; rather, the strong reading holds that if Jones is a paradigm borderline case, then both 'Jones is bald' and 'Jones is not bald' are definite cases of non-truths. Which of these views of the truth conditions of vague sentences should we regard as 'correct'? I don't think this question has a clear answer: each way of talking has considerable intuitive appeal, each has its utility, and I think the best thing to say is that we have two different concepts of truth for vague sentences, each serving different purposes. (I think it can be argued that because of our logical need for a device of infinite conjunction and disjunction, we require a notion of truth that accords with the weak view. Also, the weak notion connects up more naturally with the theory of validity. But I won't pursue either of these points.) An inflationist can easily explain the intelligibility of each of these two truth concepts, as we will see shortly. But it may at first seem as if the deflationist who retains classical logic can explain only the weak concept, since that is the one given directly by the disquotation schema. But this is incorrect: since the deflationist who adheres to classical logic needs a definitely operator anyway, to explain what it is for Jones to be a borderline case, nothing can prevent him from using it to explain a secondary truth concept: 'p' is true* iff Def('p' is true); or equivalently, iff 'Def(p)' is true. Truth* explained in either of these equivalent ways is just the strong truth defined above. Of course, if you disallow the deflationist who adheres to classical logic the use of the definitely operator, then he can't introduce talk of strong truth. But that is the least of his problems; the more fundamental problem is that he can't then distinguish vague from non-vague language. I argued above that the use of a weak truth-predicate is compatible with vagueness; but the problem is that nothing said with the weak truth predicate alone, without the definitely operator,
enables you to express that your language contains vague terms, or to express the way that they are vague: nothing allows you to say that Jones is a borderline case. So we return to the key question: can the deflationist make sense of the definitely operator? In order to answer this, it is best to ask whether an inflationist needs a definitely operator, and if so, how she can make sense of it. How does an inflationist make sense of Jones being a borderline case of baldness? Here's one way: to say that Jones is a borderline case is to say that neither 'Jones is bald' nor 'Jones is not bald' is true, where we utilize a strong (non-disquotational) notion of truth. However, the typical inflationist wants to explain the truth 8 I'm continuing to ignore views that abandon de Morgan's laws or the laws of double negation, or which allow the
assertion of explicit contradictions. end p.229
value of complex sentences in terms of the truth values of simpler sentences, and ultimately in terms of the semantic features of predicates. The usual way of doing this for non-vague predicates has it that 'Jones is F' is true iff the object referred to by 'Jones' satisfies 'is F', and 'Jones is not F' is true iff that object doesn't satisfy 'is F'. But if we apply this account of truth to 'bald' and assume classical logic, we get the conclusion that either 'Jones is bald' or 'Jones is not bald' is true, contrary to what we wanted. And giving up the law of excluded middle is not enough to give the desired conclusion that neither 'Jones is bald' nor 'Jones is not bald' is true: that conclusion requires that something neither satisfies nor fails to satisfy 'bald', i.e. it requires a counterinstance to excluded middle, which as I have noted is ruled out even on the most plausible nonclassical views. So in both the classical case and (the plausible version of) the nonclassical case, the above account of truth would yield only the weak truth predicate that is obtainable from the disquotation schema. 9 Where then does the inflationist get her strong truth predicate? One possibility of course is to mimic the deflationist and employ a definitely operator not explained in terms of truth: 'strongly true' just means 'definitely weakly true', with 'weakly true' explained in terms of compositional semantics as in the last paragraph (rather than in terms of the disquotation schema, as the deflationist would have it). But in either the classical or nonclassical logic version, this requires an independently meaningful definitely operator not explained in terms of truth; so an inflationist who takes this stand would be in the same boat as the deflationist. But (if we stick to classical logic anyway, as I will do for the next four paragraphs barring footnotes) there is an alternative inflationist approach: super-valuational semantics. 10 Here we employ as a primitive the idea of a candidate extension (or partial extension) of 'bald': roughly, a set of objects is a candidate extension of 'bald' if there is a legitimate way of making 'bald' precise that would give it this set as its extension. Then a sentence containing 'bald' is true (in the strong sense of 'true') iff it comes out true relative to each of the candidate 9 It might be objected that the inflationist can't intelligibly employ the concept of satisfaction in connection with vague
predicates, so that the above account doesn't yield even the weak truth predicate, and that we must use a disquotational schema to get that. But I think that is incorrect: the inflationist must simply recognize that the satisfaction predicate and the weak truth predicate that results from it have all the vagueness of object level terms like 'bald' built into them. 10 There is also an analog if we adopt the kind of nonclassical logic contemplated before, in which instances of excluded middle are neither asserted nor denied. The analog is to regard objects not as simply satisfying or failing to satisfy atomic predicates, but as satisfying them to different degrees, where these 'degrees' aren't necessarily linearly ordered, but form a distributive lattice with an additional operation corresponding to negation that obeys de Morgan's laws and the double negation law (i.e. it is an involutive anti-automorphism). [For instance, the degrees for 'bald' or 'happy' might be k-tuples of real numbers between 0 and 1, one such number for each factor that affects baldness or happiness; use the obvious partial order, and let the operation corresponding to negation take
extensions.11 So we can say that Jones is a borderline case without using a 'definitely' operator, by saying that Jones is in some but not all of the candidate extensions of 'bald'. The use of the truth-theoretic notion of candidate extension can thus be viewed as enabling us to avoid the 'definitely' operator; alternatively, it could be viewed as providing an explication of it, in that to say that Jones is definitely bald is to say that Jones is in all the candidate extensions of 'bald'. 12
But one might well doubt that the definitely operator has really been avoided here. In my informal explication of 'candidate extension' I used the notion of a legitimate precisification of 'bald': and of course what I meant was, a way of making 'bald' precise without putting any object in its extension that is now definitely not in its extension, or leaving anything out of its extension that is now definitely in its extension. So it looks like our attempted explanation of the definitely operator in terms of (strong) truth, combined with the compositional account of (strong) truth in terms of 'candidate extension' and our explanation of that, has really only succeeded in analyzing the definitely operator in terms of itself. 13 This argument against the view that the definitely operator is to be explained in terms of an inflationist notion of (strong) truth may not be completely decisive. I do think it beyond serious doubt that we must regard the concept of 'candidate extension' as explained in terms of an antecedently understood definitely operator (or, somewhat less naturally, in terms of the concept of strong truth). Nonetheless, an inflationist could say that the metaphysical order of things is the other way around: strong truth is metaphysically explained in terms of candidate extension (i.e., it is 'reduced to' the relation is a candidate extension of and other parallel concepts); and the definitely operator is then metaphysically explained in terms of truth and thus indirectly in terms of candidate extension. Is this defensible? I think it is hard to see much point in reducing the definitely operator to the relation is a candidate extension of if that must in turn be taken as a metaphysically primitive relation between sets and predicates. But an inflationist need not so take it: she can try to reduce it to non-semantic terms, via causal 11 If there are several vague terms in the language, you have to be careful to work this idea out so that 'penumbral
connections' are preserved: see Fine 1975. This applies in particular if 'definitely', or 'true' in either the weak or strong senses, is in the language. See the treatment of 'higher order vagueness' on pp. 293-4 of the Fine paper. (I think there is something a bit unnatural about the supervaluational treatment of higher order vagueness, and regard it as an advantage of taking the definitely operator as basic that you avoid this, but I will not pursue the matter.) 12 To turn this into a general definition of 'definitely' of course you'd have to give a more general account that dealt with the problems about multiple vague terms mentioned in the previous note. 13 Similarly, on the approach to a nonclassical logic of vagueness in fn. 10, the worry is whether 'b satisfies P to the maximum degree' isn't just another way of saying that b definitely satisfies P; in which case the claim to have explained the notion of definitely in terms of something less problematic is illusory. (That there should be an analogy between the semantics of fn. 10 and supervaluational semantics is not surprising: supervaluational semantics can be viewed as a kind of Boolean-valued semantics— the Boolean value of a formula being the set of all those (legitimate combinations of) candidate extensions in which it comes out true— and thus as a special case of the kind of latticevalued semantics in fn. 10.) end p.231
connections or whatever. 14 And then, as long as the definitely operator is not employed in that reduction, nothing blocks combining that reduction with the proposed reduction of 'definitely' to 'candidate extension', so as to get an account of the definitely operator in non-semantic terms. But the underlined clause is significant: it is not unreasonable to think that even if 'extension of' as applied to precise terms could be explained in non-semantic terms, any way of extending this to an account of 'candidate extension of' for vague terms would have to use a definitely operator. Whether or not this is so is not something I can pursue here; but if it is, then even the inflationist would do better to take the definitely operator as primitive rather than explaining it in terms of truth. In any case, the idea that we should take the definitely operator as primitive seems to me to have considerable appeal. Taking conceptual primitiveness first, it seems to me that we don't come to understand 'definitely' by having been told a definition of it (in terms of 'true' or 'candidate extension' or anything else), but in the way we have come to understand operators like negation: by having implicitly learned rules of use. Moreover, I see no reason to think, in either the negation case or the 'definitely' case, that the central rules of use involve the interaction of the operator with a truth predicate; though of course in both cases the rules of use for a disquotational truth predicate tell us how the truth predicate and the operator interact. 15 As for the metaphysical primitiveness (irreducibility) of the definitely operator, I confess some qualms; but on the whole it seems to me that the best thing to say is that 'definitely' is, like 'not', a logical notion, and that there is no more reason to reduce the one to more primitive terms than there is to reduce the other. 16 ('Definitely' certainly seems very different from 'true' and related notions if these latter are understood as semantic, that is, as not ultimately disquotational: the only way to plausibly maintain that 'true' is anything like a logical notion is either to give it the properties (1) and (2) that were characteristic of disquotational truth, or to define it in terms of a notion that has those characteristics together with a definitely operator.) Reducibility questions aside, let's return to the claim that 'definitely' is conceptually independent of any truth predicate: that is, that the central rules of use do not involve the interaction of the operator with a truth predicate. If this 14 A deflationist will see no need for such reductions: the deflationist view is that truth and reference make sense
primarily for our own language, and are essentially logical concepts governed by the disquotation schemas and not requiring reduction. (On this view, the significance of the compositional principles connecting truth to reference and extension is much less than it is on inflationist views: see Ch. 4, sect. V.) An inflationist needn't see a need for such reductions either, but it seems to me hard to motivate the combination of a lax attitude on this with a feeling that the definitely operator can't be metaphysically primitive. 15 That is, they tell us that 'not p' is true in the weak disquotational sense if and only if 'p' is not true in that sense, and that 'definitely p' is true in the weak disquotational sense if and only if 'p' is definitely true in that sense. 16 A deflationist needn't rule out a reduction of definitely to, say, epistemological terms. Timothy Williamson has recently pointed out to me that there are more subtle ways to invoke epistemological factors in an account of the definitely operator than the ways Horwich considers. Because of this, I hesitate to rule out an epistemological reduction; the more cautious stance is that none is required. end p.232
view is to be maintained, the task remains of saying what the rules are that govern our understanding of 'definitely', and on this I have less to say than I would like. Some of the rules of use (on the assumption that we keep classical logic for vague predicates) are the modal-like laws I have already mentioned: these entail that 'Definitely p' is a strengthening of p, that it distributes over conjunction, and so forth. (There are also rules of invalidity, e.g. rules disallowing the distribution of 'definitely' over disjunction or negation, disallowing the inference from 'Definitely p' to 'Definitely definitely p', and so on.) Other rules of use for 'definitely', I think, are learned when we learn vague terms: it is part of the learning of 'bald' that (relative to the contextually appropriate standards— see footnote 7 ) certain sorts of people are definitely bald, certain other sorts definitely not bald, and certain other sorts neither. (A fuller account would have more to say about how 'bald' and 'definitely' shift together with the context.) There are also important connections between the definitely operator and various similarity and ordering predicates: e.g. we learn that if b is more bald than c and c is definitely bald, so is b; and that if it is indefinite whether an object is red, that is because it is indefinite whether it is similar enough to objects that are taken as paradigms of 'red' in our training in the use of the word.17 And we learn that we can reduce the indefiniteness of our assertions by restating them in terms of similarity and ordering and the like: replacing 'Jones is bald' by 'Jones is balder than X but less bald than Y'. Also, the role of asserting 'p' when one doesn't believe 'Definitely p' is different than when one does: for instance, when one doesn't believe 'Definitely p', one is likely to regard someone who asserts 'p' as not genuinely disagreeing. (One might even restrict the concept of assertion, so that to honestly assert 'p' requires believing 'Definitely p' rather than just 'p'.) There are also connections to our notions of knowledge and of dependence on the physical facts: from 'It is indefinite whether p' we can infer both 'It is not knowable whether p' and 'The physical facts don't determine whether p'; but the converse inferences are not acceptable (without a physicalist premise, in the second case, or a strong premise about our intellectual powers in the first). If it is indefinite whether p, there is not only no point in trying to find out, there is no point in even speculating: it is totally arbitrary what one says. It is also worth noting that we believe that as science advances, its language will become more and more determinate; this seems relevant to our conviction that vagueness is a 'deficiency in language', not a 'deficiency in the world'. Much more detail is needed about all of this. For instance, a fuller account ought to say in a clearer and more motivated way what it is about how we use the operator that rules out the sort of epistemological/physicalist reading that Horwich recommends, and that rules out other clearly unintended readings that my skeptical readers will devise. Also, the reader may well worry that spelling out the rules of use in detail would amount to introducing a strong truth predicate of an inflationist sort, that is, one not simply definable from 7 That is, a clear case of being a borderline case, as opposed to a borderline case of being a borderline case. I
should note that vague predicates are typically used according to contextually varying standards— someone who counts as bald in the context of a search for an actor on an Alberto VO5 commercial may count as not bald in the context of a search for a Yul Brynner look alike. So when I speak of a paradigm borderline case, I mean a borderline case given the contextually determined standards. 17 Here I have appealed to the conceptual role of 'red', but said nothing about the extension of the word. end p.233
'definitely' and disquotational truth but one that is genuinely semantic. I doubt that this worry is correct, but certainly more details would be needed in order to dispel it. It might seem that we would avoid all this hard work of giving the rules of use for 'definitely' if we adopted the line that 'definitely' is to be conceptually explicated in terms of a strong truth predicate. But of course we don't ultimately avoid anything at all by this supposition: analogous questions arise when it comes to giving an account of our understanding of the strong truth predicate. For instance, what is it about our use of that that rules out the reading 'could be established by the totality of physical information'? The question seems no easier to answer than the question about the definitely operator, which means that the difficulty cannot fairly be taken as an objection to the claim that the
definitely operator is primitive. In trying to evaluate the costs of deflationism for an understanding of vagueness, I have presupposed that vagueness does not require us to use a logic in which instances of excluded middle are actually to be denied. How would the discussion be affected if we were to adopt such a logic? (For logics on which they are neither asserted nor denied, see footnotes 10 and 13 .) On this radically nonclassical logic, we have no need of a definitely operator (or a distinction between weak and strong truth): Jones is a borderline case iff it is not the case that he is either bald or not bald. Since this is so, we now have no worry about how a definitely operator is understood. It may seem, then, that a deflationist could save a lot of trouble by defending such a logic. Unfortunately, this sanguine view is incorrect: the trouble is only relocated. For if we assume this logic, the rules of use of logical connectives like 'not' and 'or' are much more complicated than they are in the classical case, in that sentences of the form 'p or not p' can't always be asserted, but rather are sometimes asserted, sometimes denied, and presumably sometimes neither asserted nor denied. A deflationist needs to be able to spell out such rules of use, without relying on the notion of truth in the spelling. This fact might suggest that indeed the adoption of such a logic of vagueness would make deflationism less plausible than it is if we assume classical logic, but this too seems to me mistaken. For again, difficult as it may be to explain the rules of use of these radically nonclassical connectives without invoking the notion of truth, it really doesn't help very much to invoke truth-theoretic notions in the explanation: the problems are going to be largely relocated into explaining the rules of use for these truth-theoretic notions; and even with that given, the account may presuppose it known whether instances of excluded middle at the meta-level (e.g. 'Either "bald" is true of Jones or "bald" is not true of Jones') are to be asserted, denied, or neither. I haven't worked through the details of all this— indeed, the details are going to depend on the precise nature of the nonclassical logic and the proposed semantics— but I doubt that the prospects for defending deflationism over inflationism are much affected either way by which logic for vague sentences we adopt. 10 There is also an analog if we adopt the kind of nonclassical logic contemplated before, in which instances of
excluded middle are neither asserted nor denied. The analog is to regard objects not as simply satisfying or failing to satisfy atomic predicates, but as satisfying them to different degrees, where these 'degrees' aren't necessarily linearly ordered, but form a distributive lattice with an additional operation corresponding to negation that obeys de Morgan's laws and the double negation law (i.e. it is an involutive anti-automorphism). [For instance, the degrees for 'bald' or 'happy' might be k-tuples of real numbers between 0 and 1, one such number for each factor that affects baldness or happiness; use the obvious partial order, and let the operation corresponding to negation take
2. Referential Indeterminacy The discussion of vagueness, while attempting to defend deflationism against a charge of inadequacy, also raises a serious question about how deflationism is to be understood. The question concerns how the deflationist should deal with the threat of referential indeterminacy. A simple example of what I mean by referential indeterminacy is provided by the word 'mass' as used in Newton's time: given the way the world actually is (in particular, given special relativity), there are at least two different physical quantities that seem about equally good candidates for what the word as used in Newton's time referred to. You could take 'mass' as having referred to rest mass. Of course, on this construal not all of Newton's assertions involving 'mass' would come out true, but obviously there is no reasonable construal of his term that makes all of his assertions involving 'mass' come out true, since his theory was false. Part of what makes this a prima facie good construal of his term 'mass' is that it makes many of his assertions involving 'mass' come out true, and many others come out approximately true, in the sense that the numerical values come out very close to correct at low velocities. Indeed, this
construal would seem obviously 'the right one' were it not for the fact that there is another construal that is about equally good. The other construal takes 'mass' as having referred to the frame-dependent quantity relativistic mass (rest mass divided by (1 − v 2 /c 2 ) 1/2 , where v is the velocity of the object relative to the frame of reference). This too makes many of Newton's assertions come out true— maybe slightly more of his assertions than the other one does— and makes many others come out approximately true. In one way this construal is slightly less neat than the previous one, in that relativistic mass is a frame-dependent quantity; but this can hardly be disqualifying, since the referent of many of Newton's theoretical terms (for instance, 'velocity', 'energy', and 'momentum') were uncontroversially framedependent. 18 It seems to me that there are no clear grounds for preferring one construal to the other. But one shouldn't conclude that the term lacked reference: it isn't at all like 'phlogiston'. (For more on this last point, see Chapter 6, pp. 184-6). Rather, it is much better to say that it referred, but that it is indeterminate which of the two quantities it referred to. 18 This last point seems to be missed by Earman and Fine (1977). On other points they raise: (1) Fine makes a big
point of the fact that Einstein in his later writings decided that it was better in formulating relativity to use 'mass' for rest mass than for relativistic mass (in contrast, I might add, both to his original usage and to the usage of many textbooks). I personally agree that this later policy makes for a more aesthetically pleasing presentation of special relativity; but this seems at best loosely related to the question of what we take Newton as having been referring to. (2) Earman's remark 'If . . . "relativistic mass" denotes a new kind of mass, then for every distinct kind of noninertial coordinate system there will be yet a new kind of mass' (p. 537) seems to me highly misleading. We don't say that there is a new kind of energy for each kind of coordinate system, but that energy is a frame-dependent quantity: in effect, a function from objects and coordinate systems to numbers. Similarly for relativistic mass. end p.235
If this is right then 'mass' as used by Newton was in many ways rather like 'bald', in that 'bald' can be construed as having various precise properties as equally legitimate candidates for its referent. The analogy is in fact a pretty strong one, in that the options for the truth conditions of sentences containing referentially indeterminate terms seem to be precisely the same as the options for the truth conditions of sentences containing vague terms: for instance, a supervaluational account is a plausible option in each case. There are of course some obvious differences between 'mass' as used by Newton and 'bald': I think the most important is that everyone knows that there is no fact of the matter as to whether certain people are bald, but no one in Newton's time knew that there was no fact of the matter as to how mass and momentum were related: referential indeterminacy as illustrated in the 'mass' example is rather like an unexpected sort of vagueness. In the paper where I originally introduced this example I had a thoroughly inflationist perspective: I spoke of 'mass' as having 'partially denoted' both rest mass and relativistic mass, and I made clear that I regarded partial denotation as an objective relation between the word and the quantities that in no way depended on how we translate the seventeenthcentury term into our twentieth-century language of rest mass and relativistic mass, and that in no way depended on our disquotation schema. And I argued that from this perspective it looks like the existance of indeterminacy in many earlier scientific terms (for of course 'mass' was not atypical) provides a pretty good inductive argument that many of our own scientific terms are probably referentially indeterminate as well, even though we may have no idea how. 19 Maybe later scientists will discover this, and propose alternative translations of our terms into theirs; but even if it isn't ever discovered, the indeterminacy of our terms would still be objectively there. Of course in saying this I meant to be provocative. What I now want to ask is, should a deflationist accept the conclusion that we have reason to regard our own terms as referentially indeterminate? It may seem at first as if deflationism is committed to denying that our own terms can be referentially indeterminate: after all, the disquotation schema tells us that 'x has charge y' is true of
overall theory is false: the individual theories that make up our overall theory (e.g. quantum theory and general relativity) don't fit together coherently, and indeed the individual theories themselves often harbor inconsistencies. So to argue for indeterminacy, we need only argue that theories that are false overall tend to have indeterminate terms (or, what implies this, that it is generally somewhat indeterminate which sentences are false when a theory is false overall). And that claim seems to me a pretty plausible one. Perhaps the argument for it is partly inductive, but even so, I think it is an induction that raises fewer doubts than the direct induction for the indeterminacy of terms in current theories from the indeterminacy of terms in past theories. end p.236
quite compatible with it being indefinite whether a given object has a given charge. I don't deny that the idea that our current term 'charge' might be indeterminate is more puzzling than the idea that our current term 'bald' might be indeterminate: as noted earlier, we presumably learn to use terms like 'definite' in tandem with terms like 'bald', learning for instance that people like Jones are neither definitely bald nor definitely non-bald. Certainly nothing in this training directly establishes that our current term 'charge' has no definite application; if anything, it would seem to justify the contrary conclusion. But of course what we learn to assert in our training with a word is not incorrigible, and the key issue is, does the inductive argument that many of our scientific terms are probably indeterminate (or the slight improvement over the direct induction, offered in footnote 19 ) have less force if we adopt a deflationist perspective? Perhaps it does: a natural deflationist move is to say that what the 'mass' example shows is simply that there is no best translation of Newton's word into our current scientific language. It might seem then that the strongest inductive conclusion that could be drawn from this and similar examples is that many of our terms will have multiple good translations into future theories; and it may not be evident how to get from this conclusion to the conclusion that there is any indefiniteness in the extension of these terms. In fact though I think we can get a slightly stronger inductive conclusion from examples like the 'mass' example: we get the conclusion that for many of our terms, there are possible improvements of our present best theory involving that term, such that the term is translatable into each of these improved theories in more than one way. (These theories are improvements over ours in many ways: more observationally adequate, more comprehensive, more coherent, etc. Indeed we can even say that they are truer than our theory, in that more of their sentences that are in the language of our current theory are disquotationally true.) 20 The gap between this conclusion and the conclusion that some of our terms have no definite extension does not seem so large, so I tentatively think that the deflationist should reach the same conclusions as the inflationist on the issue of indeterminacy in current scientific terms. I now want to briefly mention a rather different sort of controversial example of referential indeterminacy. Many people have wondered whether certain complicated logical operators, e.g. unrestricted second order quantifiers, can have determinate truth conditions: various considerations suggest that they don't, for instance, the fact that everything one says with them could be reinterpreted as involving only more restricted property (or set) quantification. Of course, it is no help to say 'When I say "for all properties" I mean for all properties': the unquoted use of 'all' is itself subject to reinterpretation. 19 Actually there is a better argument for the conclusion. First, we don't need an induction to argue that our current
overall theory is false: the individual theories that make up our overall theory (e.g. quantum theory and general relativity) don't fit together coherently, and indeed the individual theories themselves often harbor inconsistencies. So to argue for indeterminacy, we need only argue that theories that are false overall tend to have indeterminate terms (or, what implies this, that it is generally somewhat indeterminate which sentences are false when a theory is false overall). And that claim seems to me a pretty plausible one. Perhaps the argument for it is partly inductive, but even so, I think it is an induction that raises fewer doubts than the direct induction for the indeterminacy of terms in current theories from the indeterminacy of terms in past theories. 20 The need for a restriction to the sentences in our language is probably not that significant, since the effects of new vocabulary can usually be roughly captured less economically in terms of old vocabulary plus higher order quantification— cf. Ramsey sentences. end p.237
It may seem that a deflationist has a simple reply to such arguments for referential indeterminacy: after all, the disquotation schema simply tells us that 'For all properties P, X(P)' is true iff for all properties P, X(P). But again, the analogy between referential indeterminacy and vagueness, together with the discussion of vagueness in the previous section, shows that things are not so clear. For in addition to the notion of truth, we can employ a definitely operator; and we can then say that while the disquotation schema trivially gives truth conditions to sentences containing 'for all properties P', still it doesn't settle under what conditions sentences containing these locutions are definitely true. Is this a reasonable position? That is a big issue which I won't try to settle here. (Settling it would involve looking in some detail at the arguments for the indeterminacy.) My own tentative view is that it is reasonable: that the shift to the deflationary perspective does not really do very much to undermine the force of the arguments for the indeterminacy of powerful logical operators like second order quantifiers. Whether this conclusion tends to make deflationism more plausible, or throws away one of its apparent advantages, is something I leave for the reader to judge. 21
3. Non-Factual Discourse: Introduction In the rest of the chapter I will be discussing discourse which can plausibly be held not to have a primarily 'fact-stating' role. Examples of such discourse include normative discourse— ethical claims, epistemological claims, and so forth—
and also various other sorts of discourse, including indicative conditionals, subjunctive conditionals, 'because' statements, and many more. Many people are less sympathetic than I am to the view that such sentences aren't fully factual; but it seems to me that such a view is at least intelligible, and a deflationist ought to be able to make sense of the view. The question is, how? A somewhat superficial way of putting the problem that such statements raise is to say that a non-factualist view of them requires that they lack truth value, whereas as we've seen the disquotational schema together with classical logic entails that every sentence is either true or false (where 'false' means 'has a true negation'). Put this way, the problem is not obviously compelling. In the first place, it is by no means obvious that an account that gives these utterances a role that is not primarily fact-stating must say that the epithets 'true' and 'false' do not apply to them. (Compare the use of the weak truth predicate by one who recognizes vagueness.) In the second place, it is also by no means obvious that a deflationist can't find a way to make sense of a use of 'true' according to which such statements aren't true or false. (Compare the deflationist's introduction of 'true* ' for vague utterances.) But even if one or both of these lines of defense against the somewhat superficial objection can 21 I discuss the arguments for the indeterminacy of powerful logical operators in Field 1994 [and in chapters 9 and 12
of this volume], and argue that the conclusion of these arguments is not so hard to live with. end p.238
be made out, a deeper worry may remain, namely that the deflationist simply can't make sense of the distinction between discourse that is fully fact-stating and discourse that isn't. I will discuss the superficial and the deeper worry, both in the case of evaluative sentences and in the case of indicative conditionals; but first I want to mention a slightly simpler case in which similar worries may arise. (I will use certain features of the discussion of this simpler case in the later discussion of the evaluative case.) According to special relativity, there is no fact of the matter as to the time order of events in certain cases (namely, when the events are spacelike separated); in these cases, we can only make sense of time order relative to a frame of reference.22 This does not of course prevent the advocate of special relativity from using the unrelativized locution 'A happened earlier than B' (even in application to spacelike separated events): it's just that one must regard a relativization to some frame as implicit in the context (much as an utterance of 'John was kissing Mary' must be understood as saying that he was kissing her at a certain time implicit in the context). In the kissing case, everyone agrees that there is this timerelativity. In the time-order case, this has not always been so: Einstein's non-factualism about an absolute (linear) time order was once a controversial doctrine. And the question might arise whether a deflationist can make sense of it. The worry is that even if we are treating 'earlier than' as indexical, i.e. as implicitly frame-relative, still the disquotational schema allows us to say (6 T ) 'A was earlier than B' is true iff A was earlier than B. So the disquotational schema doesn't settle whether the sentence 'A was earlier than B' is being used indexically. So what account can a deflationist offer of what distinguishes the person who uses 'earlier than' indexically from the person who doesn't? The general outlines of an answer to this question are actually quite easy to provide, but before I give them I had better make explicit how I am supposing a deflationist will deal with uncontroversial examples of indexicality, e.g. 'John was kissing Mary'. If we have a sentence involving indexical elements, the clearest way to proceed is to modify the disquotation schema for it slightly: (7 K ) (For any time t) (any token of) 'John was kissing Mary' is true relative to time t if and only if John was kissing Mary at time t. Here we have made the time relativity that is only implicit in the quoted sentence explicit in the right hand side of the biconditional, and introduced an explicitly time-relativized truth predicate to make the left hand side jibe with the right. But we don't actually need to give up the more usual disquotation schema (6 K ) 'John was kissing Mary' is true if and only if John was kissing Mary. 22 Here and in what follows, I mean by 'time order' linear time order: I'm not considering the non-frame-relative partial
order given by the light cone structure. end p.239
Indeed, there are two distinct ways to understand this so that it is acceptable; the first takes 'John was kissing Mary' on the left as referring to the sentence type, the second takes it as referring to a certain sort of token of that type (or to a class of tokens narrower than the whole sentence type). In both cases, the idea is that the right hand side of (6 K ) contains an implicit relativity to time; so we can make (6 K ) come out acceptable by supposing that 'true' on the left hand side also contains an implicit relativity to time. If we take the implicit relativity on the left and right to be to an
unspecified time, then both sides in effect contain free variables: in that case, (6 K ) is simply shorthand for (7 K ), and we have a reading of (6 K ) that takes it as about a sentence type. But we can also take the implicit relativity to be to some specific time that we who are using (6 K ) have in mind.23 In that case, we should understand (6 K ) in such a way that what is implicit in the right hand side, and in 'true' on the left, is a singular term with no explicit variables (but which may itself be indexical): e.g., 'John was kissing Mary' is true relative to that evening when I was having dinner with Joan if and only if John was kissing Mary on that evening when I was having dinner with Joan. This gives a reading of (6 K ) appropriate to those tokens of 'John was kissing Mary' in which the implicit indexical 'then' is adequately translated by the implicit indexical 'then' on the right hand side of (6 K );24 and this reading is an instance of (7 K ). The analog in the relativity case is clear: the nonfactualist about absolute time order— that is, the advocate of special relativity— does indeed accept (6 T ). He also accepts both of the following: (8 T ) 'A was earlier than B relative to F 0 ' is true if and only if A was earlier than B relative to F 0 (where 'F 0 ' is any singular term with no explicit variables), which applies only to sentences in which the framerelativity is made explicit; and (7 T ) 'A was earlier than B' is true relative to F if and only if A was earlier than B relative to F, which handles sentences that leave the frame of reference implicit, by building the frame-relativity into the truth predicate. And we can explain (6 T ) (in 23 Of course, a deflationist (and most nondeflationists too) hold that I can have a specific time in mind only by
mentally employing some symbol that refers to that time— disquotationally refers, for the deflationist. 24 I take it that the standards of translation of one's own recent indexical utterances are fairly objective: when one uses an indexical (or any other word) with a different 'intended referent' from one occurrence to the next, this is reflected in one's inferential procedure. (We hook up the different occurrences of the indexical with different sets of beliefs and desires, and we disallow certain inferences that we normally allow.) In the case of tokens used by different speakers, or even the same speaker at very different times, it may be less clear that talk of 'correct translation' of one person's indexical into the other's language makes objective sense; if it makes no objective sense in a given case, then we must regard my talk of disquotational truth for your utterances as relative to a translation. end p.240
both a type interpretation and a token interpretation) in terms of (7 T ), in complete analogy to (6 K ). Getting back to the question of how to tell (if you are a deflationist) whether someone is a nonfactualist about absolute time order or a factualist, the answer should be clear, at least in outline: you won't get the distinction by looking at (6 T ), since that is acceptable to both; rather, you must look at what else the person accepts, and what inferences the person finds acceptable. Indeed, if the task were only to distinguish the nonfactualist from a factualist like Newton, we could make the distinction simply in terms of whether the person sometimes employs the explicitly frame-relativized concept. Newton accepted only (6 T ), not (7 T ) or (8 T ), since he didn't have the relativized concepts that the latter two employ. An adherent of special relativity on the other hand does have those concepts; and he can say if he wants that when A and B are spacelike separated, (6 T ) is justifiable only as a less explicit way of stating (7 T ), or as a consequence of (7 T ) valid only for a limited class of tokens of 'A was earlier than B'. Really though the basic difference between the nonfactualist and Newton is antecedent to the use of the truth predicate: it arises because the nonfactualist states his overall view using a relativized predicate that Newton did not understand. Things become slightly trickier when we consider factualists about time order who do use frame-relativized concepts (perhaps because they are acquainted with special relativity but want to deny it). There are two different cases to consider. In the less interesting case, the factualist takes the relativity to F to be trivial, in that time-order relative to one frame always coincides with time-order relative to any other. Here there is no difficulty in distinguishing the view from the nonfactualist view given by special relativity, without employing inflationist truth concepts in making the distinction: the distinction comes out precisely in that time order is sensitive to the frame for the nonfactualist but not for the factualist. In the more interesting case, the factualist takes the relativity to be nontrivial in just the way the special relativist does— he advocates the Lorentz transformation— but he regards some of the frames as 'privileged' in that time order relative to them coincides with 'genuine time order'; whereas the nonfactualist holds that there is no 'genuine time order', but only time order relative to a frame. The crucial question for deflationism is whether this distinction can be made— or made as well as an inflationist can make it. I think that the distinction between the nonfactualist about absolute time-order and even this sort of factualist can be
drawn in terms of what sentences they accept and of what inferences they find acceptable. A rough and ready way to draw the distinction would be to say that the factualist is willing to use the unrelativized notion of 'earlier than' in his serious utterances, even in connection with events that he knows are spacelike separated; whereas although the nonfactualist often uses 'earlier than' without explicit relativization, he would always be willing to replace his locutions with locutions involving the frame-relativized concept. The main problem with this way of drawing the end p.241
distinction is that the factualist might also be willing to take the relativized notion as the only basic one: he could do this by replacing talk of unrelativized time order with talk of time order relative to a cosmically privileged frame. How then are we to distinguish between the nonfactualist and the sort of factualist who accepts a non-trivial relativity to reference frames and employs only relativized locutions in serious discourse, but still regards some reference frames as 'cosmically privileged'? Again I think that there is not much difficulty making sense of the distinction between the two positions (in those cases where the distinction does indeed make sense). Here, we must ask why the factualist says that one frame is cosmically privileged. If he says this because he thinks he needs the notion of a cosmically privileged frame to state certain laws of nature which resist Lorentz-invariant formulation, then this will give clear content to the idea of a cosmically privileged frame; but it will also give a clear way of distinguishing his views from the nonfactualist's, independent of either the notion of cosmic privilege or truth-theoretic concepts. If he says it because he thinks he needs the notion of a cosmically privileged frame to explain something nonscientific, then again this gives a way to distinguish his views from the nonfactualist's. I must admit that if the factualist were to say that the notion of a cosmically privileged frame plays no role in explaining anything whatsoever, scientific or nonscientific, then we could not distinguish his view from the nonfactualist's in this way: the only distinction then seems to be that the factualist enjoys using the phrase 'cosmically privileged'. But I think it is reasonable to wonder if that sort of 'factualism' would have any content. And if it has no content, there is no need for a deflationist to worry about explaining its content.
4. Evaluative Discourse Now let's turn to an apparently more serious problem, the case of evaluative utterances like 'Voting for the lesser of two evils is a bad idea' or 'Belief in quarks is justified on current evidence'. As I've noted, a non-factualist needn't say that such utterances can't legitimately be called true or false: a non-factualist could just as well say that it is perfectly legitimate to call the claims true, it's just that the attributions of truth have the same not-fully-factual status as do the claims themselves. (That of course is what the disquotation schema dictates that one say.) But if one does say this, and if one's only notion of truth is the disquotational one, how is one to convey what is 'not fully factual' about the utterances? A rather unpersuasive answer would be to introduce an 'it is a fact that' operator, analogous to the 'definitely operator' used in the case of vagueness. (I'm embarrassed to say that in Field 1986 I made this suggestion, without any elaboration to make it more persuasive.) What makes this unpersuasive is both that it is less obvious than in the case of 'definitely' that this is an operator that people employ, and also that it is quite unobvious without further explanation end p.242
what the rules of use of such an operator would be. It may be that we can make sense of a 'factuality operator' in the end, but I think that the most straightforward way to make the distinction between factual and not-fully-factual discourse will be in other terms. Another attempt that I think inadequate is the one in Horwich 1990: after arguing that a deflationist should adhere to the disquotation schema in the case of ethical sentences, Horwich says that we can still make sense of emotivism: . . . the emotivist might attempt to characterize the unusual nature of certain ethical propositions by supposing, very roughly speaking, that the meaning of 'X is good' is sometimes given by the rule that a person is in a position to assert it when he is aware that he values X . . . (p. 88). The worry about this is that it doesn't seem to capture very much of emotivism at all, in that an ethical realist would agree that there is a connection between asserting 'X is good' and valuing X. Earlier I mentioned that Horwich's method of handling vagueness seems tantamount to denying that there is vagueness, and saying that we just can't know whether Jones is bald because we can only know about his physical characteristics and baldness isn't determinable from them. In the evaluative case too his remarks are compatible with the view that 'good' denotes a perfectly factual property, but one with the peculiar feature that our valuing something somehow gives us evidence that that thing instantiates this property, and/or that our belief that something has this property somehow leads us to value it. Perhaps such a position is better than a non-factualist position (though I doubt it); but our task was to show how a deflationist
can at least make sense of non-factualism, and on Horwich's account it is quite unobvious wherein the non-factualism lies. To see how we might do better, let's forget about deflationism for the moment, and ask what a non-deflationist account of the sense in which evaluative utterances are not fully factual might look like. The best non-deflationist account of this that I know of is the one offered in Gibbard 1990. (In calling it non-deflationist I don't mean to say that the denial of deflationism plays any central role in the account offered— indeed, I will argue that it doesn't play such a central role— but simply that the general tenor of Gibbard's discussion is much more in line with the inflationist tradition than the deflationist.) Unlike cruder writings in the emotivist tradition, Gibbard grants (what I think should be obvious) that evaluations have a factual component; the trick is to combine factual and non-factual components in a single analysis. He does this by giving a more general framework for semantics than the usual possible worlds framework; we semantically evaluate utterances not simply in terms of the set of worlds in which they are true, but in terms of the set of pairs <w,n> of possible worlds and complete norms of evaluation such that the utterance would be true in that world according to that norm. Here a possible world is a complete specification of factual information; where this is end p.243
not assumed to contain any 'normative facts'. 25 A complete norm, for instance a precise version of a utilitarian norm, will in effect associate with each evaluative predicate a non-evaluative 'equivalent' ('maximizes pleasure' or whatever); to say that a sentence is true in <w,n> is to say that the sentence that results from substituting for each evaluative term in it the non-evaluative 'equivalent' provided by n is true. If a sentence contains no evaluative terms, then for any n, truth in <w,n> just reduces to the usual notion of truth in a world w: the norms don't matter, and we can call such a sentence fully factual. If the sentence does contain evaluative terms, then (except in special cases) it will be normsensitive: the norms will matter. 26 The facts— the possible world w— matter too of course, if the norms are at all reasonable: surely for any reasonable norm n, whether 'We should intervene in Bosnia' is correct in <w,n> depends on the facts about what is going on in Bosnia in w. No person's actual norms are complete: actual norms are mental representations of which complete norms are maximal refinements. But if N is a norm in this broader sense, we can say that a norm-sensitive utterance is true in <w,N> iff it is true in <w,n> for each complete norm n that refines N. (In effect we are identifying a norm in the broad sense with the set of complete norms that refines it, and then supervaluating.) If a sentence is true in <w,n 1 > and false in <w,n 2 > when n 1 and n 2 both refine N, then it will be neither true nor false in <w,N>. I assume that each norm N is consistent; since people aren't consistent, this assumption prevents the normative components of a person's mental state from being represented by a single norm: it must be represented by a collection of different norms, some in conflict with others. (Indeed, this would be necessary even if consistency were required in a person's normative state: for norms that are consistent can conflict given factual beliefs, and even norms that don't conflict given one's factual beliefs can come into conflict when those factual beliefs change.) For the distinction between complete norms and possible worlds to be interesting, there must be a psychological distinction between adhering to a norm and having a belief. (At least, between adhering to a norm and having a belief in the most straightforward sense; I will explain the qualification a few pages from now.) For if adhering to a norm were simply a matter of believing something (e.g., if it were believing that acting in accordance to the norm had a special property like correctness) then information relevant to the truth of that belief (e.g. information about that special property) ought to be included in each possible world w, and there would be no need to include the norm as a separate ingredient needed for evaluation of the mental state. (Or to put it 25 A nonfactualist goes further, and says that the actual world doesn't contain 'normative facts'. (It is better not to
count the non-instantiation of normative properties as a normative fact, so as to cleanly separate nonfactualism from an 'error theory'.) Whether she allows for other possible worlds that include normative facts seems more or less a matter of modal taste. But if we want to supply a semantic framework acceptable to a factualist too, we ought to allow for the epistemic possibility of normative facts in the actual (and other possible) worlds. 26 Besides logical truths, the special cases where the norms don't matter include explicitly relativized sentences like 'We should intervene in Bosnia according to such and such norms': these are not norm-sensitive, but fully factual. end p.244
another way: leaving information about the special property out of the possible worlds and putting it into the norms instead would be a cheap trick if we thought that adhering to a norm is simply a matter of having a belief about a normative property; it would suggest that what we're calling 'possible worlds' are really only the non-normative aspects of possible worlds, and what we're calling norms should be thought of as the normative aspects of possible worlds, i.e. the collections of basic normative facts that hold in that world.) So for the norms to serve a role (and not simply be an aspect of the possible world misnamed), adhering to a norm can't simply be a matter of belief (in the straightforward sense of that term). Spelling out exactly what the psychological role of adhering to a norm is and exactly how it differs
from the psychological role of believing is a difficult matter, which I will not enter into; what follows is premised on the supposition that it can be done. 27 (Some of Gibbard's discussion is suggestive in regard to spelling out the difference, as is some of the recent discussion of the 'desire as belief' thesis (Lewis 1988 and 1996); but more needs to be said.) Even if the supposition that the psychological distinction can be drawn is granted, the nonfactualist is of course not done: he or she still needs to argue that there is no need to include facts about special normative properties in the possible worlds. That is, the nonfactualist needs to argue that we can make sense of evaluative discourse and evaluative argument in terms of the acceptance of norms but without supposing that it makes sense to speak of a norm as correct. I believe that this nonfactualist view is correct, but it is not my goal here to adjudicate it. My goal, rather, is to see whether (this version of) non-factualism depends as heavily on an inflationist notion of truth as this initial presentation may make it appear. I have followed Gibbard relatively closely in my initial formulation of non-factualism, but before discussing the issue of whether a deflationist can be a nonfactualist I would like to reformulate nonfactualism slightly: instead of speaking of a sentence as true in the pair <w,N>, where N is either a complete or an incomplete norm, I will speak of it as true in w, relative to N. Then when w is the actual world @, I will speak of the sentence as true relative to N, rather than true in <@,N>. So the distinction between fully factual and norm-sensitive utterances is that factual utterances are straightforwardly true or false, independent of any norm; whereas norm-sensitive utterances aren't straightforwardly true or false in this sense, they are true or false only relative to (complete or incomplete) norms. That's what a nonfactualist says, anyway. A factualist would want to restrict attention (in connection with possible world w) to those norms that aren't 'objectively wrong (in w)', i.e. those that aren't in conflict with the normative facts (in w). The factualist can allow that our norms 27 Actually, it isn't strictly necessary for my purposes that the distinction between norm-adherence and believing exist
in the psychologies of ordinary people or of committed factualists: a distinction in the psychology of the nonfactualist would be sufficient for making sense of evaluative practices without having beliefs in special 'evaluative properties'. But I have little doubt that the distinction can be made for ordinary people as well. end p.245
have a non-objective component that settles issues unsettled by the normative facts, so more than one norm might still pass the test of not being objectively wrong. But 'straightforward truth' in the factualist's sense would be truth in all norms that aren't objectively wrong. The nonfactualist can agree to this too, it's just that she thinks the italicized clause doesn't exclude anything. The factualist/nonfactualist debate about evaluatives, then, is the debate about whether evaluatives are straightforwardly true. This way of putting the issue is just a notational variant of the way Gibbard puts it, but I find this way of putting things clearer and more intuitive. One advantage of this formulation is that it emphasizes the prima-facie inflationist presuppositions of the factualist/nonfactualist distinction: certainly 'straightforward truth' is not the same as disquotational truth, so the question will arise how a deflationist can make sense of it. I will argue though that just as a deflationist can recognize definite truth as a strengthening of disquotational truth in connection with vague and indeterminate utterances, so also he can make sense of straightforward truth as a strengthening of disquotational truth in connection with evaluative utterances. To see this, it is useful to notice that the reformulated Gibbard view is formally rather analogous to an uncontroversial view about indexical sentences and a once-controversial view about sentences about time-order: just as 'John was kissing Mary' is true only relative to certain times, and 'A was earlier than B' may be true only relative to certain reference frames, so 'Intervention in Bosnia would be good' is true only relative to certain norms. Despite this formal analogy there are huge pragmatic differences between the evaluative case and the indexical case. One important difference arises from the fact that because norms as I have defined them must be consistent, each person has not a single norm but a multitude of competing norms that are in various degrees of conflict; there is no important analog of this in the other cases. This pragmatic difference is quite important, for I think that a plausible nonfactualist account of moral or epistemological debate will turn on the interplay of distinct norms within a person and between people, in ways that I cannot explore here. An even more important pragmatic difference between the evaluative case and the other cases is that what we count as disagreement differs drastically: in the indexical case (including the 'earlier than' case), agreement as to what is true relative to an assignment of values to the indexical elements counts as agreement, whereas this is certainly not so in the normative case. If we agree that we should intervene relative to certain specific norms and not intervene relative to others but one of us accepts norms of the first sort while the other accepts norms of the second sort then we disagree, despite being in factual agreement. The deflationist takes this as simply part of the conceptual and social role of evaluative terms, needing no explanation in terms of truth conditions. As a consequence of this second pragmatic difference, a certain distinction that is unimportant in the case of ordinary
indexicals (including 'earlier than') becomes crucial: the distinction between an implicitly relativized assertion end p.246
and the corresponding assertion in which the relativization is made explicit. (Here for simplicity I slur over the fact that because of the first pragmatic difference, there is no unique corresponding explicitly relativized assertion.) When relativization is made explicit, sensitivity to context or frame of reference or norm is lost. The rule of agreement in the case of ordinary indexicals is such that this loss of sensitivity to context or frame of reference has little effect: two people will disagree in their context- or frame-sensitive assertions if and only if they disagree on their explicit relativizations to the respective people's contexts or frames, at least when each knows what contexts or frames each party is relativizing to. But under the rule of agreement in force in the evaluative case, this is not so: when there is no norm-sensitivity in a sentence, there is no way for disagreement in norm to count as disagreement. Consequently, we need to sharply distinguish between a person's utterances of 'Intervention in Bosnia would be good' and 'Intervention in Bosnia would be good according to my norms'. (However, the conceptual role of 'good' will be such that we can infer from either of these claims to the other.) Since the corresponding distinction was unimportant in the 'earlier than' example in the previous section, we must be careful in extending the morals drawn there to the evaluative case. But despite this important pragmatic difference between these cases, and the previous difference too, still the formal analogy is there, and I will soon exploit it. Both the formal analogy and the pragmatic differences emerge in the framework for sentence-tokens or utterances. Let's look at the indexical case first. Suppose Sam says 'John was kissing Mary', and I take him as meaning that John was kissing Mary on a certain Saturday night. Sam's utterance is disquotationally true for me (that is, disquotationally true on my understanding of it) iff John was kissing Mary on that Saturday. In other words, his utterance is disquotationally true for me iff it is true relative to the time that I regard as appropriate to associate with the implicit indexical in it. (I say it is disquotationally true for me in these conditions, since someone else might interpret it differently, and we are sidestepping the question of which interpretation is 'correct', and of whether talk of 'correctness' of an interpretation of another person even makes literal sense.) Similarly, we can say that an evaluative utterance is disquotationally true for me iff it is true relative to the norms I regard appropriate to associate with the evaluative terms. That is the formal analogy. The first pragmatic disanalogy is that the norms I regard as appropriate to assign will never be close to unique; this often leads to there being no fact of the matter as to whether evaluative assertions are disquotationally true even relative to my own norms (taken as a whole). The second pragmatic disanalogy is that in the indexical case the object I will normally regard as appropriate to assign to an indexical is the one I think the speaker intended, whereas in the evaluative case the norms I will typically relativize to are not the ones I think she had in mind, but rather, my own. (I should add that in modal contexts, it is typically my own actual norms that matter: I typically treat my 'counterparts' in the way I treat other people. Thus the sentence 'If my norms had been like those of Hitler, racist attitudes end p.247
would have been acceptable' comes out false on the norms I relativize to, that is, on my actual norms.) The formal analogy and the pragmatic differences together suggest how the deflationist can handle evaluatives in a nonfactualist fashion. The proper version of the disquotation schema for an evaluative utterance is (7 E ) (For any norm N) 'We should intervene in Bosnia' is true relative to N iff we should intervene in Bosnia relative to N.28 If we somewhat indeterminately select one of our own norms to relativize to, and leave the relativization implicit, we get: (6 E ) 'We should intervene in Bosnia' is true iff we should intervene in Bosnia. This looks straightforwardly factual; but because of the implicit relativization, it is really only factual relative to a norm. That such utterances are at least factual relative to norms should be uncontroversial: relative to (any precise version of) a utilitarian norm, it may be a difficult empirical issue whether we should intervene in Bosnia. The point of nonfactualism is that there is no further fact here— no fact about non-relativized oughts, or about which of the norms that we might relativize to is 'objectively correct'. The genuine disagreement that remains when people agree as to the relativized oughts is disagreement in attitude, not factual disagreement. This analysis makes clear that the use of the disquotational truth predicate for implicitly relativized assertions is compatible with the use of an alternative truth predicate 'straightforwardly true'; there instead of implicitly relativizing, we supervaluate over all norms not ruled out as being in conflict with 'normative facts'. (For the non-factualist, this will include all norms.) I have been explaining how a deflationist can formulate a nonfactualist view about evaluatives: it involves taking the basic evaluative predicates as all having an extra argument place for norms (much as nonfactualism about absolute time-order involves taking 'earlier than' as having an extra argument place for frames of reference, and denying the need for an 'earlier than' predicate that has no such extra place). Also, of course, just as a nonfactualist about time-
order could not invoke the notion of a 'cosmically privileged' reference frame, so too the nonfactualist about evaluatives cannot invoke the notion of an 'objectively correct' norm. This isn't to say that he can't have strong preferences for some norms over others: after all, an adherent of special relativity can prefer certain frames, e.g. frames in which his body is often at rest. Nor is it to say that he can't talk of some norms as better than others, for this will 28 We must distinguish (7 E ) from
(8 E ) 'We should intervene in Bosnia relative to N 0 ' is true iff we should intervene in Bosnia relative to N 0 , where 'N0 ' is a closed singular term. (7 E ) employs a norm-sensitive truth predicate; (8 E ) doesn't, which makes it appropriate only for factual assertions about specific norms, not for evaluative assertions. end p.248
often be licensed by his higher norms given his factual beliefs. (This is one example of how a nonfactualist can make use of the fact that we have not a single norm but a multiplicity of them, interacting in complicated ways.) Nor is it to say that he can't speak of the evaluative claims licensed by his norms as true: for given that truth is disquotational, the norms license calling an evaluative claim true to precisely the extent that they license the evaluative claim itself. But he can't invoke any notion of 'correctness' for norms not explained in some such way (whether it is explained in some other way or left unexplained). It might be thought that the last of the ways just listed for making sense of truth-talk is all that the factualist deflationist should want: if it licenses talk of norms as correct, what factualist could ask for more? But again, look at the time-order case: here too, when we adopt a frame of reference to make temporal comparisons, we adopt the declaration that those temporal comparisons are disquotationally true. The reason that this does not seem sufficient for factualism about temporal order is that it isn't the agent's beliefs that license the truth of the temporal comparison, it is the agent's adoption of a frame of reference, which goes beyond what the agent believes. Similarly in the evaluative case: it is the agent's norms rather than the agent's beliefs that license the claim that the evaluation is true. At this point, though, we must face a long-deferred complication: an ambiguity in the notion of belief. For surely there is a perfectly good sense in which an agent who sincerely utters 'Intervention in Bosnia is a bad idea' does so because she believes that intervention in Bosnia is a bad idea. I think that what a nonfactualist must say about this is that while such a notion of belief is perfectly acceptable, it is not belief in its purest sense, but a hybrid of pure belief and the acceptance of a norm. Similarly, if I accept a certain frame of reference F and believe that event A is spacelike separated from event B but prior to it relative to F, then there is a perfectly good sense in which I believe that A is prior to B although spacelike separated from it. Does this make me a factualist about time order? I don't think so, for here too the sense in which I believe these things is a hybrid of pure belief and another attitude. This shows, though, that attention to the psychological distinction between (i) pure belief (ii) non-doxastic attitudes like adoption of a frame of reference and acceptance of a system of norms, and (iii) the hybrid of (i) and (ii) that I am calling 'impure belief' is crucial to advancing the discussion further. But I see little reason to doubt that the distinction can be drawn on psychological grounds. Three paragraphs back I gave the deflationist account of what nonfactualism consists in. The deflationist explanation of factualism is then obvious: it consists in the claim that nonfactualism won't do, that we need either evaluative predicates that are not norm-relative or else the notion of an 'objectively correct' or 'cosmically privileged' norm, where this is either left as an unexplained end p.249
notion or explained in some way that is supposed to transcend the viewpoint from a norm. 29 The basic lines on which the dispute will be carried out then are clear: the nonfactualist argues that we can accommodate all the phenomena using only the norm-relative notions, together with both preference among norms and non-normative beliefs; the factualist claims that we can get only an impoverished caricature of evaluative discourse by so doing. 30 My own opinion is that the nonfactualist will win, but my aim here has just been to show that the debate makes sense from a deflationist viewpoint. What consequences does this account have for the use of evaluative terms by people whose meta-beliefs are that there are intrinsically evaluative facts? If nonfactualism is right, those people are making an error, somewhat analogous to the error made by Newton when he assumed earlier than to be a straightforward 2-place relation; or better (since in Newton's case he didn't say explicitly that time order was absolute), analogous to the error made by people today who know about special relativity but still postulate absolute time order for one reason or another. When asked about the truth conditions of such people's utterances involving 'earlier than', we are I think pulled in two directions. One direction
is to say that despite their explicit beliefs to the contrary, their sentences involving 'earlier than' have truth conditions only relative to a frame of reference; the other is to say that their sentences do have non-relative truth conditions, but that these involve an absolute earlier than relation that is never instantiated, so that most of their sentences involving 'earlier than' are false. Or one might have a mixed account, taking the first line for their more practical sentences and the second for their more theoretical ones. It seems as if there is no clearly correct answer here. Similarly in the evaluative case I think: a nonfactualist has the choice of whether to regard the evaluative utterances of the explicit factualist as norm-relative despite the factualist's explicit intentions to the contrary, or not norm relative but postulating an intrinsically evaluative property that nothing actually has, or to take some mixed account where some evaluative utterances are treated one way and some the other. Again I think it rather a matter of taste how one answers this. There is still further indeterminacy in the case of those who have not explicitly thought about the meta-ethical or metaepistemological issues: there it is likely to be indeterminate what their meta-ethical and meta-epistemological views are, as well as indeterminate to what degree any falsity in their meta-ethical and meta-epistemological views trickles down to their object level ethical and epistemological claims. 29 I suppose that someone might profess factualism while denying that there are any purposes for which we need
either unrelativized notions or a notion of 'objective correctness': he might say that even though we don't need them for anything, we have them. But as in the time-order case, such a version of factualism is not an interesting, or even a very clear one since it is then unclear what 'objectively correct' amounts to. 30 Typically, factualists about value are nonskeptical about the realm of extra facts they posit: that is, they will think that we have an ability to detect the moral and epistemological 'facts'. If so, there is a pressure for them to say something about how we do the detecting. The nonfactualist faces no such pressure; this is a further mark of the factualist/nonfactualist distinction that a deflationist can point to. end p.250
I have stressed the analogy to the case of 'earlier than' because it is intermediate between the evaluative case and the case of ordinary indexicals. It shares with evaluatives the fact that the relativity in question is, or at least once was, highly controversial, so that there can be a genuine dispute as to whether there is a domain of unrelativized facts; but in the important pragmatic respects that mark off evaluatives from indexicals, it is like indexicals. I emphasized the pragmatic differences between the evaluative case and the time-order case a while back; but for the issues discussed in recent paragraphs, it is the similarity rather than the differences between the cases that matters. There is some connection between the position I'm adopting here and what I said in an earlier paper where I invoked a 'factuality operator': I think that the idea that time order can only be made sense of as frame-relative can be used to justify saying that not explicitly frame-relative talk of time order isn't fully factual, and that similarly, the idea that evaluative discourse can only be made sense of as norm-sensitive can be used to justify saying that not explicitly norm-relativized talk of goodness and the like isn't fully factual. My complaint about my earlier way of putting things isn't that it was incorrect, but that without further explanation it was totally unpersuasive. I hope that the further discussion in this section begins to remedy that. 31
5. Indicative Conditionals It isn't only evaluative discourse that can be viewed as nonfactual: various other sorts of discourse are plausibly so viewed as well. In these cases too, one may well worry whether a deflationist can make sense of the idea that the discourse in question is nonfactual. I suspect that in every case this worry will prove answerable, but unfortunately in most of these cases it does not seem possible to answer it in quite the same way as in the case of evaluatives, and the details of how it ought to be answered in each case need to be worked out 31 This paragraph raises the question of whether for vagueness too, appeal to a 'definitely' operator might be a
superficial way of saying something that is better said in terms of some sort of relativity. 'Bald' is vague; but we can invent various fairly precise standards of baldness (e.g. crude standards like having less than n hairs on one's head, for a given n, or less crude ones that take into account things like the distribution of hair; even the crude ones aren't fully precise since there are borderline cases of 'hair' and 'on', but they are a lot more precise than 'bald'). We might say then that the vagueness of 'bald' consists in the fact that we haven't bothered to settle among the fairly precise standards; if Jones is a borderline case of baldness, that is because he is bald relative to some of the standards of baldness that we are allowing but not bald relative to others of these standards. (If you like, this reduces the vague to the evaluative.) This account may seem attractive, but it has a difficulty: if we take the relativization literally, it won't handle all cases of vagueness unless there is a fully precise language in which to state the standards, for otherwise it gives a regress. (It may be, though, that we can handle some cases of vagueness in this way, e.g. those like 'tall' that derive from comparatives. In such cases, rather than saying we are relativizing to standards we can say that we are relativizing to a border.) The relativization strategy has a similar problem in the evaluative case if you think that there
is no fully non-evaluative language, but I don't see why anyone should think that (if the reduction of vagueness to evaluativeness is not assumed): 'proton' and 'electromagnetic field' might turn out to be vague or indeterminate, but I don't see how they can plausibly turn out to be evaluative. end p.251
separately. I will discuss just one example, indicative conditionals; my treatment will be rather sketchy, but I hope at least to illustrate some of the issues involved. Although in mathematics we are taught to treat 'if A then B' as the material conditional A B, i.e. as equivalent to ¬A V B, it is well known that outside mathematics it seems to obey a quite different logic (a logic formulated in Adams1975). For instance, few of us would be willing to infer from 'Clinton won't die in office' to 'If Clinton dies in office then Danny DeVito will become president', or from 'The human population will be greater in 1997 than it was in 1992' to 'If in 1994 a nuclear war wipes out most of the human race, then the human population will be greater in 1997 than it was in 1992'; but both inferences are valid on the standard logic for 'if . . . then'. These examples seem to indicate that 'if A then B' is not equivalent to '¬A V B'; or to put it another way, 'if A then B' does not seem to have the same truth conditions as ' '. One's first reaction is to suppose that it has some other truth conditions, though it is not easy to say what these might be; 32 but further thought has led 32 David Lewis's famous triviality proof (1976) is an obstacle to this course: it seems to show that only by making truth
conditions for indicative conditionals radically indexical could we get truth conditions that respect the 'surface logic' of arguments involving such conditionals. (Lewis's own view is that we need not respect the surface logic of these arguments: we should declare the inferences about Clinton and nuclear war as truth preserving despite their intuitive unacceptability.) In more detail: let {Ci } be a set of mutually exclusive and exhaustive sentences, each compatible with the sentence A. Lewis formulates the requirement that 'if . . . then' has truth conditions that respect 'surface logic' as the requirement that (*) P(if A then C i ) = P(C i |A) holds for a class of probability functions P which includes any subjective probability function that we adhere to and any that results from one we adhere to by conditionalization. (That degrees of belief obey the laws of probability is assumed.) The first step in Lewis's proof is the transition from this to the requirement that (**)P(if A then C i |Cj ) = P(C i |A&C j ) holds for any probability function we adhere to: the argument works by applying (*) to the probability function Pj obtained by conditionalizing on C j . (That gives Pj (if A then C i ) = Pj (Ci |A), and thus by conditionalization we get (**).) The second step is to note that the linearity principle P(if A then C i ) = ∑ j P(if A then C i |Cj )·P(C j ), (which follows from the probability axioms) implies that (**) is unsatisfiable except by completely trivial probability functions. As several people have noted (e.g. Price 1988), the first step of Lewis's argument turns on 'if . . . then' having the same meaning in the context of the probability function Pj that it has in the context of the probability function P. If the meaning varies, then (*) gives only that Pj (ifpj A then C i ) = Pj (Ci |A), and so conditionalization yields only the hard-tointerpret P(if PjA then C i |Cj ) = P(C i |A&C j ), rather than the claim P(if P A then C i |Cj ) = P(C i |A&C j ) that leads to triviality. Others have noted (e.g. Edgington 1986) that such extreme context-dependence seems to destroy much of the point of ascribing truth conditions: e.g. it makes it impossible for one person to disbelieve the same conditional that another believes. In addition, it may be doubted that the first step of the argument is where those who want to minimize the significance of the Lewis argument should focus. (**) amounts to the claim that P(if A then C i |Cj ) is 1 if i = j and 0 otherwise. If we interpret this as implying that we should accept 'if A then C i ' whenever we accept C i , then obviously it is unacceptable: let C i be 'Clinton will be President next year' and A be 'Clinton will die tomorrow'. But if we interpret (**) as saying that certainty in C j should lead to certainty in 'if A then C j ', (**) looks like what we ought to expect in a theory of indicative conditionals. (As I'll note later, there is a sense in which the Lewis argument can still be challenged at the second step.) end p.252
many to question that it has truth conditions at all.33 For what the above examples suggest is that reasoning with
conditional sentences is reasoning on a supposition: to accept 'if A then B' is to conditionally accept B, that is, accept B relative to the supposition A. (If you think of acceptance probabilistically, this is regarding the conditional probability of B given A as high; that is, regarding the probability of A& B as substantially higher than the probability of A & ¬B. Adams showed that this idea of conditional acceptance justifies our intuitive convictions as to which inferences involving conditionals are correct.) If this conditional acceptance view is right it isn't at all obvious that the assertion 'if A then B' says anything about the world; all that's required (at most) is that A does and that B does. (For instance, there need be no claim C whose probability in all circumstances is the same as the conditional probability of B given A in those circumstances. Certainly the material conditional A B isn't such a C.) In short, this account explains the workings of indicative conditionals without postulating 'conditional propositions' or 'conditional facts': it is a nonfactualist view. (In the special case where the conditional probability is 0 or 1, it is arguable that the acceptability conditions are the same as those of the material conditional,34 which may explain why the material conditional account works so well in mathematical contexts.) The nonfactualist view of indicative conditionals is not inevitable: both the view that these conditionals have the classical truth conditions and the view that they have nonclassical ones have adherents. The question I want to discuss is, can the deflationist make sense of the issue? For it seems that if we are talking about disquotational truth, indicative conditionals do have truth conditions: 'If Clinton dies in office Danny DeVito will become President' is true iff if Clinton dies in office Danny DeVito will become President; equivalently, it is true iff if 'Clinton dies in office' is true then 'Danny DeVito will become President' is true. A deflationist who wants to make sense of nonfactualism must explain any notion of truth on which indicative conditionals don't have truth conditions; and if he gives up the claim that they don't have truth conditions, he must explain in other terms what makes these conditionals nonfactual. (Again I regard the latter challenge as the more important one; and I assume it is unsatisfactory to simply invoke an 'it is a fact that' operator in its solution.) I should say at the start that it isn't at all clear to me that there is in the end much of an issue between the nonfactualist and the sophisticated defender of the view that indicative conditionals have the classical truth conditions, that is, the truth conditions of ' '. Any sophisticated defender of this view has to 33 It might be objected that to say that a sentence has no truth conditions makes no sense: at most, the sentence can
never be true or false, but to say that is to give it a kind of truth conditions. This seems right; however, I think that the main claim of those who say that indicative conditionals don't have truth conditions is really that you can't explain their acceptance conditions in terms of their truth conditions (even using conventional implicatures and the like in the explanation); and that for this and other reasons, the question of what truth conditions to attach to the conditionals is idle. 34 This requires a view of conditional probability that takes it as defined and capable of taking the value 1 even when the antecedent has probability 0. end p.253
explain away the apparent invalidity of the counterintuitive inferences mentioned three paragraphs back; and when all the explaining away is done, I think that the resulting view may differ only verbally from the nonfactualist view (that is, the two views may differ only in assigning a different meaning to 'true' and perhaps 'valid'). I'll say a bit more about this later. In any case, the difference between nonfactualism and the view that indicative conditionals have the classical truth conditions is an atypical example of a dispute over factualism, in that here the factualist postulates no realm of facts that the nonfactualist doesn't believe in: everything that this 'factualist' expresses with a conditional can be expressed by a nonfactualist using negation and disjunction. This makes the distinction between this sort of factualism and nonfactualism of less interest than most factualist/nonfactualist disputes. I take the main opponent of nonfactualism to be that form of factualism (e.g. Stalnaker 1984) according to which there are conditional facts not expressible in first order logic but which are expressed by indicative 'if . . . then' sentences. How, without using 'fact' or 'true', can we distinguish the various factualist views from the view that indicative conditionals do not (and do not purport to) express such facts? The key to solving the analogous problems in the case of evaluatives was to point out that upon suitable relativization, evaluative statements become perfectly factual. Truthtalk thus has clear application to the explicitly relativized statements, and can be extended to the implicitly relativized ones in standard ways; on the other hand, the need to relativize to get a truth value (together with various features of the pragmatics of what counts as an appropriate relativization) gives good sense to these statements being nonfactual. If we are to apply this idea in the case of indicative conditionals, we must decide relative to what they are factual. One thing we might relativize to is a subjective probability function— this is a somewhat natural move, since advocates of the view that indicative conditionals do have truth conditions that respect the 'surface logic' of ordinary arguments need to invoke such relativity to a probability function: see the next to last paragraph of footnote 32 . But I don't think that invoking relativity simply to a subjective probability function is satisfactory. The main problem is that whereas it makes clear sense to talk of the truth or falsity of an evaluative statement relative to a norm, it does not make clear sense to talk of the truth or falsity of an indicative conditional relative to a subjective probability function: the subjective
probability function gives only a conditional probability to associate with the conditional, not a truth value. It might be possible to get around this problem by relativizing to some further ingredients in addition to the probability function— e.g. to a norm of evaluation that tells you how high a conditional probability must be for a conditional to count as 'true' (relative to that norm and probability function); but I am not at all sure that this or anything like it 35 is an attractive way to proceed. 32 David Lewis's famous triviality proof (1976) is an obstacle to this course: it seems to show that only by making truth
conditions for indicative conditionals radically indexical could we get truth conditions that respect the 'surface logic' of arguments involving such conditionals. (Lewis's own view is that we need not respect the surface logic of these arguments: we should declare the inferences about Clinton and nuclear war as truth preserving despite their intuitive unacceptability.) In more detail: let {Ci } be a set of mutually exclusive and exhaustive sentences, each compatible with the sentence A. Lewis formulates the requirement that 'if . . . then' has truth conditions that respect 'surface logic' as the requirement that (*) P(if A then C i ) = P(C i |A) holds for a class of probability functions P which includes any subjective probability function that we adhere to and any that results from one we adhere to by conditionalization. (That degrees of belief obey the laws of probability is assumed.) The first step in Lewis's proof is the transition from this to the requirement that (**)P(if A then C i |Cj ) = P(C i |A&C j ) holds for any probability function we adhere to: the argument works by applying (*) to the probability function Pj obtained by conditionalizing on C j . (That gives Pj (if A then C i ) = Pj (Ci |A), and thus by conditionalization we get (**).) The second step is to note that the linearity principle P(if A then C i ) = ∑ j P(if A then C i |Cj )·P(C j ), (which follows from the probability axioms) implies that (**) is unsatisfiable except by completely trivial probability functions. As several people have noted (e.g. Price 1988), the first step of Lewis's argument turns on 'if . . . then' having the same meaning in the context of the probability function Pj that it has in the context of the probability function P. If the meaning varies, then (*) gives only that Pj (ifpj A then C i ) = Pj (Ci |A), and so conditionalization yields only the hard-tointerpret P(if PjA then C i |Cj ) = P(C i |A&C j ), rather than the claim P(if P A then C i |Cj ) = P(C i |A&C j ) that leads to triviality. Others have noted (e.g. Edgington 1986) that such extreme context-dependence seems to destroy much of the point of ascribing truth conditions: e.g. it makes it impossible for one person to disbelieve the same conditional that another believes. In addition, it may be doubted that the first step of the argument is where those who want to minimize the significance of the Lewis argument should focus. (**) amounts to the claim that P(if A then C i |Cj ) is 1 if i = j and 0 otherwise. If we interpret this as implying that we should accept 'if A then C i ' whenever we accept C i , then obviously it is unacceptable: let C i be 'Clinton will be President next year' and A be 'Clinton will die tomorrow'. But if we interpret (**) as saying that certainty in C j should lead to certainty in 'if A then C j ', (**) looks like what we ought to expect in a theory of indicative conditionals. (As I'll note later, there is a sense in which the Lewis argument can still be challenged at the second step.) 35 To make talk of the 'truth' of a conditional 'If A then B' more appropriate, we should actually consider not the conditional probability of B given A, but the conditional probability of B given A&R, where R is the totality of all truths (known or unknown) of a certain type K; this would allow conditionals asserted only because of mistaken background beliefs to count as mistaken. The type of truths K to be included in R would presumably depend on the interests of the evaluator of the conditional; this type should then be taken as a third item that we'd need to relativize to, if we wanted to pursue the relativization line. Also, there would be no need to suppose that the norm of evaluation simply gives a fixed number v such that 'If A then B' counts as true if and only if P(B|A&R) is at least v: it could allow that the threshold for a particular conditional depends on certain holistic considerations. end p.254
Perhaps then we should forget about relativization in this case. Is there another way to explain the difference between 'factualists' about indicative conditionals and 'nonfactualists' that a deflationist can appeal to (that is, that doesn't rely on a notion of truth or some related notion)? I think so: a factualist and a nonfactualist will be committed to using conditionals in different ways. Consider first the factualist who believes that 'If A then B' has the same truth conditions as '¬A V B'. The distinction
between such a factualist and a nonfactualist would seem to be obvious: the former does not accept the counterintuitive inferences mentioned in the second paragraph of this section, while the latter would seem committed to accepting them. Of course, a sophisticated advocate of the material conditional view will have a story to tell about why such inferences seem unacceptable (and in some sense, are unacceptable). The best such story I know is in Jackson 1979. Jackson grants the unassertability of the conclusions of these inferences, but holds that these conclusions are nonetheless true; and he explains the unassertability in such a way that the classical truth conditions play some sort of role, though not an ineliminable one. (Basically, he gives a story about why there is a conventional implicature that when we assert 'If A then B', not only is P(A B) high but P(A B|A) is high; violating this conventional implicature is highly misleading. And the requirement that P(A B|A) is high is equivalent to the nonfactualist's requirement that P(B|A) is high, so Jackson gets just the same results about assertability as the nonfactualist.) One standard objection to such attempts (e.g. Edgington 1986) is that it seems that we don't just not assert things like 'If Clinton dies in office then DeVito will become President', we don't believe them either. I take it that the Jackson response to this objection would have to be that the conventional implicature makes these inappropriate to assert even mentally; and the felt invalidity is then that these inference forms don't preserve mental assertability, even though they preserve truth. So we get both a 'surface logic' that tells us which inferences preserve acceptability, which is just the Adams logic that the nonfactualist offers; and a 'deep logic' that tells us which inferences preserve truth, which is the standard logic of ' '. A worry about any such view is whether the 'deep logic' is doing any work: if even our mental processing is in terms of the surface logic, what is the point of postulating the deep logic? Perhaps an answer can be given: perhaps for instance at the deepest levels we do reason classically with our 'if . . . then' constructions, and the special conventions of correct assertibility come in only later. If a factualist postulates such a distinction of levels, he has made end p.255
sense of the distinction between an inference being fully unacceptable and its being at a deep level acceptable but seeming unacceptable due to a violation of these special conventions. But if this is how he distinguishes his view from the nonfactualist's, then a deflationist can make sense of the distinction in the same way: the distinction has been drawn without the use of 'true' or 'fact'. The same holds for any other way of making the distinction intuitively meaningful that I have been able to think of. In general, a deflationist should have no trouble distinguishing the nonfactualist view from any view that postulates a logic different from the Adams logic, as long as the postulated logic plays a role in our reasoning. Now let's turn to factualist views (e.g. Stalnaker 1984) that (when restricted to unembedded conditionals) yield the Adams logic, that is, which are indistinguishable from the nonfactualist view in terms of which inferences are deemed correct. Can the deflationist distinguish nonfactualism from such versions of factualism? One difference arises if there are sentences 'A' that represent logical and metaphysical possibilities but not epistemic possibilities. On the nonfactualist view, there can be no issue of whether to accept 'If A then B' when 'A' is not regarded as even a minimal epistemic possibility: the whole point of conditionals depends on taking their antecedents to be epistemically possible, and the account yields that all conditionals whose antecedents aren't epistemically possible are equally acceptable. But for someone like Stalnaker, there is a factual issue in this case as to whether the conditional is true, and (as long as the antecedent isn't also logically or metaphysically impossible) some conditionals with epistemically impossible antecedents will be true and others false. So one test of whether someone adheres to this sort of factualism is whether he takes debates about the acceptability of such conditionals seriously. Another point of distinction between factualism (of either the Stalnaker or the material conditional kind) and nonfactualism is that on a factualist view of indicative conditionals it would seem to be possible to embed indicative conditionals arbitrarily, so that claims like 'Either it is the case that if Oswald didn't act alone then the CIA assisted him or it is not the case that if the Red Sox win the pennant they will lose the World Series' will be meaningful. On a nonfactualist view on which acceptability conditions for unembedded conditionals are based on conditional probabilities (or on reasonable changes of epistemological state to accommodate new information), there will be natural ways to extend the acceptability conditions to handle certain embeddings of the conditionals, but it is not at all evident that sense can be made of arbitrary embeddings of conditionals. So a possible test for deciding whether someone is a factualist about indicative conditionals might be to look at what embeddings of conditionals he regards as meaningful. This test is not fully satisfactory, however— for one thing, it doesn't distinguish the nonfactualist from the factualist who ad hocly restricts the allowable embeddings, and for another, it is not clear that a nonfactualist couldn't find a way of extending the acceptability conditions to handle all embeddings. end p.256
Note however that if a nonfactualist does extend the acceptability conditions to handle all embeddings, the Lewis triviality proof (footnote 32 ) shows that the usual probabilistic laws can't apply to such embeddings (contrary to Lewis's
tacit assumption). This I think is the most fundamental difference between the factualist and the nonfactualist: the nonfactualist must hold that the acceptability of sentences containing conditionals is not governed by the usual probabilistic laws that govern the acceptability of 'factual' sentences. One way in which the laws of acceptability of indicative conditionals might be nonstandard is simply that the conditionals not be embeddable. However, an analysis of the second step of the Lewis proof reveals that in fact even unembedded conditionals behave nonstandardly, at least with respect to conditional degree of belief. In particular, I think that the proof makes clear that the linearity law (L) P(D) = P(D|B) · P(B) + P(D|¬B) · P(¬B) should not be acceptable to the nonfactualist when D is of form A→C: a nonfactualist will want to read the left hand side as equivalent to P(C|A), and the right hand side as P(C|A & B) · P(B) + P(C|A & ¬B) · P(¬B); these are equivalent on the further assumption that B and A are independent (P(A&B) = P(A) · P(B)), but are not equivalent in general.36 Note that this failure of a law for conditional degree of belief arises even for unembedded conditionals: issues about the embeddability of conditionals are secondary. Of course, if conditional degree of belief is defined in terms of absolute degree of belief in the usual way, the failure results from a failure of the law P(D) = P(D & B) + P(D & ¬B) for conditional D, which would be automatically disallowed if we could not embed conditionals even in conjunctions. But conjunctions with simple conditionals as conjuncts are intuitively clear, and moreover it is natural to take conditional degree of belief as primitive; for each of these reasons, the failure of the classical probability laws for degrees of belief in conditionals seems to be independent of issues about the extent to which the embedding of conditionals makes sense. Of course, there is a sense in which the two kinds of factualist must also accept the failure of the probabilistic laws, given the Lewis proof. An advocate of the material conditional reading of conditionals grants that the probabilistic laws fail for the 'surface logic' of conditionals, but adheres to the laws for the postulated 'underlying logic'. And an advocate of the Stalnaker approach postulates that the probabilistic laws seem to fail, but blames the apparent failures on contextual variations in the meaning of the conditional: see footnote 32 . In other words, the Stalnaker approach requires a distinction between the 'deep' laws of acceptability, which it holds are probabilistic, and the non-probabilistic laws of acceptability that are visible on the surface. As with the material conditional view of conditionals (where the 'deep' v. 'surface' 32 David Lewis's famous triviality proof (1976) is an obstacle to this course: it seems to show that only by making truth
conditions for indicative conditionals radically indexical could we get truth conditions that respect the 'surface logic' of arguments involving such conditionals. (Lewis's own view is that we need not respect the surface logic of these arguments: we should declare the inferences about Clinton and nuclear war as truth preserving despite their intuitive unacceptability.) In more detail: let {Ci } be a set of mutually exclusive and exhaustive sentences, each compatible with the sentence A. Lewis formulates the requirement that 'if . . . then' has truth conditions that respect 'surface logic' as the requirement that (*) P(if A then C i ) = P(C i |A) holds for a class of probability functions P which includes any subjective probability function that we adhere to and any that results from one we adhere to by conditionalization. (That degrees of belief obey the laws of probability is assumed.) The first step in Lewis's proof is the transition from this to the requirement that (**)P(if A then C i |Cj ) = P(C i |A&C j ) holds for any probability function we adhere to: the argument works by applying (*) to the probability function Pj obtained by conditionalizing on C j . (That gives Pj (if A then C i ) = Pj (Ci |A), and thus by conditionalization we get (**).) The second step is to note that the linearity principle P(if A then C i ) = ∑ j P(if A then C i |Cj )·P(C j ), (which follows from the probability axioms) implies that (**) is unsatisfiable except by completely trivial probability functions. As several people have noted (e.g. Price 1988), the first step of Lewis's argument turns on 'if . . . then' having the same meaning in the context of the probability function Pj that it has in the context of the probability function P. If the meaning varies, then (*) gives only that Pj (ifpj A then C i ) = Pj (Ci |A), and so conditionalization yields only the hard-tointerpret P(if PjA then C i |Cj ) = P(C i |A&C j ), rather than the claim P(if P A then C i |Cj ) = P(C i |A&C j ) that leads to triviality. Others have noted (e.g. Edgington 1986) that such extreme context-dependence seems to destroy much of the point of ascribing truth conditions: e.g. it makes it impossible for one person to disbelieve the same conditional that another believes.
In addition, it may be doubted that the first step of the argument is where those who want to minimize the significance of the Lewis argument should focus. (**) amounts to the claim that P(if A then C i |Cj ) is 1 if i = j and 0 otherwise. If we interpret this as implying that we should accept 'if A then C i ' whenever we accept C i , then obviously it is unacceptable: let C i be 'Clinton will be President next year' and A be 'Clinton will die tomorrow'. But if we interpret (**) as saying that certainty in C j should lead to certainty in 'if A then C j ', (**) looks like what we ought to expect in a theory of indicative conditionals. (As I'll note later, there is a sense in which the Lewis argument can still be challenged at the second step.) 36 I am assuming the 'certainty interpretation' of the conditional degrees of belief, mentioned in fn. 32. Perhaps there is an alternative interpretation on which the fault lies instead with the assumption that we modify probabilities by conditionalization, used in the first step of the proof. 32 David Lewis's famous triviality proof (1976) is an obstacle to this course: it seems to show that only by making truth conditions for indicative conditionals radically indexical could we get truth conditions that respect the 'surface logic' of arguments involving such conditionals. (Lewis's own view is that we need not respect the surface logic of these arguments: we should declare the inferences about Clinton and nuclear war as truth preserving despite their intuitive unacceptability.) In more detail: let {Ci } be a set of mutually exclusive and exhaustive sentences, each compatible with the sentence A. Lewis formulates the requirement that 'if . . . then' has truth conditions that respect 'surface logic' as the requirement that (*) P(if A then C i ) = P(C i |A) holds for a class of probability functions P which includes any subjective probability function that we adhere to and any that results from one we adhere to by conditionalization. (That degrees of belief obey the laws of probability is assumed.) The first step in Lewis's proof is the transition from this to the requirement that (**)P(if A then C i |Cj ) = P(C i |A&C j ) holds for any probability function we adhere to: the argument works by applying (*) to the probability function Pj obtained by conditionalizing on C j . (That gives Pj (if A then C i ) = Pj (Ci |A), and thus by conditionalization we get (**).) The second step is to note that the linearity principle P(if A then C i ) = ∑ j P(if A then C i |Cj )·P(C j ), (which follows from the probability axioms) implies that (**) is unsatisfiable except by completely trivial probability functions. As several people have noted (e.g. Price 1988), the first step of Lewis's argument turns on 'if . . . then' having the same meaning in the context of the probability function Pj that it has in the context of the probability function P. If the meaning varies, then (*) gives only that Pj (ifpj A then C i ) = Pj (Ci |A), and so conditionalization yields only the hard-tointerpret P(if PjA then C i |Cj ) = P(C i |A&C j ), rather than the claim P(if P A then C i |Cj ) = P(C i |A&C j ) that leads to triviality. Others have noted (e.g. Edgington 1986) that such extreme context-dependence seems to destroy much of the point of ascribing truth conditions: e.g. it makes it impossible for one person to disbelieve the same conditional that another believes. In addition, it may be doubted that the first step of the argument is where those who want to minimize the significance of the Lewis argument should focus. (**) amounts to the claim that P(if A then C i |Cj ) is 1 if i = j and 0 otherwise. If we interpret this as implying that we should accept 'if A then C i ' whenever we accept C i , then obviously it is unacceptable: let C i be 'Clinton will be President next year' and A be 'Clinton will die tomorrow'. But if we interpret (**) as saying that certainty in C j should lead to certainty in 'if A then C j ', (**) looks like what we ought to expect in a theory of indicative conditionals. (As I'll note later, there is a sense in which the Lewis argument can still be challenged at the second step.) end p.257
distinction arose at the level of the logic rather than the theory of acceptability), we may reasonably be skeptical as to whether the deep laws are doing any work; but any theory according to which they indeed do work presumably postulates mental operations that a nonfactualist about conditionals wouldn't accept, and the postulation of such mental operations would serve to distinguish between the factualist and the nonfactualist views independent of any theory of truth. I think that this discussion justifies the conclusion that insofar as the difference between the factualist and the nonfactualist is genuine, it arises independently of their use of the epithets 'true' and 'false' (and their use of 'fact'). If this is so, then making the distinction (insofar as it is genuine) poses no difficulty for the deflationist.
When it comes to the question of how the deflationist nonfactualist should apply the term 'true' to indicative conditionals, there are various options. One option is to use the word in the purely disquotational sense. On this option, one says (*) 'If Clinton dies in office Gore will become President' is true iff if Clinton dies in office Gore will become President; so anyone who accepts that conditional accepts the claim that it is true. But since (I've argued) I don't need the term 'true' to display my nonfactualism about the conditionals, I can display in my behavior the fact that I am treating the right hand side and hence the left hand side of (*) as 'factually deficient'. A second option is to use the factual deficiency of indicative conditionals to motivate a not-purely-disquotational use of 'true': 'straightforwardly true' or whatever. This would coincide with disquotational truth for sentences not containing conditionals (or evaluatives, etc.), but would differ on conditionals. Actually the kind of factual deficiency we've seen in indicative conditionals could motivate any of several different nondisquotational truth predicates. The simplest (and the one most clearly suggested by the claim that conditionals don't have truth conditions) would have it that indicative conditionals are never true or false. But perhaps a more natural option is to regard them as having no truth value when their antecedents are false, but say that when their antecedents are true, they have the same truth value as their consequents. A third option would be to also allow a conditional to have truth value when its antecedent is consistent and entails either the consequent or the negation of the consequent. It is clearly a matter of uninteresting convention which of these nondisquotational truth predicates (or, predicates of 'straightforward truth') to prefer. Similarly, it is unimportant whether one employs one of the nondisquotational truth predicates or the purely disquotational one. What is important rather is how independent of truth talk one is to convey the 'factual defectiveness' of the conditionals; and I think the discussion above shows in outline how this can be done. 37 37 I am grateful to Stephen Schiffer, Marian David, and the editors, for helpful comments on an earlier version. end p.258
9 Some Thoughts on Radical Indeterminacy Abstract: Discusses some issues about indeterminacy of reference and truth, from two points of view about reference and truth: that of a correspondence theory and that of a disquotational theory. It is argued that a correspondence theorist can continue to accept the usual disquotation schemas for reference and truth, despite the indeterminacy. And it is argued that the disquotationalist can accept indeterminacy even in his own conceptual scheme. Together, these claims mean that the two views on truth are much closer in their treatments of indeterminacy than one might have thought. Also discusses indeterminacy in our logical and mathematical vocabulary. Keywords: correspondence theory of truth, disquotation, disquotationalism, finiteness, indeterminacy, logical connectives, mathematics, reference, truth
Hartry Field
I A natural question to raise about words—or about their mental analog, concepts—is: in virtue of what facts do they refer to whatever it is that they refer to? In virtue of what does the word 'insanity' refer to insanity, the word 'entropy' refer to entropy, and so forth? There is a view called disquotationalism according to which this question is misconceived. I'll have something to say about that later. But putting disquotationalism aside for now, it would seem that this question about words or concepts needs an answer. And the only kind of answer we can take seriously is a naturalistic answer: the Brentanian answer that words or concepts refer by virtue of irreducibly mental connections between the mind and the world is not a serious contender. What if no naturalistic answer is possible? If we put disquotationalism aside, the only conclusion to be drawn is that the assumption that 'insanity' and 'entropy' determinately refer to insanity and to entropy is an illusion. Of course, it surely isn't a complete illusion: even if there are no naturalistic facts that fully determine the referents of these words or concepts, surely there are naturalistic facts that partially determine the referents. (Surely facts about our use of 'insanity' determine that the word doesn't stand for lubricating jelly.) But it is hard to see what could give us grounds for confidence that the naturalistic facts determine a unique referent for these words. And though I won't argue the matter, there are considerable grounds to doubt that a unique referent could be determined. (Indeed, ordinary vagueness should be enough to establish the point, though in my view there are more interesting kinds of
indeterminacy as well.) If there are no facts that fully determine the reference of 'entropy' or 'insanity', then from an objective point of view the best we can say is that such a word 'partially refers' or 'indeterminately refers' to each of a range of things (viz., the eligible candidates for being the referent), and all we can expect of a theory of reference is that it explain how the naturalistic facts determine the partial referents of our words or concepts.1 1 It might be objected that naturalistic facts can't determine even the partial referents of our concepts, on the grounds
that it is somewhat arbitrary which things you count as partial referents of a concept. But this just shows that 'partially refers' (and associated terms like 'determinate' and 'borderline case of') themselves refer indeterminately. 'Higher order indeterminacy' is just indeterminacy in our semantic terms. For more on this, see the Appendix to Chapter 7. (This treatment accommodates not only second-order indeterminacy, but also indeterminacy of higher orders: e.g. if 'borderline case of' is indeterminate in application to a predicate of form 'borderline case of F' (where F is nonsemantic) we have third order indeterminacy.) end p.259
Does this mean that we might have to give up such claims as that 'entropy' refers to entropy and to nothing else? Not necessarily (even continuing to put disquotationalism aside). We can continue to say these things, provided we posit a suitable sort of referential indeterminacy in the word 'refer' itself. In Chapters 6 and 7 I said that in light of the possibility of indeterminate reference we should regard 'refers' itself as indeterminate: nothing in our practice settles whether in cases of indeterminate reference we should say that the term refers to all the candidates for the reference or to none of them. 2 But in fact the machinery introduced in those chapters allows a better account of the way in which 'refers' is indeterminate. The idea is that indeterminacy needn't operate on a word-by-word basis. For instance, consider the vagueness of color words (vagueness being a special kind of indeterminacy): we want to say that for certain 'bluegreen' shades, nothing in our practice dictates whether they are blue or dictates whether they are green, but still our practice dictates that they are one or the other and not both. Fine (1975) nicely described this holistic sort of indeterminacy by saying that there is a 'penumbral connection' between 'blue' and 'green'. A similar holism can arise for types of indeterminacy other than vagueness (see Chapter 7 for examples); there talk of 'penumbral connections' isn't appropriate, and in the more general context, I used the locution 'correlative indeterminacy'. What I failed to see there is that it is easy to use the machinery of correlative indeterminacy to tie the reference of 'refers' to that of indeterminate 'ground-level' concepts, such as perhaps 'entropy' and 'insanity'. So if 'entropy' partially refers to E 1 and to E 2 , then we can say that relative to an assignment of E 1 to 'entropy', 'refers' refers to a relation that holds between 'entropy' and only one thing, viz. E 1 ; and analogously for E 2 . In this way we can get the result that even if 'entropy' partially refers to many things (and hence doesn't determinately refer to anything), still the sentence ' "Entropy" refers to entropy and to nothing else' comes out true. (Indeed, determinately true: true on every acceptable combination of the partial referents of 'entropy' and 'refers'.) The advocate of indeterminacy can still 'speak with the vulgar'. To summarize: if we reject disquotationalism, we should expect an answer or at least a partial answer to questions like, 'In virtue of what does "entropy" refer to entropy?' But it may well be that only a partial answer is possible. If so, we should conclude that 'entropy' is indeterminate, that it partially refers to more than one thing. But we can reconcile this conclusion with our continuing to accept ' "Entropy" refers to just one thing, namely entropy'. That we can continue to use the term 'refers' in this way should reduce the resistance many have felt to positing indeterminacy. 2 'Candidate for the reference' may itself be indeterminate, as observed in the previous footnote; this would of course
give rise to further indeterminacy in 'refer'. end p.260
II It seems to me that from a nondisquotationalist viewpoint it is bound to be the case that there is a substantial level of indeterminacy in our concepts, though there is room for debate as to just how high that level is. One question that has been of some interest to me is the extent of indeterminacy in our logical concepts. I think it is clear that if we must take seriously the question of in virtue of what 'entropy' refers to entropy, we must equally take seriously the question of in virtue of what 'not' refers to the usual truth function and 'there is' refers to the function that maps nonempty sets into the True and empty sets into the False. Of course, we needn't accept this Fregean way of talking, according to which logical connectives refer; but the point survives the shift to other ways of talking. On a more sober way of putting things, the question is: in virtue of what facts about the use of 'not' and 'there is' do they contribute to truth-conditions in the way that the standard Tarskian accounts say that they do? A natural approach to answering this question is in terms of the conceptual role of the logical concepts: especially their role in deductive inference, though perhaps their role in probabilistic inference as well. (Indeed, perhaps the former is
simply a special case of the latter.) This approach in terms of conceptual role has the effect of dividing the original question into two subquestions: first, in virtue of what facts about the use of 'not' and 'there is' do they have the conceptual roles they do? and second, in virtue of what do these conceptual roles determine these contributions to truth conditions? The first subquestion may sound trivial, but as Wagner 1981 observed, it is not trivial if conceptual role is taken as an idealized or normative notion ('From not both A and B one ought not to infer not A, except when one sees an entailment from A to B'). There is a serious question as to how such an idealized conceptual role is determined from the use of our words (if uses are described in a less idealized and less normative fashion). The second subquestion is even more obviously nontrivial: even given an idealized inferential role for 'not' or 'there is', there seems a big gap between that and contribution to truth conditions. Of course there is a natural strategy for trying to fill the gap: the contribution to truth conditions should be such that (insofar as possible) the inferences licensed in the conceptual role come out necessarily truth preserving, and the inferences taken as illegitimate in the conceptual role account don't. But it isn't at all obvious that this strategy will lead to a unique answer for the contribution to truth conditions of 'not' or 'there is', even if the idealized conceptual role of these terms is assumed. 3 The opportunities for indeterminacy 3 Obviously only one two-valued truth-function will preserve truth for the standard inferences involving 'not'; but how are we to argue that the "referent" of 'not' must be a truth function at all, let alone a two-valued one? In the case of 'there is' the threat of nonuniqueness in the truth conditions that would legitimize the standard inferential role is greater: for instance, on certain ways of restricting quantification, the standard inferential role would presumably be preserved. (In any case, even if we knew of no candidates for an alternative 'referent' for the logical symbols, that wouldn't constitute much of an argument that there wasn't such an alternative.) end p.261
multiply when one considers the problems of getting from brute usage to idealized conceptual role. (Among other things, the famous skeptical problem in Kripke 1982 is, I think, best seen as a problem about getting from brute use to conceptual role; though this view of Kripke's argument is nonstandard.) So the 'conceptual role approach' to the question of what determines the contributions of our logical words and concepts to the truth conditions of the sentences and thoughts that contain them has a considerable danger of leading to the conclusion that even our logical concepts have a substantial degree of indeterminacy. Some may feel that this argues against the conceptual role approach. But it is hard to imagine an alternative nondisquotationalist approach with any chance of doing better. If we are forced to acknowledge such indeterminacy in our logical concepts, how upsetting should this be? At first blush, a proponent of such indeterminacy might seem forced to deny the truth of (N) The sentence that results by putting the word 'not' in front of sentence S is true if and only if S is not true. But this is not so: again it is possible to reconcile the acceptance of indeterminacy with 'speaking with the vulgar', by another use of the strategy of postulating a correlative indeterminacy in our semantic terms. This time, what we must say is that the word 'true' has an indeterminacy correlative to that of 'not' and 'there is'. More fully, what we should say is that claim (N) is partly definitive of the notion of truth. For each of the 'partial referents' of 'not'—that is, each of the legitimate contributions to truth conditions you posit for 'not' (as it occurs on the right hand side of (N))—the acceptance of (N) gives rise to an associated constraint on the partial extension of 'true'. If for simplicity we ignore any indeterminacy in terms other than 'not' and 'true', we can say that each partial extension of 'not' is associated with a partial extension of 'true'.4 Just as it is inappropriate to evaluate 'That is green and blue' with partial extensions of 'green' and 'blue' that aren't associated with each other, so too it is inappropriate to evaluate (N) with partial extensions of 'not' and 'true' that aren't associated with each other. And if you use associated partial referents, (N) is guaranteed to come out correct despite the indeterminacy. 5 As a result, postulating an indeterminacy in logical concepts like 'not' may be less disruptive to ordinary semantic reasoning than one might think. I realize that this 'correlative indeterminacy' strategy for minimizing the impact of indeterminacy deserves a fuller discussion. For instance, the proposal works so well at 'hiding' the indeterminacy that the indeterminacy only shows itself at the semantic level in fairly technical concepts like partial reference; 4 If we don't ignore the other indeterminacies in the language, the right thing to say is that each partial extension of
'not' is associated with many partial extensions of 'true', but no two partial extensions of 'not' are associated with the same partial extension of 'true'. 5 'Correct' is a metalinguistic notion of (determinate) truth. But one can connect the object language and the metalanguage suitably, so that the sentence will come out true on any of the partial referents of the object level 'true'. end p.262
and it could well seem worrisome that you need a special technical vocabulary to articulate the distinction between
determinate and indeterminate language. I think in the end this can be argued not to be so worrisome; but to the extent that this or other worries about the approach I've suggested seem compelling, I think they argue for a more disquotationalist approach to indeterminacy (on which I will comment later). For to repeat, I think that the chance of significant indeterminacy even in our logical notions is serious on a nondisquotationalist approach, and that what I have proposed is the least disruptive way to accommodate it.
III Before turning to disquotationalism, I want to say a bit about a further consideration that seems to indicate that there is a possibility of considerable indeterminacy in our logical vocabulary. But as a prelude to considering our logical vocabulary, let's consider our mathematical vocabulary. (And in discussing that, I will assume for the sake of argument that there is no indeterminacy in the logical vocabulary of standard first-order logic.) In particular, let's consider the question of whether we have a determinate notion of an ω-sequence. An ω-sequence is a linear ordering which has a first member and no last member, and where each member has only finitely many predecessors. Put another way, the ordering is just like that of the natural numbers. These explanations would seem to determine the notion of an ω-sequence completely. But the first explanation does so only if the notion of finiteness is determinate; and the second does so only if the notion of natural number is determinate (more precisely, only if it is determinate up to isomorphism).6 We could explain the notion of finiteness or natural number in terms of the notions of set and membership (or characterize the notion of an ω-sequence directly in such terms); but such characterizations supply the requisite determinacy to the notions of finiteness and ω-sequence and natural number only if the notions of set and membership are sufficiently 6 That 'natural number' is at best determinate up to isomorphism is influentially argued in Benacerraf 1965. I accept
his argument, but want to put it aside here, for it is irrelevant to the issues I am concerned with. (In my view the kind of indeterminacy Benacerraf's argument shows is rather banal.) Strictly speaking, the same considerations that show that 'natural number' is at best determinate up to isomorphism also show that 'ω-sequence' is at best determinate up to an isomorphism of the theory of linear orderings (or of set theory, if you take linear orderings to be by definition sets of a certain kind). That is, any system of abstract objects with the same structure as the system of all linear orderings (or of all sets) is an alternative candidate for satisfying the predicates of the theory of linear orderings (or of set theory), so for this reason it is guaranteed to be indeterminate which abstract objects satisfy the predicate 'ω-sequence'. Similarly, Benacerrafian considerations show that the predicate 'finite' is at best determinate up to an isomorphism of the structure of sets. Still, the quantifier 'only finitely many' is not shown indeterminate by Benacerraf-type considerations. What I am really interested in is the issue of its determinacy; this is equivalent to the determinacy up to isomorphism of the predicates 'finite', 'ω-sequence', and 'natural number'. For ease of exposition, though, I will be careless about these matters in the text. end p.263
determinate. My question is, are any of these notions sufficiently determinate to make the notion of an ω-sequence determinate? To answer this, we need to ask what facts there are about the use of these mathematical notions that could serve to determine their referents, that is, their contributions to the truth conditions of the sentences that contain them. The only obvious answer is that their referents are somehow determined by their conceptual roles; and their conceptual roles would seem to be determined entirely by the most comprehensive theory of pure mathematics involving these concepts that we have adopted. But if that is so, it would seem that a substantial indeterminacy in these concepts is inevitable. For it is well known that any attempt to axiomatically characterize the notion of an ω-sequence (together perhaps with associated mathematical concepts like finiteness, natural number, set, and function) has nonstandard models. 7 In such a nonstandard model, the things that count as ω-sequences start out like 'genuine' ω-sequences—a 0 , a 1 , a 2 , and so on; but after all of these 'standard members' with only finitely many predecessors, they have additional 'nonstandard members' with infinitely many predecessors. Nonetheless the sentence that appears to say that each member of the sequence has only finitely many predecessors comes out true in the model, for the model is nonstandard for 'finite' as well as for 'ω-sequence': quantifications of the form 'for only finitely many x, A(x)' come out true even though infinitely many objects in the model satisfy 'A(x)'. Similarly, the sentence that says that the members of the sequence are ordered like the natural numbers also comes out true, for in these models the 'natural numbers' have the same strange order.8 I have said that these models are 'nonstandard'. To call them nonstandard is to say that the common sense view of things is that they are 'unintended' models. But on the nondisquotationalist assumptions we have been making, it
7 I assume that the axiomatization must be expressed in a compact logic, that is, a logic that is like first order logic in
that the only way that a set of sentences can be inconsistent is for some finite subset of it to be inconsistent. This seems reasonable: a determinate grasp of a non-compact logic would presumably have to involve a determinate grasp of proofs which rely essentially on infinitely many premises, and such grasp seems impossible. Given compactness, we don't even need to assume that the 'axiomatization' is recursively enumerable: any set of sentences, no matter how computationally complex, will admit the nonstandard models if it admits the standard ones. 8 The argument that any theory of finiteness, natural number, etc. will have such nonstandard models is simple enough to be worth stating for those not familiar with it. Let T be a theory expressing all of our assumptions about finiteness. It may contain other mathematical vocabulary such as 'natural number', 'function', and 'set'; I assume only that it is expressed in a compact logic. (See preceding footnote.) If T is minimally adequate, then it will be consistent with T that there are infinitely many objects ordered like the natural numbers: a 0 , a 1 , a 2 , a 3 , . . . in that order, with each object in the ordering having only finitely many predecessors. Let T+ be the result of adding to T all truths about such an ordering that are expressible in the language, including the claim that everything in the ordering has only finitely many predecessors; by assumption, T+ is satisfiable by the standard sequence a 0 , a 1 , a 2 , a 3 , . . . But now add a new name 'c' to the language of T+ , and expand T+ to a larger theory T* by adding each sentence 'There are at least n predecessors of c', where n is any ordinary numeral. Each finite subset T*0 of T* is obviously consistent, since given a finite subset T*0 , one can make it true of the natural numbers by interpreting 'c' as standing for any number larger than all of the n for which 'There are at least n predecessors of c' occurs in T*0 . By compactness, the whole of T* must then be consistent. But that means that it is consistent to simultaneously assume T+ and assume the existence of an object c with at least n predecessors for each n; that is, with what we would ordinarily regard as infinitely many predecessors. end p.264
only makes sense to think of these models as 'unintended' if we think there could be some account of in virtue of what they are unintended: an account of what it is about our practice with 'ω-sequence', 'finite', 'natural number', 'set', and so forth that rules these models out. And the fact that these models make true every sentence we accept that contains these concepts seems to preclude such an account. 9 So it seems at first blush as if these models can't be 'unintended' in any clear sense, in which case the notion of an ω-sequence and the concept of finiteness are highly indeterminate. 10 The conclusion that nonstandard models of finiteness and ω-sequence and related notions are in no clear sense 'unintended', and hence that these notions are highly indeterminate, is intuitively a highly surprising one. Most of us are inclined to resist it. But, I repeat: on the nondisquotationalist assumptions we have been making, the only way to do this is to find facts about the use of our mathematical notions like 'ω-sequence' and 'finite' and 'natural number' and 'set' that make these models unintended. In my view, the problem of finding facts about usage that rule out these models is completely insoluble if we look only to the uses of these notions within pure mathematics. However, it is also my view that the problem can be solved by consideration of the uses of these notions in connection with the physical world. That's the good news. The bad news is that this solution depends on somewhat controversial empirical assumptions about the physical world; so that if we are 'unlucky' and these empirical assumptions are false, then 'finite' and 'ω-sequence' do indeed have the high degree of indeterminacy that the above argument suggests. (And as I will argue later, that would probably lead to even more indeterminacy in ordinary logical concepts like 'not' and 'there is' than anything contemplated in the previous section.) Let's look first at the good news. My claim is that the empirical uses of mathematics can make notions like 'finite' and 'ω-sequence' determinate, even though the purely mathematical uses fail to do so. At first blush this idea may seem to be a nonstarter. For if the problem with using mathematics to eliminate the indeterminacy was the existence of nonstandard models, this problem arises in the empirical case too: even if we include empirical predicates in our overall theories, those theories will still have nonstandard models (models that give everything we say the same truth value it has on standard models, but give an intuitively bizarre interpretation to 'finite'). 9 We could even add a complete (and therefore non-axiomatizable) set of sentences that goes beyond what we
accept: if what we add is true of the standard model it is true of some nonstandard models. 10 It would of course be absurd to hold that when we use the terms 'ω-sequence' and 'finite' and 'natural number', we
intend to use these terms to have the extensions they have in the nonstandard models. But that isn't the issue: the issue is whether we have succeeded in intending anything definite at all. As usual when contemplating radical cases of indeterminacy, we must use the very indeterminate term in question to specify the nonstandard interpretations of it. But that's no problem, because an indeterminacy argument can be conceived as a kind of reductio: on the assumption that the term in question has one referent, we can show that another is an equally good candidate for the referent. end p.265
But this argument overlooks the fact that the existence of nonstandard models in this sense is not intrinsically a sign of indeterminacy, it is a sign of indeterminacy only when there is no prospect of specifying the facts about the use of words in virtue of which the nonstandard models could count as 'unintended'. My claim is that certain empirical notions are included in our overall theory of finiteness, and that this allows us to show that on certain empirical assumptions, the 'unintendedness' of the nonstandard models of this overall theory will be explainable. The general strategy should be clear. I have assumed throughout the paper that empirical notions like 'red' and 'cat' and 'longer than' are far from being wholly indeterminate: 'red' definitely fails to be true of green things, and 'cat' definitely fails to be true of cherries. This seems a reasonable assumption: for it is natural to suppose that our practice of accepting and rejecting sentences containing such predicates determines that paradigm red things are in the extension of 'red' but paradigm green things aren't; and that supposition is natural because our practice with words like 'red' includes not only the acceptance of general theoretical principles, but includes also an observational practice which causally ties the extension down. Other physical predicates, say 'neutrino', are less tied to observational practice, but the theoretical principles governing them include words that are more tied to observational practice, and this does a lot to fix their extension. We can of course come up with grossly nonstandard models in which 'red' is true of green things and 'neutrino' of billiard balls, but that creates no particular philosophical problem because we can point to features of our practice with these words that explains why these models are 'unintended'. If there really is a problem with the mathematical notions, it is that we can't do the analogous thing in those cases. My solution is to argue that we can in fact do the analogous thing in the finiteness or ω-sequences case, if we are 'empirically lucky'. (Even with 'empirical luck', we won't be able to tie down the extensions of 'set' and 'member' very much, but we can tie them down enough so that when 'finite' and 'ωsequence' and 'natural number' are defined in terms of them those defined notions will be determinate (up to isomorphism).) More fully, the idea is that what determines the acceptable referents of mathematical terms like 'ω-sequence' and 'finite' is indeed the mathematical theory we accept; but that this theory includes impure set theory, that is, set theory that allows for sets of physical objects and (more importantly) allows for the use of physical vocabulary in specifying sets (e.g. 'the set of all objects with mass more than 20 grams'). And since the physical vocabulary has its reference fairly well tied down (directly or indirectly) through our observational practices, the question is going to be whether there are 'physically standard' models of our impure set theory that make 'finite' and 'ω-sequence' nonstandard. (By a 'physically standard' model I mean one not ruled out by the constraints on our physical vocabulary.) My claim is that if we are empirically lucky, there are no such 'bad' models of impure set theory: no models of it that are physically standard but nonstandard with respect to 'finite' and 'ω-sequence'. Our end p.266
acceptance of impure set theory will then serve to rule out nonstandard extensions for 'finite' and 'ω-sequence'. 11 There are a number of basically equivalent ways to work out the details of this. A simple method 12 is based on the observation (easily verified) that if a model of (pure or impure) set theory contains at least one genuine ω-sequence, then it must be standard with respect to 'finite' and 'ω-sequence': that is, all and only the genuine ω-sequences in the model must satisfy the formula 'is an ω-sequence', and all and only the genuinely finite sets in the model must satisfy 'finite'. 13 So all we need to do is argue that with empirical luck, each physically standard model of impure set theory must contain at least one genuine ω-sequence. What the 'empirical luck' requirement amounts to is basically this: that our language contains a determinate description of a physical ω-sequence (determinate both as to which objects are included in it and how they are ordered). If our language contains such a determinate description, then any physically standard model of impure set theory is bound to have that ω-sequence in the model (it is bound to contain the set of objects in the physical ω-sequence and the physically-described ordering of them). To illustrate, suppose space is infinite in each direction, and that it is Archimedean (each bounded interval is of finite length). Then we could determinately pick out an ω-sequence as follows: we single out a particular point P of physical space and a particular ray extending from it in one direction (call it 'eastward'); then there is a unique set of points that (i) contains only points on the ray, (ii) contains P, and (iii) is such that whenever it contains any point x, it contains a point one mile east of x but no points in between. This set, ordered west to east, is to be our ω-sequence. If physical space is infinite and Archimedean, this description D determinately singles out a genuine ω-sequence: there isn't enough physical space for there to be a non-ω model of it (or of a set theory that contains it) that is physically standard. There are of course non-ω models of description D that aren't physically standard: for instance, you can get one by interpreting the physical predicate 'one mile apart' to apply to successive pairs of a converging series of points (thereby leaving plenty of room for nonstandard points), or by interpreting the physical predicates 'point of space' and 'one mile apart' as having natural numbers and pairs of natural numbers in their extension (so that 'room' requirements simply don't arise). But we can explain why these nonstandard models are 11 In using the notion of model in this way, I presuppose that the standard operators of first order logic determinately
contribute to truth conditions (in the way given in the definition of truth in a model). That the determinacy of the first
order logical operators must be presupposed is to be expected: we can hardly expect to argue for more determinacy in a complicated quantifier like 'only finitely many' than in the standard operators of first order logic. (To allow for the possibility that our first order logical vocabulary is indeterminate, one could say that what is being argued here, on the assumption of empirical luck, is that relative to each candidate for the 'referents' of the first order logical vocabulary there is only one 'referent' for the finiteness quantifier.) 12 For a superficially different approach, see Ch. 12 (written earlier than the present Chapter). 13 I'm putting this slightly loosely: see the 2nd and 3rd paras. of fn. 6. (E.g., what it would pin us down to is not actually the genuinely finite sets, but the objects x in the model for which there are only finitely many things y in the model such that
'unintended', if as I have assumed our observational practices (and perhaps other causal facts) explain how our physical vocabulary applies to all and only what it does. That is why these models can be excluded from consideration as 'not physically standard'. It might be objected that we can't determinately single out a specific point, or a specific direction, since our means of singling out points of space all suffer from imprecision. True enough, but the argument is easily generalized to avoid the difficulty: after all, each candidate for the ω-sequence we have described is still a genuine ω-sequence, if space is infinite and Archimedean, and this is enough for the basic line of argument to go through. Similarly, the argument really isn't affected by the fact that predicates like 'one mile apart' are somewhat indeterminate; they would have to have a huge and grossly implausible degree of indeterminacy for a nonstandard ω-sequence of physical objects to be a candidate for the referent of description D. It is of course merely an empirical assumption that space is infinite and Archimedean. But even if this assumption is false, all is not lost: perhaps time is infinite and Archimedean, in which case it will serve to yield the determinate characterization of an ω-sequence required to run the above argument. (It's worth noting that the natural intuitive explanations of finiteness assume that time is infinite and Archimedean. For instance, 'a finite set is one that could be counted (completely, and at a set rate, say one per second) if we went on long enough': if time were non-Archimedean we could count infinite sets in this way, and if it were finite in extent we couldn't count all finite sets in that way.14 ) And there are many other possibilities for determinately singling out a physical ω-sequence (or rather, singling one out in a sufficiently determinate manner: see the preceding paragraph). Note also that for the argument to work it is not required that we know of a determinate description of a physical ω-sequence: because of our acceptance of impure set theory (and of principles that connect up set-theoretic notions with the notions of finiteness and ω-sequence and natural number), the mere fact that the language contains a determinate description of a physical ω-sequence serves to pin down our notion of set sufficiently to make 'finite' and 'ω-sequence' and 'natural number' determinate (up to isomorphism). Still, it is not a priori obvious that any physical ω-sequences can be singled out in a (sufficiently) determinate way. For instance, if the physical universe is finite in the large and small, there are no physical ω-sequences at all, hence certainly none that can be determinately singled out. And that brings us to the bad news: in that eventuality, my line of argument suggests that there is nothing whatever about our use of the term 'finite' or of 'natural number' or 'set' or any other mathematical terms that could serve to make the nonstandard 14 Unless 'could be counted' allows the consideration of non-actual time structures; but then the problem about the
possibility of counting infinite sets is worsened. (A slightly different explanation in terms of counting, that allows counting to go arbitrarily fast as long as it maintains the same rate throughout the counting, allows arbitrary big finite sets to count as finite even if time is finite in extent, as long as time is densely ordered. But this too worsens the problem of infinite sets counting as finite: that could now happen if there were infinitesimally close points of time.) end p.268
models of finiteness (or of ω-sequences) 'unintended'. 'Finite' and 'ω-sequence' would be indeterminate, in a very surprising way. And that would pose a further problem for the determinateness of ordinary logical notions like 'not' and 'there is'. For I've argued that as long as we continue to put aside the disquotationalist view, we must take seriously the demand to explain in virtue of what our concepts contribute to truth conditions in the way that they do. To the extent that we think this unexplainable in principle, we should conclude that the concepts are indeterminate. And in the case of ordinary logical concepts, the only obvious way to even hope to explain their contribution to truth conditions is in terms of their inferential role. But now the worry is that the notion of inferential role will depend on the notion of finiteness. For isn't an inference a finite string of sentences, each related to some of its predecessors by certain rules? If so, then if it is indeterminate which strings are finite, it will presumably be indeterminate which strings are inferences; and then in some cases it may be indeterminate if there is a possible inference from a set of sentences Γ to a sentence B. (It may
even be indeterminate which sets of sentences imply contradictions.) And that presumably means that the notion of inferential role becomes indeterminate, in a way beyond any that was considered before. (Indeed there is a more basic problem: sentences themselves are certain finite strings of alphabet symbols; if finiteness is indeterminate it becomes indeterminate what strings are sentences.) I don't say that it is completely clear that this problem arises for any possible notion of inferential role; nor that the problems that arise this way really go beyond the problem considered earlier of how inferential role is determined by brute facts about usage; nor that if there is an additional problem, it gives rise to a level of indeterminacy that is impossible to live with. All I really want to claim is that there is a worry here which deserves serious discussion.
IV I have suggested that a nondisquotational approach to truth and reference makes a high degree of indeterminacy in our terms fairly likely (even if 'empirical luck' enables us to avoid the massive indeterminacy contemplated in the previous section). And though I have tried to minimize the extent to which this conclusion should be upsetting, I have no doubt that many will find it so. This raises the question of whether we might avoid the conclusion if we thought of truth and reference disquotationally. Attractive though this suggestion is, I think that ultimately the introduction of disquotational truth and reference doesn't change much. To view truth and reference disquotationally is to view them as not really semantic notions at all, especially in their most central applications. The most central applications of 'true' and 'refers' (according to the disquotationalist) are to our own idiolect; and in these central applications they function as end p.269
logical terms. Ignoring certain complications involving ambiguity and indexicality and the like, we can say the following: the claim Some sentences of our language that are F are true is in effect simply an infinite disjunction: it is the disjunction of all sentences of our language of the form F('p') and p.15 (To get the intuitive sense of this, instantiate 'F' with 'will never be rationally believable'.) In particular, S is true (since it is equivalent to 'Some sentence of our language that is identical to S is true') 16 is equivalent to the disjunction of all sentences of the form 'p' = S and p; and hence anything of the form 'q' is true is equivalent to the corresponding q. Reference, both for predicates (truth of) and for singular terms, is similar; for singular terms, we have that Some terms of our language that are F refer to x is in effect the disjunction of all formulas of our language of the form F('t') and t = x; which implies as before that any formula of form 's' refers to x is equivalent to the corresponding formula s = x. In general, truth and reference (in their most central senses) are devices for expressing certain otherwise inexpressible disjunctions (and conjunctions: see footnote 15 ). So viewed, there is no room for an interesting account of in virtue of what a given term refers to a given object: the claims that 'rabbit' refers to all and only the 15 In addition,
Some sentences that are F are untrue is in effect the disjunction of all sentences of the form F('p') and ¬p; equivalently, All sentences that are F are true
is in effect the conjunction of all sentences of the form If F('p') then p. 16 Here I use the fact that the notion of truth in question applies only to sentences of our own language. 15 In addition,
Some sentences that are F are untrue is in effect the disjunction of all sentences of the form F('p') and ¬p; equivalently, All sentences that are F are true is in effect the conjunction of all sentences of the form If F('p') then p. end p.270
rabbits, and that 'entropy' refers to something if and only if it is entropy, are simply logical truths, when 'refers' is understood disquotationally. It might be thought that if the fundamental notion of reference is disquotational there is simply no room for indeterminacy, but this would be a mistake. For there is certainly room for indeterminacy of translation; and this gives rise to something reasonably called indeterminacy of reference, albeit not of purely disquotational reference. Let's consider a simple example (which is adapted from Brandom 1996). Imagine that a community of English speakers was separated from the rest of us prior to the development of the theory of complex numbers, and that they independently developed that theory. However, they developed different symbols than ours for the square roots of −1: instead of calling them 'i' and '−i', they call them '/' and '\'. (Of course they know that / is −\ and hence that \ is −/, but still they use both symbols.) This presents a problem of translation: which of their two symbols should we equate with our symbol 'i' and which with our '−i'? It seems clear that there is no right answer here, for (unlike 1 and −1) the numbers i and −i are structurally identical. That is, whereas 1 differs structurally from −1, for instance in being its own square, no such difference distinguishes i and −i: if you take any mathematical sentence whatever and substitute '−i' for 'i' in all occurrences, the resulting sentence has the same truth value as the original. Because of this complete mathematical symmetry between i and −i, it is hard to see how any possible facts about the mathematical behavior of the other community could give reason for preferring one translation over the other. 17 And it seems clearly incorrect to think that this is purely an epistemological limitation: it isn't that there is a subtle fact as to 'the correct translation' that we can never know, it is that there is simply no fact of the matter: the whole idea of a unique 'correct translation' is misconceived. 18 I have presented this example as an example of indeterminacy not of reference but of translation: there are two pairings of the other community's expressions with ours that equally well preserve stimulus meanings, inferential roles, indication relations and suchlike. Suppose now that we bring in the idea of disquotational reference. Then (assuming we accept the existence of mathematical objects) we know that 'i' disquotationally refers to i and '−i' to −i. How about '\' and '/'? Well, prior to our translating these terms (and hence in effect incorporating them into our own idiolect), they don't disquotationally refer to anything, since the notion of disquotational reference applies only to terms of our own idiolect. Still, we can say that '/' disquotationally refers to i relative to 17 If the other community has not only developed complex number theory but started to apply it to the physical world
—e.g. to electromagnetic phenomena—then the argument becomes a bit more complicated, but the conclusion is not altered. 18 Incidentally, this example seems to me to create a serious problem for those versions of mathematical structuralism according to which mathematical terms have determinate reference, but to positions in structures instead of to objects of a more usual sort. That view is possible for ordinary numerals, since they purport to stand for objects with objectively different positions in their associated structure; but for 'i' there is an indeterminacy even as to which position is denoted. And the failure of the view for 'i' reduces its attractiveness for ordinary numerals. Several other forms of structuralism remain viable: see Ch. 11, sect. 6. end p.271
one of the adequate translation schemes, and disquotationally refers to −i relative to the other one. From a disquotational point of view, 'reference' for terms not in our idiolect can only mean disquotational reference relative to an adequate translation scheme; so the reference of '/' and '\" is indeterminate. But then how about our own terms 'i' and '−i'? Aren't they objectively just like '/' and '\'? If so, it would seem that our own terms must be referentially indeterminate too; but how could they be, if reference in our own language is
disquotational? Quine addressed this dilemma, in Quine 1968a. His attempt to solve it (involving the idea that reference is 'relative to a background language' as well as to a translation manual, and involving an analogy between reference and location as location is conceived on the relational theory of space) seems to me quite incoherent if taken literally (for the reasons I gave in Chapter 7). But I now think that Quine was gesturing at something perfectly intelligible. One way to put his point is that it is perfectly possible to 'look at ourselves from the outside' and ask about ways of 'translating' our language nonhomophonically into itself. (That is, using a translation other than the identity translation.) We often do this in certain counterfactual contexts. If we had used the word 'white' as we now use 'purple', would the sentence 'Snow is white' have been true? Well, it would have been disquotationally true, for that just means that snow would have been white; but it would not have been true relative to the appropriate translation of the counterfactualEnglish into our actual English. But even outside counterfactual contexts, there is nothing to keep us from considering not only pure (unrelativized) disquotational truth and reference but also disquotational truth and reference relative to nonhomophonic translation schemes. And to the extent that there are adequate nonhomophonic translation schemes, we have an indeterminacy of (not purely disquotational) reference in our own language, quite analogous to what we have in other languages. The translation scheme that takes 'i' to '−i' is one such adequate scheme. The use of the notion of translation here may be misleading: normally we think of a translation of an expression as an aid in understanding it, and nonhomophonic translations of our own language don't serve this role. But the point can be put differently. The point is that on a disquotational view, the important structure that language has isn't to be conceived of in referential terms, but in terms of such things as stimulus meanings and inferential role and indication relations. Everything of objective semantic importance in language is a reflection of such things. For that reason, any mapping of a language into itself that leaves such things fixed preserves everything of objective semantic importance. Since the mapping of 'i' to '−i' leaves such things fixed, there is no objective semantic difference between them. They do of course differ in their disquotational reference, or their contribution to disquotational truth conditions; but as I said from the start, disquotational truth and reference aren't really semantic notions, they are logical notions. We see then that disquotationalism allows for semantic indeterminancy in one's own language. It allows for the reflection that even though 'i' disquotationally end p.272
refers to i, still this fact floats free of all that is semantically important, viz. stimulus meaning and inferential role and indication relations and the like. We could express this by saying that even though 'i' refers to i and not to −i, all that is determinate is that it refers to one or the other of the square roots of −1. The complex number example is an especially simple one in many ways. For one thing, it gives rise to no indeterminacy in truth value. (It is thus a case of what the Quine literature calls 'inscrutability of reference' as opposed to full-fledged indeterminacy of translation.) For another thing, the example is one where we have in our language the means to fully express the nonhomophonic translation on which the ascription of indeterminacy is based. In more complicated examples of indeterminacy this will often not be so; instead, the translation will only be expressible relative to a parameter. For instance, in the case of 'finite', what the nonstandardness argument tells us is that there are weird models M of set theory such that 'is in the extension of 'finite' in M' serves as a "translation" of 'finite' that preserves the inferential role in pure mathematics of everything we say. (And it serves the inferential role outside mathematics too, if finiteness isn't physically characterizable.) But 'is in the extension of 'finite' in M' is a parameterized expression: what we are doing is "translating" the 1-place predicate 'finite' with the 2-place predicate 'is in the extension of 'finite' in x' together with the instructions to fix the value of x on a model M with the needed characteristics. Of course if we could fully describe a relevant M then we could use the description to get an ordinary 1-place translation; but we may be unable to describe such an M in complete detail, but simply prove that at least one with the desired properties exists. But as far as I can see, the fact that we need to use parameters in expressing the translation is of little philosophical significance. Parameterized translations are natural things for disquotationalists to appeal to in other contexts too (for instance, in allowing us to speak in a natural way of truth for languages that aren't translatable into ours except with parameters). And it would be easy to expand our language to include new names (themselves having indeterminate reference) corresponding to the parameters—e.g. we could simply introduce a new name c stipulated to be a name for a model with the needed characteristics—in which case 'finite' would be translated without parameters as 'is in the extension of "finite" in c'. It seems evident that if you can argue for a philosophically significant indeterminacy in the language that contains such a name 'c', you can equally argue for it in the language without such a name. The need of parameters in translations simply doesn't matter. I have argued that disquotationalism allows for the possibility of significant indeterminacy in one's own language; and though I originally argued for this in the case of a simple example where the indeterminacy is based on a nonhomophonic mapping of the language to itself that does not make use of parameters, I have just argued that the case where we need parameters is not philosophically different. If what I have been arguing is correct, then it is hard to see why the extent of indeterminacy should be a whole lot less on a
end p.273
disquotational view of truth and reference than it is on a nondisquotational view. Of course, disquotationalism allows us to 'tame' indeterminacy by letting us use the standard disquotation schemas. But I argued earlier on that a nondisquotationalist too can accept the usual disquotation schemas in the face of indeterminacy, by taking 'true' and 'refers' to have a kind of indeterminacy correlative with that of the nonsemantic predicates of one's language. The conclusion seems to be that a disquotational approach to truth and reference, attractive though it may be, doesn't fundamentally alter either the possibility or the import of radical indeterminacy. 19 19 I am grateful to Ned Block and Stephen Schiffer for comments on an earlier draft that led to improvements. end p.274
Postscript 1. My argument that we should take seriously the possibility of indeterminancy in our logical connectives flies in the face of an argument by Vann McGee (2000) against any such indeterminancy. In the case of the material conditional, McGee argues as follows: the material conditional is governed by two rules, an introduction rule and an elimination rule. But any two connectives 1 and 2 in our language governed by these rules are provably equivalent. The argument for their equivalence is simple: A, A 1 B B, by 1 -elimination, so A 1 B
A 2 B, by
2 -introduction;
and the converse analogously. I think it is correct that we can't maintain two distinct conditionals in our language, both governed by unrestricted introduction and -elimination. (By 'unrestricted' I mean that they apply even to sentences containing the other conditional.) But from this we can not infer either of two further conclusions: (i) that the fact that a person employs a connective called ' ' that is subject to both rules is sufficient for us to conclude that he means the same by ' ' as we do. (ii) that our word ' ' is determinate. (McGee's claim is (ii). (i) is a claim that might be used in an attempt to bridge the gap between what was proved in the previous paragraph and (ii).) (i) in fact seems clearly false, for the advocate of intuitionistic logic and classical logic both accept -introduction and elimination, but their concepts of differ. (E.g., the inference from (p q) p to p is valid classically but not intuitionistically.) The argument above shows that if we were to combine classical and intuitionistic connectives within a single language we would have to restrict the application of at least one of the rules for one connective as applied to sentences containing the other, but that doesn't show that their unrestricted use by people who have only one of the concepts fully determines the meaning of the concept. That (ii) doesn't follow from what has been proved can be seen by considering a person less logically sophisticated than the advocate of either classical logic or intuitionistic logic. Let's suppose that this person reasons in accordance with -introduction and -elimination; but that he has not reflected on the validity of inferences like those above that would distinguish classical from intuitionist . Suppose in addition that he does not have any concept of end p.275
negation. (If he had a concept of negation that obeyed classical laws, the -rules applied to sentences containing it would imply that his ' ' was not intuitionistic.) Let us also suppose that there is nothing in his informal explanations of the conditional, or of associated notions such as assertion, that settle whether his concepts are intuitionistic or classical. This seems like a clear case of indeterminacy. Of course, our own concept of the material conditional is not nearly as indeterminate as this person's. Maybe it isn't indeterminate at all. But my point has simply been that its determinacy does not follow from the acceptance of the rules, despite the proof above. 2. The attempt in the last section of the chapter to use nonhomophonic translations of a language into itself to make sense of indeterminacy in the home language (on a disquotational view) was not well thought out, and I doubt that sense can be made of it. As I noted, talk of 'translation' here needs a special gloss: we don't need to translate our own language to understand it. To gloss it, I noted that on a disquotational view the important structure that language has isn't to be conceived of in referential terms, but in terms of such things as stimulus meaning and inferential role and indication relations. . . . Any mapping
of the language into itself that leaves such things fixed preserves everything of objective semantic importance. Since [a mapping of the language that differs from the identity only by taking 'i' to '−i'] leaves such things fixed, there is no objective semantic difference between these terms. This seems correct, but what exactly does it have to do with indeterminacy? It does connect to indeterminacy on a nondisquotational view according to which the reference of terms is determined by the factors of objective semantic importance (say, stimulus meanings, inferential role, and indication relations): if they can't make a difference between referring to i and referring to −i, nothing can, and so reference is objectively indeterminate. But what was under consideration was a disquotationalist view. I now think that the way to make sense of indeterminacy in the home language on a disquotational view is very different: see the next chapter. And while this new way to make sense of indeterminacy would allow us to make sense of even the very radical indeterminacies (in our logical constants and in number theoretic concepts) that are contemplated in sections ii and iii of this chapter, I do not think that the argument for the radical indeterminacies is nearly as compelling from the disquotational perspective as it is from the nondisquotational. So I now prefer to think of this chapter as a challenge to an orthodox theorist who wants both (i) to reject a disquotational account of truth and reference, and (ii) to believe in the determinacy of logical and number theoretic concepts without basing the latter belief on such empirical assumptions as that the universe is not finite both in the large and small. Indeed, there is a challenge even to the antidisquotationalist who is willing to base his belief in determinacy on such cosmological assumptions: for the cosmological assumptions are relevant only to the indeterminacy argument of end p.276
section iii, they don't help with the worries about indeterminacy raised in section ii. 1 1 Section ii directly raises a problem only for the determinacy of the logical constants, not for the determinacy of
number theory. But if the logical constants are not assumed determinate, even the cosmological assumptions don't guarantee the determinacy of number theory: the model-theoretic argument of section iii presupposes the determinacy of the logical constants. end p.277
10 Indeterminacy, Degree of Belief, and Excluded Middle Abstract: Discusses the question of how to reconcile the acceptance of indeterminacy in one's own language with the acceptance of a minimal notion of truth. Argues that regarding a sentence of one's own language as indeterminate involves adopting non-standard laws of thought for it: not necessarily a non-standard logic, but non-standard degrees of belief that do not obey the laws of classical probability. (A postscript gives an alternative, somewhat similar in spirit, where the laws of logic are revised as well.) The view is compared to a recent suggestion by Stephen Leeds, that we not recognize any indeterminacy in our own language other than indeterminacy as to how to translate it into a privileged sub-part of the language, the vocabulary we take seriously even in our most serious theorizing. Keywords: continuum hypothesis, degree of belief, excluded middle, indeterminacy, Stephen Leeds, probability, supervaluation, vagueness
Hartry Field
I Referential indeterminacy (for instance, indeterminacy as to what a singular term stands for or what a general term has as its extension) is a widespread phenomenon. Ordinary vagueness is a special case of indeterminacy: for instance, it is indeterminate whether the word 'rich' has in its extension certain moderately rich people,1 and it is indeterminate precisely which atoms are parts of the referent of 'Clinton's body' at a certain time. But there are more interesting examples as well. One kind of example arises in the context of belief in false theories. Consider the use of the term 'heavier than' by preNewtonians. Did it stand for the relation of having greater mass than, or for the relation of having greater weight than? In pre-Newtonian physics there was no distinction between the weight of an object and its mass; and since the term 'heavier than' was applied almost exclusively in the context of objects at the surface of the earth where there is a nearperfect correlation between mass and weight, there is little in the pre-Newtonian use of the term that could have settled the matter. Some pre-Newtonian utterances may to some extent favor one interpretation over the other; unfortunately, they were probably about equally matched in importance by utterances favoring the other interpretation. I suppose we
could say that the term didn't stand for either the relation more weighty than or the relation more massive than. But then what did it stand for? There is no third relation that is a better candidate for what their term stood for; and it seems unsatisfactory to say that it didn't stand for anything at all: that would seem to imply that they never said anything true when they said that one thing was heavier than another, which is hard to swallow. The best conclusion seems to be that their term 'sort of stood for' each of the two relations, but didn't determinately stand for either. This means for instance that if A is an object on the moon that is more massive than but less weighty than an object B on earth, there is no determinate fact of the matter as to whether A is heavier than B, on 1 Here and in what follows I ignore the fact that words like 'rich' are highly context-sensitive; I will imagine a context of
use that settles many questions about who is rich. But any realistic such context will still leave many questions about who is rich unsettled. end p.278
their use of 'heavier than'. (So if we want referential indeterminacy in a singular term, 'the heavier of A and B' is an example.) On the other hand, if A is both more massive than and more weighty than B, then we should regard it as determinate that A is heavier than B on their use of 'heavier than'. An interesting feature of this example is that just as Newton in effect discovered that 'heavier than' was indeterminate between being more massive and being more weighty, Einstein in effect discovered that 'is more massive than' is itself indeterminate, between having more rest mass and having more momentum per unit velocity. And for all we know, future physicists may find distinctions that we miss, giving rise to indeterminacy that we can't yet be aware of. Or so it would seem. But Steve Leeds (1997, section IV), while granting that it makes sense to ascribe indeterminacy to terms in earlier theories, has denied that it makes sense to ascribe indeterminacy to our own terms. The underlying rationale for this seems to be a disquotational view of reference, on which reference for our own singular terms is pretty much defined by the schema (R) If b exists then 'b' refers to b and to nothing else (and analogously for general terms, etc.); talk of reference for other people's terms makes sense only relative to a correlation of their terms to ours, and in cases like the pre-Newtonian 'heavier than' there is no best translation to use.2 But whatever the rationale, the view is prima facie surprising, among other things because there are circumstances where it seems quite reasonable to suspect indeterminacy in specific terms in our currently best theory. A possible example: it seems to be generally believed that the various tensor fields that Einstein introduced into gravitational theory make no physical sense on a sufficiently small scale and that the quantum gravitation theory of the future will have to replace them; but as far as I know our best theories of gravitation today still employ them, because we don't yet have a clear enough idea of what the appropriate replacement terminology might be. This is the sort of circumstance where indeterminacy may be suspected: terms like 'the Ricci tensor' are unlikely to straightforwardly refer, but are also unlikely to be straightforwardly denotationless. Of course examples like this depend on the fact that we know specific ways in which our currently best theory is problematic; but if you grant that it makes sense to suspect indeterminacy in those cases, I don't see why you shouldn't grant that it makes sense to suppose it in other cases where we have no such specific knowledge. For surely all of us would concede the possibility that our current theory is false, even if (contrary to fact) we had no 2 A more moderate view would be that we can have an unrelativized notion of reference extending beyond our terms,
it will just be highly indeterminate. But the more moderate view is not available to one who wants to deny indeterminacy in our own terms, for it postulates an indeterminacy in our term 'refers'; that is why I have tentatively ascribed to Leeds the view that 'refers' is inapplicable to other terms except by relativization. Note that this less moderate view requires that we have no notion of inter-theoretic synonymy that could be used to break the relativization: inter-theoretic synonymy is not merely indeterminate, it makes no sense. (Indeterminacy of translation would be allowed, but it would have to be viewed as indeterminacy of how to translate, not as indeterminacy of 'the synonymy relation'.) end p.279
specific knowledge of its being problematic in certain ways; and examples like 'heaviness' and 'mass' make clear that often when a theory is false, some of its terms are indeterminate. A different kind of example is the one from Brandom 1996 discussed in the previous chapter (p. 271): the community of mathematicians whose symbols for the square roots of −1 are '/' and '\', in contrast to the symbols 'i' and '−i' that we were taught to use. This gives rise to a problem of translation: which of their two symbols should we translate as 'i' and which as '−i'? It seems clear that there is no right answer here, for (unlike 1 and −1) the numbers i and −i are structurally identical: if you take any mathematical sentence whatever and substitute '−i' for 'i' in all occurrences, the resulting sentence has the same truth value as the original. Because of the complete mathematical symmetry between
i and −i, it is hard to see how any possible facts about the mathematical behavior of the other community could give reason for preferring one translation over the other. And it seems absurd to suppose that this is purely an epistemological limitation. It isn't that there is a subtle fact as to 'the correct translation' that we can never know, it is that there is simply no determinate fact of the matter: the whole idea of a unique 'correct translation' is misconceived. I have presented this example in the first instance in terms of translation, but (assuming platonism about mathematics, as I shall) we could equally present it as a problem about reference for terms in the other language: there seems to be no determinate fact of the matter as to which square root of −1 the term '/' refers to. 3 Of course, their typical utterances (for instance, '/ 2 = −1') will come out true under either assignment, hence determinately true. This example is of interest in connection with Leeds' position, because it certainly seems at first glance that if the foreign terms '/' and '\' are referentially indeterminate, then so is our own term 'i': wouldn't it be grossly chauvinistic to suppose that we have the ability to determinately single out one of the square roots of −1, but no one else can have this ability? The most obvious way to evade the charge of chauvinism is to say that it isn't that our term 'i' is better than their term '/', but simply that our word 'refers' has determinate application in the case of our terms but not in the case of theirs. But that way of avoiding the chauvinism charge is unavailable to anyone who wants to deny indeterminacy in our own language since it assumes an indeterminacy in our word 'refers'. Moreover, we could modify the example to make it even clearer that there is indeterminacy in our own language, by extending our language to include '/' and '\' as well as 'i'. We could now express the indeterminacy 3 A not implausible view for the semantics of the language of natural numbers is that it is completely indeterminate which ω-sequence of abstract objects our terms '0', '1', '2', etc. stand for. If this view is accepted, it presumably extends to complex numbers, so that any abstract object whatever is a candidate for the reference of 'i' when conceived as part of a structure isomorphic to the complex numbers. However, there is a way to oppose the claim of indeterminacy in the natural number case: one can claim that each numeral determinately refers to a position in an ωsequence. What is interesting about the example of the square roots of −1 is that this structuralist move won't work: the position occupied by i is not structurally different from that occupied by −i, so even the move to structures won't restore determinacy. end p.280
at the object level: there is no fact of the matter as to whether / = i. The obvious explanation of there being no fact of the matter is that both '/' (in this extension of our present language) and 'i' are referentially indeterminate (and their indeterminacies are independent of each other). Again, the conclusion that 'i' is indeterminate doesn't conflict with the determinate truth of our mathematical claims, since they come out true relative to either assignment of a root of −1 to 'i'; 4 only 'don't cares' like '/=i' have indeterminate truth value. One worry about allowing indeterminacy in our own language is that this may appear to conflict with the acceptance of such 'disquotation schemas' as (R) If b exists then 'b' refers to b and to nothing else and (T) For any x, 'P' is true of x if and only if P(x); and these principles seem central to our understanding of the notions of reference and being true of. A possible response would be to somehow restrict the application of the schemas to determinate language, but I think that this is not entirely appealing. Perhaps, then, the idea of indeterminacy in our own language should be abandoned? But how can it be, without falling into chauvinism? This is essentially the puzzle about indeterminacy that is raised in Quine 1968a. (Quine's answer to the puzzle is a bit obscure.) Fortunately the choice between abandoning the disquotation schemas and accepting the incoherence of indeterminacy in our own language is unnecessary: there is a very natural account that allows both. The account requires a combination of two ideas.5 The first idea is that indeterminacy can be holistic. For instance, in the '/' and '\' example, the candidates for the reference of the two terms are correlated: since the speakers accept the principle '/ ≠ \', their practices dictate that neither square root of −1 can simultaneously count as / and as \, even though their practices don't dictate which of the square roots of −1 to assign to which term. The second idea is that semantic terms like 'refers' can themselves be indeterminate. How do we combine these ideas in such a way as to make the schema (R) come out true? The idea is simply to suppose that our acceptance of the disquotation schema (R) creates a holistic connection between 'refers' (as applied to our own language) and each of the singular terms of our language. In particular, there is a holistic connection between the interpretations of 'i' and 'refers': any acceptable interpretation that assigns a mathematical object x to the term 'i' assigns to 'refers' a set that includes the pair <'i',x> but doesn't include the pair <'i',y> for any y other than x.
Such a holistic connection between 'refers' and 'i' guarantees that '(If i exists then) "i" refers to i and to nothing else' comes 4 And of course we can avoid the indeterminate language here if we feel compelled to, by replacing talk of i and −i by
sentences of the form 'Whenever x is a root of −1 and y its conjugate, . . . '.
5 Both ideas were articulated in Ch. 7, but I missed the obvious point that they could be combined in a way that
preserves schema (R). That point is spelled out in slightly more detail in Ch. 9. end p.281
out determinately true, and similarly for all other instances of the disquotation schema (R). This solution does what we want: it gives a natural account of how ' "i" refers to i' can be determinately true even though the apparently analogous claim ' "/" refers to i' isn't. The reason for the asymmetry is that in learning to use the term 'refers' we learn to accept (R), and this sets up a connection between the word 'refers' as applied to our term 'i' and our term 'i'; it doesn't set up any connection between 'refers' as applied to '/' and 'i'. We get this asymmetry without chauvinism: our term 'i' is just as indeterminate as the foreign term '/', it's just that the indeterminacy is 'hidden' in our ordinary semantic claims because there is a compensating indeterminacy in our ordinary semantic vocabulary. I believe that this account removes one main worry about positing indeterminacy in our own language. But one should not overestimate what it accomplishes. Earlier I mentioned that Leeds' denial of the possibility of indeterminacy in our own language was probably due to the acceptance of a disquotational view of reference, according to which the notion of reference for our own language is defined by the disquotation schemas. One might think that if that is Leeds' view, then the account just sketched shows that it is confused: Of course disquotationalism doesn't rule out indeterminacy in our terms. If 'i' is indeterminate, then when we define disquotational reference in terms of it and the other terms of our language, we get an indeterminate notion of disquotational reference. The fact that it is determinate that 'i' refers to i doesn't show that 'i' is determinate, it is compatible with 'i' and 'disquotationally refers' both being indeterminate. But it is this imaginary rebuttal of Leeds that would be in error. The part after the first sentence is correct, but it merely shows how a prior indeterminacy in 'i' would give rise to an indeterminacy in 'disquotationally refers'. But presumably Leeds' argument is that there is no way to make sense of the prior indeterminacy in 'i', if we recognize no notion of reference beyond the disquotational. One way around Leeds' conclusion would be to deny the premise that reference must be defined in terms of the disquotation schema. But I will try to show that there is another way around his argument, which makes sense of the indeterminacy in our own language independently of the theory of reference and therefore independently of the issue of whether it is defined disquotationally. Putting disquotationalism aside, the idea that we should deny the existence of indeterminacy in our own language would in any case appear almost hopeless: for surely ordinary vagueness is a kind of indeterminacy, and surely vagueness is ubiquitous? But vagueness itself can seem problematic: indeed, there is a central problem about vagueness, much discussed in recent years, that puts Leeds' worries into sharper focus. I'll call it 'Williamson's puzzle', since he has been a main proponent of it (Williamson 1994). There are people who believe this puzzle to be so serious as to cast doubt on whether the phenomenon end p.282
of vagueness, or of indeterminacy more generally, can be genuine. It will be the subject of the rest of this paper, until the final section when I return to Leeds.
II Williamson's puzzle is that for any question whatever, there is a simple and straightforward argument for the conclusion that it has a determinate, objective, factual answer. Applied to an ordinary vagueness case, the argument goes as follows: 1. Joe is rich or Joe is not rich. 2a. If Joe is rich, then it is a (determinate, objective) fact that Joe is rich. 2b. If Joe is not rich, then it is a (determinate, objective) fact that Joe is not rich. 3. So it is a (determinate, objective) fact that Joe is rich or it is a (determinate, objective) fact that Joe is not rich. This amounts to saying that there is a determinate, objective fact of the matter as to whether Joe is rich. An analogous
argument can of course be given against any claim of the form 'there is no determinate, objective fact of the matter as to whether p'; so what we have here generalizes into an argument that there can be no such thing as referential indeterminacy. Indeed, the form of argument generalizes to cases that aren't obviously cases of referential indeterminacy. Consider the familiar idea that certain evaluative debates, or certain debates about indicative or subjunctive conditionals, are 'nonfactual'. One is tempted to say for instance that there is no objective fact of the matter as to which kind of ice cream is better, chocolate or coffee, and that there is no objective fact of the matter as to whether Bizet and Verdi would have been French rather than Italian had they been compatriots. If the argument is right, neither these nor any other claims of nonfactuality make any sense. We have here a very powerful form of argument. There are philosophers who accept this argument across the board; Williamson himself seems to be one, though as far as I know he has never discussed its implications except in the case of vagueness. Suppose that we have enough information about Joe's income, his assets, his liabilities, the economy of his society, and so forth, to be confident that no further such information could help us decide whether he is rich. Even so, Williamson and other 'epistemic theorists' hold, there is a fact of the matter as to whether he is rich; it's just that we can never know. Put another way: facts about richness, insofar as they outrun facts about assets, liabilities and the like, are epistemically inaccessible to us, but they are facts nonetheless (facts which an omniscient god would presumably know even though we can't). Similarly, I assume, there is an objective though epistemically inaccessible fact as to whether it was mass or weight that pre-Newtonian uses of 'heaviness' stood for. end p.283
To my mind, this position is beyond belief. Epistemic theorists sometimes accuse those who say there is no fact of the matter of being verificationists. But this charge is totally off the mark: in fact, what is wrong with the epistemic position is that like verificationism it blurs the important distinction between the unverifiable but factual and the nonfactual. That distinction has been crucial to science: for instance, Lorentz and Einstein agreed that questions about the absolute simultaneity of spacelike separated objects are unverifiable, but disagreed as to whether they were factual; and nearly everyone has assumed this difference in their positions to be substantive. (The difference can be scientifically important: for instance, John Bell (1987) tentatively proposed reviving Lorentz's theory some years back, to provide a more satisfactory interpretation of quantum mechanics.) Similarly, an important debate in the interpretation of quantum theory has been whether a particle has determinate position when a momentum measurement is made; here it is agreed on all sides that if it does, its position at that time is unverifiable. Such disputes aside, there are plenty of examples where our theories dictate that certain intuitively factual questions could never be answered: e.g., certain questions about the details of the interior of a specific black hole in an indeterministic universe.6 It seems beyond belief that the question of whether Joe is rich, or whether the pre-Newtonians referred to weight rather than mass, is anything like that. I have occasionally heard proponents of the epistemic (no-indeterminacy) view of vagueness concede that there is a difference between vagueness cases and the scientific examples, but say that this doesn't go against the epistemic view: they say that the difference between the two sorts of examples is that in the scientific cases it is merely physically impossible (or impossible according to our scientific theories) for us to find the answer, whereas in the vagueness cases it is conceptually impossible for us to do so. But I don't think that the epistemic theory can be defended on this basis. To see this, we need to ask just what is supposed to be conceptually impossible in the vagueness case. Obviously the claim can't be that it is conceptually impossible for us to know whether a certain person is rich: imagine our discovering that the person has billions hidden in Swiss bank accounts. The three most likely alternatives are: (A) that it is conceptually impossible for us to know whether a person is rich given that the person is a borderline case of being rich; (B) that it is conceptually impossible for us to know whether a given person is rich given that the person's financial situation is . . . (where the blanks are of course to be filled in in such a way that we would intuitively regard anyone for whom those details were true as a borderline case of being rich); and 6 The kind of details I have in mind are about what is going on far enough inside the boundaries of the black hole so
that any individual unfortunate enough to have entered the black hole would have been destroyed by the tidal forces long before coming into a position to observe them; and they are details that could not be predicted from the prior state of the universe outside (or near the border of) the black hole, given the assumed indeterminism. end p.284
(C) that it is conceptually impossible for us to know whether a given person is rich given that the person's financial
situation is . . . and given that we don't have any way of ascertaining richness except via financial situation and given that we don't have any way of determining which side of the division between the rich and the nonrich contains financial situation. . . . In each of these, the locution 'p is conceptually impossible given q' should be interpreted as meaning that the conjunction p&q is conceptually impossible. (A) has it that what is conceptually impossible is conjunctions like: (#) Joe is a borderline case of being rich and we know whether he is rich. But the claim that (#) is conceptually impossible is rather uncontroversial, and of no use to the epistemic theorist: its explanation is (i) that 'borderline case of p' just means 'case that isn't determinately p or determinately not p'; and (ii) that it is a conceptual requirement on knowledge that one can't know that p unless it is determinate that p. (The more commonly cited conceptual requirement that one can't know that p unless p is a special case.) This explanation of the conceptual impossibility of (#) is available to the non-epistemic theorist, for it does not require an epistemological explanation of 'determinately'; it does nothing to support the epistemic theory. Another way to put the point is to notice that even someone who thought that you could explain the notion of a borderline case in terms of the physical impossibility of knowing would recognize the trivial conceptual impossibility of (#): (#) would hold because of the analysis of borderline case in terms of the physical impossibility of knowing and the trivial conceptual impossibility of (##) It is physically impossible for us to know whether Joe is rich and we know whether he is rich. (Note that you could replace 'Joe is rich' by a statement about the interior of a black hole in (##) without losing the conceptual impossibility, so that the alleged distinction between the two kinds of examples would not arise on this interpretation.) The case of (C) is similar: sure, it is conceptually impossible that we know that Joe is rich given the absence of any faculties by which we could find out, but similarly we have no way of knowing the details of the interior of the black hole given the absence of any faculties by which we could find out (for instance, given that we have no direct faculty of perceiving the interior of the black hole by extra-physical means). If there is any hope for using the distinction between conceptual and physical impossibility to defend the epistemic theory, it must be by taking the relevant conceptual impossibility of knowledge to be of the sort (B). But the only way for (B) to be conceptually necessary is for it to be conceptually necessary that we not have the faculties mentioned in (C). Could that be a conceptual necessity? Perhaps: maybe it is conceptually necessary that if Joe's financial situation is . . . then there is no fact (or no determinate fact) as to whether Joe is rich, in which case we could use the conceptual necessity noted under (A) end p.285
to argue that it is conceptually impossible to know whether Joe is rich (or to have faculties for knowing it). In other words, certain kinds of non-epistemic theorists could hold it conceptually impossible that we have such faculties. But how could an epistemic theorist hold this? After all, the epistemic theorist takes the question of whether people in Joe's financial situation are rich to be a matter of determinate fact; the claim that we have no means to detect such a fact must then be viewed as a medical limitation on our part, not a conceptual necessity. Of course, this is a hopelessly unintuitive way to view the limitation, but that is just to say that the epistemic view is hopelessly unintuitive.
III If what I have said is right, then the initial argument 1-3 has a false conclusion, so it must go wrong somewhere. But where? There are two main options. One is to say that Premise 1 is wrong: instances of the law of excluded middle fail when the disjuncts lack determinate truth value. I'll call this the no excluded middle option. The other main response is to keep excluded middle even as applied to vague or indeterminate language, indeed to keep classical logic generally for such language, but to reject premise (2): 'it is a determinate fact that p' and 'it is a determinate fact that not p' are genuine strengthening of 'p' and 'not p', and in cases of indeterminacy neither strengthening holds even though excluded middle holds. I'll call this the classical determinately operator option. There is also a third option: to keep both excluded middle and Premise (2), but give up the inference from 'p or q', 'if p then r' and 'if q then r' to 'r'. Giving up that inference makes perfectly good sense if we read 'if p then r' in certain nonstandard ways: for instance, as Dp r or Dp Dr, where 'D' means 'it is a determinate fact that' (and 'q s' abbreviates
's or not q'). In that case, the failure of the inference is due to the gap between 'p' and 'Dp': because of that gap, 'if p then r' in the stipulated sense doesn't imply 'p r', so we shouldn't expect the inference to hold. But clearly the rejection of the inference based on such a nonstandard reading of 'if . . . then' isn't really a third option, it is just the second option (the existence of a gap between 'p' and 'Dp') in disguise. (I'll mention another nonstandard reading of 'if . . . then' later, which also allows both excluded middle and Premise (2), and which emerges very naturally in the development of the second option.) It is formally possible to reject the inference even given the standard reading of 'if p then r' as 'r or not p', but (at least in the context in which excluded middle is accepted) I think that this has little appeal. Consequently, in the rest of the chapter I will focus entirely on the first two options. Much of the rest of the chapter will be devoted to assessing their respective merits. There seems initially to be a great deal to be said for keeping classical logic, including the law of excluded middle. For one thing, there are a great many different alternatives to classical logic on which excluded middle is end p.286
renounced, and if we are to renounce the use of excluded middle in the context of indeterminacy then we will have to decide between them. Second, if we are to give up excluded middle for language that is vague or indeterminate, then presumably we should give it up as well for language that we think might be vague or indeterminate: we should reason in a way that doesn't prejudge the issue of its determinacy. Of course it is reasonable to suppose that if, while reasoning in a broad logic that doesn't presuppose excluded middle, we conclude that certain parts of language are determinate, then this reasoning will license the use of excluded middle in those parts. But still, the broader logic that doesn't presuppose excluded middle will need to be taken as basic whenever there is a serious possibility of vagueness or indeterminacy in the language. A third point is an extension of the second: it is arguable that there are no sentences at all for which serious worries about vagueness and indeterminacy can be excluded; if so, giving up classical logic in the case of vagueness and indeterminacy would seem to require taking the view that no instance of excluded middle AV¬A is ever strictly valid (except when A is valid or ¬A is valid), but always requires justification that is to be given in a logic that doesn't use excluded middle anywhere. And somewhat independent of the third point there is a fourth: if we are to take seriously the idea that vagueness or indeterminacy is a quite widespread phenomenon, then we should consider the possibility that the language in which we discuss the semantics of vague and indeterminate language will itself be vague or indeterminate; and then if classical logic can't be used with vague or indeterminate language, we won't even be able to use classical logic in meta-theoretic reasoning about the logic of vague or indeterminate language. I don't say that any of this is decisive, but it provides some motivation for taking the classical logic option. But there are two difficulties with it that must be overcome before the classical determinately-operator option can be deemed fully satisfactory. Before discussing them, let us set aside a verbal issue. Proponents of the determinatelyoperator option differ as to their preferred use of 'it is true that p': some take it as equivalent to 'p' (call this the weak reading), others take it as equivalent to 'it is a determinate fact that p' (the strong reading). Obviously nothing can hang on whether we use the term 'true' in its weak or its strong sense. Perhaps the safest policy is to introduce two distinct words, 'true W ' and 'true S ', for these notions. Of course, if we use the strong notion of truth, we don't need the determinately operator in addition: 'it is true that p' is just another way of saying 'it is a determinate fact that p', on the strong reading of true. (So questions about the interpretation of 'determinately' can equally be regarded as questions about the interpretation of 'true', on the strong reading of 'true'.) The first difficulty with the classical determinately-operator option is a completely obvious one: the operator 'it is a determinate fact that' ('determinately', for short) would seem to require some sort of explanation. (Using the term 'it is true that' in place of 'determinately' obviously would solve nothing: we'd be invoking a sense of 'true' in which 'it is true that p' isn't simply end p.287
equivalent to 'p', so an explanation of this seems in order.) Moreover, the explanation of 'determinately p' can't be anything like 'p, and we might find out that p', for that would collapse the determinately-operator view into the epistemic view that we have rejected. But then, what is the explanation? It doesn't seem at all easy to provide. (Of course we can partly explain it, by citing certain laws that it must obey: for instance, stipulating that it obeys the laws of the modal system T, or S4, or S5. But obviously this is not nearly enough to settle its meaning uniquely.) The most popular way to try to explain 'determinately' (or 'true' in the strong sense) is the supervaluational approach: 'determinately p' holds iff p is true in all admissible interpretations of the language. But which are the legitimate interpretations? The simple-minded view is that there is only one, the one in which 'rich' stands for rich things, 'Clinton's body' stands for Clinton's body, and so forth; if so, 'determinately p' becomes equivalent to 'p', and there is no indeterminacy. To avoid this, we must apparently say something like this: J is a legitimate interpretation of L iff either it is the correct interpretation of L or there is no determinate fact of the matter as to whether it is the correct interpretation
of L. But if we say this we need an antecedent grasp of the idea of no determinate fact of the matter in explaining legitimate interpretation, so that when we then use the latter to explain the former we are going in a circle. The circularity can be disguised a bit more than I have done, but I don't see how to eliminate it. For instance, we might say that the legitimate interpretations are those which assign precisifications of predicates in the original language. Here, a precisification of a predicate is any set that contains everything that that predicate determinately applies to and contains nothing that it determinately fails to apply to; things of which the predicate is indeterminate (that is, of which it neither determinately applies to nor determinately fails to apply to) will be in some precisifications but not others. I've just explained 'precisification' in terms of 'determinate', and no other explanation is obvious, so the explanation of 'determinate' in terms of 'precisification' (via the intermediate notion of 'legitimate interpretation') is correct but circular. That's the first worry about the attempt to posit indeterminacy while keeping classical logic. I think it is a fairly serious one. Indeed, I believe that if one supposes that the only way to resolve the first worry is to provide a reductive explanation of determinateness, one will have to conclude that the first worry is totally irresolvable. But I think that the demand for a reductive explanation is unreasonable: after all, we can't give a reductive account of negation, but that doesn't mean we don't thoroughly understand it. What we ought to want, I think, is an account of the conceptual role of the notion which in some loose sense that I will not try to make precise 'fixes its meaning close to uniquely'. (I say 'close to' because we ought to allow that the notion itself be indeterminate.) But the first problem has certainly not disappeared: it is not at all clear how to give the needed specification of the conceptual role. The modal laws that govern this operator are far from fixing the sense uniquely, and it seems unclear what to add to them. (I should note that what we must end p.288
explain isn't just the conceptual role of assertions of form 'it is determinate that p', but of other constructions in which 'determinately p' is embedded: most notably, of 'it is not determinate that p'.) Let's leave the matter here for now, and go on to the second worry. The second worry is that even if we had a clear understanding of just how 'determinately p' is supposed to strengthen 'p', it is not at all clear that we would be done: one could still raise the question of how the purported fact that it is neither determinately the case that p nor determinately the case that not-p is supposed to show why it's misguided to even speculate whether or not p. And this problem seems especially acute given that by classical logic (which the view assumes) either p or not-p. One idea for trying to address both of the worries at once (suggested in Chapter 8) is to simply postulate that part of the conceptual role of 'determinately' is that we regard it as misguided to speculate about questions that we take to have no determinate answers. As it stands, this seems feeble. One could try to disguise the feebleness by using the locution 'true that' in place of 'determinately' (in the strong sense, on which 'it is true that p' is not equivalent to 'p'). In this terminology, the idea would be that though either p or not p, still it is neither true that p nor true that not p, and what we are postulating is that it is misguided to speculate about questions that have no true answers. Although the postulate may sound better when put this way, I think that it can't be: if the postulate is feeble with one terminology it is feeble with the other. (The law of conservation of feebleness.) Presumably the view sounds better when put this way only because the analogous claim about weak truth is so uncontentious. Why exactly does it seem feeble to simply build into the conceptual role of 'determinately' that it is misguided to speculate about questions that we take to have no determinate answers? The reason, I think, is that neither of the original worries seems fully answered. With regard to the first worry, the problem is that unless more is said about the way in which we regard it as misguided to speculate about whether Joe is rich, we don't capture the sense in which this question is indeterminate. (There is a sense in which we may regard it as misguided to speculate about questions whose answers can clearly never be discovered, but if that were the sense in question we would be back in the epistemic view. In another sense, we may regard it as misguided to speculate about questions that people will laugh at us for asking, but that would be even worse at capturing the sense of indeterminacy we want.) It may well seem that the only hope for explaining the relevant sense of 'misguidedness' is by saying 'misguided because any answer is going beyond the determinate facts'. Obviously if that were the best we could do then it would be grossly circular to use the misguidedness in an attempt to clarify 'determinately'. The second worry doesn't seem fully answered either. True, if we could fill out our response to the first worry along the lines suggested we would have built into the idea of indeterminacy that it is misguided to speculate on the answers to questions whose answers are indeterminate. But we would not have end p.289
addressed the worry that this sits ill with the acceptance of classical logic (in particular excluded middle, especially in conjunction with non-contradiction). Doesn't accepting that either Jones is bald or Jones is not bald, but not both, somehow pull the carpet out of the view that in the intended sense it is misguided to speculate whether Jones is bald?
Despite these doubts, I think there is some hope for the idea of building into the conceptual role of 'determinately' that it is misguided to speculate about the answers to questions one regards as having no determinate answers. I will make an effort to do so in sections V and VI, but first I want to say something about the extent to which our problems would be lessened were we to abandon excluded middle for language that is or might be vague or indeterminate.
IV A full discussion of the 'no excluded middle option' would be difficult, because there are many different nonclassical logics in which excluded middle is abandoned as a general principle, and different considerations apply to different ones. But let us make one important division: between logics in which it is possible to reject certain instances of excluded middle without hopeless inconsistency, and those in which that is not so but nonetheless certain instances are not assertable. I'll call the former 'radically nonclassical' and the latter 'moderately nonclassical' (though these labels could be misleading in a number of ways). In one respect the radically nonclassical logics are more natural for dealing with vagueness: they give a neater explanation of the idea of indeterminacy (or of a borderline case). To say that Joe is a borderline case of richness is simply to say that it is not the case that either Joe is rich or Joe is not rich. But there is a cost: denying instances of excluded middle in this way requires a fairly radical revision of logic: either one that disallows the inference from not-(p or q) to not-p and/or the corresponding inference to not-q, or one that allows the simultaneous assertion both of r and of not-r. (Reason: from not-(p or not-p), the inferences give us not-p and not-not-p; take r to be not-p.) I don't say that such a radical revision of logic is out of the question. The first of the two subpossibilities, in the strong form of denying the inference from not-(p or q) both to not-p and to not-q, seems initially quite natural in the context of vagueness: an assertion of 'A or B' can with some plausibility be taken to mean something like 'it is either determinate that A or it is determinate that B', in which case it is clear that the inference from not-(p or q) to not-p (and also to notq) should fail. But if we literally propose this as a reading of 'or' (even in embedded contexts), the view is only terminologically different from the classical determinately operator view: it is in effect simply the proposal that instead of taking 'A or B' as meaning ¬(¬A & ¬B), we should take it as meaning ¬(¬DA & ¬DB), where D is the determinately operator. (Note that it will do no good to then kick away the ladder of the determinately operator, end p.290
and simply use 'or' as if it were defined in this way: for we could reintroduce 'D' by defining DA as 'A or A', and we could reintroduce classical disjunction as well by defining it in terms of negation and conjunction.) Clearly a genuine alternative to classical logic requires messing with 'not' or 'and' (or both) as well as with 'or'. Once these points are appreciated, the first of the two sub-possibilities mentioned in the last paragraph looks far less promising, and the best hope (for the radical nonclassicist) would seem to be with allowing the simultaneous assertion of both p and not-p. As much recent discussion has shown (e.g. Priest 1998), it is possible to develop interesting logics in which asserting both p and not-p (and even, asserting their conjunction) is not 'hopelessly inconsistent', that is, where this doesn't imply everything. As far as I am aware however, no such 'paraconsistent logic' has found very useful application in connection with vagueness. There is certainly more to be said here, but I will not pursue the matter further in this chapter. Turning to the moderate views, how do they explain the idea of Joe being a borderline case of richness? A moderate view will have it that while we shouldn't accept the disjunction 'Joe is rich or Joe is not rich', we shouldn't deny it either; and it may explain the inappropriateness of our asserting this by saying that it isn't true. But for this explanation to make sense, 'true' must mean something like 'determinately true': if it meant 'true' in the classical sense in which ' "p" is true' is equivalent to 'p', then we couldn't deny the truth of the disjunction without denying the disjunction itself. Similarly, the view will likely hold that we should deny the disjunction ' "Joe is rich" is true or false', where 'false' is taken to mean 'has a true negation'. Again this shows that for either 'Joe is rich' or its negation or both, the attribution of truth to the claim is not being regarded as equivalent to the claim itself; 'it is true that' is just a determinately operator under another name. If renaming 'it is determinate that' as 'it is true that' solved the philosophical problems, we could have done that in the classical case as well. So if the goal is to avoid a special nonclassical notion of truth that amounts to a notion of determinate truth, each instance of excluded middle must go hand in hand with the claim that that instance is true, and also with the claim that one of its disjuncts is true: either we don't deny any of the three or we deny them all. But the moderate theorist can't deny all three: not denying excluded middle was what made him moderate. And it seems wholly unsatisfactory to say that in cases of indeterminacy we don't deny any of the three: after all, the only obvious way to assert that the example is an example of indeterminacy, without using a primitive notion of determinateness, is to assert that neither it nor its negation is true. (A person's unwillingness to assert 'Joe is rich or Joe is not rich' wouldn't convey to us that he regards
'Joe is rich' as indeterminate: it would convey only that he doesn't know that 'Joe is rich' is determinate.) The moral is clear: if one gives up excluded middle in merely the moderate way, one has just as much need of a determinately operator as does the (non-epistemicist) advocate of excluded middle. end p.291
Should we conclude from this that there is no point in abandoning excluded middle for vagueness without denying instances of excluded middle? That would be too quick. We can safely conclude, I think, that the need to explain the determinately operator (or if you prefer, the true that operator) is just as great on the moderate no-excluded-middle view as it is on the classical logic view. But perhaps the explanation will be easier? One might think this impossible: moderate nonclassical logic is presumably a weakening of classical logic, and if you can't explain the operator with a stronger logic at your disposal, how could you expect to with only a weaker logic? But that argument over-looks the fact that the determinately operator one needs in the nonclassical case is slightly different than the one that is needed classically: in particular, in the nonclassical case the operator will obey the law D(A V B) iff DA V DB. If the logic admits the deMorgan laws and double negation elimination and the conditional can be defined away, then this law, together with the corresponding law for conjunction (which holds for the classical operator as well), allows us to put every sentence into an equivalent normal form with the following feature: ' ' doesn't occur at all, and no occurrence of either '¬' or 'D' has an occurrence of either '&' or 'V' in its scope. (In other words, if a generalized atomic sentence is one built from atomic sentences using only D and ¬, then a normalized sentence is one built by conjunction and disjunction from generalized atomic sentences.) Indeed, we can normalize a bit further, by disallowing consecutive occurrences of '¬'; and if the analog of the S4 law is assumed, as I will, then we can also disallow consecutive occurrences of 'D'. Given this, we can fully explain assertions and denials of determinateness if we can explain assertions of the form DA, ¬DA, D¬A, ¬D¬A; D¬DA, ¬D¬DA, D¬D¬A, ¬D¬D¬A, and so forth, where A is atomic. So we have reduced the problem of explaining the notion of determinateness quite considerably. But we certainly haven't eliminated it: even if we put the 'higher order' sentences aside, there is still the problem of explaining what it means to assert that ¬D (Joe is rich) and ¬D¬ (Joe is rich). (Also, what it means to conjecture that the corresponding things might be the case for Sam.) And here I think that the moderate7 no-excluded-middle theorist is no better off than the classical theorist. Admittedly, the classical theorist had another problem: the apparent oddity of saying that it is misguided to speculate whether Joe is rich (when we believe Joe to be a borderline case) while at the same time saying that of course by classical logic either he is rich or he isn't. The no-excluded-middle theorist obviously avoids this problem, for he doesn't hold that either Joe is rich or isn't. I take this to be a serious motivation for giving up excluded middle in the context of vagueness and indeterminacy. 7 Something like this is arguably so for the radical theorist as well: although he has no need of a special
determinately operator, he does use a different set of logical constants, and in the most plausible form of the view that I know of he must appeal to special contextual shifts in the 'degree of truth' that is required for instances of excluded middle and noncontradiction to be acceptable; and it may be that an analogous problem arises in explaining either the new constants or the shifts in thresholds. I will not explore this. end p.292
Still, the problem of explaining assertions and conjectures of determinateness and of indeterminateness remains; and as stressed earlier, the nonclassical logic option has its own costs. So let us now go back to the classical determinately-operator option, and see if we can overcome the problems we have found with it.
V Let us return to the idea briefly mentioned earlier: that it is part of the conceptual role of 'determinately' that it is misguided to speculate about the answers to questions that one regards as having no determinate answers. I argued before that saying just this, without saying anything else, is feeble. What I want to do now is to add to it in a way that will make it less feeble. The approach that I will adopt to doing this is influenced by Schiffer 1998, but my proposal will be very different from his in details. My approach (like Schiffer's) will be to look at a model of an idealized epistemic agent, and to explain how such an agent might have different attitudes toward sentences or propositions that he regards as potentially indeterminate than toward sentences or propositions that he regards as determinate but difficult or impossible to answer. (I speak of 'potentially indeterminate' rather than 'indeterminate' because a sentence like 'Sam is rich' needs to be handled carefully even when the agent regards it as merely possible that Sam is a borderline case.) To simplify things, I'll suppose that the agent's language L is quantifier-free: it is built up from atomic sentences by the truth functional
operators plus the operator D. What are our idealized epistemic agents to be like? One standard idealized model of an epistemic agent is that provided by the crude Bayesian picture, according to which an idealized agent has point-valued degrees of belief in every sentence of his language and they satisfy the laws of probability.8 This makes sense (as a crude idealization) when the agent does not recognize any potential for vagueness or indeterminacy in his own sentences, but I don't think it obvious that the recognition of the possibility of vagueness or indeterminacy should leave it unaffected. I propose a generalization of it, which will have the Bayesian theory as a special case for sentences that the agent treats as not even potentially indeterminate: more exactly, the Bayesian theory will hold in the sublanguage generated from sentences A such that the 8 If the language contains context-sensitive elements one must speak instead of probability assignments to sentences
relative to a context; or probability assignments to propositions, where those are viewed as something like pairs of sentences and contexts. In what follows, I will assume a fixed context, so that we can speak simply of an assignment to sentences. When we come to vagueness, we won't be assigning probabilities to sentences, but we will be assigning other kinds of degrees of belief to sentences; again, this makes sense as long as the context is held fixed. (As remarked earlier, most vague terms are highly context-dependent, so the need to hold the context fixed is especially important.) If you like you can talk instead of assigning degrees of belief to propositions, as long as you don't employ a heavy-duty notion of proposition on which 'Sam is rich' only expresses a proposition if Sam is not a borderline case. end p.293
agent fully believes 'Either it is determinate that A or it is determinate that ¬A'. 9 It is both heuristically useful and mathematically simpler to start with a probability function P and construct a function Q from it that need not obey the laws of probability; I will take the resulting Q to be appropriate as a degree of belief function. (I will leave open whether there may be other appropriate degree of belief functions not constructible in this way.) P should be thought of as simply a fictitious auxiliary used for obtaining Q. (The degrees of belief that Q assigns will be very different from the 'vagueness related degrees of partial belief' that Schiffer has proposed.) It should go almost without saying that in talking about a probability function P in the present context, we should assume that the probability function is constrained in the obvious way by the logic of the operator D, which (for reasons that will emerge) I will take to be S4, or more cautiously, a normal modal logic M that is at least as strong as S4. So more explicitly, I assume (I ) If B is provable in M then P(B) = 1. Also, it is natural to assume (II ) If P(A) = 1 then P(DA) = 1. It follows from the first assumption that if A 1 , . . . , A n entail B in the strong sense that A 1 & . . . & A n B is valid in M, then P(A 1 & . . . & A n ) ≤ P(B). It follows from the two together that if A 1 , . . . , A n entail B on the weaker notion of entailment in which A entails DA, and if P(A 1 & . . . & A n ) = 1, then P(B) = 1. What features do we want Q to have? First, we will want (I ) and (II ) to hold for Q as well as P. Second, we will want it to be the case that whenever P takes A not to be potentially indeterminate [that is, when P(DA) = P(A) and P(D¬A) = P(¬A)], then Q(A) = P(A) and Q(¬A) = P(¬A); and conversely, that when P takes A to be potentially indeterminate then either P(A) departs from Q(A), or P(¬A) departs from Q(¬A), or both. But there is a completely obvious way to achieve these goals: simply suppose that for any sentence B of the language, Q(B) is P(DB). What are the consequences of this? Let A be 'Sam is rich' (or the proposition that it expresses in a given context—see previous note), and suppose that the agent's probability function P assigns degrees q+, q−, and q 0 respectively to 'DA', 'D¬A', and '¬DA & ¬D¬A'. Since these three sentences are exclusive and exhaustive, the three numbers must add to 1. But then since Q(A) = P(DA) and Q(¬A) = P(D¬A), we will get that Q(A) + Q(¬A) = 1 −q 0 . If the agent is fairly confident that Sam is a borderline case, q 0 will be close to 1, and we will have a strong deviation from classical probability. If the agent doesn't know Sam very well, or knows him to be secretive about his finances, q 0 will be moderate and the deviation from classical probability will be less extreme. And if for 9 Schiffer's theory, by contrast, has two distinct kinds of partial belief, operating according to different laws, one for
where vagueness is allowed and one where it isn't; and the latter is not a special case of the former. For various reasons I think this unsatisfactory. end p.294
some reason the agent is completely confident that Sam is definitely rich or definitely not rich, even if he doesn't know which, then there will be no deviation from classical probability at all.
But though this view licenses departures from classical probability, it does not license departures from classical logic. For since A V ¬A is a classical logical truth, D(A V¬A) is provable in our modal logic, so our probability function will give it value 1. So Q(A V ¬A) will also be 1, whatever the three q-values are; that is, A V ¬A gets degree of belief 1 even when the degrees of belief in the disjuncts sum to less than 1. The classical law for the probability of disjunctions with mutually exclusive disjuncts does not hold for degrees of belief in the context where vagueness is allowed. The disjunction of the mutually exclusive claims 'DA', 'D¬A', and '¬DA & ¬D¬A' is also a tautology, so the Q-value of this disjunction is 1; but again the disjuncts needn't sum to 1 (unless the modal logic is strengthened to S5). The S4 law does require that the first two disjuncts get Q-values q+ and q− respectively, but the Q-value of '¬DA & ¬D¬A' can be less than q 0 . For in S4, the possibilities in which ¬DA & ¬D¬A hold subdivide as follows: (B+) ¬DA & D¬D¬A & ¬D¬DA ['Borderline positive'] (B−) ¬D¬A & D¬DA & ¬D¬D¬A ['Borderline negative'] (D0) D¬DA & D¬D¬A ['Definitely indeterminate'] (HI) ¬D¬DA & ¬D¬D¬A ['Hopelessly indeterminate'] (In case (HI) Sam is neither definitely rich, definitely not rich, nor definitely borderline; this seems to me a good description of many situations.) If r B+ , r B− , r D0 , and r HIare the probabilities that P assigns to these, then they must sum to q 0 . Moreover, each of the four statements is consistent in S4, so each of the r-values can be nonzero. But Q(¬DA & ¬D¬A) is P(D(¬DA&¬D¬A)), which by S4 must be P(D¬DA & D¬D¬A), i.e. r D0 . So the Q-values of our three disjuncts add to 1 − r B+− r B− − r HI . Similarly, Q(¬DA) is P(D¬DA), which inspection of the above reveals to be 1 − q + − r B+− r HI ; so we have Q(¬DA) ≤ 1 − Q(A), with the inequality strict unless both r B+and r HIare 0. So though Q(DA V ¬DA) is of course 1, Q(DA) + Q(¬DA) is merely 1 − r B+− r HI . Analogously, Q(D¬A) + Q(¬D¬A) is only 1 − r B− − r HI . What this suggests, of course, is that for an agent to treat A as potentially indeterminate is for him to have degrees of belief in it and its negation that add to less than 1. We can also call this potential first order indeterminacy, and then generalize: an agent treats A as potentially (k + 1) st order indeterminate iff he treats either DA or D¬A as potentially k th order indeterminate. It is obvious that potential (k + 1) st order indeterminacy requires potential k th order indeterminacy, and easy to see that if there is potential indeterminacy in a sentence B with a chain of k occurrence of 'D', each in the scope of the previous member of the chain, then there must be at least k th order indeterminacy in one of the end p.295
atomic sentences in B. Note carefully that with merely first-order indeterminacy in A, we have that Q(¬DA) is just 1 − Q(A): the first-order indeterminacy comes out in these being less than Q(¬A). But with second-order indeterminacy, we typically have that Q(¬DA) < 1 − Q(A) and Q(¬D¬A) < 1 − Q(¬A), though we may on occasion have just one of these. It is trivial to verify that if A 1 , . . . , A n entail B in S4 (in the strong sense mentioned above, corresponding to a derivation where the rule A/DA is applied only to the axioms of S4), then Q(A 1 & . . . & A n ) ≤ Q(B). And the conclusion of the Williamson argument, DA V D¬A, has Q-value 1 − q 0 . Consequently, the conjunction of the premises A DA and ¬A D¬A can have a Q-value no higher than 1 − q 0 ; indeed, it is easy to see that the value is exactly 1 − q 0 . (The values of the two conjuncts isn't determined by the q's and r's: we can only say that Q(A DA) is in the interval [1 − q 0 , 1 − q 0 + r B++ r HI ], and Q(¬A D¬A) is in the interval [1 − q 0 , 1−q 0 + r B− + r HI ].) It is instructive to define conditional Q-values in analogy with conditional probability: Q(B|A) = Q(A&B)/Q(A), provided that Q(A) is not 0; undefined or 1 if Q(A) = 0. It follows that Q(B|A) is just P(DB|DA). Suppose that we add to the language a 'generalized Adams conditional' →, allowed only as a main connective, governed by the principle that the degree of belief in A→B should always be the conditional degree of belief in B given A (at least when that is defined). 10 Taking conditional and unconditional degree of belief to be represented by conditional and unconditional Q, we get that the degree of belief in A→B should be Q(B|A). As is well known, the ordinary Adams conditional, where degrees of belief are assumed to obey the laws of probability, does much better than the truth functional conditional at capturing 'if . . . then' in English, at least if we restrict ourselves to sentences in which 'if . . . then' appears only as the 11
main connective: consider 'If I try out for the Yankees I will make the team'. So it might seem reasonable to conjecture that the generalized Adams conditional is appropriate for vague sentences. If so, then the degree of belief in A→DA should be Q(DA|A) (at least when Q(A)≠0), which is 1. (Note the contrast between Q and probability functions: for them, the conditional probability is never higher than the probability of the material conditional.) This is noteworthy in connection with the Williamson argument (1)-(3): all the premises of the argument get Q-value 1 if 'if . . . then' is read in accordance with the generalized Adams conditional.12 Of course, the classical rule of inference used in the argument is not valid on this reading of 'if . . . then'. 10 If we had started with a conditional probability function P(X|Y) defined even when P(Y|ZV¬Z) = 0, as in Popper
1959 (appendices iv and v), we could have defined Q by the rule Q(X|Y) = P(DX|DY); this would allow Q(X|Y) to be defined even when Q(Y|ZV¬Z) = 0. The Popper approach, extended to the present context, requires that P(X|Y) be 1 when Y is inconsistent with the modal system M, but allows other values when P(Y|ZV¬Z) is 0 without Y being inconsistent; so the same will be true of Q. 11 If one assumes that the probability of my trying out for the Yankees is strictly 0 rather than merely less than 10 −10 , one needs the generalization mentioned in the previous footnote. 12 I should note that the conclusion that the Q-values of the Adams conditionals is 1 would not have been forthcoming had I started from a typical probability function based on a modal logic weaker than S4. end p.296
I might remark that the generalized Adams conditional is also illuminating in the context of the Sorites argument: if we understand the Sorites premises (in the many-premise version of the argument) as generalized Adams conditionals they are all highly believable, even for the clearest of borderline cases; but the Sorites argument does not preserve believability on this reading of the premises. I introduced Q by starting from a classical probability function P, but have suggested that P not be taken seriously: except where it coincides with Q, it plays no role in describing the idealized agent. (Obviously the same Q can be generated from many different P's, unless Q(A) + Q(¬A) is 1 for all A.) There are really two distinct parts to this idea: first, that Q is a perfectly legitimate belief function, and second, that P is not a legitimate belief function when it generates a Q distinct from it. The second component of this would be hard to motivate if the process that led from P to Q could be iterated to yield a Q* distinct from Q, but that is not the case: if we define Q*(A) as Q(DA), Q* simply is Q. That is my principal motivation for using S4. The use of S4 might seem objectionable, on either of two grounds. First, it might be thought odd to have the degree of belief in DA always equal to the degree of belief in A, given that DA is stronger than A. It seems to me, though, that the feeling of oddity goes away once one realizes that the degree of disbelief in DA can be higher than the degree of disbelief in A (and typically will be when A is judged indeterminate): that is, Q(¬DA) can be higher than Q(¬A), and this is symptomatic of the added strength of DA over A. With classical probability the degree of disbelief in A is determined by the degree of belief in A; when, as with Q-type degrees of belief, this is not so, we must remember to consider degrees of disbelief as well as degrees of belief when comparing an agent's attitudes toward two propositions. (Without modifying the basic theory we could modify the notion of degree of belief so that it does determine degree of disbelief: say by using the phrase 'the degree of belief of A' to stand not simply for Q(A) but for the pair
that the logic of 'determinately' is S5 would seem to be quite compatible with taking 'determinately' to itself be vague, and simply requiring that the admissible sharpenings of the notion of 'determinately' be required to obey S5. This thought is expressed in the language of admissible sharpenings, which I have held to be not ultimately illuminating (though not illegitimate either); I will say a bit later how this might be captured in the language of idealized agents. For now let me just say that still another component of the intuitive idea of 'higher order vagueness' is that 'appropriate degree of belief' is vague. In the current framework, that is connected with the idea that there is no one right way to assign Q-values to sentences. This sort of higher order indeterminacy needn't be accommodated explicitly in the model
of idealized agents on offer, because each legitimate choice of Q-values will correspond to one way of making 'degree of belief' precise; if the model is appropriate for each such choice, it is appropriate overall. We could consistently suppose that the S5 law should hold for these legitimate choices of Q-values without denying that there is indeed a multiplicity. But let's avoid further discussion of these issues by sticking to S4. If, as I have suggested, the classical probability function P should be viewed as having no 'psychological reality', but is simply a device for generating the genuine degree of belief function Q, then it would be nice if we could develop the laws governing Q-functions without reference to probability functions. To see how to do this, note that for any sentence B whatever, Q(DB V D¬B) = Q(B) + Q(¬B); and more generally, (*) Q(DB V DC) = Q(B) + Q(C) − Q(B&C). Proof: Q(DB V DC) = P[D(DB V DC)]. But in S4, D(DB V DC) is equivalent to DB V DC, and by the disjunction law for probability functions, we get P(DB) + P(DC) − P(DB & DC). But also, DB & DC is equivalent to D(B&C), so we get P(DB) + P(DC) − P[D(B&C)]; that is, Q(B) + Q(C) − Q(B&C). A consequence is (#) Q(BVC) ≥ Q(B) + Q(C) − Q(B& C); we get an inequality where the classical theory has equality. Actually we can do considerably better than this: the proof of (*) generalizes to a proof of (**) Q(DB 1 V . . . V DB n ) = Σ i Q(B i ) − Σ i,j distinct Q(B i &B j ) + Σ i,j,k distinct Q(B i &B j &B k ) − . . . ± Q(B 1 & . . . &B n ), where the '±' is + if n is odd and − if n is even.13 And this yields (# #) Q(B 1 V . . . V B n ) ≥ Σ i Q(B i ) − Σ i,j distinct Q(B i &B j ) + Σ i,j,k distinct Q(B i &B j &B k ) − . . . ± Q(B 1 & . . . &B n ), 13 Indeed, we can do better still: any one of the occurrences of 'D' in the term on the left-hand side can be dropped. end p.298
which doesn't follow from (#) alone. Glenn Shafer (1975) constructed an elegant theory of belief functions on the basis of condition (# #) (together with the assignment of 1 to logical truths and the stipulation that the degrees be real numbers in [0, 1]). Shafer was motivated by considerations not having to do with vagueness, and some of the apparatus he develops doesn't have any obvious application to vagueness; 14 still, the above derivation of (# #) shows that his basic theory of belief functions can be taken over. Where are we? We've seen that if we start from a classical probability function P on a language L that includes a determinately operator (and where P respects the S4 laws for D), then the function Q P that we obtain from it must be a Shafer function on L. This means, of course, that its restriction to the D-free part of L must also be a Shafer function. And this suggests that talk of the determinately operator may have been, in a sense, unnecessary: the key to regarding a sentence as indeterminate is simply adopting degrees of belief (in D-free sentences) that accord with something like the Shafer laws; having attitudes toward sentences involving 'determinately' isn't necessary. Still, a more reflective agent can have attitudes toward sentences that contain 'determinately': his degree of belief function for the language that contains 'D' will extend his degree of belief function for the D-free part, and accord with the law that Q(DA) always equals Q(A). And we get the nice result that our two readings of 'regards A as determinate' coincide: his degree of belief in DA V D¬A will always equal the sum of his degrees of belief in A and in ¬A. (This follows from (*) on the previous page.) I close this section by mentioning three technical questions that I haven't investigated; their answers could well be obvious. First, can any Shafer function on the D-free part of the language be generalized to a Shafer function on the full language (that meets the constraints on D)? I would hope so, but if not, it implies at most that an agent with an unextendible Q-function on the D-free part of his language needs to revise his degrees of belief a bit if he is to introduce the term 'definitely'. Even that may be misleading, for I have not ruled out that there be other laws on degree of belief functions beyond the Shafer laws that I have derived. That suggests the second technical question: can any Shafer function on the D-free part of the language be obtained from a probability function on the full language? If not, then there are laws that hold for the special Shafer functions that are so derived that don't hold for all Shafer functions, and we then need to decide whether to impose all, some, or none of those laws on 'degrees of belief'. I would not want to impose any such laws merely on the basis of their being required in order that the degrees of belief 'ultimately derive from' a classical probability function, but some such laws might seem independently desirable. A third technical question is, when a Shafer function on the D-free part of L is extendible to a Shafer function on the full L, is this extension unique? I would assume not: I would assume for 14 In particular, most of his theory concerns how belief functions are to be combined ('Dempster conditionalization'),
and this seems to have no bearing on vagueness or indeterminacy. end p.299
instance that the Q-values of many sentences of form DA B and A DB when A is D-free won't be fixed by the Qvalues of D-free sentences. This is relevant in connection with the question I raised earlier, of what means would be available for recognizing higher order vagueness if for some reason we were to take D to obey the laws of S5. One way to do this, I would think, would be to suppose that there was a large element of arbitrariness in the extension of the Q-function from the D-free language to the language with D. I will say no more about this, though; higher order vagueness is not my topic, and besides, I see no reason to adopt a logic stronger than S4. It's time to review the material of this section. I have tried to give a model of an agent with the feature that despite the agent's adherence to classical logic, he treats sentences that he regards as potentially indeterminate significantly differently from ones that he regards as certainly determinate: the degree of belief in the indeterminacy of a sentence A is measured by the extent to which the probability of it and its negation sum to less than 1. There has been no attempt here to explain why the agent adopts this differential attitude between sentences he takes as probably indeterminate and sentences he takes as probably determinate: more on that in the next section. But still, the model seems to be one on which 'belief' in sentences that one treats as indeterminate 'works differently'—obeys different laws—than belief in sentences that one treats as determinate. This impression might be thought to be due simply to the crudity of the model for factual belief from which we started. That model, recall, took off from the assumption that when indeterminacy isn't in question, the agent can be taken to assign classical point probabilities to every sentence. But even when indeterminacy is not in question, the assumption of classical point probabilities is unduly restrictive (as Isaac Levi has often emphasized: Levi 1974). A better model would start with a (convex nonempty) set S of probability functions on the language. For each P in S, we could construct a degree of belief function Q P , in just the way done before; we could then take the set of all Q P for P in S as what represents the agent's degrees of belief. At first glance, this complication doesn't alter the important point: sentences viewed as indeterminate will differ in how they function from sentences viewed as determinate. On second glance, this may not be quite so clear. For from the nonempty set S of probability functions used to represent the agent's degrees of belief in determinate sentences, one can generate a numerical degree of belief function G S : for any sentence A, G S (A) is taken to be the greatest lower bound of all P(A) for A in S. If we do this, then G S (A) + G S (¬A) will typically be less than 1, and so we will have 'degree of belief functions' that look superficially like the Q-functions I have constructed, quite independent of vagueness or indeterminacy. So the worry is that by moving from a single probability function to a set of many probability functions even when indeterminacy is not at issue, we are in effect simply moving to a nonclassical degree of belief function G S whose nature might mask any further nonclassicality due to indeterminacy. end p.300
I can't fully treat the matter here, but there are two reasons to think that this worry is not serious. First, the function G S constructed from a nonempty convex set S of probability functions will not typically obey the Shafer law (# #) or even (#). 15 Second, G S isn't adequate as a representation of the degrees of belief in determinate propositions anyway, for it gives only partial information about the set S of probability functions from which it was obtained, and the information lost is crucial. 16 These points should do a great deal to allay the worry. Moreover, even if this worry about the Levi model of idealized agents were to prove correct, I think that a still fuller extension of the model of idealized agents would restore the distinction in function between beliefs that the agent regards as potentially indeterminate and beliefs for which this is not so. Recall the initial intuition: that it is misguided to speculate as to whether Joe is rich (where Joe, unlike Sam, is someone I know enough about to be quite sure that he is a borderline case (as I am currently using 'rich')). In the case where there is no indeterminacy, I don't think that precise degrees of belief are misguided, I just don't think that they are mandatory. So a fuller psychological model needs something that evaluates our degrees of belief and the degrees of belief of others, and in this way the psychological distinction should be restored.
VI I've gotten into a lot of detail, but now it's time to return to the worries about the determinately-operator approach to vagueness and indeterminacy. When we left the matter at the end of section III, we had a version of the approach on which it was somehow built into the meaning of 'determinately' that it was misguided to even speculate on questions when the answers were thought indeterminate; or to put it another way, where to regard a question as indeterminate consists in part in our regarding it as misguided to even speculate on the answer to it. The first problem was that the
relevant sense of speculation being 'misguided', or at least the relevant sense of its being 'regarded as misguided', needs specification: without that, the view seems to collapse into something like the epistemic view. Does the kind of idealized model I have discussed help with this problem? I think it does. In terms of the idealized 15 Let A be 'This coin will come up heads on the next flip', and C be any sentence that is intuitively independent of A.
If the agent takes A and C as independent and takes the coin as fair, his set of probability functions on the sublanguage generated by A and C will be the set of P such that (i) P(A) = ½, (ii) α≤P(C)≤β for some α,β (his 'lower and upper probabilities'), and (iii) P(A&C) = P(A)·P(C); Let B be A≡C. For each P in the set, we have that P(B) is 1/2, P(A&B) = 1/2P(C), and P(AVB) = 1 − 1/2P(C); so G S (A) and G S (B) will each be 1/2, G S (A&B) will be α/2, and G S (AVB) will be 1 − β/2, which violates (#) when β is strictly greater than α. 16 Suppose that A and C are intuitively independent sentences, but that this time the agent can put no lower or upper
bounds (except 0 and 1) on the probability of either. That is, the agent adopts the set of probability functions (on the sublanguage generated by A and C) meeting condition (iii) of the previous note. In this case, for sentences E in the sublanguage, G S (E) is 1 if E is a tautology and 0 otherwise, just as it would have been if condition (iii) hadn't been imposed: the shift from S to G S throws away the crucial information that the agent takes A and C to be independent. end p.301
models we don't have any analog of the idea of speculation actually being misguided, but I don't think we need that. What the models do is to build something like the idea of 'regarding speculation as misguided' into the core of the account of proper psychological functioning. That is, it is part of the idealized psychology that we simply refuse to adopt classical point-valued probabilities in certain cases (and in the fuller psychological model, recommend that others not do so either); this is taken as a natural state of affairs, not requiring special explanation. In terms of it, we can explain what it is to regard a sentence indeterminate: we regard a sentence indeterminate to the extent that the nonclassical probabilities we assign to it and its negation sum to less than 1. One might object that this explains what it is to regard a sentence indeterminate, but that what we should want is an explanation of what it is for a sentence to be indeterminate. But let's look at a somewhat similar issue: the notion of objective chance. As is widely recognized (e.g. Lewis 1980, Leeds 1984, Skyrms 1984), it seems impossible to provide a non-circular reductive explanation of the notion of objective chance; all we can do is specify its conceptual role. And the central component of the conceptual role of objective chance is provided by the connection between opinions about objective chance and degrees of belief in other matters. Indeed, at least in many important cases, an idealized agent's degrees of belief in the objective chances are completely determined by their degrees of belief in specific outcomes: cf. diFinnetti's theorem and its many extensions.17 I think it is fairly uncontroversial that the meaning of 'chance' is determined by its conceptual connection to degrees of belief in matters other than chance, in roughly this way. Different opinions are possible on what if anything this shows about the metaphysics of chance: one can try to combine it with a 'realist' view of chance, or with a view that is in one way or other 'anti-realist' (e.g. a view on which attributions of chance are 'mere projections' of our degrees of belief about other matters). Some 'anti-realists' will hold that talk of degree of belief in chances isn't strictly appropriate, since statements of chance are not straightforwardly factual, and that one should speak instead about degree of acceptance of chance-sentences; others will take a more relaxed attitude toward the use of 'belief'. I don't propose to enter into these issues, or even to decide whether they are meaningful. But I think that whatever view one takes on those issues as regards objective chance, one can take the same stand as regards determinateness. Here too, we can explain degree of 'belief' about determinateness in terms of its connection with degrees of belief in ordinary sentences: at least if we ignore complications arising from the Levi model, we can say that to 'believe' that a sentence has a determinate truth value just is to have degrees of belief in it and its negation that sum to 1. Whether this is adopting an 'anti-realist' view of determinateness, and whether it is appropriate to talk of our attitudes toward statements of determinateness as literally 'beliefs', need not be settled here. 17 See Skyrms 1984, Ch.3. end p.302
So I think that the idealized model does a good deal to explain the meaning of 'determinately'. But what about the second worry: that there seems to be something odd about thinking it misguided to speculate whether or not Joe is rich, given that one accepts that either Joe is rich or Joe is not rich. Here too I think the model helps. First, the model provides a clear picture of epistemic principles that license full belief in excluded middle even while dealing with sentences that are taken to be indeterminate. I think it was largely the absence of such a model that made it seem odd to adhere to excluded middle while recognizing indeterminacy. Second, the feeling that this is odd stemmed in part I think from the assumption that if we assume that Joe is rich or Joe is not rich, we can't regard it as misguided to assign a classical probability to the claim that Joe is rich. Assigning a classical probability does seem intuitively doubtful in this case; moreover, it might be argued that if we can assign a classical probability to the statement that p then we can speculate as to whether p. (A crude way to argue this would use the supposed connection between classical probabilities and betting behavior, and the fact that it is not misguided to wonder whether one has won one's bets.) So
by rejecting the assumed connection of classical logic with classical probability, the model helps relieve the feeling of oddity in assuming classical logic for indeterminate sentences.
VII One of the features of the account I have given is that it takes attributions of indeterminacy to arise primarily for sentences rather than for terms: terms are to be regarded as indeterminate only because sentences that contain them are to be so regarded. This may seem to raise a problem for one of the examples given in section I: I claimed that there was no fact of the matter as to which square root of −1 'i' referred to, but we don't in our language have means of referring to square roots −1 that aren't conceptually tied to 'i', so we don't have indeterminate sentences, so what can it amount to to say that 'i' is indeterminate? But this isn't really a problem: for since we think there are two roots of −1, we can always introduce new names for them, not linked to 'i': for instance, '/' and '\'. The indeterminacy consists in the fact that if we do that, our degrees of belief in '/ = i' and '\ = i' should each be 0, even though the degree of belief in the disjunction should be 1. (It should be clear that this account of the indeterminacy does not rely at all on the theory of reference: thus it is available to anyone who thinks of reference in one's own language as defined by the disquotation schema, and thus solves the problem for the disquotationalist mentioned near the end of section I.) I now want to apply the ideas of the two previous sections to another example of indeterminacy. As is well known, the size of the set of real numbers is radically unsettled by all the usual axioms of set theory (and by any known way of consistently extending these in a non-ad-hoc fashion): the axioms imply that the set of reals is uncountable, but it is consistent to suppose its size end p.303
to be 1 or 817 or ω+23or nearly any other uncountable cardinal you like. A possible view is that even though our axioms don't settle the matter (nor do any extension of the axioms that we find at all 'evident'), still our set-theoretic concepts are perfectly precise, so that there is an objectively correct answer that we will probably never know. But an alternative view, which I find far more plausible, is that our set-theoretic concepts are indeterminate: we can adopt any one of the above claims about the size of the continuum we choose, without danger of error, for our prior set-theoretic concepts aren't determinate enough to rule the answer out.18 Prior to our choice there is no determinate fact of the matter. In my view, this does not require that we give up classical logic when doing set theory. As I once put it, standard mathematical reasoning can go unchanged when indeterminacy in mathematics is recognized: all that is changed is philosophical commentaries on mathematics, commentaries such as 'Either the continuum hypothesis is determinately true or its negation is determinately true'. (Ch. 12 of this volume, p. 337) Leeds objected to this view in the paper that I began with: his position, as one might guess, is that it is simply incoherent to accept set theory while regarding its concepts as indeterminate: especially (I think he would add) when one regards classical logic as fully applicable to them. And I have to agree that there can seem something fishy in the idea: if the philosophical commentaries about determinateness are as divorced from mathematical practice as the quotation may suggest, then they seem suspiciously idle. I think though that the position developed in this paper shows that the commentaries about determinateness needn't be divorced from mathematical practice: the belief in indeterminateness will come out for instance in whether the mathematician's degree of belief and degree of disbelief in the continuum hypothesis sum to 1. This in turn will lead to further differences: a mathematician for whom the degrees sum to 1 may wonder 'Is the CH correct?', and try to gather mathematical evidence in support of one answer or the other; a mathematician for whom the degrees sum to 0 will think that talk of evidence is misguided, and will think that each possibility is worth developing. Of course, practical and aesthetic considerations may make some answers more interesting than others; but these are not evidence. (For one thing, practical considerations may favor using a set theory that gives one value to the continuum in one context and using another in a different context. There is no problem in doing this as long as the contexts are kept distinct, but this makes clear that the utility of a mathematical theory in applications is quite independent of its 'truth'. Similarly, distinct aesthetic criteria may point in different 18 If one thinks that the only possible indeterminacy here is in 'set', one may suppose that we can avoid postulating
indeterminacy by taking the extension of 'set' to be 'as large as possible': see Leeds 1997, n. 24, for such a suggestion. But in fact, an indeterminacy in the size of the continuum would be due at least as much to an indeterminacy in the membership predicate as to an indeterminacy in 'set'. After all, the idea behind the argument for indeterminacy is that there is little to determine what the set-theoretic primitives are true of besides the axioms we accept, and ' ' is one of those primitives. (I say 'little' rather than 'nothing', for reasons discussed in Ch. 12. But the other determinants don't help in making the size of the continuum determinate.) end p.304
directions: if we keep truth out of the picture there is no need to ask which of two values of the continuum is prettier overall, anymore than there is a need to decide whether Bach or Beethoven is prettier overall.) I should close by mentioning that Leeds himself offers a solution to Williamson's puzzle (quite different from the one I have suggested). His discussion is brief, but the position seems to be that while some sense can be made of saying that it is indeterminate whether a given atom is part of Clinton's body or whether a given person Joe is rich, what we are doing when we say that is (i) excluding the terms 'Clinton's body' and 'rich' from the subpart of our language that (in that context) we take to be 'first class', and (ii) restricting the primary (disquotational) use of 'refers' and 'true of' to that subpart of our language. The indeterminacy of 'rich' and 'Clinton's body' is then viewed as simply a matter of there being no uniquely best way to translate those terms into the subpart of our language to which we are applying the disquotation schema. I think this idea for trying to reduce all indeterminacy to indeterminacy of translation while recognizing vagueness in our ordinary language is ingenious, though I am skeptical of the possibility of ultimately making sense of it. 19 But even if it is workable, and preferable to the solution I have advanced, I doubt that it serves his ultimate aim, which involves excluding the possibility of there being any indeterminacy in our own terms that is more exciting than vagueness. Without further clarification in his proposal, I don't think it is clear whether it excludes some or even all of the indeterminacy one might otherwise imagine in our own scientific terms. But in the case of the Brandom example I think it is clear that indeterminacy is not excluded: we can take our 'first class language' to simply say that −1 has two square roots, without introducing a name like 'i' to 'stand for one of them'. Leeds actually does not discuss the Brandom example, so I am not sure that this conclusion would be unwelcome to him. But he certainly thinks that indeterminacy claims make no sense in the continuum example; and yet I think that something similar to what I just said for the Brandom example can be said there. More fully, the idea is that we can in a sense accept set theory without taking the language of set theory as 'first class'. We can do so even while being platonists in the sense of believing that there are non-physical eternally existing objects, and taking set theory to be talking about some of 19 Perhaps the most central problem that the approach faces is that of distinguishing between indeterminate terms
and terms that we would all be inclined to say are perfectly determinate but simply differ in extension from terms in the 'first class' part of our own language. (The problem is especially severe when the determinate terms have multiple 'approximate translations' into our first class language, none clearly best, and these 'approximate translations' differ in extension among themselves.) The obvious way for a disquotationalist to try to handle terms in other languages that differ in extension from terms in our language involves imagining adding those terms (or others with the same conceptual role) to our language, thereby allowing us to disquote. (See Quine 1953b: 135; and Ch. 4, pp. 129-30). But if we can imagine adding (analogs of) intuitively determinate foreign terms to our 'first class language', we can equally imagine adding (analogs of) terms like '/' and '\', or terms like 'rich'. The proposed solution to the problem of saying what is defective about '/' and '\' and 'rich' threatens to disappear. end p.305
them. The idea would be to take as fundamental a theory T that doesn't employ set-theoretic vocabulary but merely says that there are infinitely many non-physical eternally existing objects, and that postulates the consistency of basic set theory. (Consistency would have to be regarded as a primitive notion governed by its own axioms, rather than explained in set-theoretic terms. But there are independent reasons for adopting this attitude: see for instance Field 1991.) We then translate the language of set theory into T by taking 'set' to be true of some or all of the non-physical eternally existing objects and by interpreting 'member of' in any way that makes the usual axioms come out true. There are multiple ways of doing this, and different ones make different sentences about the size of the continuum come out true. So we get a multiplicity of translations between the language of set theory and the fundamental platonistic theory, and the continuum hypothesis comes out lacking in determinate truth value.20 One conclusion then is that even if we were to concede to Leeds his own solution to the Williamson problem, we cannot exclude the possibility of substantial indeterminacy in our own language, going far beyond mere vagueness. But I don't see any reason to concede to Leeds his solution: I think that the model of vagueness and indeterminacy that I suggested in earlier sections is more attractive, and that it answers the skeptical doubts of both Leeds and Williamson. 21 20 There is a complication: to handle the applications of mathematics to nonmathematical domains like physics and psychology, we need that these set theories are not merely consistent but jointly consistent with our consistent nonmathematical theories. And we must also suppose that each theory T of physics or psychology or whatever formulated in our ordinary set theory could be reformulated using the alternative platonistic theory. One way to do so would be to replace it by a nominalistic theory. Another would be to replace it by the claim that all of the nominalistic consequences of T-plus-some-chosen-set-theory are true. This second route may seem like a cheap trick. Note
though that on the second route, the chosen set theory needn't be one that decides the size of the continuum, though it could if that turned out to be useful in the particular physical or psychological theory in question. Also, the set theory chosen for one physical or psychological theory needn't be compatible with the one chosen for another: this makes clear that the truth of the set theory is not being assumed in the superior conceptual framework, only its instrumental utility in a particular application. 21 I am grateful to Stephen Schiffer and Paul Horwich for a number of discussions, and to Schiffer and Tim Maudlin for helpful comments on a distant ancestor of this paper. Previous versions tentatively defended a different solution to 'Williamson's puzzle'. I am especially grateful to an editor and a reviewer at Nous for their skeptical response to that solution; their skepticism provoked the solution offered here. And David Barnett pointed out an error, and suggested several improvements, in the near-final draft. I'm also grateful to Kit Fine for helping with a technical problem that arose when I was trying out a slightly different approach to the non classical degree of belief functions than the approach I finally adopted. end p.306
Postscript 1. It's worth noting that the general idea behind what I'm suggesting in this paper can be carried out for nonclassical logic as well as for classical: what I'm proposing is not intended to show that there is an advantage for classical logic over nonclassical in dealing with indeterminacy, but to remove an apparent disadvantage, namely the reliance on an unexplained notion of determinateness. More particularly, what we would need to do to carry out the program for, say, the logic of Kleene's strong 3-valued truth tables (understood as in section 2 of the postscript to Chapter 4) is to first consider an appropriate generalization of probability function for such a logic. This is not particularly problematic. Indeed, one can use the degree of belief functions Q discussed in the paper to generate degree of belief functions R appropriate for Kleene logic. (Let an elementary sentence of a propositional language L be a conjunction of one or more atomic sentences and negations of atomic sentences. Then the idea is to stipulate that R agrees with Q on elementary sentences; but to determine the Rdegrees of more complex sentences from these, in such a way that classical laws like R(AVB) = R(A) + R(B) − R(A&B) and its generalizations are restored. If this route is followed, the degrees of belief in a sentence and its negation will sometimes add to less than 1 provided that the original Q was nonclassical.)1 1 In a bit more detail: let a normalized sentence of L be a disjunction of one or more elementary sentences. Given a
function Q on L obeying the Shafer laws, define a function R on the set of normalized sentences of L by (#) If E1 , . . . , En are elementary (n≥1), then R(E 1 V. . .VEn ) is . That is, (1) For elementary E, R(E) = Q(E); (2) For elementary E and F, R(EVF) = Q(E) + Q(F) − Q(E&F); (3) For elementary E, F and G, R(EVFVG) = Q(E) + Q(F) + Q(G) + Q(E&F&G) − Q(E&F) − Q(E&G) − Q(F&G); and so forth. (Since conjunctions of elementary sentences are themselves elementary, (1) guarantees that (#) would remain true if Q on the right hand side of it were replaced by R.) It is easy to see that normalized sentences that are equivalent in Kleene logic must get the same R-value; so since every sentence of L is equivalent to at least one normalized sentence, we can define the R-value of a sentence as the R-value of its normalizations. It is not hard to show that sentences A1 , . . . An Kleene-imply a sentence C iff for all functions R so generated from Shafer functions, R(C) ≥ R(A 1 &. . .& An ). The law
holds even when the Ai s are not elementary; in particular, R(AVB) = R(A) + R(B) − R(A&B), as in classical probability but unlike with Shafer functions. end p.307
Since many if not all Shafer functions arise from classical probability functions on a language with a definitely operator
appropriate to classical logic, this shows that Kleene degree of belief functions are also obtainable from classical probability functions with definitely operators appropriate to classical logic. (Alternatively, one can doubtless construct them from classical probability functions on a language with the different kind of definitely operator more naturally associated with Kleene logic.) These results are suggestive in connection with the issue of whether the apparent disagreement between the advocate of classical logic and the advocate of Kleene logic is more than verbal, but I will not pursue this. 2. In my discussion of nonclassical logic in section IV, I implicitly identified rejecting an instance of excluded middle with accepting its negation; this was probably unfortunate, for 'reject' is a term of art that can be understood in different ways. I should have defined a 'moderate nonclassicist' without talk of rejection: a moderate nonclassicist is someone who doesn't accept all instances of excluded middle, but doesn't accept the negation of any such instance either. The problem I posed is: what it is for such a person to regard Joe as a borderline case of baldness. Is it for her to 'reject' 'Joe is bald', 'Joe is not bald', and their disjunction? Perhaps there is a sense of 'reject' in which that is so, but what is the sense? It must be weaker than accepting the negation, for the moderate nonclassicist doesn't allow accepting the negations of any of these; but it must be stronger than not accepting, for we want to capture the distinction between regarding Joe as borderline and having no idea what his hair situation is. We could define rejecting p as accepting that it is not the case that determinately p, but that would prove my claim in that section: the moderate nonclassicist needs a determinately operator. Actually the claim in that section was intended as only tentative: my real point was that the moderate nonclassicalist has as much need of a determinately operator as does an advocate of classical logic. I argue later that for an advocate of classical logic, the determinately operator isn't really fundamental: what is fundamental is nonclassical degrees of belief. This viewpoint is available to the moderate nonclassicist too, and suggests a fourth option for understanding the notion of rejection. For accepting a proposition seems intimately related to having a high degree of belief in it (though a strict identification is probably unwise if the threshold degree is the same for all propositions and short of 1, because you will lose anything like logical closure); similarly, we can take rejecting a proposition as intimately related to having a low degree of belief in it, where 'low' is stronger than 'not high'. More specifically, if accepting proposition p is something like having a degree of belief in it over a threshold t, rejecting it could be viewed as something like having a degree of belief in it under the 'co-threshold' 1 − t. If the threshold is greater than ½, as I will assume, this account makes rejection stronger than non-acceptance. And if degrees of belief in a sentence and its negation can add to less than 1 (as they do, for instance, if given by the R-functions discussed in the previous section of this postscript), it makes rejection weaker than acceptance of the negation. end p.308
Provided that the advocate of moderate nonclassical logic goes this way, then she can understand a suitable notion of rejection independent of the notion of determinateness. She would also, I think, be able to use the notion of rejection, or the nonclassical degrees of belief from which it comes, to give a conceptual role to a suitable notion of determinateness (or strong truth). In short, she is no worse off than the advocate of classical logic, barring considerations of simplicity in reasoning; but I don't see that she is better off either. 3. Near the end of section VI I say that (with a certain qualification), judging a sentence indeterminate is just a matter of having degrees of belief and disbelief in it that sum to less than 1. It would have been better to have said: judging a sentence indeterminate is just a matter of judging it appropriate to have degrees of belief and disbelief in it that sum to less than 1 (or a matter of judging it inappropriate to have degrees of belief and disbelief that sum to 1). For one thing, real degrees of belief never obey neat mathematical laws like the probabilistic laws or the Shafer laws: judgements of propriety are always involved when degrees of belief are modelled in this way. But aside from that, the evaluative element seems important to capturing the attitude I have when I regard something indeterminate: if I judge the continuum hypothesis indeterminate I deem it inappropriate to give it a classical probability value. (The analogy to chance supports this: not only do I have a low degree of belief in your ticket winning a thousand-ticket lottery in which the tickets are thoroughly shuffled and about the same size and drawn blindly, I judge it inappropriate for you to have a degree of belief that is any higher.) It may be objected that the reason we think it inappropriate to have degrees of belief and disbelief in 'p' that add to 1 is that we think it is objectively indeterminate whether p; and that for this explanation of the inappropriateness to have any content we need to give an account of objective indeterminacy independent of the appropriateness of degrees of belief. Analogously, it may seem plausible that the reason why we think it inappropriate to have a high degree of belief that a given ticket will win the lottery is that we think the objective chance is very low, and that for this explanation of the inappropriateness to have any content we need to give an account of objective chance independent of the appropriateness of degrees of belief. But in the chance case, the supposed explanation is illusory: the content of the claim that chance of a ticket winning is r can not be explained except in terms of the appropriate degree of belief in that ticket winning. Similarly, I think, in the indeterminacy case: there is no hope of explaining the indeterminacy except in terms of how it ought to affect degrees of belief. Talk of indeterminacy, like talk of chance, gives the illusion of giving
an objective explanation that isn't really there. 2 2 In both the chance case and the indeterminacy case, objective facts about the situation do seem relevant to
explaining the inappropriateness of inappropriate degrees of belief: facts about the method by which the tickets are drawn in the lottery case, facts about Joe's assets and liabilities in the case of 'Joe is rich'. My point is only that these explanations needn't use the notions of chance or indeterminacy; use of those notions seems to serve largely to create an illusion of adding objectivity to the explanation. end p.309
4. The chapter claims that the lack of 'full factualness' in many sentences with vague and indeterminate terms is intimately related to the need to employ nonclassical degrees of belief for such sentences. It's worth noting that this is exactly the same conclusion that I reached for indicative conditionals in Chapter 8: I said that the Lewis proof shows that if we adhere to the thesis that the degree of belief in a conditional is the conditional degree of belief in the consequent given the antecedent, then we must modify the usual probabilistic laws for conditional sentences; and that this is intuitively related to treating those sentences as 'not fully factual'. This raises the question of whether nonfactualism about evaluative sentences might similarly be clarified by means of a nonclassicality in our degrees of belief toward evaluative sentences. (This would be one way of clarifying the distinction between 'pure' and 'impure' belief in Chapter 8.) The probabilistic arguments against the 'desire as belief' thesis (Lewis 1988 and 1996) superficially suggest a positive answer, but I haven't explored this matter. 5. The chapter mentions the issue of whether notions like reference should be viewed in a 'deflationist' spirit, but tries to avoid taking a stand. But I should have mentioned that even if (as I argue) a deflationist view leaves room for indeterminacy, it might undermine the motivations for indeterminacy in some cases. And in fact in some cases I think it probably does: see the postscript to Chapters 9 and 12. But not, I think, in the cases under discussion. For instance, I take it that our doubts about whether the tensor fields that we postulate make physical sense on a very small scale are sufficient to motivate judging it inappropriate to have degrees of belief and disbelief that add to 1 in the case of certain sentences involving the notations for such tensor fields, so indeterminacy in our own terms is well motivated in this case. In the case of indeterminacy in earlier terms we likewise have a motivation for positing indeterminacy: even if we ourselves use 'heavier than' determinately on each occasion (which I doubt), we can introduce a new term into our language stipulated to behave like the pre-Newtonian 'heavier than'; the arbitrariness we would feel in deciding whether it coincides in extension with 'more massive than' or 'more weighty than' would give grounds for having degrees of belief and disbelief that add to 0 for certain sentences involving this new term. And the reflection that such degrees of belief often are appropriate in the case of false theories, together with the truism that many of our current beliefs are likely to be only approximately true, i.e. false strictly speaking, make it natural to have degrees of belief and disbelief in claims of current physical theory that add to somewhat less than 1, even where we don't have the kind of specific grounds that we have in the tensor field example for suspicion of the term. In the size of the continuum example, the felt arbitrariness of any 'decision' as to the value—the lack of any constraints on the choice—seems to me to be good grounds for adopting nonclassical degrees of belief. (If it is protested that the actual value of the size of the continuum is such a constraint, that's like saying that the actual fact as end p.310
to whether Joe is rich serves as a constraint in the case of ordinary vagueness.) And the brief discussion of the complex number example at the start of section VII of the chapter is sufficient to motivate indeterminacy there even for a deflationist. end p.311 end p.312
Part 3 Objectivity end p.313 end p.314
11 Mathematical Objectivity and Mathematical Objects Abstract: Focuses on an issue about the objectivity of mathematics—the extent to which undecidable sentences have determinate truth-value—and argues that this issue is more important than the issue of the existence of mathematical objects. It argues that certain familiar problems for those who postulate mathematical objects, such as Benacerraf's access argument, are serious for those with highly 'objectivist' pictures of mathematics, but dissolve for those who allow
for sufficient indeterminacy about undecidable sentences. The nominalist view that does without mathematical entities is simply one among several ways of accomplishing the important task of doing without excess objectivity. There is also a discussion arguing for one kind of structuralism but against another. Keywords: Paul Benacerraf, continuum hypothesis, finiteness, indeterminacy, mathematical objects, mathematics, nominalism, objectivity, Platonism, structuralism, undecidability
Hartry Field Mathematics appears to be a highly objective discipline: there seem to be clear standards of rightness and wrongness in mathematics. One argument for the existence of mathematical objects, and for their having a nature that is independent of human opinions about mathematics, is that this is required for mathematics to have that kind of objectivity. Somewhat less vaguely, the argument might go like this. First, if mathematics is to be objective, then when we try to answer a mathematical question, we must be trying to figure out which answer to it is objectively correct, that is, objectively true: anything less than this would be a sacrifice of objectivity. So for instance, if we are trying to figure out the order of the Galois group of some polynomial (over the rational numbers), then of the possible answers A 0 : G has order 0, A 1 : G has order 1, A 2 : G has order 2, etc., one of them must be objectively true and the others objectively false. But second, if there are no mathematical objects, then the term 'G' (that is, 'the Galois group of p(x)') simply doesn't refer (and neither do '0', '1', '2', etc.); so there can be no more sense to the question of which of these answers is objectively true than there is to the question of exactly how many hemoglobin cells there are in Santa Claus's body today. So there must be mathematical objects, if mathematics is to be an objective discipline. For a variety of reasons, this isn't a very good argument, and I mention it now just to indicate a way that the two topics of this survey, mathematical objectivity and mathematical objects, have often been linked. My principal focus in the early part of the survey will be with mathematical objectivity, not with mathematical objects. But since I have mentioned the argument, I should indicate at the start one possible response to it. (Not in the end the best response, I think.) The response I have in mind is that the argument overlooks the possibility of understanding mathematical claims at other than face value. A widespread view in the philosophical literature is that a mathematical sentence which seems to make a claim about mathematical objects (groups, end p.315
polynomials, numbers, etc.) really does no such thing. How then is it to be understood? That differs from author to author, and maybe from one part of mathematics to another as well; but one possibility (see Putnam 1967, Hellman 1989) is that any such mathematical sentence is to be understood as a complicated kind of possibility statement, whose details I will not try to give. This might suggest that we can have mathematical objectivity without mathematical objects: even if there are no mathematical objects, why couldn't it be the case that there is exactly one value of n for which A n modally interpreted is objectively true? I'll come back to this idea, but first I want to look a bit more closely at the idea of mathematical objectivity.
1. Logical Objectivity and Specifically Mathematical Objectivity The idea that mathematics is an objective discipline is an idea with several facets.
A One way in which mathematics seems on its face to be completely objective is that it seems on its face that there are completely objective standards of mathematical proof. Perhaps standards of mathematical proof weren't always objective—that seems to be the moral of Lakatos 1976—but since about the time of Frege it has been required that proofs be formalizable. On current standards, to mathematically prove something one must state all nonlogical assumptions explicitly as axioms, and one must argue from one's axioms to the claim to be proved in a way that could be turned into a formal derivation given sufficient effort.
Of course, there are sometimes disputes as to whether an informal derivation could be turned into a formal derivation: but this is no serious qualification on the objectivity of mathematics, since such disputes are settlable. Is there any possibility of a more serious challenge to the view that standards of mathematical proof are completely objective? Yes, at least two challenges are possible. First, even if the above makes it an objective question what is a genuine derivation in a given derivation procedure, one might hold that there is no objective fact as to whether a genuine derivation in procedure P should count as a proof, by holding that there is no objective fact as to whether the logical inferences licensed by P are logically correct. More radically, one might hold with Wittgenstein 1956 (see also Kripke 1982) that there is something unobjective even about the drawing of consequences in a formal derivation procedure. Both of these positions are challenges to the objectivity of logic, and hence to the objectivity of mathematical proof. end p.316
I mention such challenges only to put them aside. From now on I will be assuming that logic, hence mathematical proof, is fully objective. And because proof is so important in mathematics, this concedes most of what we may have had in mind in calling mathematics objective. It ought to be obvious that if mathematics is objective only in this sense, then the link between mathematical objectivity and mathematical objects, contemplated at the start of the chapter, is wholly illusory: you don't need to make mathematics actually be about anything for it to be possible to objectively assess the logical relations between mathematical premises and mathematical conclusions.
B But there are further respects in which it might be asked whether mathematics is an objective discipline. These further issues concern the objectivity of mathematics per se, as opposed to logic: we can ask not about the objectivity of proof from given axioms, but rather about the objectivity of the choice of axioms. One possible position is that the 'correctness' of a mathematical statement is simply a matter of its being derivable from explicitly or implicitly accepted axioms (and of its negation not being derivable from those axioms—a qualification we can omit if we assume that the accepted axioms are consistent). We might call this 'extreme anti-objectivism' (though it really isn't that extreme, in that it casts no doubt on the objectivity of proof). More fully, the view is (i) that even when a mathematical statement is 'correct', this will only be as a result of our having explicitly or implicitly adopted axioms from which it is derivable; and consequently, (ii) that for sentences that our mathematical theories (including implicit axioms) can't prove or refute, neither they nor their negations are objectively correct. In the rest of this section and the next, my focus will be on evaluating aspect (ii) of the view. A popular example to illustrate aspect (ii) of anti-objectivism concerns the size of the continuum (i.e., the set of real numbers). In standard set theory (with the axiom of choice) one can prove that there are infinitely many sizes that infinite sets can have, and that these sizes fall into a simple order 0 , 1 , 2 , and so on. (Here, 'and so on' goes beyond the finite ordinals into the transfinite.) And there is a famous proof that the size of the continuum is bigger than 0 . But how much bigger? It turns out that virtually any answer you want to give to this question is consistent with standard set theory. (And indeed, with any expansion of standard set theory to include other axioms that are typically regarded as at all 'evident'.) That is, for all finite values of α other than 0, and for all but a few isolated transfinite values, the claim that the size of the continuum is α is consistent with everything we accept; one can't even put an upper bound on all the possible values. (This follows from celebrated end p.317
results of Gödel, Cohen, and Solovay.) And given these facts, it is very natural to wonder if it makes any sense to suppose that there is any such thing as an 'objectively correct' answer to the size of the continuum. The anti-objectivist says 'no': any answer to the question (aside from the few answers that are inconsistent with axioms we already accept) is equally good, and could be added on as a new axiom. (It wouldn't be 'evident', but being 'evident' is presumably only a requirement on axioms if your goal is to get axioms that are objectively correct.) There are other cases where the extreme anti-objectivist position seems much less plausible. For instance, Gödel showed not only that our mathematics (if it is consistent) leaves some sentences unsettled, he showed that it leaves unsettled some sentences of the simple form (*) For all natural numbers x, B(x) in which B(x) is a decidable predicate, hence a predicate such that for any numeral n we can either prove B(n) or
prove ¬B(n) (and by very uncontroversial proofs). But it is plausible to argue that any undecidable sentence of this form must be objectively correct. For the undecidability of (*) shows that there is no numeral n for which ¬B(n) is provable. (A proof of ¬B(n) would yield a proof of the negation of (*), contrary to its undecidability.) So by the supposition about B(x), it must be that for each numeral n, B(n) is provable (by a very uncontroversial proof), hence presumably objectively correct. And that seems to show that the generalization (*) is also objectively correct. This argument is not beyond controversy—for instance, it assumes at the last step that it is objectively the case that there are no natural numbers other than those denoted by numerals, which could conceivably be questioned—but it is surely very compelling. If so, an anti-objectivist should probably moderate his position to accommodate it. Indeed, nearly everyone believes that the choice between an undecidable sentence and its negation is objective not only for the simple sorts of number-theoretic statements just discussed, but for elementary number-theoretic statements more generally. This belief would be a very hard one to give up, since many claims about provability and consistency are in effect undecidable number-theoretic claims, so that an anti-objectivist about elementary number theory would need to hold that even claims about provability and consistency often lack objectivity. That is a position that few will want to swallow. Still, it isn't obvious that if one grants specifically mathematical objectivity to elementary number theory one must grant it to higher reaches of mathematics.
2. Mathematical Objects Without Specifically Mathematical Objectivity One might suppose that the issue of specifically mathematical objectivity (issue B above) is intimately connected to the issue of the existence of mathematical end p.318
objects (numbers, functions, sets, etc.) that mathematical theories are about. (a) In one direction, one might argue that if there are entities such as sets or numbers for mathematical theories to be about, then there is an objective answer to the question of which theory about those entities is true. (b) In the other direction, one might argue that this is the only way that we could get specifically mathematical objectivity. The argument for (b) might run something along the lines of the argument that opened the chapter. The argument would be slightly better now, in not suggesting that mathematical objects were required merely for the objectivity of mathematical proof; but it would still not be very good I think. But let's put (b) aside for now, and ask whether (a) is correct. Hilary Putnam has given a powerful argument (1980) that makes it very hard to see how the question of the size of the continuum could have an 'objectively correct' answer, even if there is a single fixed universe of mathematical objects. The argument, in sketchy outline, is that there are lots of properties and relations that the mathematical objects in this universe can stand in; and there isn't a whole lot to determine which such properties and relations we should take our mathematical predicates to stand for, beyond that they make the mathematical sentences we accept true. (Mathematical predicates, after all, are not causally tied down to their extensions in any direct way, in the way that observational predicates are; and unlike theoretical predicates in empirical science, they don't even seem to be very strongly tied down to their extensions in an indirect way.) So it looks like whatever choice α we care to make for the size of the continuum (as long as it's consistent with the rest of our set theory), we can find properties and relations for our set-theoretic vocabulary to stand for that will make that choice true and the other choices false; and there is nothing in our use of set-theoretic predicates that could make such an interpretation of the set theoretic vocabulary 'bizarre' or 'unintended'. That is Putnam's argument, in broad sketch; and if correct (as I believe it is), it makes it hard to get much objectivity as to the choice of set-theoretical axioms even assuming the existence of mathematical objects. (With minor variation, Putnam's argument can also be brought to bear against the objectivity of undecidable sentences involving higher-order quantification.) 1 Of course, there is a worry as to how far Putnam's argument can be pushed: does it equally show that there is no objectivity even for undecidable sentences in number theory? If so, that is reason to suspect his argument, for I noted before that it is hard to take seriously the supposition that there is no objectivity there. Fortunately, Putnam's argument does not extend to number theory: see Field 1994 or Chapter 12 of this volume for a more careful account of Putnam's argument that shows why this is so. Basically the moral of those 1 Boolos 1984 gives an attractive account of monadic second order quantifiers according to which they don't range over special entities, but instead are 'plural quantifiers'. This helps make (monadic) second order quantification acceptable to those with ontological qualms. But as far as I can see it does not help alleviate the force of Putnam's argument against the determinacy of second order quantification. end p.319
discussions is that Putnam's argument doesn't preclude that we have a determinate notion of finiteness2 that defies
formalization; and indeed, there is a natural account, consistent with Putnam's argument, of what makes the notion determinate. Given this, a moderate anti-objectivist position is that the 'correctness' of a mathematical statement is simply a matter of its being a consequence of accepted axioms (and of its negation not being a consequence of those axioms), in an objective but not-fully-formalizable sense of consequence that goes a bit beyond first order consequence in including the logic of the quantifier 'only finitely many'. This is enough to give rise to complete objectivity in number theory, because number theory is completely axiomatizable in the logic of finiteness (using the axiom that every natural number has only finitely many predecessors). I think that it is anti-objectivism of this moderate sort, rather than extreme anti-objectivism, that Putnam's argument really suggests. Let's use the notion of 'consequence' in the slightly broad sense just indicated, and use 'consistent' correspondingly: a set of sentences is consistent if not every sentence is a consequence of it in that broad sense of consequence.3 Then according to moderate anti-objectivism, mathematicians are free to search out interesting axioms, explore their consistency and their consequences, find more beauty in some than in others, choose certain sets of axioms for certain purposes and other conflicting sets for other purposes, and so forth; and they can dismiss questions about which axiom sets are true as bad philosophy. I suspect that many mathematicians would find this position highly congenial. If you like you can summarize the position by saying that there is no objectivity in mathematics beyond the objectivity in logic. But this summary requires two qualifications: first, the 'objective logic' here is not formalizable, it includes the logic of finiteness (though not higherorder logic); second, the summary is inaccurate if it is taken to imply that in choosing one system of axioms over another we can't take into account such factors as interestingness, utility, beauty, and concordance with one's concepts. The point of this 'anti-objectivism' is merely that truth adds nothing as a further constraint: it is too easy to achieve. When mathematicians decided to accept the axiom of choice (assuming for the sake of argument that they hadn't implicitly accepted it all along), they refined their pre-existing conception of set so that the axiom became true of it; if we were to give up the axiom of choice in favor of some alternative such as the axiom of determinacy, we would be revising our conception of set in such a way that the axiom of choice is false of it. Once you have consistency (in the expanded sense I've indicated), your advocacy of the axioms will be enough to make them true as you intend them. That in effect is what Putnam's argument suggests. 2 More exactly, of the quantifier 'only finitely many'. 3 In the case of impure mathematics, one really needs to broaden the notion of consistency in a different direction as
well, to what I've called conservativeness: see Field (1989/91: 55-8; n. on 96-7). This qualification doesn't affect anything of substance in the present paper, so I will ignore it in what follows. end p.320
3. The Prospects for Mathematical Objectivity Without Mathematical Objects If the view of mathematics just sketched is correct, then the argument that opened the paper—the argument that purported to derive the existence of mathematical objects from the objectivity of mathematics—went wrong prior to where the advocate of non-face-value interpretations challenged it. The argument from objectivity to objects had two steps. The first step was the claim that, if mathematics is to be objective, then when we try to answer a mathematical question, we must be trying to figure out which answer to it is objectively true. The second step was the claim that mathematical truth can only be made sense of in terms of mathematical objects. It is the second step that advocates of non-face-value interpretations of mathematics challenge. But whether or not they are correct about the second step, I have argued that the first step is already highly questionable: we can account for whatever mathematical objectivity there is quite independent of any assumptions we make about mathematical truth, using just the objectivity of logic in the slightly expanded sense given above. And not only can we do so, we must: for Putnam's argument shows that at least on the standard view of mathematical truth, where it is explained in terms of mathematical objects, truth is too easy to achieve to constrain our choice of mathematical axioms.4 Could the idea of specifically mathematical objectivity be saved by shifting to a non-face-value interpretation of mathematics? At first sight this seems plausible. Consider for instance the view that mathematics should be understood modally (Putnam 1967, Hellman 1989). Even if there are no mathematical objects, why couldn't it be the case that there is exactly one value of α for which C α ('The size of the continuum is α ') modally interpreted is objectively true? There are two reasons why it is doubtful that any objectivity is achieved in this way. The first is that there may be more than one scheme for modally translating set theoretic claims. Why not suppose that under one scheme for modally translating set theory, C 23 comes out true and the others false, whereas on another scheme it is C 817 that is the true
one? The second and probably more important difficulty is that modal translations, and non-face-value translations of mathematics more generally, employ powerful logical notions whose objective status is itself questionable. For instance, they certainly employ modal operators; and usually (as in both the cases of Hellman and Putnam) they also employ higher-order quantification, for which questions of interpretation arise that are very closely analogous to questions about the interpretation of quantification over sets. Indeed, the argument of Putnam (1980) that if there are mathematical objects there is no determinate answer 4 More fully, Putnam's argument shows that truth is too easy to achieve as long as there are (infinitely many)
mathematical objects. I'd add that if there aren't, then truth is too hard to achieve to be a useful constraint. end p.321
as to just which ones we are picking out when we speak of 'all sets' carries over completely to the case of secondorder quantification, with the result that the size of the continuum is no more objective on the 'mathematics as modal logic' picture than on the 'mathematical object' picture (or than on the fictionalist picture that will be discussed in section 4). We seem then to have an argument that on any view of mathematical truth, whether in terms of mathematical objects or not, mathematical truth does not give rise to specifically mathematical objectivity, and plays no role in accounting for the kind of mathematical objectivity that there is. But there are two places where some may feel that mathematical truth is playing a role behind the scenes, even on the picture of (non-specifically) mathematical objectivity I have enunciated. The first place mathematical truth might be argued to be playing an unannounced role is in the notion of logical correctness: it may be proposed that the idea of correct logical inference must somehow be based on a prior notion of mathematical truth. This proposal would be extremely hard to defend if applied to the inferences of first-order logic: how on earth could one develop mathematics prior to such logic? But it might seem better as applied to inferences involving the quantifier 'only finitely many': one might think that the correctness or incorrectness of such inferences must be based on mathematical truths about the natural numbers or about cardinality (hence on facts about natural numbers or sets, on the mathematical object picture). But even if this view were accepted (which I don't think it should be), it would not undercut the main point that I have argued. What I have argued is that in choosing among typical competing mathematical claims {A 1 ,. . . ,A n } (say about groups or topological spaces or whatever), the objectivity of the choice is unaffected by considerations about the truth value of the A i s. The view now under consideration would imply at most that issues about the truth value of certain other mathematical claims may be involved in the decision. The other place that mathematical truth might be argued to be playing an unannounced role is in my mention of utility. I have allowed (as surely anyone must) that considerations of utility play a role in our selecting some mathematical axioms over others. But it might be argued that if utility plays a role in the decision, truth is indirectly playing a role: for the utility of a mathematical theory can only be explained (or anyway, is best explained) in terms of its truth. But this strikes me as an unpromising view. One problem with it is that utility is relative to the purposes at hand: a given set theory could perfectly well be useful in one context while a 'competing' one (say, one that attributes a different size to the continuum) was useful in another. Of course, when we find use for prima-facie competing theories, we typically say that the theories don't really compete after all: an arithmetic in which 68+57 = 5 needn't count as false, it can be taken as a correct theory of, say, the numbers modulo 120. But this move is available, whatever the prima-facie competing theories (as long as they are consistent); since it is always available, lack of truth can never be used to rule out a mathematical theory, which is just end p.322
another way of saying that calling a consistent mathematical theory true seems completely unexplanatory of any utility it may have.5
4. The Existence and Nature of Mathematical Objects Those who have recognized the limitations on the objectivity of mathematics have tended to draw as a moral some 'anti-platonist' view about the nature of mathematical objects. For instance, one view often associated with a denial of objectivity is that mathematical objects are mind-dependent, or dependent on the beliefs of the mathematical community: that view of mathematical objects certainly suggests that any mathematical claim that we accept will be true providing it is consistent with the other mathematical claims we accept, and that is just what the limitations on objectivity we have seen also suggest. 6 Another view that seems to have the same anti-objectivist consequence is Dummett's suggestion (1959) that mathematical objects and mathematical facts pop into existence as we probe. There is another view with this anti-objectivist consequence which is less happily characterized as anti-platonist: it is the 'full
blooded platonism' of Balaguer 1995 and 1998, according to which there isn't just a single universe of sets, but many different ones existing side-by-side: some in which the continuum has size 23 , some in which it has size 817 , and so on. Also, as we've seen, Putnam's argument has it that even a view on which there is a single universe of sets, independent of the mind and existing prior to our probing, ultimately must yield the same anti-objectivist consequence as do these other views: for even if there is a single universe of pre-existing sets, there are multiple relations on it that are candidates for what we mean by membership, so that the effect of many universes is achievable in a single universe. 'Multiple universe' views and 'mind-dependent objects' views merely have the virtue of making the antiobjectivist consequence manifest. One more view with the same anti-objectivist consequence is the fictionalist view, on which there are no mathematical objects at all. Again, this gives rise to the same limitation on objectivity: without mathematical objects, anything goes, as long as it meets the requirements of consistency (in our broad sense); though of course some fictions are more useful, beautiful, etc. than others. The fictionalist view sounds at first more radical than the other forms of anti-platonism, in that on the most straightforward view of mathematical truth, there can't be mathematical truth in any interesting sense if there are no mathematical objects. A fictionalist may if he likes choose to avoid this 5 This last point also seems to me to cast doubt on the idea (contemplated in the previous paragraph) that the
correctness of logical inference must be based on the truth of mathematical claims. For if any consistent mathematics can count as true, which mathematics is it that constrains our logic? 6 Provided of course that consistency is understood in the strong sense discussed in section 2. end p.323
consequence by adopting a non-face-value view of mathematical language, according to which 'there are prime numbers bigger than a billion' really doesn't assert the existence of anything, but simply makes (say) a modal claim. Assuming that an acceptable non-face-value interpretation of mathematical language can be worked out, would this form of fictionalism be better than the form that simply says that mathematics isn't true? I think that the issue between these two forms of fictionalism would be wholly uninteresting: both agree on the metaphysics, they disagree simply on the semantics of ordinary language. (The dispute seems uninteresting even if we assume that ordinary language has a semantics clear enough to settle the issue; and that assumption is dubious.) It might indeed be doubted that there is a significant difference between fictionalist views and some or all of the other anti-objectivist views I've mentioned. The doubt is clearest for the anti-platonist views: can there really be an important issue between, say, the view that there are only mind-dependent mathematical objects and the view that there are no mathematical objects at all? There are various ways in which one might try to hold these views only verbally different— differing, say, only in the meaning they assign to 'exist'. An analogous doubt can be raised about the difference between anti-platonist views and Balaguer's 'full-blooded platonism'; Balaguer himself raises such a doubt near the end of Balaguer 1998. Indeed, it might even be argued that the difference between the anti-platonist views and the standard platonist view of a single universe of mind-independent mathematical objects collapses, if the standard platonist accepts Putnam's argument and recognizes the futility of thinking that those mathematical objects will supply a kind of objectivity that is unavailable on the anti-platonist views. So in the rest of the chapter when I discuss platonism, I will primarily have in mind the kind of platonism that does not accept Putnam's point: the kind of platonism according to which there is a uniquely correct answer to the size of the continuum, difficult as it may be to know what that answer is.
5. The 'Access' Argument Probably the most influential argument against platonism has been that it is hard to see how we could have epistemological access to mathematical objects as the platonist conceives them. This argument received its most influential articulation in Benacerraf 1973. The Benacerraf articulation of the argument was in terms of a causal theory of knowledge, and defenders of platonism have sometimes seized on the doubtfulness of any such theory of knowledge as their response to Benacerraf. But such a response is completely superficial: the point Benacerraf was making surely goes deeper. One way to make Benacerraf's point is in terms of the principle that a theory tends to be undermined if it needs to postulate massive coincidence. Consider the following two claims: end p.324
(1) John and Judy have run into each other every Sunday afternoon for the last year, in highly varied locales: in opera houses, at hockey games, in coffee houses, in sleazy bars, and so forth.
(2) John and Judy have no interest in each other and would never plan to meet; nor are they both in a club that has met in these varied locales, nor is there any other such hypothesis that could explain the perfect correlation between their locations on these 52 consecutive Sundays. (Of course, if the universe is deterministic there must be some sort of explanation of any correlation that might exist between John's and Judy's location: for each of their locations is separately explainable from the laws of physics and the initial conditions of the universe. What (2) is intended to rule out is the possibility of explaining the correlation in any more 'unified' way.) It seems clear that (1) and (2), though not jointly inconsistent, stand in strong tension: a system of beliefs that contained both (1) and (2) would be highly suspect. Put another way: if you believe the correlation in (1), you better believe that there is some unified explanation of it. But if this is so, won't platonism also be highly suspect, unless it postulates some explanation of the correlation between our mathematical beliefs and the mathematical facts (that is, some explanation of why it is that we tend to believe that p only when p, for mathematical p)? And it is hard to see how to explain such a correlation without postulating something extremely mysterious: a causal influence of mathematical objects on our belief states, a god who predisposes us to recognize the basic truths of the mathematical realm, or whatever. A platonist can partially answer this challenge by pointing out that there are logical interconnections between our mathematical beliefs. Indeed, if one considers only mathematics in the modern era when it has become highly axiomatized, one could argue that the task of explaining the correlation between our mathematical beliefs and the mathematical facts reduces to (a) explaining why we tend to infer reliably, and (b) explaining why we tend to accept p as a mathematical axiom only if p. This ameliorates the problem slightly, but certainly doesn't eliminate it, for the question arises as to how the reliability in (b) is to be explained if not by some non-natural mental powers or some beneficent god. One common response to the Benacerrafian argument as I have outlined it is that it 'proves too much': the claim is that if the argument were valid it would undermine a priori knowledge generally, and logical knowledge in particular; the unacceptability of the latter consequence would then show that there has to be something wrong with the argument. I have addressed this elsewhere (Field 1996, sect. V), arguing that there is a fundamental difference between the logical and the mathematical cases. I also argue there against the idea that the 'metaphysical necessity' of mathematics can be used to block the Benacerrafian argument. To my mind, the Benacerrafian argument is thoroughly convincing against any form of platonism that pretends to much 'specifically mathematical objectivity'. But it seems to me not to have a great deal of power against Balaguer's end p.325
'full blooded platonism',7 or against the Putnamian view that there is a single mind-independent mathematical universe but that the mathematical sentences we accept so directly determine their content that they are bound to come out true as long as they are consistent. Once again it is not the mathematical objects per se, but the claims about mathematical objectivity implicit in most standard platonism, that seem to give rise to the most serious problems.
6. The Structuralist Insight Another influential argument against some forms of platonism was presented in another classic paper of Benacerraf's (1965). The article begins by considering the significance of alternative reductions of elementary number theory to set theory: for instance, there is Zermelo's reduction, according to which 0 is the empty set and each natural number n>0 is the set that contains as its sole member that set that is n−1; and von Neumann's reduction, according to which each number n is the set that has as its members the sets that are the predecessors of n. It seems clear that there is no fact of the matter as to which of Zermelo's and von Neumann's identifications and any of the other identifications one might come up with 'gets things right': there isn't anything to get right. Just what broader implications to draw are controversial. One possibility, of course, is that numbers are simply sui generis entities, distinct from sets, so that all of the alleged identities are just false. It seems clear that Benacerraf didn't want to draw this conclusion, but he didn't want to draw the conclusion that numbers are definitely sets either: rather, his conclusion was that the sui generis option was just another option, on par with the Zermelo and von Neumann identifications: there is no fact of the matter as to whether numbers are sets, as well as there being no fact of the matter which sets they are if they are sets. The core idea—which I'll call the structuralist insight—is that it makes no difference what the objects of a given mathematical theory are, as long as they stand in the right relations to one another.8
If one focuses just on Benacerraf's example, it might seem attractive to deny the structuralist insight: given that we had the idea of natural number before the idea of set, the proposal that the ordinary idea of natural number is about 7 Indeed, the desire to get around the Benacerraf argument was the explicit motivation of Balaguer 1995. 8 More exactly, there is no choosing between two isomorphic models of a mathematical theory. Earlier in the paper I
advocated Putnam's model-theoretic argument. This accepts the structuralist insight and extends it. First, it extends it from isomorphic models to certain cases of elementarily equivalent models, that is, certain cases of models that differ in structure in a subtle enough way that they give rise to the same truth value for every sentence in the mathematical language. (Indeed, I think it extends the conclusion to all models that are elementarily equivalent in an expansion of first order logic that includes the finiteness quantifier standardly interpreted.) Second, it extends it to certain cases of models that aren't elementarily equivalent but give rise to the same truth values for all of our mathematical beliefs. end p.326
sui generis objects is certainly much more plausible than that it is about sets, and so the possibility broached at the start of the preceding paragraph is not unattractive here. But even if we were to accept this and consequently reject a full-blown structuralism, it is important to realize that the Benacerraf example is just the tip of the iceberg: throughout mathematics one is constantly defining kinds of objects, and almost every time one does so there is considerable arbitrariness in just how one does so. (Should one, for instance, define a 2-place function as a set of ordered triples, or as a function from objects to 1-place functions, or in some third way? Should one define a lattice as an ordered pair of a set and a certain kind of partial ordering on it, or as an ordered triple of a set and two operations of meet and join?) It doesn't seem at all attractive to regard all of these objects as sui generis, but here too it seems completely arbitrary which identification one makes. And an adequate philosophy of mathematics needs to account for this. One possible account of the Benacerraf phenomenon is the fictionalist one (see Wagner 1982): numbers are fictitious objects anyway, and while the fiction in which they standardly figure tells us that 0 and 1 each precede 2, it doesn't tell us which if any objects are members of 2; so asking what the members of 2 are is like asking what Little Red Riding Hood had for lunch the day before she visited her grandmother. An alternative account—the one that Benacerraf himself proposed, and is also proposed in Hellman 1989—is that arithmetic should be construed at other than face-value. In Benacerraf's version, it doesn't really treat of the numbers 0,1,2,. . . , but instead treats of arbitrary (actual or possible) progressions (ω-sequences) of distinct objects. As Kitcher (1978) points out, this really doesn't help as it stands, since the Benacerraf phenomenon arises for ω-sequences as much as for numbers. Hellman's response to this problem is to restate the ω-sequence idea in second-order logic, without use of special objects. It would take us too far afield to try to evaluate this here, or to investigate the prospects for an analogous treatment of other examples where the Benacerraf problem arises (lattices, topological spaces, tensor products, and so forth). Another alternative, similar in spirit to the Benacerraf—Hellman line but not requiring a non-face-value interpretation of mathematics (or second-order logic), is to treat mathematics as referentially indeterminate: our singular terms '0', '1', '2', etc. purport to single out unique objects, but fail to do so; similarly, our general terms like 'natural number' and '<' and 'is the sum of' fail to single out unique classes of or relations among objects. It isn't hard to develop this line in a way that allows the standard number-theoretic truths to come out true, but that makes there be no fact of the matter as to whether '2 is a set' is true. 9 (It is this 'ontological platonism coupled with substantial 9 I did so in Ch. 7. Kitcher 1978 says that his objection to the previous alternative applies to this one too, on the
ground that you need set theory to develop the account of truth for indeterminate mathematical sentences; but I think this extension incorrect, for I think the set-theoretic metalanguage can itself be regarded as indeterminate. end p.327
indeterminacy line' that the Putnamian argument of Section 2 extends: see footnote 8 .) Still a fourth alternative is that numbers are objects that are somehow 'incomplete': 2 has properties like preceding 3 and being prime, but simply has no property that determines whether it is a set. (This view is sometimes put by saying that 2 is simply a position in a structure, and that it has no properties other than its structural ones: see Resnik 1981, Shapiro 1989.) This view is rather like the view that there is vagueness in the world rather than in language. But I am skeptical that the view can live up to its motivations. I assume that the prima facie attraction of the view is that it avoids indeterminacy in language: the symbol '2' determinately refers to an incomplete object with only number-theoretic properties. This seems to work fine not only for '2' but for the terms we use to describe structures in which there are no symmetries; but symmetries create a problem. Consider an example from Brandom 1996. 10 In the theory of complex numbers, −1 (like every other nonzero complex number) has two square roots; the term 'i' is standardly introduced for one of them (−i, of course, being the other), and is standardly used in many calculations. But even assuming that we have somehow fixed which objects are the complex numbers, which subset of them are the real numbers, and which functions on them are addition and multiplication, our usage must leave completely undetermined
which of the two roots of −1 our term 'i' refers to: for there is no way to distinguish i and −i in the theory of complex numbers, no predicate A(x) not itself containing 'i' that is true of one of them but not the other. 11 So even if one says that 'i' is just a position in the system of complex numbers, there is indeterminacy, for the complex plane contains two structurally identical positions for roots of −1, with no distinguishing features. I doubt, then, that 'structuralism' in the sense of the fourth view is the best way to capture the structuralist insight.
7. Mathematical Objects and the Utility of Mathematics Finally a few words on Putnam 1971, an article with a rather different tenor from Putnam 1980. Putnam 1971 is the locus classicus for the view that we need to regard mathematics as true because only by doing so can we explain the utility of mathematics in other areas: for instance, its utility in science (e.g. for stating fundamental scientific laws) and in metalogic (e.g. for theorizing about logical consequence). And although Putnam earlier held that we can use modality instead of mathematical objects to explain mathematical truth, it is 8 More exactly, there is no choosing between two isomorphic models of a mathematical theory. Earlier in the paper I
advocated Putnam's model-theoretic argument. This accepts the structuralist insight and extends it. First, it extends it from isomorphic models to certain cases of elementarily equivalent models, that is, certain cases of models that differ in structure in a subtle enough way that they give rise to the same truth value for every sentence in the mathematical language. (Indeed, I think it extends the conclusion to all models that are elementarily equivalent in an expansion of first order logic that includes the finiteness quantifier standardly interpreted.) Second, it extends it to certain cases of models that aren't elementarily equivalent but give rise to the same truth values for all of our mathematical beliefs. 10 Brandom uses the example for a different, though not unrelated, purpose: for arguing against Frege's logicism. 11 Slightly more strongly, the function that takes x+iy into x−iy (for x, y real) is a real-algebra-automorphism of the complex numbers (i.e. a field-automorphism that leaves the reals fixed). It is also pretty clear that the applications of the complex numbers don't serve to distinguish i from −i, though I won't take the trouble to make this claim precise. end p.328
not at all clear that we can explain the applications of mathematics to contingent disciplines such as physics in ordinary modal terms. (That we can't do this in any reasonable way was the conclusion of Field 1989/91: 252-69. Hellman 1989 has a proposal for how to do it, but using a much more controversial kind of modality than the kinds needed elsewhere in his theory.) So this may provide an argument for mathematical objects as well as for mathematical truth. The general form of this Putnamian argument is as follows: (i) We need to speak in terms of mathematical entities in doing science, metalogic, etc.; (ii) If we need to speak in terms of a kind of entity for such important purposes, we have excellent reason for supposing that that kind of entity exists (or at least, that claims that on their face state the existence of such entities are true). There are two strategies for disputing the argument. The bold (some would say foolhardy) strategy involves substantially qualifying premise (i). The idea is to try to show that in principle we don't need any assumptions that seem on their face to postulate mathematical entities in formulating theories in science, metalogic, or elsewhere: we can in principle do these disciplines 'nominalistically'. Even if this 'nominalization project' can be carried out, we still need mathematical entities to do science, metalogic, etc., in a practical way: even if they are 'theoretically dispensable' they are practically indispensable. And so we need to explain the practical indispensability of mathematical objects. But the advocate of the bold strategy says that to explain that, given their theoretical dispensability, we need only show how mathematical entities serve to facilitate inferences among nominalistic premises. And (he continues) if facilitating inferences is the only role of mathematical entities then (ii) fails: something much weaker than truth ('conservativeness') suffices to explain this limited sort of utility. That's the idea of the strategy. Unfortunately, the nominalization project is nontrivial. I did a certain amount of work trying to carry it out some time ago. 12 I won few converts, but I'm a stubborn kind of fellow who is unwilling to admit defeat. The less bold (more likely to succeed?) strategy is to challenge premise (ii) of the argument in a more thorough-going way: to deny that we can get from the theoretical indispensability of existence assumptions to the rational belief in their truth. Putnam calls arguments based on (ii) 'indispensability arguments', and vigorously defends them, but there is a good bit of recent work in the philosophy of science arguing that they need some sorts of qualification, and many have argued that the sorts of qualifications needed rule out the application to the mathematical case. The most frequent basis for arguing this latter claim has to do with the fact that mathematical entities don't seem to be causally involved in producing
12 See for instance Field 1980 for the applications of mathematics to physics, and Field 1991 for the applications to
metalogic. end p.329
physical effects. This response has considerable plausibility. One worry about it is that if mathematical entities are theoretically indispensable parts of causal explanations (as (i) contends), there seems to be an obvious sense in which they are causally involved in producing physical effects; the sense in which they are not causally involved would at least appear to need some explanation (preferably one that gives insight as to why it is reasonable to restrict (ii) to entities that are 'casually involved' in the posited sense). I suspect, though, that a close look at the reasons that make it hard to theoretically dispense with certain mathematical entities in certain contexts would enable one to sharpen the intuition that the role of these entities is not causal and does not support a very convincing indispensability argument. 13 (For another way to argue that indispensability arguments are less plausible in mathematics than elsewhere, see Hawthorne 1996.) I will say no more here about the prospects for either the more bold or the less bold strategy for responding to Putnam 1971. But suppose both strategies fail: suppose that Putnam's argument forces us to believe in mathematical truth, and perhaps in mathematical objects. Would this force us to revise the conclusions earlier in the paper about mathematical objectivity? One might naturally suppose that it would: for if we need to regard mathematics as true in order to explain its utility for science or metalogic, then couldn't issues about the truth of disputed hypotheses such as the size of the continuum make a difference to science or metalogic? (For instance, since our theory of physical space has it that straight lines have basically the structure of the real numbers, mightn't the size of the continuum make a difference to the theory of physical space?) If so, that would presumably give an objective basis for deciding issues like the size of the continuum. Despite its superficial plausibility, I think this argument can be shown to be thoroughly misguided. The reason is that the part of the role of mathematical entities in theorizing that is not easily shown dispensable is their role as exemplars of possibilities: mathematics provides rich structures that are not found in the physical world but that are nonetheless highly useful in describing the physical world, and in describing logical inference patterns also. (See Shapiro 1983.) But in its role as a source of rich structures, set theory with one choice of continuum size and set theory with another choice are equal: if mathematics with one choice for the size of the continuum were used in an application, one could use mathematics with another choice for the size as well (if need be, by constructing a model for the second mathematics within the first).14 Even supposing that it is an objective matter whether physical lines 13 To test this out, suppose that the role of sets in physical theory was simply to allow us to assert the local
compactness of physical space; or suppose that its role was simply to allow us to accept (Cp ) rather than (Cs ), where these are as in Field 1989/91: 136-7. 14 'Construct' must be understood as a bit looser than 'define', because of well-known limitations on 'inner model' proofs. The second model could for instance be the result of collapsing a Boolean valued model (which is explicitly definable in the first model, but is not a model in the usual sense) by an appropriate ultrafilter; appropriate ultrafilters provably exist in the first model, but none will be explicitly definable in it. end p.330
contain 23 points, it doesn't follow that the acceptability of a mathematics that says that the continuum has size 23 should turn on this: for physical lines are one thing and the mathematical continuum (the set of real numbers as defined via Dedekind cuts) another. In other words, even if talk of the cardinality of physical lines makes objective sense and that cardinality differs from the cardinality one accepts for the mathematical continuum, it doesn't follow that one's mathematical theory is unsuited to the description of physical space: it's just that the structure of physical lines may not be quite that of the real numbers, but might be that of some other ordered field that 'looks a lot like' the real numbers. And that, it seems to me, is a conclusion we ought to regard as possible on independent grounds. The conclusion I have been trying to suggest, here as elsewhere in the chapter, is that what is of primary importance in the philosophy of mathematics isn't the issue of mathematical objects but the issue of mathematical objectivity. This is not an original view: it is one that Putnam attributes to Kreisel (Putnam 1975: 70). Kreisel, though, was presumably saying it in defense of a view on which mathematical objectivity far transcends logical objectivity; whereas I have been saying it in defense of the view that (barring the qualification about finiteness made in section 2) logical objectivity is all the objectivity there is. end p.331
12 Which Undecidable Mathematical Sentences Have Determinate Truth Values?
Abstract: Argues that typical undecidable sentences of set theory (e.g. about the size of the continuum) can have no determinate truth-value, since nothing in our practice can determine which 'universe of sets' we are talking about. But it gives an account of how typical undecidable number-theoretic sentences can have determinate truth-value. The account has it though, that, whether these undecidable number-theoretic sentences have determinate truth-value, turns on assumptions about the physical world that could fail, and it explores the consequence of their failure. It concludes with a critique of an argument that at least the Gödel sentence of our fullest mathematical theory must be determinately true: the critique is that the argument depends on the same sort of reasoning that leads to the semantic paradoxes. Keywords: continuum hypothesis, finiteness, Gödel, incompleteness theorem, indeterminacy, objectivity, second-order logic, semantic paradoxes, truth, undecidable sentences
Hartry Field
1. Metaphysical Preamble I will begin by contrasting three metaphysical pictures about mathematics. The first, which I will have little to say about in what follows, is the fictionalist picture. This says that strictly speaking, there are no mathematical entities; still, we can perfectly well reason from premises that postulate such entities, and systematic reasoning from such premises is both intrinsically interesting and highly useful in our practical affairs. On this view, mathematical theories aren't literally true. Of course, a fictionalist needs to say something about why these theories are useful if they aren't true: in disciplines other than mathematics, the utility of a theory is generally taken as good reason to believe that the theory is at least approximately true, and if it isn't good reason in the mathematical case then we need to know the relevant differences between the mathematical and the nonmathematical. Myself, I think that there are such relevant differences, and that the fictionalist view can be defended,1 but I will almost completely ignore it in what follows. The second picture about mathematics is standard platonism. On the usual version of this view, mathematical theories like number theory and set theory and the theory of real numbers are each about a determinate mathematical domain; or at least, a determinate mathematical structure, for there is no need to suppose that isomorphic domains (domains that have the same structure) are distinguishable from a mathematical point of view.2 Even if a certain sentence in set theory or whatever couldn't be decided in any mathematical theory that we could have reason to accept, still there is a fact as to whether it is true in the relevant domain or structure. If it is, then it is determinately true, and if not it is determinately false, so it has a determinate truth value despite its being undecidable. 1 An interesting recent paper relevant to its defense is Hawthorne 1996. 2 I do not mean to commit the standard platonist to assuming that 'domains' or 'structures' are sets: the view that set
theory is true of some determinate domain is not supposed to involve a commitment to a set of all sets. end p.332
The third picture about mathematics might be called 'plenitudinous platonism'.3 In one sense of 'platonism' it is an extremely platonist view; in another sense, it is the antithesis of platonism. It is extremely platonist in that it postulates lots of mathematical objects: at least as many as standard platonism does, and in some loose but intuitive sense, many more. But Kreisel famously remarked that the interesting issue isn't mathematical objects but mathematical objectivity, and on the objectivity issue, plenitudinous platonism is virtually indistinguishable from fictionalism. So what exactly is 'plenitudinous platonism'? Well, roughly what it is, is the view that whenever you have a consistent theory of pure mathematics (that is, a consistent theory that neither postulates non-mathematical objects nor employs non-mathematical vocabulary), then there are mathematical objects that satisfy that theory under a perfectly standard satisfaction relation.4 It is of course simply a theorem of set theory that whenever you have a consistent (first order) theory (pure mathematical or not) then there are mathematical objects that satisfy it under some satisfaction relation or other: that's the completeness theorem. What plenitudinous platonism adds is that as long as the consistent theory is purely mathematical, then you don't need to cook up some artificial model out of sets to satisfy the theory; rather, the theory is trivially satisfied by entities that the theory is about and that are guaranteed to exist purely by virtue of the consistency of the theory. Indeed, the plenitudinous platonist will add, the idea that you need to cook up some model out of sets to satisfy the theory is only well-motivated if one thinks that there is a uniquely privileged notion of set. But there are many internally consistent set theories that conflict with each other (differing for instance over the size of the continuum); for each one, there are mathematical entities satisfying it; and there is no point in supposing that there is a privileged notion of set such that the entities satisfying each of these theories are all constructed out of the entities satisfying the privileged theory. I don't pretend that this is a claim of extreme precision, but I do think it has a fairly clear intuitive content. Another imprecise way to put it is as the view that all the consistent concepts of set and membership are instantiated side by
side. 'Instantiated side by side' is intended to convey the idea just stressed: we refuse to single out one instantiation as privileged and to regard all others as merely 'unintended models' generated by the completeness theorem. But it also suggests that we could regard all quantifiers over mathematical entities in 3 Mark Balaguer (1995) calls it 'full blooded platonism'. An earlier version of Balaguer's paper (a chapter from his
dissertation) helped convert me to the attractions of the position. (I had hinted at something like it earlier, but without following through on its implications: see Field (1989/91: 275-8), especially the discussion of whether there is a broadest possible notion of set.) David Papineau (1993) calls the position 'postulationism'; he argues against it. Penelope Maddy called it 'plentiful platonism' at the same conference at which this was presented, which suggested the term in the text. 4 More generally, one might hold that whenever you have a theory that postulates mathematical objects, then as long as the theory is 'nominalistically correct' in the sense that all its consequences that don't postulate such objects and don't use specially mathematical vocabulary are correct, then there are mathematical objects that satisfy the theory under a perfectly standard satisfaction relation. end p.333
a mathematical theory as implicitly restricted by a predicate to which all other predicates of mathematical entities in the theory are subordinate. In different mathematical theories the overarching predicate is different; so mathematical theories that appear to conflict with each other when written without their overarching predicates don't really conflict. (It isn't necessary to say that there is anything to preclude meaningfully quantifying over all mathematical entities at once, without an overarching predicate; one can say just that there isn't anything interesting and true to say about so plenitudinous a realm.) The phrase 'instantiated side by side' suggests that nothing is included under more than one overarching predicate. Actually though it is better to say that there is no mathematical interest to the question of whether things falling under one overarching predicate also fall under another, and the matter can be conventionally decided either way. If sets 23 are entities that satisfy standard set theory plus the claim that the size of the continuum is 23 , and sets 817 are entities that satisfy standard set theory plus the claim that the size of the continuum is 817 , then it is mathematically uninteresting whether the sets 23 are included among the sets 817 , or the sets 817 are included among the sets 23 , or neither inclusion holds. (If an inclusion does hold, or there is overlap, the membership relations ε 23 and ε 817 need not coincide on the common domain.) If we want to decide the matter by convention, the best convention is probably to say that neither inclusion holds, simply in order to emphasize that for mathematical purposes neither sets 23 nor sets 817 have privileged status. There is a strong contrast then between plenitudinous platonism and Quinean platonism. Quinean platonism takes as basic some one conception of set, and constructs out of sets so conceived all other mathematical objects: natural numbers, real numbers, and if we want, sets corresponding to other conceptions. Most platonists think the Quinean insistence on viewing natural numbers and real numbers as 'really sets' is perverse and arbitrary: why not regard them as perfectly good objects existing on their own? Plenitudinous platonism just goes this anti-Quineanism one better: we should also regard sets that satisfy conceptions other than our own as perfectly good entities in their own right, in no way requiring an explanation in terms of 'our' sets. Ontologically speaking, then, plenitudinous platonism is highly platonistic, indeed more platonistic than standard platonism: roughly, it postulates multiple mathematical universes where standard platonism (especially Quinean platonism) postulates only one. But methodologically speaking, plenitudinous platonism is quite anti-platonistic (or as I prefer to say, anti-objectivistic). To illustrate: The usual platonist view is that even after we know that the continuum hypothesis is undecidable from the standard axioms, there is still a serious question as to whether it is true, and we can still find indirect evidence for its truth. The plenitudinous platonist view is that there is no such intelligible question: set theory with the continuum hypothesis and set theory with various alternatives to it are all consistent, so all true of their appropriate domains; and the 'indirect evidence' is simply a matter of exploring the logical implications end p.334
of each set theory and making aesthetic judgements about their attractiveness and practical judgements about their utility based on these implications. Obviously this is methodologically identical to the fictionalist view, that takes each such set theory to be a fiction and evaluates the fictions on aesthetic and practical grounds. Actually in saying that plenitudinous platonism has this 'anti-objectivist' methodological consequence, I'm being a bit quick. After all, one might adopt the ontological position that there are multiple 'universes of sets' and hold that nevertheless we have somehow mentally singled out one such universe of sets, even though anything we say that is true of it will be true of many others as well. But since it is totally obscure how we could have mentally singled out one such universe, I take it that this is not an option any plenitudinous platonist would want to pursue.
I have tried to spell out plenitudinous platonism in a way that makes it look attractive, but it really doesn't matter for the rest of the chapter whether you think it even makes sense. 5 For my real interest isn't plenitudinous platonism per se, but the associated issue about objectivity just distinguished from it. And this issue can arise even in the usual platonist picture of a single universe of sets. That is, if we now take 'standard platonism' to mean simply the usual 'single universe' picture, then the advocate of standard platonism could find himself wondering how we have managed to single out the full universe as opposed to a subuniverse, and the standard membership relation as opposed to some nonstandard one. This is just the question that Hilary Putnam raised in (Putnam 1980), from a 'single universe' perspective. Putnam argued in fact that there is no way that we can have managed to determinately single out the full universe of sets and the membership relation on it, and that the incomplete content we have succeeded in giving to 'set' and 'member of' is not enough to determine the truth value of all set-theoretic sentences. In this way, he drew the same anti-objectivistic methodological consequence that the plenitudinous platonist arrives at more directly. In other words, though the usual version of standard platonism has built into it that we do have the settheoretic universe and the set-theoretic membership relation determinately in mind, this is really not part of standard platonism per se; and if Putnam is right, this usual version of standard platonism cannot be maintained. I'm inclined to think that Putnam is right: that the 'anti-objectivist' methodology is in the end the right conclusion for anyone to draw, whatever their ontological views. Perhaps the main advantage of plenitudinous platonism over standard platonism is that (like fictionalism) it leaves little room for disguising this. 5 For a response to one doubt about the coherence of plenitudinous platonism, see the Appendix to Field 1994.
(Many of the other main ideas of the present chapter are discussed there as well.) Tony Martin raised another interesting doubt after the conference which I do not have the space to pursue here. end p.335
2. The Objectivity Issue The issue that I will be concerned with, then, is the objectivity issue. If we put the fictionalist option aside, we can formulate the issue as: which of our mathematical sentences have determinate truth values? I will assume that the mathematical sentences we accept are all determinately true (or rather, that those we would continue to accept when any logical errors are filtered out are all determinately true). This seems reasonable, given that the content of our mathematical sentences is determined in large part by which ones we accept, and that there is no reason (independent of fictionalism anyway) to think that such independent determinants of content as there may be exert any pressure toward taking accepted mathematics as untrue. (Roughly speaking, then, accepted mathematics is 'true by convention', or true by the logical consequences of our conventions.) 6 ,7 Given this, we can rephrase the question as: (DTV) Which undecidable mathematical sentences 8 have determinate truth values? By 'undecidable mathematical sentences' I mean 'mathematical sentences such that neither they nor their negations follow in first-order logic from our fullest mathematical theory'. Platonists in Kreisel's sense—objectivists, I'm calling them—assume the answer to be 'all' (all the ones well formulated in our currently best mathematical language, that is); the extreme anti-objectivist position is that the answer is 'none'. Although the remarks at the end of Section 1 might suggest that I favor extreme anti-objectivism, I actually tend to favor an anti-objectivism not quite so extreme; for we'll see that our 6 Quine pointed out long ago that there is no hope of making sense of the idea that mathematics and logic together
are true by convention (since logic is required in determining the consequences of one's conventions); but the difficulty doesn't arise if one confines the claim to mathematics alone. 7 I am assuming in this paragraph that the mathematics we accept is consistent: were we to accept inconsistent mathematical claims, we wouldn't want to take them all as true. (Probably the best thing to say is that if our overall mathematical theory is inconsistent, then anything implied by all the most natural consistent replacements of it is true by convention, whereas anything implied by some but not all of the natural consistent replacements gets no determinate truth value from our inconsistent convention.) (I take consistency as a logical notion rather than a mathematical one (see Field 1991), so that taking our mathematics as true merely by convention does not make consistency claims true merely by convention. I don't doubt that our mathematical conventions could indirectly affect the notion of consistency; but they do not do so in a way that makes it trivial that our mathematical conventions come out consistent.) 8 By a mathematical sentence I don't mean any sentence that mentions mathematical objects or employs mathematical vocabulary: that would allow mathematically formulated sentences of physics to count as mathematical, and I don't intend (DTV) to cover them. On the other hand, 'sentence that neither postulates non-mathematical objects nor employs non-mathematical vocabulary' is too restrictive: it's a reasonable definition of pure mathematical sentence, but I think 'There is a set of all grapes' should count as (impurely) mathematical. Roughly, by a
mathematical sentence I mean the sort of sentence that would be appropriate to settle by mathematical considerations alone. But it is too much trouble to try to make this precise, and for present purposes it won't do much harm to restrict attention to pure mathematical sentences. end p.336
conventions can sometimes make determinately true certain sentences of pure mathematics that are not logical consequences of those conventions.9 Two points of clarification. First, when I say that certain mathematical sentences might lack determinate truth value, I do not intend to suggest that we must abandon classical reasoning in connection with those sentences. In my view a great many concepts involve some sort of indeterminacy—for instance, vagueness—and as a result many sentences containing them lack determinate truth value. It would cripple our ability to reason if we were prevented from using classical logic whenever indeterminacy might arise. Fortunately, it is not necessary to do so: we can perfectly well say that everyone is either bald or not bald, as long as we add that not everyone is either determinately bald or determinately not bald. What's crucial to the logic of vagueness isn't that we give up classical logic but that we add to it a new 'determinately' operator—in effect, a notion of a sentence being determinately true. The same holds in the case of other sorts of indeterminacy. Consequently, standard mathematical reasoning can go unchanged when indeterminacy in mathematics is recognized: all that is changed is philosophical commentaries on mathematics, commentaries such as 'Either the continuum hypothesis is determinately true or its negation is determinately true.' Second clarification: it might be protested that it is unclear what should count as 'our fullest mathematical theory', and that this makes 'undecidable' in (DTV) unclear. I agree, but I don't think it matters much for question (DTV): for on any reasonable construal of what counts as our fullest mathematical theory, there will be sentences undecidable in it, 10 and the question is which (if any) such sentences have determinate truth value. We should construe the extreme antiobjectivist as asserting that if a sentence is undecidable on every reasonable candidate for our fullest theory (and some mathematical sentences surely will meet this condition), then it definitely has no determinate truth value; whereas if it is undecidable on some reasonable candidates but decidable on others then there may be no clear fact of the matter as to whether it has a determinate truth value. I think that anti-objectivism has considerable plausibility for the typical undecidable sentences of set theory. It has much less plausibility for the undecidable sentences of elementary number theory: these strike almost everyone pretheoretically as having determinate truth value, though we may not know what it is. I suspect that this feeling arises from the feeling that we have a 9 A simple example of how this could happen in principle: suppose that our fullest mathematical theory did not imply
the axiom of infinity (taken as: the axiom that there are infinite pure sets), but did include both the axiom that there is a set of all physical things and a replacement schema that applies to physical vocabulary as well as mathematical. Then if the determinate truths of physics include the claim that there are infinitely many physical things discretely ordered under some physical relation, then the axiom of infinity follows from true physics plus the mathematics, but not from the mathematics alone. (The point is that set theory with the denial of the axiom of infinity isn't 'conservative' see: Field 1989/91: 56-7, for an elaboration.) A similar illustration might in principle arise for the axiom of inaccessibles, but only if the claim that there are inaccessibly many physical objects could in principle achieve the status of a determinate truth about the physical world. 10 Assuming it to be consistent. end p.337
determinate notion of finitude (that is, of 'finite set', or of the quantifier 'only finitely many', which I abbreviate ' '). Let N + be elementary number theory plus 'Every natural number has only finitely many predecessors'. Our fullest mathematical theory includes N + ; and if 'finite' is determinate, all models of this that aren't definitely 'unintended' are isomorphic, so all give the same truth value to any number theoretic sentence A, so A determinately has this truth value by virtue of our commitment to N + . The incompleteness of formal arithmetic results from the incompleteness of the theory of ' '; but if the latter is held determinate despite the undecidability of certain sentences in it, the same will hold derivatively of the concept of natural number. If we do assume the determinacy of finitude, then we can give determinate sense to a conception of semantic consequence for -sentences which builds in that ' ' gets 'the correct' truth conditions. Then if we call a sentence -decidable iff either it or its negation is a semantic consequence (in this sense) of our fullest mathematical theory, any -decidable sentence will get a determinate truth value. So the key question is: can the assumption that ' ' is perfectly determinate be maintained? There is certainly reason to hope that it can: the notion of finiteness is after all a central ingredient in many key notions, such as that of a sentence in a given language (sentences being finite strings of symbols meeting certain conditions) and that of a proof in a given system (proofs being certain finite strings of sentences). Any indeterminacy in
the notion of finiteness would doubtless infect the notions of sentence and proof, and of logical consequence and logical consistency, 11 and perhaps would indirectly infect even the first-order logical constants themselves (since our understanding of them depends on proof or consequence). Whether such a conclusion would be completely devastating is a question I will return to. But it is clear that we would like to be able to maintain the determinacy of 'finite', and thus of 'natural number'. And I think we can: I will argue that there is a natural account of how our practices might give determinate content to undecidable but -decidable sentences, an account that does not extend to the typical undecidable sentences of set theory.
3. Putnam's 'Models and Reality' and the Concepts of Finiteness and Natural Number In the first half of his 1980 article, Hilary Putnam gave what I think is a compelling argument against the objectivist position in set theory: he argued that even accepting the component of standard platonism that says there is a single set-theoretic universe
are related to proof and model indirectly by a 'squeezing argument': see Field 1991. end p.338
consequence relation in second-order logic, even if that is interpreted as in Boolos 1984.)12 In basic outline, the argument is this: (ia) There is nothing in our inferential practice that could determine that our term 'set' singles out the entire settheoretic universe V rather than a suitably closed subpart of V. (ib) Even on the assumption that it singles out the whole set-theoretic universe V, there is nothing in our inferential practice that could determine that our term 'ε' singles out the membership relation E on V as opposed to some other relation on V that obeys the axioms we have laid down. (ii) The indeterminacy in (ia) and (ib) is sufficient to leave indeterminate the truth value of typical undecidable sentences of set theory. (Actually (ib) by itself would suffice. In many cases (ia) by itself would also suffice.) My interest in this part of the paper is in the scope of this argument: assuming it basically correct, just what undecidable statements does it cover? What are the limits of the semantic facts that our inferential practices might determine? I will try to show how our inferential practices might determine the semantics of 'set' and 'ε' well enough to make determinate the quantifier 'only finitely many' (when defined in terms of 'set' and 'ε' in one of the standard ways), and hence determine the truth value of -decidable but undecidable sentences, including the undecidable sentences of number theory. Some readers of Putnam have thought that his argument for indeterminacy is based simply on the existence of nonstandard models. 13 But if that were so, it would apply as much to non-mathematical language as to mathematical. It seems pretty clear that that is an unattractive consequence: the natural picture is that our practice of accepting and rejecting sentences containing predicates like 'red' and 'horse' and 'longer than' largely determines which of the objects and pairs of objects that we quantify over satisfy these predicates. This seems plausible because our inferential practice with these words includes not only general theoretical principles, but an observational practice which causally ties their extensions down. Other physical predicates, say 'neutrino', are less tied to observational practice, but the theoretical principles governing them include words that are more tied to observational practice, and this does a lot to fix their extension. The prima-facie problem in the mathematical case is that the theoretical principles of pure mathematics don't tie down predicates like 'ε' even in this indirect way. This might suggest that in the case of mathematical predicates, indeterminacy of extension follows simply from the existence of nonstandard models. If this were so, there would be no hope of exempting the finiteness quantifier from indeterminacy. Let a grossly nonstandard model of set theory be one in which 12 So attempts to use the second order consequence relation to evade Putnam's argument are question-begging. For
further discussion, see Weston 1976, and Field 1994. 13 See for instance Lewis 1984. That reading is not without textual support, both from other writings of Putnam at about the same time as his (1980), and even from the second half of that paper; nonetheless it seems to me to miss the interesting argument in the first half of that paper. end p.339
there are objects y that satisfy 'finite set' even though for infinitely many objects x, the pair <x,y> satisfies 'ε'. (This is equivalent to a non-ω model.) If set theory has models at all it has grossly nonstandard ones, and in these ' ' gets a
nonstandard interpretation. But our inferential practice with terms like 'set' and 'ε' extends beyond pure mathematics: we use these notions in physical applications. That is, anyone who has learned the notions of set and membership will apply them (and related notions like that of a function, that are usually defined in terms of them) to the physical world. These physical applications of 'set' and 'ε', not just the mathematical applications, are available to help determine their extensions (given that the extensions of the physical predicates is reasonably determinate). And perhaps these physical applications make determinate the extension of 'finite' and hence the contribution that ' ' makes to truth conditions. I will argue that if our physical theory includes certain reasonable but not totally obvious 'cosmological assumptions', and if these assumptions are correct, then the contribution of ' ' to truth conditions will indeed be determinate, so that -decidable sentences will get determinate truth values. The cosmological assumptions I will use are these: (A) time is infinite in extent; (B) time is Archimedean. More precisely, define Φ(Z) to mean Z is a set of events which (i) has an earliest member and a latest member; and (ii) is such that any two of its members occur at least one second apart. Then assumption (A) is that there is no finite bound on the size of sets that satisfy Φ; assumption (B) is that only finite sets satisfy it. In accordance with the discussion above, I am going to assume that our physical vocabulary is quite determinate. Indeed, for simplicity, I will assume that it is completely determinate, at least relative to a choice of domain of quantification. (I will not need to assume that the domain of our quantifiers is determinate, even when the quantifiers are restricted to physical entities.) More precisely, I will take an interpretation or model of our own language to be 'unallowable' or 'objectively unintended' unless the extension of 'cow' in the model includes precisely those members of the domain of the model that are cows; and similarly for every other physical predicate of the language. As I've said, this is doubtless a bit more determinacy than we actually have, but I don't think the oversimplification is harmful in the present context. Reasons for not assuming the analogous determinacy in the mathematical case were given earlier in this section. Now, let S consist of the above cosmological assumptions plus set theory. (Impure set theory; that is, set theory that postulates sets whose members are non-sets, including a set of all nonsets, and including sets defined using physical vocabulary.) From S we can infer the following: end p.340
(*)
xA(x) ≡ Y Z f[Φ(Z) & Y contains precisely the x such that A(x) & f is a function that maps Y 1-1 into Z].
I claim that if we accept S, then this consequence of it allows us to extend the determinacy in the physical vocabulary to the notion of finiteness. I don't deny of course that the theory S has nonstandard models, in which certain infinite sets satisfy Φ(Z) and hence in which certain infinite sets satisfy the predicate 'finite'. However, if assumption (B) is true, any such model must assign a nonstandard extension to the formula Φ(Z); and in particular, it must either contain things that satisfy 'event' which aren't events, or it must contain pairs of events which satisfy 'earlier than' or 'at least one second apart' even though the first isn't earlier than the second or the two aren't one second apart. Either way, the model will violate the constraint on the interpretation of the physical vocabulary, viz. that the extension of such a predicate in the model can only contain things that actually have the corresponding property. In sum: if assumption (B) holds then no model of S in which 'event' and 'earlier than' and 'at least one second apart' satisfy the constraints on the interpretation of the physical vocabulary can be one where any infinite sets satisfy Φ(Z). 14 But it's easy to see that no pair of an infinite set and a finite set can satisfy ' f (f is a function that maps Y 1-1 into Z)' in any model; so no allowable model of S can be one where any infinite sets satisfy 'finite'. And of course there are no models of S where any finite sets fail to satisfy 'finite', since S includes set theory and genuinely finite sets satisfy 'finite' in every model of set theory; so (if assumption (B) holds) in every allowable model of S, a set satisfies 'finite' if and only if it is genuinely finite. As a corollary, every number-theoretic sentence gets the same truth-value in every allowable model of S. This would be cold comfort if assumption (A) weren't also true: if it were false, no model meeting the constraints on the interpretation of our physical vocabulary could satisfy set theory plus (*) (since finite sets bigger than any sets that actually satisfy Φ would be constrained to be both finite and infinite); in that case, the interpretation of 'finite' couldn't be
fixed by (*). (B) without (A) is not enough.15 The key to the argument, of course, is the assumption that the physical world provides an example of a physical ωsequence that can be determinately singled out. The cosmological assumptions (A) and (B) are really unduly restrictive, since they entail a specific way in which it might be possible for a physical ω-sequence to be determinately singled out. Other ways are possible too: for instance, it might be that while time is non-Archimedean, a certain 14 By 'infinite set' here I really mean 'object y such that for infinitely many x, <x, y> satisfies "ε" in the interpretation'. 15 If we modified (ii) in Φ(Z) to say merely that there is some unit of time such that any two members of Z occur at
least that unit of time apart, the infinitude assumption becomes easier to satisfy (though still nontrivial): it is now true as long as time is either infinite in extent or infinitely divisible. But the Archimedean assumption would be correspondingly harder to satisfy, given the possibility of points that are infinitesimally close. Since it might in any case be doubted that there is a definite physical fact as to whether there are infinitesimals, I think it safer to stick with a formulation for which they are irrelevant. end p.341
kind of matter exists only in the initial ω-sequence of it (but exists arbitrarily late through that ω-sequence); or it might be space rather than time that provides the physical ω-sequence. However the physical ω-sequence is determinately singled out, it is easy to then use the bridge between the physical and the mathematical that is given in standard (impure) mathematics to make the notion of finiteness fully determinate even in its mathematical applications. 16 It might be thought objectionable to use physical hypotheses to secure the determinacy of mathematical concepts like finiteness. I sympathize—I just don't know any other way to secure their determinacy. It might be thought especially objectionable to use physical hypotheses to secure the determinacy of mathematical concepts like finiteness when those hypotheses are themselves expressed in terms of those mathematical concepts. But recall that on my view, we needn't first secure the determinacy of a concept before we use it in reasoning: if that were required, reasoning could never get started. Rather, we can reason classically with our concepts from the start. Indeed, I have claimed that such classical reasoning isn't to be called into question should we later discover by such reasoning that the concepts employed in it lack determinacy; it certainly isn't to be called into question when (as envisioned here) we discover by such reasoning that the concepts are determinate.
4. Extreme Anti-Objectivism I think it's pretty clear that the sort of considerations just given in the case of the theory of natural numbers do not extend to typical undecidable statements of set theory, such as the continuum hypothesis.17 This does not seem to me in the least disturbing: I don't see any pretheoretic reason why such statements should be assumed to have determinate truth value. Their lack of determinate truth value seems to me to be fully compatible with accepted methodology in mathematics. I have already pointed out that the recognition of indeterminacy in no way forces us to give up classical reasoning. Also, we can still advance aesthetic criteria for preferring certain values of the continuum over others; we must now view these not as evidence that the continuum has a certain value, but rather as reason for refining our concepts so as to give the 16 Although the argument as presented assumes the complete determinacy of the physical vocabulary (relative to a
choice of domain of quantification), I think it is clear that relaxing this a bit wouldn't undermine the force of the argument. 17 Given a reasonable assumption about the limits on how determinate quantification (even restricted to physical things) can be, this follows from a slight generalization of the usual relative consistency proofs to the case of impure set theory, coupled with a downward Skolem-Löwenheim argument. (I won't give the details, but pp. 415-16 of Field 1994, in addition to Putnam 1980, should help the reader see the general idea.) But even if the assumption about the indeterminacy of quantification is relaxed, the prospects for giving sentences about the size of the continuum determinate truth value are quite bleak. Perhaps we could exclude as 'unintended' any model in which the mathematical continuum doesn't have the cardinality of physical space; but even if so, it doesn't rule out an indeterminacy in what functions (from pure sets to pure sets) there are, and that will give rise to an indeterminacy in what the cardinality of physical space is. (With fewer functions, there are more intermediate cardinalities, so the cardinality of physical space is higher.) For a bit more on these points, see pp. 419-20 of Field 1994. end p.342
continuum that value, but I don't see this as in violation of any uncontroversial methodological demand. What might be more disturbing about the considerations advanced here is that the case for the determinacy of finiteness was based on certain 'cosmological hypotheses'. Not, to be sure, on the specific cosmological hypotheses
(A) and (B): as noted, these could fail and other cosmological hypotheses be used in their place. But if neither (A) and (B) together, nor any suitable surrogate for them, were correct, then the argument that undecidable but -decidable statements have determinate truth value would break down. And this may seem disturbing: most of us feel that we have a perfectly definite conception of finiteness, which gives a definite truth value to -decidable statements even if we don't know what it is. The consequences of giving this up seem quite radical: I observed earlier that if we don't have a determinate conception of finitude, then we don't have a determinate conception of formula of a given language, or theorem of a given system, or consistency of a given system. Should we really conclude that our conviction that we do have such determinate conceptions ought to depend on the belief that either cosmological hypotheses (A) and (B) or some suitable surrogate for them are correct? I'm not sure what the answer to this is. It would be nice if the mere possibility of cosmological hypotheses like (A) and (B) should somehow be enough to ensure the determinacy of 'finite' and thus of -decidable sentences. But there is a considerable obstacle to seeing how this would go: for instance, we can't expect a generalization of (*) in which a possibility operator precedes the ' Z', since then the possibility of infinite sets satisfying Φ would make infinite sets come out finite. (It is possible for the universe to be nonstandard—in whatever sense it is possible for it to be infinite if in fact it is finite.) But I don't think it completely obvious that we couldn't live with the idea that our conception of finitude (and hence of formula, proof, consistency, etc.) is not fully determinate. For even if there is no way to make sense of the idea that "nonstandard" models of 'finite' are "objectively unintended", still that doesn't show that 'finite' isn't determinate enough to give determinate truth values to typical sentences containing it. So, for any string of less than (say) 10 20 symbols, it would seem to be a perfectly determinate matter of fact whether it was a formula of a given language; and if it is humanly provable that a certain string of symbols is infinite, then that string is definitely not a formula of the language. Perhaps this is all the determinacy that we have reason to be confident of.
5. Does Gödel's Theorem Refute Extreme Anti-Objectivism (1)? But even if one is willing to swallow that our conception of finitude is somewhat indeterminate, one may think that some undecidable sentences must have determinate truth values, contrary to 'extreme anti-objectivism'. In end p.343
particular, one might think that the Gödel sentence of our fullest mathematical theory must certainly have determinate truth value: it must be determinately true if our fullest mathematical theory is consistent, and determinately false otherwise. I am going to argue that this argument is question-begging, but before doing so, I would like to deal with an objection to the way I have formulated it: some may feel that the right way to put what Gödel's theorem shows is that there is no such thing as 'our fullest mathematical theory', or that if there is such a thing, it isn't recursively enumerable and therefore has no Gödel sentence. More fully, the view is that no consistent recursively enumerable theory M could be our fullest mathematical theory, since 'our fullest mathematical theory' would have to be closed under Gödelization: if it included M, it would have to include M's Gödel sentence G M .18 If we accept this, and exclude the possibility that 'our fullest mathematical theory' is inconsistent, we are left with the conclusion that 'our fullest mathematical theory' either doesn't exist or isn't recursively enumerable. This objection seems to me misguided: the most one can get from Gödel's theorem is that the phrase 'our fullest mathematical theory' is vague, and that for each consistent and recursively enumerable theory M that is a pretty good candidate for its denotation, M {G M } (or a theory that includes that) is also not bad as a candidate for its denotation. Compare 'bald': under the usual crude idealization that baldness depends only on the number of hairs on the head, we know that if {x | x has less than n hairs} is a pretty good candidate for the extension of 'bald', then {x | x has less than n + 1 hairs} isn't a bad candidate either. But of course one can't conclude from this that the extension of 'bald' is closed under the addition of a hair, for that would imply that everyone is bald (as long as they don't have infinitely many hairs). Similarly, the fact that M {G M } is about as good a candidate as M for the denotation of 'our fullest mathematical theory' doesn't imply that our fullest mathematical theory is closed under Gödelization. And that conclusion would be thoroughly 18 Strictly speaking, 'the Gödel sentence of M' is ill-defined: the Gödel sentence depends not just on M but on a way
of defining 'axiom of M' and 'rule of inference of M' (as well as on various choices that can be made once for all theories in a given language, such as a Gödel numbering). To set things right, the discussion in the text (here and in subsequent paragraphs) should really be done not in terms of (candidates for) our fullest mathematical theory M, but in terms of (candidates for) our fullest theory-predicate S: where a theory predicate is an RE-formula (in one variable)
which numerates the axioms and rules of some theory M. (I'm using the terminology of Feferman 1962.) For example, the sentence to which this note is attached should read: the view is that no theory-predicate S that numerates a consistent mathematical theory could be our fullest mathematical theory-predicate, since 'our fullest mathematical theory-predicate' would have to be closed under Gödelization: if what it numerates included S, it would have to include S's Gödel sentence G S . (The correction here is of little significance for finitely axiomatized theories, but of more importance when M isn't finitely axiomatized. And it is arguable that in that case, theory-predicates rather than theories are more directly 'psychologically real': that the theory predicate is the means by which an infinitely axiomatized theory is represented in our finite heads.) For readability I have chosen to speak sloppily of theories instead of theory-predicates in the text, though in a footnote I will mark one place where it is of philosophical interest to note the more correct formulation. end p.344
implausible, at least if we stipulate that by 'our fullest mathematical theory' we mean something like 'the set consisting of all our explicit mathematical beliefs, plus perhaps those mathematical sentences we could easily be brought to explicitly believe, plus perhaps their logical consequences'.19 If we mean anything like this by 'our fullest mathematical theory', then there can be no question that on any precisification of the phrase, the set exists and is recursively enumerable, and hence (if consistent) is not closed under Gödelization. It is just that there is no maximally inclusive precisification. How then must the extreme anti-objectivist position be formulated, so as to make explicit the vagueness of 'our fullest mathematical theory?' The position is that insofar as M is a good candidate for our fullest mathematical theory, sentences undecidable in M have no determinate truth value. So if A is a sentence undecidable on all reasonable candidates for our fullest mathematical theory, it definitely has no determinate truth value; whereas if it is decidable on some but not all of the reasonable candidates for our fullest theory, it is indeterminate whether it has determinate truth value. We have a second order indeterminacy, since 'indeterminate' is tied to 'fullest mathematical theory' (according to the philosophical principles here under consideration) and hence is just as indeterminate as that is. Returning now to the argument against extreme anti-objectivism that I claimed was question begging: the argument, modified slightly to allow for the second order indeterminacy just noted, was that the Gödel sentence of any candidate M for our fullest mathematical theory must certainly have determinate truth value: it must be determinately true if M is consistent, and determinately false otherwise. As I've said, this is question begging. Let's grant for now that if M is consistent, its Gödel sentence G M is true, and that if it is inconsistent then G M is false. It only follows that G M is either determinately true or determinately false if we assume that M is either determinately consistent or determinately inconsistent. And while of course that will be the case if 'finite' is determinate, we are now exploring the possibility that 'finite' isn't determinate. Indeed, we are exploring the possibility that the only mathematical claims that are determinately true are those that are provable in M. (And some of those may not be determinately true either, if their negation is also provable in M.) But if CON M is the standard formalization of the consistency of M, CON M isn't provable in M, unless ¬ CON M is too; so insofar as the extreme anti-objectivist identifies consistency with the standard formalization thereof, he will say that it is not determinate whether M is consistent and hence it's not determinate what truth value its Gödel sentence has.20 (Here I have in mind the case where M contains no humanly recognizable inconsistency. If it contains a humanly recognizable inconsistency, it is natural to hold CON M and G M determinately 19 We would probably want to refine this a bit, to reduce the chance that all reasonable candidates for our fullest
theory are inconsistent; but this won't affect the case for recursive enumerability. 20 On the other hand, if T is a theory that (like Peano arithmetic) can be proved consistent in all reasonable candidates for our fullest theory, then G T and CONT will be determinately true. end p.345
false; in a moment I'll show that this verdict can be reconciled with extreme anti-objectivism.) 21 I don't deny that it is a somewhat awkward feature of extreme anti-objectivism that it may be indeterminate whether a candidate for our fullest theory is consistent (or more accurately, indeterminate whether a candidate for our fullest theory-predicate picks out a consistent theory—see footnotes 18 and 21 ). To investigate how serious this awkwardness is, let's look first at what an extreme anti-objectivist should say about the possibility that our fullest theory is definitely inconsistent: say, that it contains a humanly recognizable inconsistency. Probably the best thing to say (as observed in footnote 7 ) is that the adoption of an inconsistent theory would make determinately true just those sentences A that are implied by all of the most natural consistent replacements for that theory.22 Presumably, if a candidate M for our fullest theory contains a humanly recognizable inconsistency, the inconsistency lies outside Peano
arithmetic (P), and every natural consistent replacement for M implies both P and CON P (as does every consistent candidate for our fullest theory). So P and CON P come out determinately true on all reasonable candidates for our fullest theory, even the definitely inconsistent ones. On the other hand, if M contains a humanly recognizable inconsistency, then presumably every natural consistent replacement of M implies ¬CON M (as does every consistent candidate for our fullest theory); so CON M would then be determinately false. And so too would M itself. These conclusions seem to be what we would intuitively want, so the possibility of definite inconsistency in candidates for our fullest mathematical theory does not seem especially problematic for extreme anti-objectivism. But what are the consequences for extreme anti-objectivism of its not being determinate whether a candidate M for our fullest theory is consistent? Presumably Peano arithmetic and CON P still come out determinately true, since I've argued that they come out true whether or not M is consistent; but presumably CON M and M now lack determinate truth value, since they come out determinately false if M is inconsistent and not determinately false if it is consistent. And these conclusions too seem to be what we ought to expect. So it is hard to see how to reduce to absurdity the position that it is indeterminate whether the candidates for our fullest theory are consistent. I don't claim that that is an attractive position; only that it might be the least unattractive option to adopt if all cosmological hypotheses like those considered in section 3 were to prove false. 21 As I observed in fn. 18, the sort of formulation given in this paragraph is rather sloppy: to set things right, I should
speak not of a theory M, but of a theory-predicate S that picks out a theory. This is of philosophical interest in the present context, since it points up the fact that intuitively, the indeterminacy as to the truth of G S and CONS may be partly due to the fact that when 'finite' is not assumed determinate, there can be an indeterminacy in which theory the predicate S picks out. 18 Strictly speaking, 'the Gödel sentence of M' is ill-defined: the Gödel sentence depends not just on M but on a way of defining 'axiom of M' and 'rule of inference of M' (as well as on various choices that can be made once for all theories in a given language, such as a Gödel numbering). To set things right, the discussion in the text (here and in subsequent paragraphs) should really be done not in terms of (candidates for) our fullest mathematical theory M, but in terms of (candidates for) our fullest theory-predicate S: where a theory predicate is an RE-formula (in one variable) which numerates the axioms and rules of some theory M. (I'm using the terminology of Feferman 1962.) For example, the sentence to which this note is attached should read: the view is that no theory-predicate S that numerates a consistent mathematical theory could be our fullest mathematical theory-predicate, since 'our fullest mathematical theory-predicate' would have to be closed under Gödelization: if what it numerates included S, it would have to include S's Gödel sentence G S . (The correction here is of little significance for finitely axiomatized theories, but of more importance when M isn't finitely axiomatized. And it is arguable that in that case, theory-predicates rather than theories are more directly 'psychologically real': that the theory predicate is the means by which an infinitely axiomatized theory is represented in our finite heads.) For readability I have chosen to speak sloppily of theories instead of theory-predicates in the text, though in a footnote I will mark one place where it is of philosophical interest to note the more correct formulation. 21 As I observed in fn. 18, the sort of formulation given in this paragraph is rather sloppy: to set things right, I should speak not of a theory M, but of a theory-predicate S that picks out a theory. This is of philosophical interest in the present context, since it points up the fact that intuitively, the indeterminacy as to the truth of G S and CONS may be partly due to the fact that when 'finite' is not assumed determinate, there can be an indeterminacy in which theory the predicate S picks out. 7 I am assuming in this paragraph that the mathematics we accept is consistent: were we to accept inconsistent mathematical claims, we wouldn't want to take them all as true. (Probably the best thing to say is that if our overall mathematical theory is inconsistent, then anything implied by all the most natural consistent replacements of it is true by convention, whereas anything implied by some but not all of the natural consistent replacements gets no determinate truth value from our inconsistent convention.) (I take consistency as a logical notion rather than a mathematical one (see Field 1991), so that taking our mathematics as true merely by convention does not make consistency claims true merely by convention. I don't doubt that our mathematical conventions could indirectly affect the notion of consistency; but they do not do so in a way that makes it trivial that our mathematical conventions come out consistent.) 22 Anyone who thinks that my use of 'inconsistent' and 'consistent' here assumes the determinacy of the notions should recall the second paragraph of sect. 2. But if you like, you can replace 'inconsistent' by 'definitely inconsistent' here, and likewise for 'consistent'. Similar remarks apply to much of the discussion in the next few paragraphs. end p.346
6. Does Gödel's Theorem Refute Extreme Anti-Objectivism (2)?
There is another kind of argument against extreme anti-objectivism based on Gödel's theorem. It claims that we know that the Gödel sentences of the candidates for our fullest mathematical theory not merely have determinate truth value, but are true. The argument is a mathematical induction: all the logical and nonlogical premises of M are true; the rules of inference preserve truth; so all the theorems must be true; so the theory must be consistent, so the Gödel sentence must be unprovable and hence true. Of course, the induction can't be formalized in M; but it is often felt that it is somehow 'informally valid'.23 I have doubts about the intelligibility of this idea of 'informal validity', but would rather not rest on them: I will argue rather that the induction is not simply unformalizable, it is fallacious, in that it relies on the incorrect principles about truth that are responsible for the semantic paradoxes. My argument will presuppose that 'true' as applied to mathematical sentences is a perfectly good mathematical notion. Of course, though restrictions of this notion (e.g. to sentences that don't themselves contain 'true' and that have only restricted quantifiers) are definable in set theory, the full notion of mathematical truth is not definable in set theory or other more ordinary mathematical terms, by Tarski's Theorem. Still, it is a notion that we can axiomatize: we all implicitly accept many axioms involving it (e.g. the instances of the schema (T) True ('p') if and only if p for 'p' not containing the word 'true', and the claim that all of these instances are true); and it is possible to consistently extend these axioms in any of several attractive ways, perhaps adding some special rules of inference involving 'true' as well. Whatever axioms and rules of inference about truth we accept are part of our mathematical theory M. What I now want to argue is that the inability to formally carry out the inductive argument that all theorems of M are true needn't rely on excluding the notion of truth from the formal principle of induction. (If it did rely on that, then if we could convince ourselves that inductions involving 'true' were 'informally valid' we would have an informal argument for the truth of the Gödel sentence of M.) In addition, the inability to formally carry out the induction needn't rest on an inability to pass from asserting of each axiom that it is true to asserting that all of them are true (or analogously for the rules of inference); indeed, there is no difficulty in making such a passage if M has 23 The reader will note that this argument would if valid establish only the truth of the Gödel sentence, not its
determinate truth. The gap could be filled by the principle that if we can informally prove the truth of something, it must be determinately true (a principle that is plausible, though not beyond controversy); alternatively, we could have done the induction in terms of determinate truth in the first place (though the premises of the induction then seem somewhat less obvious). Either way, the argument for the determinate truth of the Gödel sentence is a bit shakier than the argument for its truth. But I want to respond to even the less shaky argument. end p.347
only finitely many axioms and rules, and I know of no strong reason why this shouldn't be so. (There is more discussion of this in Field (1994: 404, esp. notes 19 and 20).) Rather, the most fundamental source of the problem, I claim, is the assumption for each axiom of M that it is true and for each rule of inference that it preserves truth: some of these assumptions are not only unprovable, but refutable in M. Exactly which axiom of M fails to be true or which rule fails to be truth-preserving depends on your theory of truth, but in any adequate theory of truth, one of the axioms or rules of M will suffer this fate. For instance, in all versions of the Kripke supervaluational theory (without 'closing off') or of the Gupta-Belnap revision theory,24 the inference rules include both modus ponens and the inference from A to True('A'). But now consider the claims that these rules are truth-preserving: indeed, consider just the weak schematic forms of these claims, that is, the schemas I True('A') & True('A B') True('B') and II True('A')
True('True('A')').
No version of either the Kripke theory (without closing off) or the Gupta-Belnap theory accepts both these schemas. (The attractive versions of each accept the first schema but not the second.)25 And indeed, each version of either theory explicitly rejects certain instances of one or the other schema: adding the schemas not accepted to the theory would engender a version of the Liar paradox. In a Kripke theory obtained by 'closing off' a fixed point, the axiom schema True('A')
A
will typically hold; but then the schema True('True('A')
A')
that asserts the truth of all instances of the previous schema has counter-instances (for instance, when you instantiate
on the Liar sentence). The point made here for the Kripke and Gupta-Belnap theories can be made with great 24 Kripke and Gupta and Belnap present their accounts as explicit definitions of truth; to do this requires that the
quantifiers of the object language be interpreted as ranging over less than everything (in particular, ranging over some set D), so what they are really defining isn't truth but 'comes out true when the quantifiers are restricted to domain D'. Obviously we couldn't use such restricted truth predicates in the inductive argument under consideration. But the Kripke and Gupta-Belnap definitions are of interest in that one can investigate which principles of truth they validate, and one can then use those principles in an axiomatic theory of truth; that way the restriction to a domain is not required, and the use of the truth predicate in the inductive argument is prima facie more promising. This approach (advocated in McGee 1991) is what I am presupposing. 25 By a 'version' of the Kripke theory or the Gupta-Belnap theory, I mean a decision as to what conditions we impose on (i) the sets of sentences we supervaluate over in the Kripke case and (ii) the sets of sentences we allow in the limit stages in the Gupta-Belnap case. Attractive versions of these theories impose closure under modus ponens as one of these conditions; this guarantees Schema I, but at the same time rules out Schema II. end p.348
generality for other theories of truth. (For more details see Field 1994, note 18 .) The upshot is that the premises of the inductive argument for the truth of all theorems of the theory (and hence the consistency of the theory and the truth of its Gödel sentence) can't all be accepted; the unprovability of the Gödel sentence is not due to an inability to carry out formally an induction that is 'informally valid'. I have been arguing that the consideration of the Gödel sentence of (a candidate for) our fullest mathematical theory gives no decisive reason for thinking that some undecidable sentences have determinate truth values. But my discussion in the last few paragraphs has implications also in a different context, where we assume (either on account of cosmological assumptions like those in section 3, or for other reasons) that the Gödel sentence does have determinate truth value, and we are interested in whether it is determinately true or determinately false. In that context, my claim is that we have less of an argument that the Gödel sentence is true than many people think. Of course, to say this is not to say that we shouldn't hope that the Gödel sentence is true: hoping that is tantamount to hoping our theory consistent, which seems like a reasonable attitude. It also isn't to say that we shouldn't have a positive degree of belief that it is true: we can reasonably have positive degrees of belief in many things that we don't think even informally provable. 26 This point may be usefully combined with the point, stressed several times, that there is no uniquely best candidate for 'our fullest mathematical theory'. If M is a relatively good candidate for my fullest mathematical theory, I am likely to have a fairly decent degree of belief in G M . This will probably make M′ = M {G M } a pretty good candidate for my fullest mathematical theory too, given the way my degrees of belief work, even though G M isn't even informally provable from M. (M′ may be a better candidate than M is, if M is 'on the weak side' of the cluster of theories that are candidates for my fullest mathematical theory.) We can reiterate the addition of a Gödel sentence through the constructive ordinals, as in Feferman (1962). My position is that at some point in this process, the claim of the theories to be 'our fullest mathematical theory' begins to gradually decrease. It gradually decreases, as opposed to dropping off suddenly: this seems to me the natural attitude to take, and it is one that is facilitated by taking the considerations that favor passing from a theory to its Gödel sentence as less than 'informal proof'. These last two paragraphs have been a bit of a digression. The main theme of this section is that the consideration of Gödel's theorem gives no decisive reason for thinking that some undecidable sentences have determinate truth value. And Putnam's argument makes it hard to see how any sentences 18 Strictly speaking, 'the Gödel sentence of M' is ill-defined: the Gödel sentence depends not just on M but on a way
of defining 'axiom of M' and 'rule of inference of M' (as well as on various choices that can be made once for all theories in a given language, such as a Gödel numbering). To set things right, the discussion in the text (here and in subsequent paragraphs) should really be done not in terms of (candidates for) our fullest mathematical theory M, but in terms of (candidates for) our fullest theory-predicate S: where a theory predicate is an RE-formula (in one variable) which numerates the axioms and rules of some theory M. (I'm using the terminology of Feferman 1962.) For example, the sentence to which this note is attached should read: the view is that no theory-predicate S that numerates a consistent mathematical theory could be our fullest mathematical theory-predicate, since 'our fullest mathematical theory-predicate' would have to be closed under Gödelization: if what it numerates included S, it would have to include S's Gödel sentence G S . (The correction here is of little significance for finitely axiomatized theories, but of more importance when M isn't finitely axiomatized. And it is arguable that in that case, theory-predicates rather than theories are more directly 'psychologically real': that the theory predicate is the means by which an infinitely axiomatized theory is represented in our finite heads.) For readability I have chosen to speak sloppily of theories instead of theory-predicates in the text, though in a footnote
I will mark one place where it is of philosophical interest to note the more correct formulation. 26 It also isn't to deny that there is something awkward about believing M while simultaneously believing ¬GM : this is tantamount to believing M and at the same time believing ¬CON M , and (at least in the context of an assumption of the determinacy of finiteness, where there is no question that CONM adequately formalizes the consistency of M) the beliefs do not cohere well with each other. So, were we to believe both M and ¬GM , we would have motivation to revise one of our beliefs; but it could be M rather than ¬GM that was the prime candidate for revision, so it is hard to see how this consideration gives reason to believe G M . end p.349
undecidable in all candidates for our our fullest theory could have determinate truth value, if cosmological hypotheses like those considered in section 3 are incorrect. 27 27 Thanks to Dan Isaacson and Tony Martin for some helpful questions at the conference which affected the final
version. end p.350
Abstract: Several recent authors have tried to establish that there is no indeterminacy in number theory or set theory, on the basis of certain categoricity and quasi-categoricity arguments. Some of these arguments are based on secondorder logic; others are based on the use of schemas within a first-order framework. The postscript argues that all these arguments fail. Keywords: categoricity,indeterminacy,logic,schemas,second-order logic Postscript The chapter argues (i) that a great many undecidable questions in set theory have no determinate answers, and (ii) that the same is at least possible even in number theory. Both views have been met with considerable skepticism. One possible ground for skepticism is the view that truth and reference are at bottom disquotational. The disquotational view of truth and reference has sometimes been thought to preclude making sense of indeterminacy in our current language; but the burden of Chapter 10 was to argue that this is not so. Still, even if it makes sense to treat our current language as indeterminate from a disquotational perspective, it may be that the arguments for so regarding it are less compelling from a disquotational perspective. On the way of making sense of indeterminacy from the disquotational perspective that I am now inclined to favor-the one advocated in Chapter 10-it seems to me that the force of the argument for regarding questions about the size of the continuum as having no determinate answers is not significantly reduced: see the end of the postscript to Chapter 10. But if we accept both disquotationalism and the view of indeterminacy offered in Chapter 10, then I do think there is a substantial reduction in the force of the argument that we should regard undecidable number-theoretic questions as indeterminate if the 'cosmological assumptions' fail. So I now prefer to think of Sections 3 through 6 of this chapter as a challenge to an orthodox theorist who wants both to reject a disquotational account of truth and reference, and to believe in the determinacy of undecidable number theoretic questions without basing the latter belief on cosmological assumptions. Enough on disquotationalism. Recently a number of distinguished philosophers and logicians have argued for the impossibility of any significant indeterminacy in number theory and set theory on a quite different ground. There are two variants. One supposes that number theory and set theory are formulated in full second-order logic (second-order logic conceived model-theoretically, with the stipulation that any legitimate interpretation is 'full' in the sense that the secondorder quantifier ranges over all subclasses of the domain of the first-order quantifiers). The other supposes that number theory and set theory are formulated in a variant of full second-order logic that we might call 'full schematic first-order logic'.
A. The Second-Order Variant The claim is (i) that full second-order number theory, unlike first-order number theory, is categorical, i.e. has only one interpretation up to isomorphism end p.351
in which it comes out true; and (ii) that this shows that indeterminacy worries can't arise for it. In set theory it's a bit more complicated: full second-order set theory isn't quite categorical (if there are inaccessible cardinals), but only quasi-categorical, i.e. for any interpretations in which it comes out true, either they are isomorphic or one is isomorphic to a fragment of the other obtained by restricting to sets of rank less than a certain inaccessible. But still, the quasicategoricity is enough to give most questions about the size of the continuum the same truth value in all of its
interpretations, so the idea is supposed to be that the indeterminacy in set theory is nearly eliminated. Moreover, McGee 1997 shows how, by adding a new axiom to set theory and restricting to interpretations where the first-order quantifiers range over absolutely everything, 1 we can get full categoricity in second-order set theory; given this, the argument for determinacy in the set-theory case is completely analogous to the argument ((i) and (ii)) in the number theory case. In the chapter I mention the move of trying to use second-order logic to evade the indeterminacy argument, and say that it is question-begging since the indeterminacy argument applies to the determinacy of the second-order quantifier as well as to the notion of set. The basic point is one that was well made many years ago in Weston 1976: any argument that claims about the size of the continuum don't have determinate truth value will carry over to an argument that 'all subsets' is indeterminate, and hence that it is indeterminate what counts as a 'full' interpretation. (The conception of second-order quantifiers as plural quantifiers (footnote 1 ) does not rule out their being indeterminate: there is still the issue of 'which pluralities they range over'.) So even though relative to any specific conception of 'fullness' the full interpretation of second-order set theory is quasi-unique, that doesn't do anything to alleviate the basic indeterminacy. McGee suggests (1997: 45) that this critique of the second-order determinacy argument rests on the view that second-order logic isn't really a part of logic, but is set theory in disguise. I think that's wrong: the issue of whether second-order logic is part of logic is largely a terminological issue; even if it is a part of logic, the second-order quantifiers might be indeterminate, and that is enough to undermine the claim that second-order categoricity implies determinateness. Similarly, McGee's stipulation that one is interested only in models in which the first-order quantifiers range over absolutely everything succeeds in eliminating an indeterminacy in first-order quantification only if the use of 'absolutely everything' in the stipulation is determinate: the stipulation works only if it is unnecessary, that is, only if determinate quantification over absolutely everything is possible without the stipulation. The advocate of inflationist semantics needs to say how it is that features of our practice determine that our quantifiers range over absolutely everything. 1 Of course if the second-order quantifier ranges over all subclasses of the domain of first-order quantification,
paradox results from having the first-order quantifiers range over everything. McGee avoids the paradox by adopting the Boolos 1984 conception of second-order quantifiers as not literally ranging over classes conceived as special entities, but as plural quantifiers. 1 An interesting recent paper relevant to its defense is Hawthorne 1996. end p.352
Elsewhere, McGee makes an impressive attempt to do just that (McGee 2000). He argues that (*) we ought to rule out the hypothesis that a person's apparently unrestricted quantifiers range only over entities of type F unless the person has the concept of F; and that this rules out standard attempts to show the indeterminacy of universal quantification. This has considerable prima-facie appeal, but I suspect that the appeal is illusory. If we assume that our own quantifiers determinately range over everything, then it does seem natural to suppose that someone else's quantifiers that are governed by the same rules also determinately range over everything, and could have more restricted range only if the person had the concept of this restricted range. But the real issue is over whether anyone's quantifiers, including ours, have determinate range; and if so, how it is that our practice gives them that determinate range. In this context, it isn't even clear what it would mean to have the concept of the restricted range, for if universal quantification is indeterminate, the concepts used in restricting quantification are almost certain to be so too. (And for any candidate X for the range of the unrestricted quantifiers, we automatically have a concept that is at least a candidate for picking out the objects in X: viz., the concept of self-identity! So while in some sense (*) is acceptable even in the context when indeterminacy in our own quantifiers is allowed for, it has no teeth there.) Let's go back to second-order quantification, and to the number-theory case (where we don't have to go into the issue of absolutely unrestricted first-order quantification, or get into the issue of full categoricity v. quasi-categoricity). Then we must concede to the second-order theorist that (i) is true, given the stipulation that certain interpretations 'aren't to count'; still, our stipulations as to which ones 'don't count' may itself not be completely determinate, in which case it does nothing to make second-order number theory determinate. In short, (ii) does not follow from (i). Of course, there may be independent grounds for thinking number theory determinate, and indeed I claim in the chapter that there are; but the second-order logic argument doesn't do it. And in the case of set theory, where considerations like those that I suggest for number theory don't help much, there seems to be nothing that could do it.
B. The 'Full Schematic' First-Order Variant There is a variant of the second-order determinacy argument, based on 'full schematic theories'. Determinacy
arguments of this sort have been put forward in McGee 1997, Lavine 1994, and Lavine (unpublished); and in somewhat more qualified forms, by Shapiro 1991 and Parsons 1990. What is a 'full schematic theory'? First-order theories for natural numbers and sets are typically presented using axiom schemas, schemas legitimizing the use of each sentence of a certain form in the current language as an axiom. end p.353
But these authors (and others, e.g. Feferman 1991) have pointed out that our acceptance of schemas typically includes more than this, it typically includes a commitment to accepting all new instances of the schema as we expand our language. This certainly seems right. These authors then go on to claim that schematic theories so understood—'full schematic theories'—rule out nonstandard models in a manner similar to the way that full second-order theories have been claimed to. It is this last claim that I will dispute. As a preliminary, I should say that if Weston and I are right in our skepticism about the determinacy-inducing power of second-order theories, then it is hard to see how this claim about the determinacy-inducing power of full schematic theories could possibly be correct. The most that would seem arguable is that schematic first-order theories do as well as second-order theories in ruling out nonstandard models. For full schematic theories would seem to be, in effect, fragments of second-order theories. (A first-order schema (with the schematic variable instantiated by predicates) is in obvious correspondence with a second-order formula in which there are free predicate variables but no bound predicate variables. Asserting a schema, on this interpretation, is very much like asserting the corresponding formula with free variables, which is very much like asserting its universal closure. So we have an analog, in the schematic first-order case, of assertions of Π 1 1 second-order sentences, that is, sentences in which all the second-order quantifiers appear at the beginning and are universal (or sentences that are equivalent to those). But those are the only second-order sentences for which there is anything analogous: e.g. there is nothing analogous to the denial of a typical Π 1 1 sentence.) Since 'schematic first-order arithmetic' seems effectively equivalent to a fragment of second-order arithmetic, it is hard to see how it could do any better than second-order arithmetic in making arithmetic notions determinate. In fact, it seems highly doubtful that it can do even as well. For one of the ways that second-order arithmetic rules out some of the nonstandard interpretations of first-order arithmetic is that it contains a second-order impredicative comprehension schema, which vastly increases the number of predicates on which to perform inductions and thus vastly cuts down on the possible interpretations of the theory. But nothing like impredicative comprehension is available in schematic first-order arithmetic, so it is hard to see how full schematic arithmetic does even as well as second-order arithmetic in ruling out nonstandard interpretations. The above remarks are preliminary, and Lavine has in correspondence disputed their presupposition that full schematic first-order logic is a subtheory of (nonschematic) second-order logic. Rather than address this issue directly (it is related to the issues discussed below), I will forget about second-order logic and look directly and in detail at some of the claims that McGee, Lavine, Shapiro, and Parsons have made on behalf of full schematic theories. I will stick to the example of number theory, so as to avoid some complexities. And I will sometimes pretend that there is no case to be made for the determinacy of number theory other than the case that rests on the use of full schemas; this end p.354
pretense is contrary to what I argue in the chapter, but is harmless in the context of evaluating the claims that our authors make on behalf of full schemas in achieving determinacy. I'll start with McGee, who seems to claim at one point (1997: 57) that schematic first-order arithmetic is much stronger than ordinary first-order arithmetic. Let G be the ordinary Gödel sentence, that is, the Gödel sentence for ordinary firstorder PA (based on the standard primitives). McGee seems to say that G is provable in schematic PA: all we have to do is add a truth predicate, and use inductions on it. But this is false: what is true (and what he says in his careful statement of the result) is that we get more powerful results if we add a truth predicate, use inductions on it, and add a certain compositional theory of truth. But of course, adding a compositional theory of truth is going beyond schematic arithmetic. (I don't really suppose that McGee could have meant to claim that G is provable in schematic arithmetic— indeed, he has confirmed to me that he did not—but his wording certainly strongly suggests that claim and could easily have misled some readers.) But McGee's main argument is more model-theoretic: he argues that schematic first-order number theory fixes the extensions of number-theoretic terminology uniquely. Here's the argument: suppose our arithmetical language was indeterminate, having some nonstandard models as well as the standard one as legitimate interpretations. But there is a possible expansion of the language with a new predicate 'standard natural number', that is stipulated to have only the standard natural numbers in its extension; and inductions on this new predicate will allow us to rule out the nonstandard models.
In conversation I have heard this argument from several other distinguished logicians, but I submit that it has to be a cheat. Suppose for the moment that we consider only the ordinary Peano arithmetic, that is, with schemas understood as having instances only within the current language. And let's pretend (wrongly, in my view) that I haven't picked out a unique structure up to isomorphism. If that is the case, then merely adding new vocabulary without further axioms for this vocabulary wouldn't help, and adding new vocabulary with new axioms would only allow me to lessen the indeterminacy rather than eliminate it. In particular, adding 'standard natural number' without new axioms is of no help whatever, and adding it with new axioms governing it is of no advantage over directly adding those axioms for the predicate 'natural number', without the new predicate. I think McGee and the other logicians who favor his argument would agree that that's what the situation is for ordinary Peano arithmetic as opposed to full schematic PA. But if that's so, how can adopting McGee's rich view of schemas help secure determinacy? That view of schemas merely allows me to add an instance of induction whenever I add new vocabulary. But the relevant vocabulary for McGee's argument would seem to be 'standard natural number', and we've already seen that that is no help. Of course, it's true that if I could add a predicate that by some magic has as its determinate extension the genuine natural numbers, then I will be in a end p.355
position to have determinately singled out the genuine natural numbers. That's a tautology, and has nothing to do with whether I extend the induction schema to this magical predicate. But if you think that we might someday have such magic at our disposal, you might as well think we have the magic at our disposal now; and again, it won't depend on schematic induction. So the only possible relevance of schematic induction is to allow you to carry postulated future magic over to the present; and future magic is no less mysterious than present magic. The moral for McGee's argument is that it is a cheat to describe the expansions of the language in terms of their extensions. The reason this is a cheat is that it assumes that the predicates in the expanded language have determinate extensions, which they won't if the indeterminist is right about number theory. (Again, I don't think the indeterminist is right, but I'm pretending to here.) (Actually I have a second doubt about McGee's argument. McGee thinks that the predicates we need to consider adding to the language may be ones that it is psychologically impossible to add because of their complexity. Nonetheless, he says, the rules of our language don't prohibit our adding them. But can it really be determinate that the rules of language permit us to add predicates of a kind that it is psychologically impossible for us to add? I would have thought that the question of whether the rules of our language say 'You are allowed to add any predicates at all' or say rather 'You are allowed to add any predicates that it is psychologically possible for you to add' is about as clear an example of indeterminacy as we could want. But this doubt about the argument is secondary: the main doubt is not about the addition of predicates with extremely complex associated axioms, but about the addition of predicates with specified extensions.) Lavine and Shapiro and (with more reservations) Parsons have suggested a different argument for the determinacy of number-theory and the quasi-determinacy of set theory; the argument is quite similar to McGee's argument for the determinacy of the logical constants, discussed in the postscript to Chapter 9. As with that argument, the core of the Lavine-Shapiro-Parsons argument is a good argument for a more limited conclusion; I'll give the argument for the more limited conclusion first, and then consider the possibility of either extending the conclusion to yield determinacy, or at least of extending it part way. The conclusion of the good argument can be stated (slightly vaguely) as follows: if one has two systems of schematic Peano arithmetic in one's language, one can inductively define an isomorphism between them. Here's the idea. Suppose we have in our language two predicates N 1 and N 2 (two candidates for what the natural numbers are), and that to go with each N i we have corresponding constants O i and function symbols Suc i , + i and · i . We now inductively define a function symbol f as follows:
end p.356
This is a standard method for defining functions on the natural numbers, so it gives a function from things satisfying N 1 to things satisfying N 2 . (We're going a bit beyond Peano arithmetic here, in talking about functions; but no very strong extension of arithmetic is required.) We can also argue that the function is 1−1, by an inductive argument in the
system N 1 , using an induction property that involves this inductively defined function. 2 (This is not in the N 1 -system, so we're making use of the fact that this is schematic arithmetic.) We can argue that f is a homomorphism similarly. And finally, we can argue that f is onto, by an induction in N 2 , again using an induction property that involves the inductively defined function f. 3 The conclusion of this argument isn't very surprising: in a framework of ordinary first-order set theory, without the full interpretation of schemas, we could already argue that the natural number system (of that set theory; i.e. the natural number system allowing induction over formulas containing 'set') was unique up to isomorphism, by essentially the above argument. There, the conclusion didn't establish the determinacy of the natural numbers, but only determinacy relative to the notion of set. Here we have redone the argument with less of a dependence on a background set theory; the question is, does this give us a more convincing case for determinacy? I don't see that it does. It's one thing to have the conclusion that I can't accept two genuinely distinct systems of natural numbers (with schemas interpreted as 'full' in both cases), and another to accept that the unique system I accept is determinate. After all, why couldn't the two systems N 1 and N 2 I introduce each be indeterminate, with the proof showing simply that they must be indeterminate in precisely the same way and so be equivalent? (The predicate N 2 (x) becomes in effect like the predicate ¬¬N 1 (x): distinct, equally indeterminate, but equivalent.) So the stronger conclusion that we want—the determinacy of these predicates—doesn't follow. Parsons certainly recognizes this, and Shapiro may too. What they do is formulate an intermediate claim between the weak claim and the strong claim, argue for it, and then say that maybe skepticism about determinacy is uninteresting once the intermediate claim is established. The intermediate claim is: (IC) Any two people who accept schematic arithmetic must regard each other's theories as equivalent. 2 For if not, there is a smallest N 1 −number n such that for some smaller N 1 -number m, f(n) = f(m). But n is of form
Suc 1 (p), where p≥ 1 m, so f(n) is Suc 2 (f(p)). If m is 0 1 , f(m) = 0 2 , so f(n) = f(m) gives that Suc 2 (f(p)) = 0 2 , which violates a Peano axiom in the N 2 system. And if m is Suc 1 (q), then f(m) = Suc 2 (f(q)), so f(n) = f(m) gives that Suc 2 (f(p)) = Suc 2 (f(q)); another Peano axiom in the N 2 system gives that p = q, but then we have a p
I am skeptical both that there is reason to accept this intermediate claim, and also that if we do accept it we should regard further skepticism about determinacy as uninteresting. I will concentrate on the first of these issues, since it is this one on which Parsons and Shapiro are most clearly committed. 4 Suppose X (male) regards his own schematic arithmetic as indeterminate, but recognizes the weak conclusion that he must regard any 'copy' of it in his own language as equivalent to his. Must X regard someone else Y (female) who advocates schematic arithmetic as also accepting something equivalent to what X accepts? I don't believe so. (In my discussion I will allow X to put aside any worries about whether Y really does accept full schematic arithmetic; and I will assume that X adopts a principle of charity, so that he regards all of Y's beliefs as true.) Suppose both X and Y use the same vocabulary: 'number', 'successor', and so forth. The question at issue is whether X must translate Y's number-theoretic vocabulary homophonically; or rather, whether X must suppose that if he introduces special terms as translations of Y's vocabulary (say 'number*', 'successor*', and so forth), he has an argument for an equivalence between his own vocabulary and the translation of Y's. Can't we just apply the conclusion in the one-language case to get this result? No. For though I have had X grant that Y subscribes to the full schema (and is correct in so doing), that only means that he views Y as committed to accepting all new instances in Y's own language (and that X should be committed to their translations). Unless X can argue that for any predicate P in his language (e.g., 'number'), Y can expand her language to include a term that X should translate as P, there is no reason to assume that X should view Y's schemas as full with respect to X's language. And there is no way to argue this without begging the question. For instance, X can't argue that since Y accepts full arithmetic she must accept induction on a predicate that means the same as X's predicate 'natural number'; that's simply a variant of McGee's procedure of arguing that Y must accept induction on a predicate whose extension is the natural numbers, which we've already seen to be a cheat. (McGee's was a cheat because it assumed determinateness of extension, which was what was there at issue; this one is a cheat because it assumes determinateness of translation, which is here at issue.) I consider another question-begging attempt two paragraphs after next. Let's go to a different question, which is whether the argument in the one-language case can be adapted to the twolanguage case. Again, the answer is no, but the details are quite interesting and worth a look.
The first step in trying to adapt the argument to the two-language case is for X to define a function from the natural numbers (in his sense) to the extension of Y's predicate 'natural number'. Or to put it syntactically: X must introduce a function symbol whose domain he describes with his term 'natural 4 The discussion that follows is closely related to my discussion of McGee's argument for determinacy of translation of
the logical connectives, in the postscript to Ch. 9. end p.358
number' and whose range he describes by the predicate 'natural number*' that he uses to translate Y's term 'natural number'. I think that this step of X's argument is legitimate. 5 Moreover, X can argue by induction on the numbers (X's numbers; that is, by an induction using X's term 'number') that this function is 1−1, and that it meets a homomorphism condition; so there is an embedding of the numbers into the number*s. If X could also argue that this function is onto, he would have reached the intermediate conclusion; but to argue that, he would need to 'do an induction on Y's numbers', that is, an induction with a conclusion of the form 'For every number*, . . . '. Moreover, the induction predicate would have to use the function symbol f recently defined (for instance, the induction predicate might be 'number* not in the range of f'). But there is no reason for X to think that there is any way that Y might expand her language so that it would contain a predicate that X should translate in this way. Consequently, X needn't accept such an induction as legitimate: indeed, accepting it as legitimate would beg the question in favor of the homophonic translation. Of course, Y can reach the same conclusions about X that X reaches about Y. Can X make use of this? I don't think so. What Y concludes is 'the numbers are embedded in the number # s', where 'number # ' is Y's translation of X's term 'number'. X is committed to supposing that the translations of Y's theorems are true, and X's translation of 'number' is 'number*'; so X can get the conclusion that all number*s are numbers if he can argue that he ought to translate Y's term 'number # ' as 'number'. But how can he argue for that? The hypothetical translation of Y's language that X needs to show inadequate, after all, is one on which Y's term 'number' is strictly broader than X's term 'number'. But on any adequate translation, Y's term 'number # ' is at least as broad as Y's term 'number'; so the hypothetical translation obviously must take 'number # ' as strictly broader than X's term 'number'. That means that to assume that the term is to be translated as 'number' would simply beg the question. (Indeed, X's hypothetical translation of Y's language must be one on which no term of that language applies to all and only the natural numbers. Given that, it is no mystery to X why Y has to accept a translation of X's term 'number' that is true of more than the numbers and hence is not adequately translated back into X's language as 'number'.) It may sound as if X and Y are each committed to a chauvinist perspective: each regards their own predicate 'number' as standing for the numbers, and the other's term as broader. But this is not so. In the first place, though each 5 The first way that I put it, in terms of extensions, might encourage a bad objection even to the introduction of the
function symbol. The objection is that X is imagining indeterminacy of reference in Y's language, and so X cannot grant that Y's term has a determinate extension. This seems to me a bad objection, because we can allow ourselves to speak of the extension of the term even while recognizing that our talk of it is indeterminate in these ways. After all, X is imagining that his own term 'number' is indeterminate, but thinks he can still use number-theoretic arguments, so why should he think that his term 'the extension of Y's predicate' is illegitimate simply because indeterminate? But there is no reason to dwell on this: we can avoid the issue, by sticking to the syntactic formulation of the construction. end p.359
regards the other's number theory as adequately translated in a way that translates the foreign 'number' by a term broader than the domestic 'number', nonetheless each regards the homophonic translation as adequate too. (The issue was indeterminacy of translation, not determinate but bizarre translation.) In the second place, it is not true that each regards his or her own term 'number' as determinately standing for numbers: the appropriate lesson of indeterminacy of translation between structurally similar languages is that each of the languages is referentially indeterminate. For understanding these points, it may help to shift from a syntactic to a semantic perspective. We have agreed that if either X or Y had 'two copies' of full schematic arithmetic in their own language, then they would have reason to regard the two as equivalent. To see from a semantic perspective why it doesn't follow that X must regard Y's full schematic arithmetic as equivalent to his own, let J be an acceptable semantic interpretation of X's whole language, including the part of his language in which he talks about his own language and about Y's language. Let N J be the set that J assigns to 'natural number'; it will also be what J assigns to 'the extension of X's term "natural number" ', assuming that X accepts that his own term is true of all and only the natural numbers. Let E J be the set that J assigns to 'the extension of Y's term "natural number" ', and F J the set assigned to 'the extension of Y's term "the extension of X's term 'natural number' " '. Then the point of the last few pages is that X can in full schematic arithmetic reason to a conclusion whose truth requires that N J E J F J , but X can't reason to any conclusion that will necessitate the
reverse inclusions. In the last few paragraphs I have argued against the two-language extension (IC) of the one-language conclusion that one cannot rationally accept two copies of full schematic arithmetic without regarding them as equivalent. In closing, I will simply remark that even were (IC) established it would be far short of determinacy: it would guarantee that in any acceptable semantic interpretation of X's language, the extension of 'natural number' would be identical to the extension of 'the extension of Y's term "natural number" '. But even that unsupported conclusion wouldn't show that in any acceptable interpretation this common extension would contain no nonstandard elements. 6 6 I'm grateful to Shaughan Lavine for helpful comments on an earlier version of this postscript. end p.360
13 Apriority as an Evaluative Notion Abstract: We want our methods of reasoning to be conducive to the truth; if we have reason to think that some of our methods are not, we revise them in favour of methods we think will be more so. This chapter presents a puzzle about the application of this to our most basic methods, and argues that the proper resolution would be that we could not ever have reason to think that they are not conducive to the truth; we could have reason to think that they have not been conducive to the truth in the past, but the methods in question would be sufficiently 'self-correcting' that this discovery will qualitatively alter their future applications, and once this is seen, there would be no temptation to think they will not be conducive to the truth in the future. These most basic methods are thus a priori, in a strong sense that includes empirical indefeasibility. This point and a number of others are used to argue against standard 'naturalist' approaches to epistemology. But, rather than favouring 'non-naturalism', the chapter argues for a view on which epistemological claims are not fully factual. Keywords: a priori, indefeasibility, naturalism, naturalist epistemology, normative discourse, objectivity, relativism, rules, scepticism, self-correcting methods
Hartry Field Apriori justification is often thought mysterious or out of keeping with a naturalistic view of the world; strong forms of a priori justification that involve empirical indefeasibility are often thought especially mysterious. While this is no doubt correct for excessive claims of apriority—for instance, claims to a priori access to features of the physical world—I will argue that it is incorrect if intended as a claim about the existence of any apriority at all.1 What is mysterious in most forms of (non-excessive) apriorism isn't the apriorism itself but the background assumptions about epistemology. But in questioning these background assumptions, I will be producing an account of apriority that few apriorists will like.
1. The Concept of Apriority Let's define a weakly a priori proposition as one that can be reasonably believed without empirical evidence;2 an empirically indefeasible proposition as one that admits no empirical evidence against it; 3 and an a priori proposition as one that is both weakly a priori and empirically indefeasible. Some writers use 'a priori' in a way that imposes no empirical indefeasibility requirement, but it seems to me that that removes the main philosophical interest of apriority: traditional debates about the apriority of logic and Euclidean geometry have largely concerned the issue of whether any empirical evidence could count against them. Another reason for keeping an indefeasibility requirement will be given later in this section. The empirical indefeasibility requirement does need to be either restricted or interpreted delicately, if it is not to immediately rule out a priori knowledge. As a first illustration (emphasized in Kitcher 1983), the credibility of any proposition could be diminished by evidence that well-regarded experts don't 1 The mystery that excessive claims to apriority would create is briefly discussed in fn. 19. I believe that there is no
analogous mystery for apriorism about logic, or for apriorism about the basic features of scientific method. 2 Here and throughout, 'reasonable' will mean 'epistemically reasonable': crassly pragmatic motivations for and against
believing are not to be taken into account. 3 'Empirically indefeasible' may be too weak a term for what I've just defined: the term suggests only that there can
never be sufficient empirical evidence against it to outweigh any initial plausibility it might have. But it isn't easy to find examples that meet the weaker condition but not the stronger, and I will continue to use the term in the stronger sense. end p.361
accept it. This first illustration doesn't seem to me very interesting: perhaps it shows that we must impose a slight restriction on empirical indefeasibility to allow for defeasibility by the opinions of others;4 but surely it doesn't suggest that we should give up an indefeasibility requirement entirely (which would allow us to regard a proposition of logic as a priori while at the same time granting that experimental results in quantum mechanics could tell against it). But there is a more interesting argument against an empirical indefeasibility requirement. The argument has two steps. First, empirical indefeasibility seems equivalent to empirical unaugmentability: the condition that there can be no empirical evidence for the proposition. Their equivalence follows from the hard-to-contest principle that an experience counts as evidence for a proposition only if some contrary experience would count as evidence against it, and vice versa. But second, as has often been noted, complex and unobvious logical truths can admit empirical justification without diminishing their claims to a priori status. For instance, a proposition of form ((p q) p) p is obviously entailed by p, so someone who didn't realize it was a logical truth might empirically justify it by empirically justifying p. So a proposition that should be an extremely plausible candidate for apriority seems empirically augmentable and therefore (given the equivalence) empirically defeasible. The best way to deal with this argument, I think, is to distinguish empirical justification and empirical evidence: evidence involves something like ideal justification, ideal in that limitations of computational capacity are ignored. The idea is that reflection on the logical facts reveals that the evidence for p doesn't raise the 'ideal credibility' of the logical truth ((p q) p) p: for ideally that would have been fully credible to begin with. If an observation doesn't raise the ideal credibility of the claim, it shouldn't count as evidence for it. Similarly, an observation must lower the ideal credibility of a claim to count as evidence against it. A nice thing about this resolution of the second argument against an empirical indefeasibility requirement is that it could be employed in the Kitcher example too: while the non-ideal credibility of, say, a complex logical truth can certainly be lowered by empirical evidence that well-respected logicians didn't accept it, ideal credibility can't be lowered in this way; for that reason, the evidence about the opinions of logicians really isn't evidence against the logical truth. Whether the Kitcher examples are to be handled this way or by a slight restriction on the empirical indefeasibility requirement is something I leave to the reader. I want to say a bit more about my proposed definition of apriority, but first it would be well to generalize: it is important to consider not only the apriority of propositions, but the apriority of methodologies or rules for forming and revising beliefs. For the moment, we can take as examples of such methodologies (classical) deductive inference, (your favorite version of) inductive inference, and (your favorite rules of) perceptual belief formation, i.e. of the formation 4 One restriction that would have this effect was suggested in Field 1996. end p.362
of perceptual beliefs on the basis of experience. In analogy to the above, I will call such a methodology or rule weakly a priori iff it can be reasonably employed without empirical evidence; empirically indefeasible if no empirical evidence could undermine the reasonableness of its employment; and a priori if it meets both conditions. Again, I think the most interesting component of apriority is empirical indefeasibility. Note that I have not required that an a priori proposition can only be reasonably believed by someone who has a nonempirical justification for it: not only would that conflict with the examples above of a priori propositions reasonably believed entirely because of empirical justifications, it would also conflict with the possibility of a priori propositions reasonably believed without any justification at all. ('Default reasonable' propositions.) Similarly for the case of rules. I think that we ought to allow for the possibility of default reasonable propositions and rules;5 more on this shortly. My definition classifies default reasonable propositions and rules as, trivially, weakly a priori; so that they are a priori if and only if they are empirically indefeasible. If one were to hold that a priori justification is required for reasonable belief in an a priori proposition and for reasonable employment of an a priori rule, then default reasonable propositions and rules could never count as a priori. That would be most undesirable: surely among the most plausible examples of default reasonable propositions and rules are simple logical truths like 'If snow is white then snow is white' and basic deductive rules like modus ponens and 'and'-elimination. It would be odd to exclude these from the ranks of the a priori merely because of their being default reasonable.6 If our concept of apriority were simply weak apriority we would have the opposite problem: default reasonable propositions would automatically count as a priori. But there is no obvious reason why propositions such as 'People usually tell the truth' shouldn't count as default reasonable, and it would be odd to count such propositions a priori. Empirical indefeasibility seems the 5 One must be careful not to be led into ruling them out by pun on the word 'justified'. In one sense, a justified belief
is simply a reasonable belief; in another, it is a belief that has a justification. If it is assumed that these senses are equivalent, the exclusion of default reasonableness is automatic; but in fact their equivalence needs argument. (Note that if their equivalence is not assumed, there is no reason not to suppose that 'unjustified' beliefs in the second
sense can be essential ingredients in the justification of other beliefs.) 6 The problem of default reasonable propositions and rules is curiously overlooked in discussions of how the notion of a priori proposition and/or a priori justification is to be defined. For instance, the discussion of a priori justification in Bonjour 1998 assumes throughout that for a belief to be reasonable it must have some justification or other, if not empirical then a priori. Presumably Bonjour thinks that there are no default reasonable propositions. However, the obvious way to retain that position is to allow for circularity in the justificatory process (see the next section), and Bonjour makes a point of disallowing such circularity: that is the basis on which he argues that for induction to be reasonable it must be possible to give a justification of it that doesn't use induction. He thinks that such a non-circular justification of induction is possible; I will not discuss this, but unless he also thinks that a non-circular justification of deduction is possible, then the exclusion of circularity would seem to make the recognition of default reasonable rules of deduction mandatory if deductive skepticism is to be avoided. (I suspect that Bonjour would say that rather than being default reasonable, the basic rules of logic are justified by acts of a priori insight. But this seems like just an obscurantist redescription; in any case the only argument for it seems to rest on defining a priori justification in a way that ignores the possibility of default reasonableness.) end p.363
obvious way to distinguish those default reasonable propositions that are a priori and those that aren't. There is another possibility worth considering: I have argued against saying that a priori propositions and rules are those that require non-empirical justification to be reasonably believed, but why not say that they are those that admit non-empirical justification? The answer is that this too might exclude simple logical truths, or rules like modus ponens and 'and'-elimination. For the only obvious way to try to give 'a priori justifications' for them is by appeal to the truthtable for 'and'. But as has often been pointed out, 'justification' of 'and'-elimination by the truth-table for 'and' requires the use of 'and'-elimination (or some equivalent principle) at the meta-level: one must pass from ' "A" and "B" are both true' to ' "A" is true'. If this counts as a justification it is a circular one, 7 and it is not obvious that 'circular justification' makes sense. I'll soon discuss that issue more fully, but at least we can say that the alternative approach to defining apriority contemplated at the start of this paragraph requires the acceptance of circular justification.8 I close this section by noting that it is not built into the definitions that an a priori proposition be true or an a priori methodology reliable; much less, that its truth or reliability is somehow guaranteed by some non-empirical justification of it. We do have strong reason to think that a priori propositions are true and a priori methodologies reliable: if we didn't have reasons to think these things, it wouldn't be reasonable to believe the propositions or employ the methodologies, so they couldn't be a priori. 9
2. Default Reasonableness There is a familiar argument for the default reasonableness of certain methodologies, including deductive reasoning, inductive reasoning, forming beliefs as a result of observation or testimony or memory-impression, and so forth. (Recall that if they are default reasonable then they are at least weakly a priori.) The argument is that no justification for anything could be given without using some of these forms of reasoning.10 So if justifications are assumed to be noncircular, and if we exclude the totally skeptical possibility that no methodology for forming and revising beliefs is reasonable, then some methodologies must be reasonable without justification: they must be 'default reasonable'. 11 7 'Circular' here is taken to include 'rule-circular': the relevant sort of circularity is where we justify the claim that a rule
is truth-preserving by use of that very rule. 8 Relative to principles of justification which allow for circular justification, the alternative definition of apriority contemplated in this paragraph may be equivalent to the one I proposed. 9 A further point worth mentioning: I do not assume that it is a failure of rationality to believe of a proposition that is not a priori that it is a priori, or to believe of one that is a priori that it is not. 10 Admittedly, this might not be so if 'acts of a priori insight' are both possible and count as justifications; but let's agree to put obscurantism aside. 11 This is compatible with 'externalist' views about reasonableness (as well as with 'internalist' views). The externalist holds that a necessary condition on the reasonable employment of inductive procedures or perceptual procedures is that those procedures in fact be 'reliable' or 'truth-conducive' or whatever (where the 'whatever' covers any other intuitively 'externalist' condition that might be imposed). This is compatible with certain procedures being default reasonable: it just implies (i) that what procedures are default reasonable depends on which ones satisfy the appropriate externalist conditions; and (ii) that evidence in favor of the satisfaction of those conditions isn't also necessary for the procedures to be reasonably employed. (I doubt that the contrast between internalist and externalist conditions is altogether clear, but I will not be making much of the contrast. In fact, I will eventually argue that even if
that distinction is clear, the distinction between internalism and externalism rests on a false assumption.) end p.364
Should we exclude all circular 'justifications' of methodological rules from being genuine justifications? A number of authors have argued against such an exclusion (Black 1958, Dummett 1978, Friedman 1979, van Cleve 1984), on what appear at first glance to be reasonable grounds. Indeed, at least part of what Dummett and Friedman say seems incontestable: a deductive justification of deduction does give us some kind of rational explanation of why we should value our deductive practice more highly than alternative deductive practices we consider defective. This is doubtless of importance—more on this later. But it is not obvious that its providing this sort of explanation of our valuing the practice means that it should count as a justification. To be sure, Dummett and Friedman grant that such circular explanations are not the sort of justifications that would persuade a committed proponent of alternative methods; but I take the issue of whether they count as justifications not to be that, but rather, whether they should add to the credibility we attach to the mode of reasoning in question. In my view, the explanations can have this justificatory value only if they aren't too easy to come by: only if there was a prima facie risk of it being impossible to explain our valuing the method,12 so that the actual providing of the explanation can justify the method by showing that the risk is not genuine. I think that in the case of deduction and induction and perception there is reason to doubt that there is a significant prima facie risk, in which case it is hard to see why the circular 'justifications' should count as justifications at all. (More about this in the inductive and perceptual cases in section 4.) Even if we concede that such circular 'justifications' have justificatory value, there is a case for certain deductive, inductive and perceptual rules being 'default reasonable'. Indeed, the case for default reasonableness is explicit in most of the works just cited: the authors argue that what makes the rule-circular justifications of certain rules count as justifications is that those rules already have a kind of initial credibility. (They think that use of initially credible rules to argue for the reliability or truth-preservingness of the rules adds to this initial credibility.) Their 'initial credibility' is my 'default reasonableness'. It is however not out of the question to hold that without circular justifications there is no reasonableness at all. That is the view of a certain kind of coherence theorist. This coherence theorist holds that simple deductive, inductive, and perceptual rules don't count as 'reasonable to employ' until the 12 Indeed, only if there was a prima facie risk that in using our methods we will come to the conclusion that the
methods do not have the properties we value. end p.365
users of those procedures have argued (using some combination of deduction, induction, and perception, the combination varying from one case to the next) that those rules are reliable. But once the rules have been used to support themselves to a sufficient degree, the rules become reasonable to employ. But I doubt that this way of avoiding the default reasonableness of certain inferential rules has any substance. Presumably not just any set of procedures that are self-supporting (i.e. which can be used in arguments for their own reliability) count as reasonable to employ: consider various sorts of counter-deductive and counter-inductive methods. What differentiates those which are reasonable (e.g. ours) from those which aren't? The natural answer is that our methods have a certain proto-reasonableness, independent of empirical evidence in their favor, that counter-deductive and counter-inductive methods lack. This proto-reasonableness might be due entirely or in part to factors like truthpreservingness or reliability; or it might be just due to the fact that we find these procedures natural to employ. Either way, once we use our method to reach the conclusion that that method is reliable, the proto-reasonableness is converted to full reasonableness; counter-deductive and counter-inductive methods don't have proto-reasonableness to begin with, so they don't become reasonable upon self-support. That, I submit, is the most straightforward way for a coherence theorist to differentiate the reasonable from the unreasonable self-supporting methods. But then it is transparent that the view is basically just a notational variant of the view that there is default reasonableness: it just calls it proto-reasonableness. Of course, 'default reasonable' is supposed to imply 'reasonable', whereas 'proto-reasonable' is supposed not to imply it (and indeed, to imply that something else is needed before reasonableness is achieved); but my point is that the advocates of this view do ascribe a positive value to what they call proto-reasonableness, it's just that they adopt a higher threshold for the value that must be obtained to deserve to be called 'reasonable'. There are two considerations that favor the lower (non-coherentist) threshold. First, if as I have suggested there is less than meets the eye to deductive justifications of deduction and inductive justifications of induction, the point of elevating the standard of reasonableness in the way the coherentist proposes is obviously diminished. I'll say no more about this now. Second, I think that at least in the case of induction, it is impossible even using the rules in question to argue for the reliability of crucial features of the rules; this means that it is hard to motivate a higher (coherentist) threshold without motivating one so high that it is unattainable. Arguing this is a bit outside the main thrust of the chapter, so I
leave it to a footnote. 13 13 Here is the argument that empirical evidence for the reliability of relevant features of our inductive procedures is
simply unavailable. Suppose we have developed a comprehensive and appealing physical theory T that the evidence at our disposal strongly supports. We can always invent bizarre alternatives that no one would take seriously, but which the available evidence equally accords with: for instance, T* T holds at all times until the year 2000, at which time U holds (where U is some detailed development of a totally discredited theory, say Aristotelian physics). The reason for saying that the available evidence accords with T* just as well as it accords with T is that the available evidence all concerns what happens at times before 2000, and T and T* agree completely about that. Despite this, we would of course all base predictions on T rather than on T*: it is part of our empirical methodology to do so, and surely doing so is reasonable. But it is hard to see that we have any evidence favoring this methodology over an alternative one which favors T* over T. Someone might claim that we do have such evidence: the abundant evidence in our possession that the laws of physics haven't changed in the past. But this is a mistake: if the laws had changed in the past, that would be incompatible with both T and T*, so it wouldn't favor either over the other, and so the fact that the laws haven't changed also doesn't favor one over the other. To make this clearer, let's look at two more theories besides T and T*: V*: The current laws of physics are T; but the laws have changed every 100 years, and will continue to do so. V: The current laws of physics are T; but the laws have changed every 100 years in the past. However, the laws won't change in 2000 or thereafter. There is little doubt that if the laws had changed in the years 100, 200, . . . , 1900, that would be pretty good inductive evidence that they would also change in 2000; and that the fact that they didn't change in the earlier years is evidence that they won't change in 2000. The reason is that our methodology gives a strong a priori bias to V* over V and to T over T*. Evidence that the laws have changed in the past would rule out T and T* but leave V* and V as consistent with the evidence; given the a priori bias, V would be dismissed as highly implausible, leaving V*, which entails a change in the year 2000. Similarly, evidence that the laws haven't changed rules out V and V*, leaving T and T* as consistent with the evidence; but this time the a priori bias leads us to dismiss T* as hopelessly implausible. But at no point is the bias for T over T* or for V* over V ever supported by evidence. (At least, it is never supported prior to 2000; and it is only prior to 2000 that the bias is important to us.) end p.366
Despite these considerations, the decision on 'threshold of reasonableness' is partly verbal, and this partly verbal decision affects the scope of weak apriority as I've defined it. Consider inductive and perceptual rules (good ones; this presumably includes the ones we use). On the lower (non-coherentist) threshold, such rules come out default reasonable and therefore weakly a priori. But on the higher (coherentist) threshold according to which good inductive and perceptual rules are merely proto-reasonable, then those rules don't count as weakly a priori unless they can be given non-empirical justifications; and this is most unlikely even allowing circular 'justifications', since the premises of an inductive 'justification' of induction or an inductive-perceptual 'justification' of perceptual rules are empirical. So the issue of the weak apriority of inductive and perceptual rules is largely a verbal issue about the threshold of reasonableness. For the reasons above, I prefer the lower, non-coherentist threshold, on which good inductive and perceptual rules count as weakly a priori. The question of their full apriority then reduces to the question of whether they are empirically indefeasible. I will have something to say in support of the empirical indefeasibility of certain inductive and perceptual rules later, though we'll eventually see that this question too has a quasiterminological component. end p.367
3. Default Reasonableness and the Evaluativist Approach to Apriority So far the discussion of default reasonableness has been focused more on the 'default' than on the 'reasonableness'. To call a proposition or rule default reasonable is to hold that it is reasonable to believe or employ it without first adducing evidence or arguments in its favor. Or in other words, that it is reasonable to adhere to it as a 'default belief' or 'default rule' (a belief or rule that is accepted or employed without adducing considerations in its favor). The previous section argued (with a slight qualification) that if one is going to have very many beliefs at all one must have default rules; but to get from this to the conclusion that some rules are default reasonable and hence weakly a priori, one needs to assume that it is possible to have a sufficient array of reasonable beliefs; and to get to the conclusion that some of the rules we employ are default reasonable and hence weakly a priori, one needs to assume that some of our own beliefs are reasonable.
What is it for a default rule (or any other rule) to be reasonable? My main discussion of this will come later, in section 5, but it will help to give a brief preview now. One approach to explaining reasonableness (I'll call it 'naturalistic reductionism') has it that the reasonableness of a rule is entirely a matter of how good the rule is at producing truth, avoiding falsehood, and so forth. In the case of deductive rules, we think that ours are objectively correct in that they have complete and non-contingent reliability; and naturalistic reductionism simply identifies this objective correctness with their reasonableness. In the case of inductive and perceptual rules it is less easy to make sense of objective correctness, but we do apparently think that the ones we employ are as a matter of contingent fact reliable, and also are good at arriving at the truth, and naturalistic reductionism simply identifies the reasonableness of the rule with some combination of these and similar 'truth-oriented' characteristics. In my view, this approach is thoroughly implausible, on numerous grounds. Here is a partial list: (1) In the case of deductive rules, the notion of reliability is quite clear: and correct rules do have complete and non-contingent reliability while incorrect ones don't. So in this case, the question of whether reliabilism gives the right answer about reasonableness is equivalent to the question of whether it is always reasonable to believe correct logical rules and unreasonable to believe incorrect ones. But I would have thought the answer to be 'no': even if the correct logic for dealing with vagueness or the semantic paradoxes is a non-classical logic (perhaps one that no one has yet formulated), we who do not realize the virtues of such a revision of logic, or even know how to formulate the logic, are not unreasonable in trying to cope with vagueness or the semantic paradoxes in the context of classical logic. We are unreliable but not unreasonable. end p.368
(2) The standard 'internalist' criticism: it is implausible to hold that our methods (assuming them reliable in the actual world) would be straightforwardly unreasonable in a 'demon world' (a world designed to make those methods unreliable, but undetectably so). (3) It isn't easy for a reductionist to satisfactorily explain why a method is unreasonable if it simply builds in an a priori belief in whatever physical theory is in fact correct. (The obvious reductionist approach to explaining that is to require that the method work in certain possible worlds other than our own as well as in our own world; but specifying which other possible worlds are relevant and which aren't, and doing so in a way that isn't grossly ad hoc, seems to me extremely difficult.) (4) The application of the notion of reliability to our basic inductive methods is crucially unclear, for reasons to be given at the end of section 4; and it is hard to supply a clear replacement for the demand that our basic inductive methods be reliable that isn't either too weak to exclude obviously unreasonable methods or so strong as to further accentuate the problems in (2). (5) The motivation for reliabilism is suspect: the motivation for wanting our beliefs to be true is clear, and this might motivate an interest in the reliability of a rule as evidence of the truth of beliefs formed by the rule, but it doesn't motivate the stronger role that the reliabilist gives to reliability. More fully: there are lots of classes to which a given belief B belongs such that the proportion of truth to falsehood in that class would have an evidential bearing on the truth of B. If our interest is in the truth of B, we thus have an indirect interest in the proportion of truth to falsehood in many such classes. But the reliabilist, in trying to reduce reasonableness to a precisely defined notion of reliability, wants to single out one particular such class as having a more-thanevidential interest: it's what constitutes the reasonableness of B. Why think that this has any interest? (6) 'Reliability' is certainly not the only thing we want in an inductive rule: completely reliable methods are available, e.g. the method of believing nothing whatever the evidence, or believing only logical truths; but we don't value them, and value instead other methods that are obviously not perfectly reliable, because of their other characteristics. And reliability itself subdivides into many different notions: for instance, short term v. long term; yielding a high probability of exact truth v. yielding a high probability of approximate truth; reliability in the actual world v. reliability over a range of 'nearby' possible worlds; etc. When one thinks about the different more precise characteristics we value, and the fact that they tend to be in competition with each other, it is hard to imagine how they could be combinable into a package that could plausibly be held to constitute reasonableness. (7) Familiar worries about naturalistic reductionism in ethics carry over to the epistemological case. For instance, (i) identifying reasonableness with a natural property seems to strip it of its normative force. (ii) In cases of fundamental disagreement about what natural property is coextensive with reasonableness, it is difficult to take seriously the idea that one party to the end p.369
debate is right and the others wrong. (Indeed, that idea seems to presuppose a non-natural property of reasonableness, whose extension is up for grabs.) 14 The naturalist can avoid this by supposing that those in fundamental disagreement about what is reasonable are using the term for different natural properties; but this relativist conclusion has the consequence that they aren't really disagreeing, which seems incredible. 15 Despite all this, I don't think naturalistic reductionism wholly misguided: I hope to provide an attractive picture that captures its insights. If naturalistic reductionism is rejected, what is to take its place? Another approach is to take our own rules as completely reasonable by fiat, and to regard other people's rules as reasonable to the extent that they are similar to ours. I'll call this 'the egocentric approach'. It too strikes me as hopelessly implausible, this time because it chauvinistically takes our own rules as sacrosanct quite independent of any properties they might have. What alternatives remain? One could try to combine features of the above approaches, taking reasonableness to be a combination of reliability (and related characteristics) and similarity to one's own methods; but this wouldn't be much better than the egocentric approach as regards chauvinism, and wouldn't help with the main problems with the naturalistic approach either. A third approach ('non-naturalism') is to regard reasonableness as a primitive property of rules or methods, not explainable in terms of anything non-normative (hence presumably undetectable by ordinary perceptual processes). But what reason would there be to suppose that any rules or methods have this strange property? And even if we assume that some have it, what reason would there be to suppose that the rules or methods we employ have it? If reasonableness consists in the possession of such a curious property, shouldn't we believe that our rules and methods (and any alternative rules and methods) are unreasonable? It seems to me that we need another option. The options above had one thing in common: they assumed that reasonableness was a straightforwardly factual property. My proposal is that it is an evaluative property, in a way incompatible with its being straightforwardly factual.16 We do, I think, evaluate 14 This parenthetical point would need to be stated with care, so as not to run afoul of the fact that there are
genuinely controversial property-identities (e.g. between being in pain and being in a certain psychofunctional state), and that controversies about them do not necessarily have to be understood in terms of higher-order properties associated with the two terms of the identity, but can be explained in terms of the differing conceptual roles of the terms. But I don't think that the analogy of controversial judgements about reasonableness to controversial judgements about pain does much to raise the plausibility of evaluative naturalism. For one thing, there is a physical property centrally involved in causing our pain judgements, and it seems a fairly straightforward factual question what this is. There seems to be no such straightforward factual question in the case of reasonableness. 15 Of course, the relativist can admit that the parties disagree in attitude, but in the context of naturalism (or any sort of fully factualist view of reasonableness) this seems ad hoc: the natural notion of disagreement for a naturalist (or any sort of factualist) is factual disagreement, and on this notion the parties do not disagree. 16 The conception of evaluative properties as 'not fully factual' has been spelled out in different ways. My favorites are Gibbard 1990, and Ch. 8 of this volume. One feature of these views is that they employ a general notion of disagreement that incorporates disagreement in both attitudes and values. When straightforwardly factual matters are at issue, disagreement reduces to factual disagreement. In typical normative disagreement, it is a combination of facts and values that are in dispute. In certain cases of fundamental normative disagreement, no facts are relevant to the disagreement, only values. In this case, the disagreement is in attitudes. But note that this invocation of disagreement in attitudes is very different from the factualist's (fn. 15): on a factualist view it is factual disagreement that should be primarily important, so invoking disagreement in attitude seems ad hoc; whereas on Gibbard's or my evaluativism, there is only one notion of disagreement, and disagreement in attitude is simply a special case of it. end p.370
rules and methods in part on the basis of our judgements as to whether using them will be good at leading to true beliefs and avoiding error. We also favor our own method over other quite dissimilar ones. These two strands in our evaluation procedure are inseparable: for (I will argue) we inevitably believe that our own methods will be better than the dissimilar methods at leading to truths and avoiding errors. One shouldn't ask whether it is the conduciveness to truth or the similarity to our methods in which the reasonableness consists, for reasonableness doesn't consist in anything: it is not a factual property. The approach I'm recommending ('evaluativism') shares with non-naturalism the conviction that it is quite misguided to try to reduce epistemological properties like reasonableness to other terms. But it is very different from non-naturalism when it comes to the question of skepticism. A sensible evaluativist will think that there are no non-natural properties, or anyway none that are ever instantiated; so that if skepticism were defined as the failure to believe that any rules and methods have such a non-natural property, then the sensible evaluativist is a 'skeptic'. But the evaluativist should say
that this is a totally perverse definition of skepticism. On a more reasonable definition, a skeptic is someone who positively evaluates abstention from all belief; skepticism in that sense is idiotic, and surely doesn't follow from the noninstantiation of mysterious properties. The meta-epistemological views just sketched are important to the interpretation of default reasonableness, and of weak apriority more generally. One kind of question about these characteristics is: in virtue of what does a given proposition or method have them? In virtue of what is it reasonable to use modus ponens on no evidence? 17 The difficulty of providing an answer to this question is one of the main reasons that apriority has seemed mysterious. The meta-epistemology I've suggested requires that this question be recast: the proper question is, why value a methodology that allows the use of modus ponens on no evidence? Well, one needs some methodology, so the question can only be why favor this methodology over alternatives, and the answer will depend on what alternative methodologies are possible. The alternatives to a methodology that allows use of modus ponens on no evidence divide between those that license its use on certain empirical evidence (maybe on the evidence that snow is white?) and those that don't license its use at all (but license no deductive inference at all, or license only some particular weak logic that 17 If default reasonableness rather than weak apriority is in question, I should say 'on no evidence or argument'. But
presumably we attach little importance to the difference between a methodology that takes modus ponens to be default reasonable and one that takes it to be weakly a priori because derivable from disjunctive syllogism which is in turn taken as default reasonable. end p.371
doesn't include it). The question then reduces to showing what is wrong with particular methodologies of each type. I don't want to get into a substantive discussion of what is wrong with particular methodologies of each of these types; my point is only that that is what is involved in defending the weak apriority of modus ponens, once one adopts the evaluativist perspective. This seems to me a substantially different perspective on apriority than one gets from more fully 'factualist' meta-epistemologies, and this different perspective does a great deal to remove the mystery from weak apriority. It isn't just issues about weak apriority that evaluativism recasts; issues about empirical indefeasibility are recast as well. For an evaluativist, defending the empirical indefeasibility of modus ponens is a matter of arguing that a methodology that takes it as empirically indefeasible is preferable to methodologies that allow it to be empirically defeated by particular kinds of evidence. If an anti-apriorist charges that it would be dogmatic for a system of rules to license the use of modus ponens on any evidence whatever, the response should be 'This is better than their licensing its revision on inappropriate evidence (say, the discovery of a new kind of virus); give me a plausible example of possible evidence that would make it appropriate to abandon modus ponens! And if the possible evidence you cite isn't obviously appropriate for this purpose, then give me at least a sketch of a theory of evidence on which the evidence is appropriate!' Without even a sketch of an answer, it is hard to see why we should take empiricism about modus ponens seriously. I don't say that we ought to rule out the possibility that someone could come up with an example of possible evidence that would make it appropriate to give up modus ponens (were the evidence actual), and of a possible theory of evidence that explained why this was evidence against the adequacy of modus ponens. But ruling out that possibility is something no apriorist should try to do: however apriorist we are about logic, we ought to be fallibilist enough to recognize the possibility that new conceptual developments will undermine our apriorism. 18 Incidentally, the failure to distinguish apriorism from infallibism about apriorism seems to underlie the widespread belief that Quine has provided an alternative to apriorism about logic. Quine's view is that one should evaluate alternative logics in combination with theories of the rest of the world: given a theory of everything, including a logic, one uses the logic in the theory to generate the theory's consequences. Then we choose a theory, including a logic, on such grounds as overall simplicity and conservativeness and agreement with observations. But this description of the methodology is so vague that it is not in the least clear that it dictates that modus ponens or any other part of logic should be revisable on empirical evidence. Whether it so dictates depends on the standards of simplicity and conservativeness of overall theories: it depends on whether the decrease of simplicity and conservativeness 18 More on this and some of the other claims in this paragraph and the next is to be found in sections 2 and 4 of
Field 1998. end p.372
that results from modifying the logic could be compensated by increased simplicity and conservativeness in other parts of the theory (holding the observational predictions fixed). It is conceivable that the standards of simplicity we use, or attractive alternative standards, will be such as to say that there are possible observations that would lead to favoring a theory that includes an alternative to our logic over ours. That is enough to undermine infallibilism about apriority, but to undermine apriority one must show that there actually are attractive standards of simplicity under which possible
observations would lead to an alternative logic, and Quine has given no clue as to what those standards of simplicity might be. (Indeed there is reason for skepticism about the existence of standards that would let possible observations undermine modus ponens. For one is likely to need modus ponens in the background logic in which one reasons about what follows from each theory-plus-logic and how well it accords with observations; and it is hard to imagine that a person using this background logic could rationally come to favor a theory-plus-logic in which the logic conflicted with the background logic.) I don't pretend that this discussion settles the case for apriorism about logic: it is intended only to illustrate how evaluativism supplies a perspective for answering the question that does not turn on rational insight into the nature of non-natural epistemological properties. 19
4. An Epistemological Puzzle I turn next to the question of whether our inductive and perceptual methodologies are best viewed as empirically indefeasible; this will lead into further discussion of reliabilism and evaluativism. A good way into these issues is by way of an epistemological puzzle. 20 It comes in two parts. Part One starts with the idea that we want our empirical methods to be reliable: to lead to a fairly high proportion of true beliefs. In particular, we want 19 One of the important issues not addressed is whether Benacerraf's (1973) puzzle about how a priori mathematical
knowledge is possible extends to other alleged cases of a priori knowledge. I think Benacerraf's argument does work against many claims to apriority. (Including claims of a priori access to mathematical entities as these are conceived by most platonists. For a discussion of which platonist views might survive the argument, see Chapter 11.) For instance, the reasons for negatively evaluating a system of rules that would allow us to adhere whatever the evidence to a particular physical theory that we hold true have to do with the fact that doing so would clearly make our belief causally and counterfactually independent of the facts; and such independence from the facts would defeat the epistemological value of the considerations on which the belief was based. (I think that is what a Benacerrafian argument against apriority about physics would amount to.) It might seem that this would apply equally well to apriority about logic. The idea would be that a priori belief in logic makes logical beliefs similarly independent of the facts, and that this is equally bad. But I think that in the logical case one simply can't make sense of the question of whether logical beliefs depend on the logical facts; so we can't make sense of the claim that is supposed to defeat the evidential value of the considerations on which the belief was based, and so the logical beliefs remain undefeated. For more details, see Field 1996, sect. V, or Field 1998, sec. 5. 20 The puzzle is implicit in many epistemological writings; probably its most explicit presentation is as the argument against 'norm externalism' in Pollock 1987 (though it is close to explicit in Putnam 1963 and Lewis 1971). My resolution is not too far from Pollock's, though Pollock's view is closer to what I've called the egocentric approach than to the evaluativism I recommend. end p.373
them to be reliable in present and future applications. But we can empirically investigate whether they have been reliable in past applications; and it is surely possible that we will discover that one of our empirical methods hasn't been reliable, and that some alternative to it has been. Alternatively, we might discover that our method has been fairly reliable, but that some alternative method has done much better. If we did discover one of these things, then since we take the past to be the best guide to the future we should conclude that our method will continue to be less reliable than the alternative. But surely if we think that one of our own empirical methods will be less reliable than an alternative, then we ought to switch from our method to the other. All of this would seem to apply not only to our 'nonbasic methods'—our rules of thumb (like 'Believe what the NY Times says') that are evaluated using more basic methods (like induction); it would seem to apply to our most basic inductive method itself. That is, it would seem that we want our most basic inductive method to be reliable, and can investigate empirically whether it has been, and we will stop using it if we find that it has not. But in this case, the investigation of the most basic method can't be by another method, for by hypothesis none is more basic. Rather, the investigation of our most basic method uses that very method. So in the case where we empirically discover that method unreliable, the decision not to use the method would be based on that very method.21 Part Two of the puzzle says that the conclusion of Part One is incoherent. How can our method, in combination with evidence E (in this case, evidence of its own unreliability), tell us not to follow that very method? Our method presumably already tells us something about what is legitimate to believe and what is illegitimate to believe when our evidence includes E (say, when it consists of E&F). 22 These instructions might be empty: they might allow us to believe what we like. Or they might tell us to stop believing. Or they might tell us any of numerous other things. But
whatever they tell us it's legitimate to believe on E&F, that's what we must do if we are to follow the method. Now if the method tells me that E undermines the method, it must tell me not to always do what the method tells me to do; in other words, it must tell me to do something different, on some evidence E&F, from what it tells me to do on E&F. In other words it must offer me inconsistent instructions. It would seem that only an inconsistent empirical method can allow itself to be undermined by empirical evidence of its own past unreliability; to suppose that good empirical methods must allow themselves to be empirically undermined in 21 It has been suggested to me in conversation that our basic method is the meta-method 'employ whatever first-
order method is most reliable'; and that this meta-method couldn't fail to be reliable. But in fact the proposed metamethod is not an employable method. To make it into one, we would need to recast it as something like 'employ whatever first-order method you believe to be most reliable', or 'employ whatever first-order method your first-order methods tell you to believe most reliable'; and these can certainly fail to be reliable. 22 Our method may take into account factors other than the available evidence—for instance, it may take account of which theories have been thought of—but as far as I can see, such additional factors won't matter to the argument that follows. end p.374
this way is then to suppose that good methods must be inconsistent, which seems absurd. To summarize: Part Two is an argument that we can't possibly treat our basic empirical methods as empirically defeasible, whereas Part One is an argument that we must do so; where to take a method as empirically defeasible is to adopt standards according to which empirical evidence could count against it. Obviously something is wrong, but what? A superficially plausible diagnosis is that the key error was the presupposition that there is such a thing as 'our basic empirical method': that is, in the supposition that we employ a method that can't be undermined by any of our other methods. One might argue that this supposition is incorrect, that in fact we employ many different methods, each of which can be assessed using the others. I think myself that the assumption of a basic inductive method is defensible if properly understood, but I will save that issue for an appendix. What I want to do now is argue that the issue of whether there is a basic method isn't central to the puzzle, because there is a more straightforward error in the argument in Part One. In investigating this, it helps to be a bit more concrete: instead of talking about 'our basic empirical method' let's instead talk about a specific inductive rule. For simplicity I'll pick an extremely simple-minded rule, but one which I think will have crucial features in common with any inductive rule that might plausibly be regarded as part of our basic inductive method. The rule I'll pick for illustration is the following: (R) If in the past you have observed n ravens, and m of them have been black, you should believe to degree (m+j)/(n+k) of any raven not yet observed that it is black, where j and k are fixed real numbers, 0<j
paradox makes vivid, using the rule for some pairs requires not using it for certain others. The rule given is in effect an instance of Carnap's λ-continuum (at least when k/j is an integer ≥2): it results from taking λ = k and the number of kinds as k/j. end p.375
Since the rule is one for degrees of belief rather than for all-or-nothing belief, talk of reliability or unreliability may not be strictly appropriate; but clearly the analog of a discovery of past unreliability in the rule is the discovery that the actual proportion of blackness among ravens observed in the past has been substantially different from j/k. The argument of Part One suggests that were we to discover this, then those observations would provide evidence against the rule. But this is mistaken. There is of course no doubt that if j/k is small, then the observation of many ravens with a high proportion of blackness among them should lead us to revise the probability of blackness for an unobserved raven upwards (barring special additional information anyway). But this doesn't mean that we should modify the rule: our observation of a high proportion of blackness among ravens is something that the rule takes into account. Suppose
our initial bias was j = 1, k = 10, so that the degree of belief assigned to a given raven being black was only 0.1. And suppose we observe 20 ravens, of which 19 are black. Then the rule tells us to believe to degree 0.667 (20/30) of an unobserved raven that it is black. The rule has in a sense told us to modify our biases. In another sense, though, the biases go unchanged: the initial bias, represented by the pair <j,k> from which we started, is still there producing the new degree of belief. Of course, the initial bias produces that new degree of belief only with the accumulated evidence, and the effect of using the initial bias with the accumulated evidence is in a sense equivalent to altering the bias: it is equivalent to altering the initial bias to j = 20 and k = 30, and then dropping the observation of the first 20 ravens from our evidence. (It would be double counting to let this observation alter the bias and then in addition let it count as evidence with the new bias.) In a sense then our rule is 'self-correcting': with accumulating evidence the old rule behaves like an altered rule would without the added evidence. Because it is 'self-correcting' in this way, there is no need for it to be revised when evidence of its unreliability in previous applications is accumulated. Of course there are ways that a rule like (R) might be inductively challenged. (R) was supposed to be a rule for ravens; and if we had a great deal of experience with other birds showing that a high degree of concentration in color tends to prevail within each species, that would give us grounds for lowering the correction factor (lowering the j and the k while keeping their ratio constant) in the raven rule. But that simply shows that the original rule isn't a serious candidate for a basic inductive rule. Any serious candidate for a basic inductive rule will allow 'cross-inductions': it will allow for evidence about other birds to affect our conclusions about ravens.24 Were we not to employ a rule that allows crossinductions, we wouldn't regard the evidence about other species as relevant to ravens, and so would not see such evidence as providing any reason to lower the correction factor in (R). 24 Kemeny and Carnap investigated how to expand Carnap's λ-continuum to include a new parameter that allows for
such cross-inductions (see Kemeny 1963: 732-3; Carnap 1963: 977). The resulting formulation of an inductive method is still quite simplistic, but a step in the right direction. end p.376
So the point is that we use a more complicated inductive rule than (R) (one that 'self-corrects' to a greater extent than (R) does); using the more complicated rule, we can inductively conclude that our future practice should not accord with the more simple-minded rule. But if we had used the more simple-minded rule in doing the assessment, we wouldn't be able to conclude that our practice should no longer accord with that simple-minded rule; similarly, I suggest, if we tried to assess the more complicated rule using the more complicated rule, we couldn't ever recognize anything as evidence for the conclusion that we shouldn't use it. We could recognize evidence that our rule hadn't worked well in the past, but this would simply be evidence to be fed into the rule that would affect its future applications; we would not regard it as evidence undermining the rule. (Some apparent objections to this are discussed in the appendix.) What I have said suggests that if indeed some inductive rule is basic for us, in the sense that we never assess it using any rules other than itself, then it must be one that we treat as empirically indefeasible (hence as fully a priori, given that it will surely have default status). So in the puzzle, the error in reasoning must have come in Part One. Moreover, it is now clear just where in Part One the error was: the error was in supposing that because the rule had been unreliable in the past it was likely to be unreliable in the future. What the discussion shows is that there is no reason to extrapolate from the past in this way: for the future employment of the rule takes account of the unreliability of the past employments, in a way that makes the future applications qualitatively different from the past employments. That's so in the case of rule (R), and it will also be so in the case of any sufficiently general inductive rule that anyone would be tempted to employ. So to resolve the puzzle there is no need to deny the existence of basic rules, we must merely recognize that any plausible candidate for a person's basic rule will have the same 'self-correcting' character as (R).25 I have been discussing whether we could empirically overturn a most basic inductive rule, and concluded that we could not. If so, this also calls into question the idea that we could ever rationally accept anything as empirically supporting these rules: for it is hard to see how there could be possible observations that support the rules without possible alternative observations that undermine them. The idea that our most basic inductive rules could get inductive support has been much discussed in the context of the justification of induction. It has often been dismissed, on the ground that an inductive justification of induction is circular, though as noted earlier a number of authors have argued that the sort of circularity involved here (rule-circularity) is not vicious. I agree with these authors that the kinds of arguments that are offered as rule-circular justifications are of interest; but their worth as justifications turns on the idea that they remove a prima-facie risk, a risk that reasoning with the rule 25 Less inductively sophisticated creatures doubtless employ simpler inductive methods that are not 'self-correcting' in
this way. Such creatures could either never have evidence for the past unreliability of their methods, or could never think to extrapolate it, or would continue reasoning as before despite the belief that their methods would be unreliable. But they aren't us. end p.377
will lead to the rule's invalidation. In the case of inductive justifications of induction, what they turn on is the idea that the basic inductive rule might be inductively undermined; and that, I am suggesting, is impossible. It is that impossibility, not rule-circularity in itself, that is the real reason why the inductive justification of induction is an illusion. (Something similar holds of deductive justifications of deduction, I believe.) Before drawing further morals, it will be helpful to consider another illustration, this time involving a perceptual rule. It is natural to suppose that rules of perception can be empirically overturned. Suppose we are initially disposed to regard things that look a certain way 'red'. (I'll pretend here that how things look to us is independent of our tendencies to judge their color: it won't affect the point.) We then discover that in certain lighting, things that look that way often aren't red; and using this information, we revise our practice of making color judgements. So it looks like our initial 'rule of perceptual judgement' (P) Believe things red if they look red has been overturned. But that, I submit, is misleading: the right thing to say rather is that our initial practice was sensitive to inductive considerations that weren't built into (P), so that (P) isn't the most basic rule we were following (even before the information about the lighting). After all, if it had been the most basic rule we were following, it is hard to see how the information about the lighting could have influenced us. One way to think about what's going on in this case concedes that we do employ (P), but only as a default rule. The more basic rule is a meta-rule that says: use (P) unless it is inductively shown unreliable in certain circumstances; if that happens, stop using it in those circumstances. The meta-rule allows (P) to be inductively overturned, but it's hard to see how the meta-rule itself can be inductively overturned: we treat the meta-rule as empirically indefeasible (indeed, as a priori). We don't really need the 'default rule'/'metarule' contrast, we can build everything into the ground level rule, by taking that rule not to be (P) but rather something more like (P*) Believe a thing red if it looks red, unless there are circumstances C that you believe you're in such that you have inductive evidence that looking red is misleading in those circumstances or: (P** ) The extent to which you should believe a thing red should be such and such a function of (i) how it looks to you; (ii) your evidence about the perceptual circumstances; (iii) your background evidence about how the way that things look depends on the perceptual circumstances; and (iv) your background evidence about what color it actually is. Here the point is that inductive evidence that the rule has failed in the past feeds into the rule, in a way that alters the results of applying the rule in the end p.378
future: for instance, in (P** ) the evidence does this by affecting our beliefs of type (iii), on which the degree of belief that the thing is red depends. We have the same situation as with the inductive rule (R): the relevance of evidence of the past unreliability of the rule isn't to undermine the rule; rather, the evidence is something that the rule will itself take account of, and that will substantially modify the future applications of the rule (in a way that might be expected to make those future applications more reliable than the past ones). Of course, not every conceivable rule will 'selfcorrect' (in this sense) on the basis of evidence of its past unreliability; but those rules that we in fact take seriously do. 26 Such general rules are never really put at risk in inductive assessment; all that is put at risk is the particular manner in which they've been employed. And again, that means that the idea of inductive justification of the rules, which requires the rules themselves to be put at risk but to survive the challenge, doesn't get off the ground. The bearing of all this on the apriority (or empirical indefeasibility) of inductive and perceptual rules is not entirely direct. For one thing, the discussion has been premised on the supposition that some inductive rule is basic for us. Whether that is so is a matter I discuss in the appendix: I argue that the question has a quasi-terminological component, but that there are considerations that favor a positive answer. But even given a positive answer to this, what I have argued hasn't been that our most basic rules are a priori or empirically indefeasible; it has been that we treat them as empirically indefeasible and indeed a priori: we don't regard anything as evidence against them. For a non-evaluativist, this distinction is crucial. For instance, a non-naturalist will say that the non-natural evidence relation may well hold between some evidence and an alternative to one of our basic rules, even though we could never be persuaded to adopt such an alternative on the basis of that evidence by principles of evidence we accept; and a reliabilist will say something analogous. From an evaluativist perspective, though, the distinction is rather academic: the only question is whether we should accept any possible evidence as undermining our rule, and since the rule itself is employed in making the evaluation of what we should do, there is no chance of a positive answer. More on this perspective in the next section.
The examples of (R) and (P*) or (P** ) have an importance beyond their bearing on the empirical indefeasibility of our empirical methodology: they also create a problem of interpretation for many versions of naturalistic reductionism. According to naturalistic reductionism, the reasonableness of an epistemological rule consists in its having a certain combination of truth-oriented properties, and most advocates of naturalistic reductionism place 'reliability' 26 So even if we had been unlucky enough, or evolutionarily maladapted enough, to employ rules which gave initial
weight to our purported telepathic experiences in addition to our perceptual experiences, then as long as those rules were analogous to (P*) or (P** ) rather than the cruder (P), we would have long since discounted telepathy. (This is in response to Goldman 1980: 42; though Goldman's formulation, and the surrounding discussion, seems to depend on (i) his temporarily assuming that we choose our basic inductive and perceptual rules, and (ii) his assuming that what we are after is finding the uniquely correct epistemological rules. I want no part of either assumption.) end p.379
high on the list of the properties that a reasonable rule must have. But as applied to 'self-correcting' rules like (R) and (P*) or (P** ), it is not entirely clear what 'reliability' comes to (even if the reliability of a rule is assessed relative to the specific circumstances of application within a possible world, rather than assessed in the possible world as a whole).27 We can see this in the case of (R) by imagining that there was a fairly strong initial bias (moderately large k), and the initial degree of belief j/k differed drastically from the actual frequency of blackness among ravens: perhaps j/k is quite small whereas the proportion of blackness among ravens is very high. (For simplicity I will confine my attention to the case where that proportion is fairly stable from one epoch to another and one region to another.) And suppose that (R) is applied long enough for the initial bias to become largely swamped by the observations. On the question of whether the use of the rule counts as reliable, there seem to be three options: (i) We can say that the rule was not reliable in its early uses (prior to the swamping), but became so later (after the swamping); after all, the degrees of belief produced by late uses of the rule closely reflected the actual frequencies, but degrees of belief produced by early uses were wildly at variance with actual frequencies. (Of course the swamping is gradual, so the shift from unreliability to reliability is gradual.) (ii) We can say that the rule was not reliable in early or late uses: the fact that it initially produces degrees of belief wildly at odds with frequencies shows that it simply isn't a reliable rule, but merely gives results in its later uses that closely match the results given by reliable rules (those with a more optimal ratio j/k). (iii) We could say that it was reliable in both: that the apparent unreliability in early uses results from taking too short-term a perspective. Which option should a reliabilist adopt? Given reliabilism, (ii) would make reasonableness hard to come by: a faulty initial bias would doom us to unreasonableness forever (barring a shift in inductive policy that is not evidence-driven). I think that (i) accords best with the normal use of 'reliable'. However, given reliabilism, (i) requires that use of the rule was unreasonable at first but became reasonable as the rule was used more and more; this strikes me as somewhat counterintuitive, and it is contrary to the doctrines of at least one prominent reliabilist: see footnote 27 . Some reliabilists might then prefer the 27 Goldman (1988) offers a substantial reason for not relativizing to circumstances within a possible world in
assessing reliability: if one is allowed to do so, what is to keep one from so narrowing the circumstances that they apply only to one instance? (That would mean that whenever the rule yields a truth, however accidentally, it would come out reliable, and so following it on that occasion would count as reasonable according to reliabilist lights.) Perhaps there are ways to block carving the circumstances so finely, but it isn't in the least clear how to do so without gross ad-hocness; and so Goldman adopts for not allowing any relativization to circumstances within a world. (Indeed, he argues that even consideration of reliability within a possible world as a whole is too narrow: one must consider reliability with respect to a class of similar worlds. The motivation for doing this is so that a rule that 'happens' to yield truths about a particular world independent of evidence won't count as reliable. At this point one might raise the question of how to carve out the relevant class of worlds without gross ad-hocness, but I will not press the matter.) 27 Goldman (1988) offers a substantial reason for not relativizing to circumstances within a possible world in assessing reliability: if one is allowed to do so, what is to keep one from so narrowing the circumstances that they apply only to one instance? (That would mean that whenever the rule yields a truth, however accidentally, it would come out reliable, and so following it on that occasion would count as reasonable according to reliabilist lights.) Perhaps there are ways to block carving the circumstances so finely, but it isn't in the least clear how to do so without gross ad-hocness; and so Goldman adopts for not allowing any relativization to circumstances within a world. (Indeed, he argues that even consideration of reliability within a possible world as a whole is too narrow: one must consider reliability with respect to a class of similar worlds. The motivation for doing this is so that a rule that 'happens' to yield truths about a particular world independent of evidence won't count as reliable. At this point one might raise the question of how to carve out the relevant class of worlds without gross ad-hocness, but I will not press the matter.)
end p.380
long-run perspective implicit in (iii): even early uses of the rule count as reliable, because the rule would yield good results if applied in the long run. If 'long run' here means really long run, this would be even more counterintuitive than (i): 'dogmatic' versions of (R) with exceptionally large k that would take millions of observations to significantly swamp would count as reliable and hence reasonable. It would also blunt the force of reliabilism, in that very few rules would be declared unreliable. But a reliabilist could avoid this by adopting (iii) for the case under discussion, where k is moderately large, and adopting view (i) or (ii) in the case of exceptionally large k where it would take a very long time for swamping to occur; in effect this is to use an intermediate length of reliability as a criterion of reasonableness for the early uses of the rule. This combination gives the most intuitively appealing results about reasonableness. But it is not clear that this is in keeping with the spirit of reliabilism: for it is now a priori (relative anyway to the assumption of stability in proportions) that all versions of (R) where k is not too large are reliable from the start, whatever the value of j (greater than 0 and less than k); the idea that the facts about the actual world determine an inductive method uniquely or close to uniquely (see Goldman 1980) is completely abandoned. Something similar holds for perceptual rules like (P*) or (P** ). Imagine a world where deceptive appearances are common enough so that in the initial stages of use the rule leads to error a quite substantial percent of the time, but not so common as to prevent the rule from ultimately 'self-correcting' if appropriate observations are made. We can again ask whether uses of the rule are 'reliable' and hence reasonable before such a 'self correction' and whether they are reliable and hence reasonable afterwards. (Actually there are two important differences between this case and the induction case: first, it is likely to take much longer for the rule to self-correct; second, the self-correction is not automatic, in that whether a self-correction is ever made is likely to depend on accidents of which observations are made and which theories are thought of. I think that both of these factors diminish the chance that a reasonable longterm perspective on reliability could rule the early uses of the rule 'reasonable'.) In this case too it is unclear what a reliabilist can say that keeps the spirit of reliabilism without making reasonableness implausibly hard to come by.
5. More on Epistemological Evaluativism I propose an alternative to reliabilism, more in line with 'non-factualist' views about normative discourse. The alternative is that reasonableness doesn't consist in reliability or anything else: it's not a 'factual property'. In calling a rule reasonable we are evaluating it, and all that it makes sense to ask about is what we value. So the relevance of the reliability of a rule to its reasonableness is simply that we place a high value on our inductive and perceptual rules leading to truth in the circumstances in which we apply them; more or less end p.381
equivalently, we place a high value on a given rule to the extent that we believe it will be reliable in the circumstances in which we apply it. We saw earlier that we should inevitably believe our most basic rules to be reliable in the circumstances in which we intend to apply them. 28 If so, we will inevitably place a high value on our own inductive and perceptual rules. Is this an 'externalist' view or an 'internalist' view? The answer is that that distinction as normally drawn (for instance in Goldman 1980) rests on a false presupposition. The presupposition is that epistemological properties like reasonableness are factual. If they are factual, it makes sense to ask whether the factual property involved includes 'external' elements. 29 On an evaluativist view, it is hard to draw a distinction between externalism and internalism that doesn't collapse. Any sensible evaluativist view will be 'externalist' in that one of the things we value in our rules is (some restricted version of) reliability. A sensible view will also be 'internalist' in that we also place a high value on our own rules: indeed, those are the rules we will use in determining the reliability of any rules we are evaluating. Which is primary, the high valuation of our own rules or the high valuation of reliable rules? It is hard to give clear content to this question, since (by the previous section) we inevitably ought to regard our own rules as likely to be reliable in the circumstances in which we intend to apply them. 30 A view like this raises the specter of extreme relativism. For mightn't it be the case that different people have different basic standards of evaluation? If so, aren't I saying that there is no fact of the matter as to which standard of evaluation is correct? And doesn't that mean that no standard is better than any other, so that those who 'justify' their belief in reincarnation on the basis of standards that positively evaluate just those beliefs that they think will make them feel good about their cultural origins are no worse epistemologically than those with a more 'scientific bias'? That of course would be a totally unacceptable conclusion. But nothing I have said implies that no standards are better than others. Indeed, some clearly are better: they lead both to more truth and to less falsehood. Of course, in saying that that makes them 'better' I am presupposing a goal that is being rejected by the imaginary 'feel gooders', but so what? All evaluations presuppose goals, and of course it is my
own goals that I presuppose in making evaluations. (To paraphrase David Lewis: Better I should use someone else's goals?) Not only must I presuppose my own goals in saying that my standards are better than others, I must presuppose my own beliefs. This is most easily seen 28 More accurately, our rules license us to so believe. 29 I assume for present purposes that the contrast between external and internal elements is clear. 30 One could hope to make sense of it by considering conditional evaluations: we ask people to evaluate certain rules
on the supposition that they are reliable, or unreliable. For instance, we consider the possibility of a world where our methods are unreliable and methods we find bizarre are reliable, and ask whether our method or the bizarre method is 'reasonable' in that world. But it seems to me that when asked this question we are torn: the two strands in our evaluation procedure come apart, and what to say is simply a matter of uninteresting verbal legislation. end p.382
by contrasting my own standards not to 'feel good' standards but to the standards of those who are interested in the truth but have bizarre views about what best achieves it (e.g. they accept counterinductive methods, or believe whatever the Reverend Moon tells them). If one were to apply the methods such people accept, one would be led to the conclusion that their methods are better than scientific methods. But again, so what? What makes scientific methods better isn't that they say that they will lead to more truth and less falsehood than these other methods, it is that they do lead to more truth and less falsehood than these other methods. In saying that they do this I am presupposing the methods I accept, but that should go without saying: that's what accepting a method involves. Of course, this is circular. ('Rule-circular', anyway). Earlier I objected that in using a methodology to evaluate itself, a positive evaluation isn't worth much as a justification of the methodology unless there was a prima-facie risk that the evaluation would have turned out negative; and that with our most basic rules there is no such risk. But I conceded that rule-circular 'justifications' of our methods have another role: they serve to explain why we value our methods over competing ones. It is that point I am stressing here, and it is enough for the point at hand; for the point at hand was that it is not part of the evaluativist view in question that all methods are equally good. (I'm not now addressing the skeptical issue: to what extent are we reasonable in thinking that our methods are better than others. I'll address that soon.) Returning to the 'argument' for extreme relativism, I think we should concede that different people have slightly different basic epistemological standards: for one thing, any serious attempt to formalize inductive methods always invokes a number of variable parameters ('caution parameters' and the like), and there seems no motivation whatever for supposing that these are the same for all people. I doubt that there are many people with radically different basic epistemological standards, though there may be some: in the case of the Moonies it is hard to know what epistemological standards might have been used in producing their beliefs. But the extent of variation is a sociological issue on which I do not want to rest my views: even if there is no actual variation in basic epistemological standards, there might have been such variation—even radical variation. Given that there is possible variation in basic standards (whether moderate or radical), should we suppose that some standards are correct and others incorrect? I doubt that any clear sense could be given to the notion of 'correctness' here. If there were a justificatory fluid that squirts from evidence to conclusions, we could say that correct standards were those that licensed beliefs in proportion to the fluid they receive from available evidence; but absent that, it is hard to see what can make standards correct or incorrect. What we can say is that some standards are better than others in achieving certain goals; and to the extent that one adopts those goals, one can simply say that some standards are better than others. Even given the goals, talk of 'correct' standards is probably inappropriate: for if it means 'best' there may be no best (there could be incomparabilities or ties; and for each there end p.383
could be a better); and if it means 'sufficiently good' then it blurs relevant differences (two methods over the threshold would count as both correct even if one were better than the other). 31 We need a goal-relative notion of better standards, not a notion of correct standards. The argument for extreme relativism fails primarily in the slide from 'there are no correct standards' to 'all standards are equally good'. The position I'm advocating does allow for a sort of moderate relativism. For in evaluating systems of epistemological rules, we can recognize that certain small modifications would produce results which have certain advantages (as well as certain disadvantages) over the results ours produce. For instance, we recognize that a system slightly more stringent in its requirements for belief is more reliable but less powerful. So we recognize that a slight modification of our goals—an increase in the relative value of reliability over power—would lead to a preference for the other system, and we regard the alternative goals as well within the bounds of acceptability. Consequently we make no very strong claims for the preferability of our system over the alternative: the alternative is slightly less good than ours given our
precise goals, but slightly better on alternative goals that are by no means beyond the pale. 'Relativism' in this weak sense seems to me an eminently attractive position. (Pollock 1987: sect. 4) tries to avoid even this weak form of relativism, by proposing that each person's concepts are so shaped by the system of epistemological rules that he or she employs that there can be no genuine conflict between the beliefs of people with different such systems; as a result, the systems themselves cannot be regarded as in conflict in any interesting sense. But this view is wholly implausible. I grant that there's a sense in which someone with even slightly different inductive rules inevitably has a slightly different concept of raven than I have, but it is not a sense that licenses us to say that his belief 'The next raven will be black' doesn't conflict with my belief 'The next raven will not be black'. It seems hard to deny that there would be a conflict between these raven beliefs, and if so, the systems of rules give genuinely conflicting instructions.)32 A complaint about evaluativism that has sometimes been made to me in conversation is that it places no constraints on what one's epistemological goals ought to be: nothing makes it wrong for a person not to care about achieving truth and avoiding falsehood, but care only about adopting beliefs 31 One way to see the importance of this is to suppose that standards improve over time, and that a certain belief B
counts as reasonable on the evidence E available at t using the quite good standards S in use at t, but counts as unreasonable on the same evidence using slightly better standards S' that only become available later (but which might in turn, for all we know, eventually be superceded). Any attempt to describe this situation in the language of 'correct standards' loses something important. 32 Pollock's view is that it is our object level concepts like raven that are determined by our system of rules. A slightly more plausible view is that our epistemological concepts like reasonable are so determined: 'reasonable' just means 'reasonable according to our (the assessor's) rules'. But that view wouldn't serve Pollock's purposes: the advocates of alternative systems of rules would still be in genuine conflict about ravens, and each could raise skeptical worries about whether it mightn't be better to shift from the system that is reasonable in their own sense (viz., their own system) to the system that is reasonable in the other person's sense (viz., the other's system). end p.384
that will make him feel good about his cultural origins. But I'm not sure what sort of ought (or what sort of wrongness) is supposed to be involved. If it's a moral ought that's at issue, fine: I'm not opposing moral standards on which one ought to aim for the truth. But I assume that what was intended was not a moral ought, but some sort of epistemological ought. And that gives rise to a perplexity: on the usual understanding of 'epistemological oughts' they govern beliefs, not goals, and I have no idea what the sort of epistemological ought that governs goals could amount to. As for 'constraints' on epistemological goals, again I don't think that the intended sense of 'constraint' is intelligible. If McRoss's main goal in forming beliefs is making himself feel good about his cultural origins, well, I don't approve, and I might try to browbeat him out of it if I thought I could and thought it worth the trouble. If I thought that my telling him he ought not have such goals would influence him, I'd tell him that. Is this saying there are 'constraints' on his goals? Nothing is constraining him unless he takes my social pressure as a constraint. But if the question is whether there are constraints in some metaphysical sense, I don't think the metaphysical sense is intelligible. We don't need to believe in metaphysical constraints to believe that he's got lousy goals. (And if calling the goals lousy is evaluative rather than factual, so what?) Perhaps talk of 'metaphysical constraints' on goals is supposed to mean only that McRoss's goals shouldn't count as 'epistemological'. Or alternatively, that the so-called 'beliefs' arrived at by a system motivated by the satisfaction of such goals shouldn't count as genuine beliefs. I have nothing against 'metaphysical constraints' in one of these senses, though they might better be called 'semantic constraints': they are simply stipulations about the meaning of 'epistemological goal' or 'belief', and of course one may stipulate as one likes. Such stipulations do nothing to constrain McRoss in any interesting way: if he has goals that don't satisfy my constraints, why should he care whether I call his goals 'epistemological' or his mental states 'beliefs'? Nor is it clear what other useful purpose such stipulations might serve. As I've said, I doubt that there are many people with such radically different epistemological (or schmepistemological) goals for forming beliefs (or schmeliefs). But their non-existence has nothing to do with 'metaphysical constraints': as Richard Jeffrey once remarked, 'The fact that it is legal to wear chain mail on city buses has not filled them with clanking multitudes' (Jeffrey 1983: 145). Let's now turn to a different complaint about evaluativism: this time not about the lack of objectivity in the goals, but about the lack of objectivity in the beliefs even when the goals are fixed. One way to press the complaint is to make an unfavorable contrast between evaluativism in epistemology and evaluativism in moral theory. In the moral case, an evaluativist might stress the possibility of variation in moral goals (e.g. with regard to the respective weights given to human pleasure and animal pain), but agree that relative to a choice of moral goals some moral codes are objectively
better than others, end p.385
and that we can make useful evaluations as to which ones are better given the goals. Such evaluations are in no way circular: in evaluating how well a given moral code satisfies certain goals, one may need to employ factual beliefs (for instance, about the extent of animal suffering), but such factual beliefs can be arrived at without use of a moral code. In the epistemological case, however, the evaluation has the sort of circularity that has cropped up several times already in this chapter: in assessing how well a system of inductive or perceptual rules satisfies goals such as reliability, one needs to use factual beliefs, which in turn are arrived at only using inductive or perceptual rules. And this circularity might be thought to somehow undermine evaluativism—either directly, or by leading to a skeptical conclusion that makes the evaluativism pointless. The circularity is undeniable: it might be called the fundamental fact of epistemological life, and was the basis for the puzzle in section 4. But it doesn't directly undermine evaluativism, it leads only to the conclusion that our basic system of inductive rules (if indeed we have a basic system) is in Lewis's phrase 'immodest': it positively evaluates itself over its competitors (Lewis 1971, 1974a). Nor is skepticism the outcome: true, systems of rules that we don't accept lead to different evaluations than ours do, but why should that undermine the evaluations provided by the rules that we do accept? I concede that in dealing with people who use different standards of evaluation than ours, we typically don't just insist on our standards: we have several techniques of negotiation, the most important of which is to evaluate on neutral grounds. And to some extent we can do this with epistemological rules. For instance, the respective users of two inductive rules A and B that differ only in the value of a 'caution parameter' can agree that Rule A is more reliable but less powerful than Rule B; as a result, each realizes that a small shift in the relative value of reliability and power could lead to a preference for the other. In fact, the process of negotiating with people whose standards of evaluation differ from ours sometimes leads to a shift in our own standards (though of course such a shift is not evidence-driven). But though we sometimes negotiate or even shift standards in this way, we don't always: in dealing with a follower of the Reverend Moon, we may find that too little is shared for a neutral evaluation of anything to be possible, and we may have no interest in the evaluations that the Moonie gives. The fact that he gives them then provides no impetus whatever to revise our own evaluations, so the skeptical argument has no force from an evaluativist perspective. Indeed, a main virtue of evaluativism is that it removes the force of most skeptical arguments. Most skeptical arguments depend on assuming that reasonableness is a factual property of beliefs or of rules, and on the understandable resistance to stripping away the normative nature of reasonableness by identifying it with a natural property like reliability (for rules; or being arrived at by reliable rules, for beliefs). Given the assumption and the understandable resistance to naturalistic reductionism, there is no alternative when faced with two radically different systems that positively evaluate themselves beyond (i) end p.386
declaring them equally reasonable, (ii) postulating some mysterious nonnatural property by which they differ, and (iii) saying that one is better simply by being mine (or more similar to mine). The second position seems crazy, and raises epistemological questions about how we could ever have reason to believe that a particular system has this property; the third position seems to strip away the normative force of reasonableness much as naturalistic reductionism did (indeed it could be regarded as a version of naturalistic reductionism, but one that uses chauvinistic natural properties); and this leaves only the skeptical alternative (i). Not a bad argument for skepticism, if one assumes that reasonableness is a factual property. Evaluativism provides the way out. end p.387
Appendix: Rules and Basic Rules In the text I have tried to remain neutral as to whether a person's behavior is governed by 'basic rules', but here I would like to argue that there is something to be said for supposing that this is so. First a clarification: when I speak of someone 'following' a rule, what I mean is (i) that the person's behavior by and large accords with the rule, and there is reason to expect that this would continue under a decent range of other circumstances; and (ii) that the person tends to positively assess behavior that accords with the rule and to negatively assess behavior that violates the rule. (In the case of epistemic rules, the 'behavior' is of course the formation, retention, or revision of beliefs.) This is fairly vague, and the vagueness means that there is likely to be considerable indeterminacy involved in ascribing epistemic or other rules to a person: to ascribe a rule to a person is to idealize his actual behavior, and idealizations needn't be unique. (I will discuss the significance of this shortly.) In any case, when I
speak of rule-following I don't mean to suggest that the person has the rule 'written into his head'. There may be rules 'written into the head', but for those to be of use some part of the brain has to read them, and reading them is done by following rules; obviously these ones needn't be written in the head, on pain of regress. In particular, when I imagined as a simple-minded illustration that we follow inductive rule (R) and that no evidence could lead us to change it, I certainly didn't mean to suggest that a person has something like my formulation of (R) 'written into his head', never to be altered by evidence. Even if some sort of formulation of the rule is explicitly written into the head, it might be very different from formulation (R). For instance, it might be that at a given time t what is written is not (R) but instead (R t ) If after t you have observed s t ravens, and r t of them have been black, you should believe to degree (r t +b t )/(s t +c t ) of any raven not yet observed that it is black, where b t and c t are parameters representing the current bias, which changes over time. Following this sequence of rules is equivalent to following (R).33 (If q t is the number of ravens observed by time t and p t is the number of them that have been black, then b t and c t are j+p t and k+q t respectively; since r t and s t are just m−p t and n−q t respectively, the equivalence is transparent.) 'Following this sequence of rules' might better be described as following the meta-rule (R*) Act in accordance with (R t ), where the parameters b t and c t are obtained from earlier parameters by such and such updating process. But a psychological model could allow (R*) to be followed without being written into the head: the system is simply built to act in accordance with (R*), and to make assessments 33 As a model of what might be 'written into the head', the sequence of (Rt )s is far more plausible than (R): if (R)
were what was written in it would require the agent to keep track of all the relevant evidence accumulated since birth, which is grossly implausible, in part because the computational requirements for storage and access would be immense. Still more plausible as a model is something 'in between' (Rt ) and (R), where the agent doesn't need to remember all the evidence, but does remember some of it and retains a sense of what judgements he would make if some of the remembered evidence weren't in. (Indeed, something more like this is probably needed to handle assessments of our past inductive behavior.) end p.388
in accordance with it also. Again, no unchanging rule-formulation need be 'written into the head'. A second clarification: not only don't I mean to suggest that the rule-formulations written into the head can't change over time, I don't mean to suggest that the rules themselves can't change as a result of observations: only that a person for whom that rule is fundamental can't recognize any observations as evidence undermining the rules. There are plenty of ways that the rules might change over time as a result of observations in a non-evidential way. Besides obviously non-rational changes (e.g., those produced by traumatic observations, or by computational errors), we might imagine changes made for computational convenience. Imagine a rule-formulation in the style of (R t ), where new evidence revises some parameters, but where the agent stores rounded off versions of the new values. 34 Over time, the values produced might start to vary considerably from what they would have been if the system had never rounded off. Here the rule formulation changes to a non-equivalent rule formulation, on the impact of evidence; the rule itself changes. But this isn't a case where the accumulated observations serve as evidence against the old rule and for the new. (If we had started with a more complicated inductive rule than (R), there would have been more interesting ways for observations to lead to non-evidential changes in rules for purposes of computational simplicity.) A less trivial example of how rules might change due to observations but not based on evidence arises when the rules are valued as a means of meeting certain goals (perhaps truth-oriented goals like achieving truth and avoiding falsehood). For there are various ways in which observations might cause a shift in goals (e.g., bad experiences might lead us to increase the weight given to avoiding falsehood over achieving truth), and thus lead to shifts in the rules for evaluating beliefs. But here too the shift in rules for believing isn't evidence-based, it is due to a change in goals. (It could also be argued that the basic rule in this example isn't the goal-dependent rule, but the rule about how beliefs depend on evidence and goals. This rule doesn't even change in the example described, let alone change as a result of evidence.) Perhaps more important are cases where observations lead us to think up new methodological rules that had never occurred to us, and we are then led to use them on the basis of their intrinsic appeal. (Or maybe we observe others using them, and are led to use them out of a desire for conformity.) Here too it is transparent that the shift of rules is not due primarily to evidence against the old rules. Of course, on the basis of the new rules we might find that there is evidence against the old. But if the old rules didn't agree that it was evidence against them (and our resolution of the puzzle in section 4 of the text says that they won't agree, if the rules are basic), then the decision to count the alleged
evidence as evidence depends on an independent shift in the rule. A third clarification: to assert that a person's inductive behavior is governed by a basic rule is not to assert that there is a uniquely best candidate for what this basic rule is. To attribute a rule of inductive behavior to someone is to give an idealized description of how the person forms and alters beliefs. For a variety of reasons, there need be no best 34 One might ask, why represent the meta-rule that the agent was following as the original (R*), rather than as a
meta-rule that explicitly tells us to round off? I don't think this modified description of the agent would be incorrect, but neither do I think that a description of the agent as following the original (R*) would be incorrect: describing an agent as following a rule involves idealization of the agent's practices (especially when that rule is not explicitly represented in the agent, as it almost certainly wouldn't be for these meta-rules), and it's just a question of the extent to which one idealizes. Obviously, as the element of idealization of the agent's actual practices at revising beliefs lessens, the scope for arguing that the practices of revising beliefs can change lessens correspondingly. end p.389
idealized description. (The most important reason is that there are different levels of idealization: for instance, some idealizations take more account of memory limitations or computational limitations than do others. Also, though I think less important, there can be multiple good idealized descriptions with the same level of idealization, especially when that level of idealization is high: since a description at a given highly idealized level only connects loosely with the actual facts, there is no reason to think it uniquely determined by the facts.) So there are multiple good candidates for the best idealization of our total inductive behavior. Any such idealization counts any factors it doesn't take into account as non-rational. Insofar as the idealization is a good one, it is appropriate to take the factors it doesn't take into account as non-rational. The lack of a uniquely best candidate for one's basic rule is largely due to a lack of a uniquely best division between rational and non-rational factors. With these clarifications in mind, let's turn to the issue of whether there are basic inductive rules. Since in attributing rules one is idealizing, really the only sensible issue is whether a good idealization will postulate a basic inductive rule (which might vary from one good idealization to the next). The alternative is an idealization that postulates multiple rules, each assessable using the others. But there is an obvious weakness in an idealization of the latter sort: it is completely uninformative about what the agent does when the rules conflict. There is in fact some process that the agent will use to deal with such conflicts. Because this conflict-breaking process is such an important part of how the agent operates, it is natural to consider it a rule that the agent is following. If so, it would seem to be a basic rule, with the 'multiple rules' really just default rules that operate only when they don't come into conflict with other default rules. Of course, this basic rule needn't be deterministic; and as stressed before, there need be no uniquely best candidate for what the higher rule that governs conflict-resolution is. But what seems to be the case is that idealizations that posit a basic rule are more informative than those that don't. According to the discussion of the epistemological puzzle in section 4, no rule can be empirically undermined by following that rule. 35 But if there are multiple candidates for one's basic inductive rule, it may well happen that each candidate C for one's basic inductive rule can be 'empirically undermined' by other candidates for one's basic inductive rule; that is, consistently adhering to a candidate other than C could lead (on certain observations) to a departure from the rule C. There's good reason to put 'empirically undermined' in quotes, though: 'undermining' C via C* only counts as genuine undermining to the extent that C* rather than C is taken as the basic inductive rule. To the extent that C is regarded as the basic inductive rule, it has not been empirically undermined. I've said that the most important reason for the existence of multiple candidates for a person's basic inductive rule is that we can idealize the person's inductive practices at different levels. At the highest level, perhaps, we might give a simple Bayesian description, with real-number degrees of belief that are coherent (i.e., obey the laws of probability). 35 That argument did not depend on an assumption that (candidates for) our basic inductive rules be deterministic.
Suppose that our most basic rules dictate that under certain circumstances a 'mental coin-flip' is to be made, and that what policies one employs in the future is to depend upon its outcome. One can describe what is going on in such a case along the lines of (R) or (R*)—unchanging indeterministic rules, simply a new policy. In that case, obviously there is no change in the basic rules based on evidence, because there is no change in basic rules at all. Alternatively, one can describe what is going on along the lines of (Rt ): the rules themselves have changed. But in this case, the indeterministic nature of the change would if anything lessen the grounds for calling the change evidence-based. end p.390
At a lower level of idealization, we might give a more sophisticated Bayesian description, allowing for interval-valued degrees of belief and/or failures of coherence due to failures of logical omniscience. At a still lower level we might abandon anything recognizably Bayesian, in order to more accurately accommodate the agent's computational limitations. Eventually we might get to a really low level of idealization, in terms of an accurate map of the agent's
system of interconnected neurons, but using an idealization of neuron functioning. And of course there are a lot of levels of idealization in between. The rules of any one of these levels allow criticism of the rules of any other level as imperfectly rational: higher levels would be criticized for taking insufficient account of computational limitations, lower levels for having hardware that only imperfectly realizes the appropriate rules. But again, insofar as you somewhat arbitrarily pick one level as the 'level of rationality', then one's rules at 'the level of rationality' can't allow there to be empirical reasons for revising what is at that level one's basic inductive rule. 36 , 37 36 Presumably the rules at the very low levels in the hierarchy just described are in any reasonable sense beyond our control, whereas the higher-level rules should count as somehow 'in our control' (despite the fact that any changes made in the higher-level rules are due to the operation of the lower-level rules). One might want to stipulate that 'the level of rationality' is the lowest level of rules under our control. But 'under our control' is itself extremely vague, so this would do little to pin down a unique 'level of rationality'. 37 Thanks to Ned Block, Paul Boghossian, Stephen Schiffer, and Nick Zangwill for extremely helpful discussions. end p.391
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Index Adams, Ernest144n. , 252 , 253 , 255-6 , 296-7 Armstrong, David30n. Balaguer, Mark323 , 324 , 325-6 , 333n. belief: and attitudes toward sentences33-8 , 52-3 , 60-1 , 109n. , 158-60 ; see also language of thought closure under consequence34 , 85-6 , 90-2 , 100-3 degrees of, 293-306 , 307-11 , 349 instrumentalism about31 , 37n. , 68-9 Bell, John284 Belnap, Nuel28-9 , 95n. , 150-1 , 348-9 Benacerraf, Paul118n. , 212n. , 214-15 , 263n. , 324-8 , 373n. Black, Max365 Bonjour, Laurence363n. Boolos, George319n. , 339 , 352n. Brandom, Robert150-1 , 271 , 280 , 305 , 328 Brentano's problem30-1 , 69-72 , 87-100 Camp, Joseph28-9 , 95n. , 150-1 Carnap, Rudolph3 , 4n. , 178 , 196-7 , 375n. , 376n. Carter, Jimmy153 chance302 , 309 Chisholm, Roderick30n. Church, Alonzo160-2 Churchland, Paul69 , 75
Cohen, Paul318 computational types151-2 , 158 , 164 , 222n. conceptual role 63-5, 112-3, 158, 169 long-armed60-1 , 73-6 , 80-1 , 95-7 , 99 , 109-11 , 153-6 ; see also 'observation sentences'. conditionals144n. , 230n. , 238-9 , 252-8 , 310 conservative extensions166 , 170 continuum, size of303-6 , 309 , 310-11 , 317-19 , 330-1 , 333 , 342 , 351 Convention T14n.-15n. , 18n. Crossley, John131n. Davidson, Donald7 , 28 , 40 , 59 , 60-1 , 87 , 124 , 167 Degree of Belief and of truth, see belief, degrees of, and truth, degrees of Dennett, Daniel36 , 69 , 70n. Dummett, Michael23 , 162n. , 323 , 365 Earman, John195 , 235n. Edgington, Dorothy252n. , 255 evaluative discourse, 128n. , 223n. , 238-9 , 242-51 , 310 , 368-71 , 381-7 Feferman, Solomon141 , 142 , 344n. , 349 , 354 Feyerabend, Paul96 Fine, Arthur195 , 235n. Fine, Kitix , 231n. , 260 finiteness263-9 , 273 , 320 , 322 , 326n. , 331 , 337-43 Fodor, Jerry38n. , 56 , 68 , 73 , 85 , 90 , 219 Frege, Gottlob8n. , 23 , 104 , 112 , 178 , 184 , 261 , 316 , 328n. Hesperus-Phosphorus problem63-5 , 75 , 169 On directions166 , 170 see also Frege-Russell tradition Frege-Russell tradition104-7 , 134 , 225 Friedman, Harvey143 Friedman, Michael365 functionalism30-1 , 43-51 , 56-8 , 62 , 74-5 , 90n. , 197-8 Gibbard, Alan243-6 , 370n.-371n. Gödel incompleteness theorems318 , 343-50 , 355 Goldman, Alvin379n. , 380n. , 381 , 382 Gordon, Robert78n. Grover, Dorothy28-9 , 95n. , 150-1 Gupta, Anil348-9 Harman, Gilbert38n. , 47n. , 56 , 58-9 , 60-1 , 65n. , 75 , 85 Hawthorne, James330 , 332n. Hellman, Geoffrey316 , 321 , 327 , 329
Horwich, Paul71-12 , 107n. , 121n. , 226 , 228 , 232n. , 233 , 243 Humberstone, Lloyd131n. end p.399
indefeasibility, empirical173 , 361-4 , 367 , 372-9 indeterminacy: of 'believes'37 , 38-9 , 53 correlative191n. , 197-8 , 209-12 , 214-15 , 231n. , 260 , 262 , 274 , 281-2 in current language, 196 , 202-5 , 216-17 , 259-63 , 272 , 279-83 , 303-6 in current scientific terms, 192 , 236-7 , 279 , 305 , 310 of logical operators, 237-8 , 261-3 , 265 , 267n. , 269 , 275-6 , 277n. , 319n. , 338 , 343 , 345-6 of mathematical vocabulary, 263-9 , 303-6 , 319-20 , 323 , 336-43 , 345-6 of rules356 , 388 , 389-91 of translationviii-ix , 53 , 64-5 , 169-70 , 171n. , 196 , 203-5 , 219 , 279n. , 280 , 305-6 , 358-60 of 'true', 'refers', and other semantic terms29n. , 143 , 147 , 152 , 188n. , 215n. , 217-18 , 219 , 221 , 259n. , 260-2 , 281-2 inductive justification of induction365 , 373-8 interpersonal comparisons (of meaning etc.)28 , 57n. , 62-5 , 75n. , 87 , 128-9 , 132-3 , 151 , 158 , 169-71 , 223-4 , 279n. intuitionism226 Jackson, Frank255-6 Jech, Thomas34n. Jeffrey, Richard385 Kaplan, David166 Kemeny, John376n. Ketland, Jeffrey143n. Kitcher, Philip327 , 361-2 Krantz, David69 Kreisel, Georg331 Kripke, Saul19 , 91n. , 117-19 , 124n. , 144 , 262 , 316 , 348-9 Kuhn, Thomas178-9 , 183-4 , 195 Lakatos, Imre316 language of thought hypothesis33 , 36-9 , 43-6 , 55-60 , 68-9 , 84-6 , 90-2 , 93 , 98-9 , 109 , 111 languages, as abstract entities4n. , 125-6 Lavine, Shaughan353-4 , 356 law of excluded middle144-6 , 226-8 , 230n. , 231n. , 234 , 286-7 , 290-2 , 295 , 303 , 307-9 law of non-contradiction145-6 , 291 , 292n. Leeds, Stephen28 , 29 , 127n. , 146 , 154n. , 155n. , 162n. , 279 , 280 , 282-3 , 302 , 304-6 Leibniz's Law generalized185
Levi, Isaac300-1 , 302 Lewis, Davidix , 31 , 34 , 35 , 43 , 44 , 45-6 , 47-8 , 49n. , 57n. , 60 , 65n. , 69 , 75 , 126 , 196-8 , 203n. , 216 , 21718 , 245 , 252n. , 257 , 302 , 310 , 339n. , 373n. , 382 , 386 Loar, Brian98n. local entities170-1 logical connectives, semantics of27 , 59-60 , 73 , 76 , 93 , 112-16 , 124-5 , 142-3 , 200-1 , 210-11 , 213n. , 220-1 , 232 , 234 , 261-3 , 275-6 , 288-9 Luce, Duncan69 Lukasiewicz continuum-valued semantics144-5 Lycan, William93n. Maddy, Penelope333n. McDowell, John28 McGee, Vann275 , 348n. , 352-6 , 358 narrow psychology v. non-intentional psychology72-4 non-intentional psychology57-8 , 59-60 , 63-5 , 72-6 , 87-93 , 97 Neurath, Otto24 objectivity215 , 315-26 , 330-1 , 333-50 , 381-7 observation-sentences, theory-ladenness of74 , 96 , 110-11 Papineau, David333n. Parsons, Charles353 , 354 , 356-8 Peacocke, Christopher27 physicalism11-16 , 24-26 , 28 , 29 , 71-2 ; see also Brentano's problem. plenitudinous platonism323-4 , 325-6 , 333-5 Pollock, John373n. , 384 Popper, Karl3 , 296n. Price, Huw252n. Priest, Graham145-6 projective explanation78-9 , 154-6 , 173-4 Putnam, Hilary29 , 30n. , 54n. , 118-19 , 121n. , 194 , 219 , 316 , 319-22 , 323 , 324 , 326 , 328-31 , 335 , 338-40 , 342n. , 349-50 , 373n. Pylyshyn, Zenon68 quasi-deflationary131-3 , 151-2 , 164-5 Quine, Willard V.viii-ix , 13 , 24-6 , 28 , 31n. , 60-1 , 62-5 , 68 , 84 , 114 , 116 , 129-30 , 132-3 , 146 , 151 , 158 , 159 , 166 , 169-71 , 178n. , 187 , 192-3 , 194-6 , 199-215 , 218 , 219-21 , 272-3 , 281 , 305n. , 334 , 336n. , 372-3 end p.400
Ramsey, Frank104 Ramsey sentences196-7 , 237n. reduction, see physicalism reference, theory of18-20 , 27 , 41-2 , 93-100 , 112-14 , 116 , 117-19 , 126-7 , 136-7 , 143 , 146-7 , 149-51 , 210-15
, 220-1 , 231-2 , 259-74 , 276 rejection308-9 relativism382-5 Resnik, Michael328 Restall, Greg145 Russell, Bertrand18-19 , 104; see also Frege-Russell tradition Schaffner, Kenneth189n. schemas and universal generalizations115 , 120n. , 123 , 141-3 , 353-60 Schiffer, Stephen29 , 65n. , 78n. , 80-2 , 98n. , 160-2 , 166 , 293 , 294 Scott, Dana195 second order logic237-8 , 319n. , 321-2 , 338-9 , 351-3 , 354 Sellars, Wilfred158 , 162n. semantic paradoxes24 , 105n. , 106n. , 115 , 120n. , 124n. , 143-6 , 222n. , 347-9 , 368 Shafer functions299-300 , 301 , 307n. , 308 , 309 Shafer, Glen299 Shapiro, Stewart143n. , 147-8 , 328 , 330 , 353 , 354 , 356-8 Sheard, Michael143 skepticism371 , 384n. , 386-7 Skyrms, Brian302 Soames, Scott27-8 , 126 Solovay, Robert318 Sorites paradox37n. , 297 , 344-5 , 349 Stalnaker, Robert27-8 , 31 , 34-5 , 43-6 , 51 , 69 , 83-103 , 126 , 254 , 256-8 Stich, Stephen75n. , 79-80 , 88 , 90 , 97n. structuralism271n. , 280n. , 326-8 supervaluation190 , 191n. , 207-12 , 214-15 , 221 , 230-1 , 236 , 244 , 248 , 288 Suppes, Patrick69 Tarski, Alfred3-26 , 27-8 , 123-6 , 141-3 , 146 , 207 , 347 Tarskian semantics40 , 43 , 52 , 58 , 124 , 125-6 , 178 , 201n. , 261 Tarski biconditionals24 , 106n. , 142n. , 143 , 144 , 152 'true', meaning of23-4 truth and truth-conditions, role of22-4 , 28 , 59-62 , 72-82 , 87-90 , 104-5 , 107 , 119-23 , 133 , 137-9 , 150-1 , 152 , 153-6 truth, degrees of195n. , 203n. , 216 , 217-18 , 230n. , 292n. truth, undefinability of124 , 142 , 152 , 347 , 348n. truth-value gaps and gluts145-6 Tversky, Amos69 understanding (and knowledge)59 , 171-2
untranslatable utterances10 , 20 , 64-5 , 129 , 132 , 147-51 , 152 , 224n. , 273 , 305n. vagueness202n.-203n. , 225-34 , 251n. , 278 , 282-303 , 305-6 , 307-11 , 344-5 , 349 , 368 higher order217-8 , 231n. , 259n.-260n. , 295-6 , 297-8 , 300 , 337 van Cleve, James365 Wagner, Steven27 , 93n. , 261 , 327 Wallace, John28 , 219 Weston, Thomas339n. , 352 , 354 Williamson, Timothy232n. , 282-6 , 296 , 297 , 305 , 306 Wilson, Mark194n. Wittgenstein, Ludwig104 , 316 Wright, Crispin121n. Yablo, Stephen69