A Defence of Arbitrary Objects Kit Fine; Neil Tennant Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 57. (1983), pp. 55-77+79-89. Stable URL: http://links.jstor.org/sici?sici=0309-7013%281983%2957%3C55%3AADOAO%3E2.0.CO%3B2-A Proceedings of the Aristotelian Society, Supplementary Volumes is currently published by The Aristotelian Society.
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/aristotelian.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact
[email protected].
http://www.jstor.org Fri Dec 14 13:41:04 2007
A DEFENCE O F ARBITRARY OBJECTS Kit Fine and Neil Tennant
I-Kit
Fine
There is the following view. In addition to individual objects, there are arbitrary objects: in addition to individual numbers, arbitrary numbers; in addition to individual men, arbitrary men. With each arbitrary object is associated an appropriate range of individual objects, its values: with each arbitrary number, the range of individual numbers; with each arbitrary man, the range of individual men. An arbitrary object has those properties common to the individual objects in its range. So an arbitrary number is odd or even, a n arbitrary man is mortal, since each individual number is odd or even, each individual man is mortal. O n the other hand, a n arbitrary number fails to be prime, an arbitrary man fails to be a philosopher, since some individual number is not prime, some individual man is not a philosopher. Such a view used to be quite common, but has now fallen into complete disrepute. As with so many things, Frege led the way. Given his own theory of quantification, it was unnecessary to interpret the variables of mathematics as designating variable numbers; and given the absurdities in the notion of a variable number, it was also unwise. It was with characteristic irony that he wrote: 'Perhaps there is a seminal idea here which we could also find of value outside mathematics' ([5], p. 160). Where Frege led, others have been glad to follow. Among the many subsequent philosophers who have spoken against arbitrary objects, we might mention Russell ([lZ], pp. 90-91), Lesniewski ([8], pp. 22-3,27), Tarski ([l3], p. 4), Church ([3], p. 13), Quine ([lo], pp. 127-8), Rescher ([ll], pp. 134-7) and Lewis ([7]) p. 203. If more philosophers of the present day have not added their voices to the protest, it is probably because they have not thought it worth the bother. As Menger says ([9], p. 144), the thesis that there are no variable numbers is 'today one of the few propositions about which logicians as well as mathematicians are in general agreement'. I n the face of such united opposition, it might appear rash to
56
I-KIT
FINE
defend any form of the theory of arbitrary objects. But that is precisely what I intend to do. Indeed, I would want to claim, not only that a form of the theory is defensible, but also that it is extremely valuable. In application to a wide variety of topics-the logic of generality, the use of variables in mathematics, the role of pronouns in natural language-the theory provides explanations that are as good as those of standard quantification theory, and sometimes better. Rather than present the finished theory at the outset, we may see it as the outgrowth of the criticisms that have been directed against its cruder formulations. Each criticism, if not deflected, will lead to an appropriate change of formulation. The finished form of the theory will then emerge as the cumulative result of these various criticisms; it will be, if you like, the prize that the proponent of the naive view can carry off with him in the contest with his critics. This is not how I myselfcame to the theory; but it is perhaps the most congenial approach for those who are already sceptical. Four kinds of objection will be considered in all. Bluntly put, they might be stated as follows: there are no arbitrary objects; the principles governing them are incoherent; the theory leads to questions without answers; and it is, in any case, of no use. Let us consider each objection in turn, with the ontological worry first. Is it seriously to be supposed that, in addition to individual numbers, there are arbitrary numbers and that, in addition to individual men, there are arbitrary men? What strange sorts of objects are these? Can I count with arbitrary numbers or have tea with a n arbitrary man? Two questions may be discerned here: are there any arbitrary objects?; and, what are they like? T o deal with the first, it is necessary to distinguish, in a way that is familiar from the philosophical literature, between two senses of the phrase 'there are'. We may use the opinions of the nominalist as an illustration. I n one sense of the phrase, he is concerned to deny that there are numbers; for that is just his position. But in the other sense, he may be prepared to admit that there are numbers; for he may be prepared to speak with the mathematician or the common man and say that there is a solution to the equation ' x + 5 = 12' or that there are prime numbers greater than 5.
A DEFENCE O F ARBITRARY OBJECTS
57
If now I am asked whether there are arbitrary objects, I will answer according to the intended sense of 'there are'. If it is the ontologically significant sense, then I am happy to agree with my opponent and say 'no'. I have a sufficiently robust sense of reality not to want to people my world with arbitrary numbers or arbitrary men. Indeed, I may be sufficiently robust not even to want individual numbers or individual men in my world. But if the intended sense is ontologically neutral, then my answer is a decided 'yes'. I have, it seems to me, as much reason to affirm that there are arbitrary objects in this sense as the nominalist has to affirm that there are numbers. If this is right, then our ontological scruples should not tell against the development of a theory of arbitrary objects. The nominalist is not against number theory, but only against a certain construal of it, one that sees its truth as resting on the existence of numbers. In the same way, our critic should not be against arbitrary object theory, but only against its realistic construal. Of course, anyone who accepts a theory yet disputes its realist commitment must give some other account of what its truth consists in. But here I am happy to go along with the most ardent reductionist and have him reduce the whole theory of arbitrary objects to one that trades in more respectable entities. Indeed, the final formulation of the theory will make it clear how just such a reduction might go. My only concern, at this stage, is to insist on the legitimacy of an intermediate level of theorizing, one that provisionally accepts the ontology of arbitrary objects. The other ontological worry concerns, not the existence of arbitrary objects, but their nature: what are arbitrary objects? The two worries are connected, though; for despairing of a satisfactory answer to the question 'what are they?', philosophers have concluded that they are not. The question 'what are they?' may be taken, in an ordinary, non-philosophical way, as a request for an explanation of what objects one is talking about. One can then do no better, I think, than refer to the kind of role that arbitrary objects are intended to play. One says much what was said a t the beginning of the paper, that each arbitrary object is associated with a range of values, that it possesses those properties common to its values, and so on. Of course, any attempt to pin down the role of the
58
I-KIT
FINE
objects more exactly may result in an inconsistent theory. But an inconsistent theory need not be entirely useless in helping us to understand what objects are being talked about; for it may point in the direction of their intended role, without actually giving it. We can all get a pretty good idea of what sets are meant to be from the axioms of naive set theory, even though those axioms turn out to be inconsistent. The question 'what are they?' may also be taken, in a philosophical way, as a request for a category or kind to which the objects can be assigned. We may then supply the quite reasonable answer that arbitrary objects belong, like sets or propositions, to the category of abstractions. To be sure, it is hard to say what in general an abstraction is or what is distinctive about arbitrary objects as a kind of abstraction; but these are difficulties for all abstractions. There may be thought to be a special problem with arbitrary objects; for it seems that they must somehow be on an ontological par with the corresponding individuals. An arbitrary number is just another number, an arbitrary man just another man. It therefore appears that one can say the same sorts of things about each. So one is led to the absurd conclusion that one might count with arbitrary numbers or have tea with an arbitrary man. This supposition of ontological parity has perhaps two main sources. The first is a certain metaphysical or psychological picture that may have been suggested by the more zealous advocates of arbitrary objects. It is as if an arbitrary man were merely a defective man, one shorn of his peculiar features; and, in the same way, it is as if a general idea were merely a defective version of a particular idea. So behind every individual man is an arbitrary man, behind every particular idea a general idea. The second source is a certain argument: membership in a particular category is a common property of all the individuals; it would therefore appear that the arbitrary object should have the property and so also be in the category. This picture and argument have had their hold, but there is nothing in the theory that requires us to accept either. The picture is an idle adjunct to the theory and may simply be dropped. The argument is more serious but, on a proper formulation of the theory, turns out to be defective.
A DEFENCE O F ARBITRARY OBJECTS
59
The second objection we wish to consider is to the effect that the theory of arbitrary objects is logically incoherent. It may be granted that there is no ontological absurdity in the mere supposition that there are arbitrary objects. But, it will be argued, there is a logical absurdity in the principles that are meant to govern such objects. For the key principle, without which a theory would be unthinkable, is that each arbitrary object should have the properties common to the individuals in its range. But formulate such a principle properly and it will be seen to lead, either on its own or in conjunction with innocuous premisses, to contradictory conclusions. There are two forms this line of argument can take. One uses a logically complex property as a counter-example to the principle. The other uses a property that is not logically complex, but is otherwise special. The first line of argument goes back to Berkeley ( [ Z ] , Intro., X) and, in one form or another, has been resurrected by many subsequent philosophers. A typical version goes as follows. Take an arbitrary number. Then it is odd or even, since each individual number is odd or even. But it is not odd, since some individual number is not odd; and it is not even, since some individual number is not even. Therefore it is odd or even, yet not odd and not even. A contradiction. This particular version of the argument uses the complex property of being odd or even. Other versions of the argument use other complex properties. However, it will be clear from the treatment of this case how the treatment of the other cases should go. Crucial to this argument is a certain formulation of what I shall call the principle of generic attribution, the principle that any arbitrary object has those properties common to the individuals in its range. Let us ignore the reference to properties as not essential to our present purposes, and deal instead with conditions or open sentences. Let cp(x) be any condition with free variable x; let a be the name of an arbitrary object a; and let i be a variable that ranges over the individuals in the range ofa. (We here follow a general convention whereby a names a.) Then the required formulation of the principle is: cp's iff every individual cp's) (G1) cp(a) ~icp(i)(a
60
I-KIT
FINE
Upon letting a name the arbitrary number of the argument, the various claims made about that number can be obtained by making appropriate substitutions for the condition cp(x). For example, upon substituting Ex (x is even), we obtain: (Gl)' EaEViEi; and the claim that a is not even (-Ea) then follows from the evident truth that not all numbers are even (-ViEi). It must be granted that, once this formulation of the principle is accepted, the argument goes through. But it may be doubted whether the formulation should be accepted. For look at its consequence (Gl)' and ask whether the arbitrary number a satisfies the condition (Ex E V iEi) of being even iffall individual numbers are even. What the intuitive principle of generic attribution seems to tell us is that an arbitrary number a should satisfy the condition iff all individual numbers do. But not all individual numbers satisfy it; no even number does. So far from being a correct formulation of the intuitive principle, ( G l ) is something that the principle requires us to judge false. Of course, we may have here a further sign that the intuitive principle is contradictory. But another possibility is that it is only intended that the intuitive principle apply to the whole context in which the name of the arbitrary object appears. Under this proviso, the correct formulation of the principle is: (G2) The sentence q ( a ) is true iff the sentence Vicp(i) is true. O r using satisfaction in place of truth, the formulation becomes (G2)' a satisfies the condition cp(x) iff each individual i in the range of a satisfies the condition cp(x). From this perspective, the original formulation ( G l ) rests on the fallacy of applying the principle internally, to only a part of the context in which the name of an arbitrary object appears. For although we may affirm icp(i)G~iq(i),we cannot correctly infer cp(a)rVicp(i). In such a way, the argument from complex properties can be stymied. It has always been thought that we have in this, and related, arguments a knockdown case against the whole theory of arbitrary objects. But if I am right, such arguments depend
A DEFENCE OF ARBITRARY OBJECTS
61
upon the failure to distinguish between two basically different formulations of the principle of generic attribution: one is merely a rule ofequivalence and is stated in the material mode; the other is a rule of truth and is stated in the formal mode. Once the distinction is made, the arguments are seen to be without foundation. But even if the traditional argument against the theory of arbitrary objects breaks down, may there not be more sophisticated variants of the argument that succeed? Consider again the statement O a v Ea ('a is odd or a is even'). From the modified formulation (G2), it follows that this statement is true and yet that neither disjunct is true. So the semantical rule for disjunction fails. O r consider the statements O a and - 0 a . From G2, it follows that neither O a nor - 0 a is true. So the semantical rule for negation fails. Suppose now O a is false only when - 0 a is true. Then the Law of Bivalence also fails. Yet why should these consequences be thought to be unacceptable? There are other cases in which we are prepared to admit similar consequences. One is provided by vague language. Consider a blob whose colour is a borderline case of both red and orange. Then it is true that the blob is red or orange, yet not true that it is red or that it is orange; and it is not true that the blob is red or that it is not red. Also, it is not as if, in the absence of the classical semantical rules, we are left in the dark as to what the truth-conditions of statements concerning arbitrary objects should be. The statement cp(a), regardless of its inner complexity, simply has the same truth-conditions as b' icp(i). But perhaps the worry is not so much about indeterminacy in the truth-conditions as inconstancy in the use of the logical constants. When 'v', let us say, occurs between statements concerning only individuals, we evaluate the result according to the classical truth-tables; when 'v' occurs between statements concerning arbitrary objects, we evaluate the result according to the rule of generic attribution. Is not this to use 'v' in two radically different ways? I do not see that the theory demands that there be no shift in the sense of the logical constants, though it is certainly convenient, should there be a shift of sense, that there be no corresponding shift of symbol. But there is, in any case, a way in
62
I-KIT
FINE
which we can see the theory as leaving the sense of'v' intact. For in evaluating a disjunction y ( a ) vx(a), we first apply the rule of generic attribution. This tells us that y ( a ) v ~ ( ais)true iff each of the statements y ( i )v ~ ( iis)true for ian individual in the range of a. We then apply the standard rule of disjunction to each of the statements y(i)vx(i). Thus at the only point at which we evaluate a disjunction, we evaluate it according to the classical rule. Thus it is not as if there are two rules of disjunction, differently activated according to the subject-matter of the statement. There is only one rule, appropriately activated according to the stage of the evaluation. But is it not disturbing that the statement y ( a ) vx(a) should be apparently disjunctive in form and yet not immediately evaluated according to the rule of disjunction? It seems that we must either deny that y ( a ) v ~ ( ais)a genuine disjunction or else give up the principle that disjunctions are evaluated directly by a disjunction rule. I, for one, am happy to accept the second alternative. But for those who are not, there is an acceptable way of holding to the first alternative. For we may suppose that the statement y ( a ) v ~ ( ais) syntactically ambiguous. It may either be formed by disjoining y ( a ) and ~ ( aor) by applying the property abstract hx(y(x)vx(x)) to a. In the first case we have a genuine disjunction, in the second case not. A more perspicuous notation would use [Xx(y(x)v x(x))]a for the second case, thereby making it clear that the statement is not genuinely disjunctive in form. We may now evaluate genuine disjunctions directly in accordance with the classical rule. The extensions of property abstracts Xxcp(x) may be evaluated in the usual way for individuals and in a way analogous to (G2) for arbitrary objects. Thus the effect of the principle (G2) is achieved without any damage to the principle of direct evaluation. However, there is still a price to pay; for the principle of property abstraction, [Xxcp(x)]a=cp(a),will fail. This principle too might be retained, if the expression Xxcp(x) were not regarded as denoting a property. But then other principles would have to be given up. In general, it is impossible to achieve complete logical parity between individual and arbitrary objects; the difference in their logical, or rather meta-logical behaviour, must show up
A DEFENCE OF ARBITRARY OBJECTS
63
somewhere. But depending upon how exactly the generic attributions are expressed and interpreted, the difference may be made to show up now in this principle, now in that. It is only the most adamant logical purist who could not find one formulation to his liking. Another logical objection to the principle remains, the argument from special properties. A typical version goes as follows. Take the property of being an individual number. Then each individual number has this property. So from the principle of generic attribution, it follows that any arbitrary number has this property-which is absurd. There is no point in appealing here to our meta-linguistic formulation (G2) of the intuitive principle. The statement 'all individual numbers are individual numbers' is true. It therefore follows from (G2) that the statement 'a is an individual number' is also true. Indeed, the reformulation (G2)' itself provides a counter-example. For the reformulation reads: (G2)' a satisfies q(x) iff each individual i in the range of a satisfies q(x). Now this statement itself is a rather complex condition ~ ( xon ) the arbitrary object a. So applying (G2) to ~ ( a )we , see that (G2)' is true iff: for every individual j (in the range of a),j satisfies q(x) iff each individual i in the range of j satisfies q(x). But this is absurd. It must be recognised, I think, that this objection requires us to make yet another modification in the formulation of our principle. We must distinguish between two kinds of condition or predicate. There are first of all the generic conditions and predicates. These include all of the ordinary predicates, such as 'being odd' or 'being mortal', and all of the conditions obtainable from them by means of the classical operations of quantification and truth-functional composition. To these conditions, the principle of generic attribution, in its revised form (GZ), is applicable. There are then the classical conditions or predicates. These include certain special predicates, such as 'being an individual number' or 'being in the range o f , and the various conditions obtained with their help. T o these, the principle (G2) is not applicable.
64
I-KIT
FINE
The principle of generic attribution should therefore now receive the following formulation: (G3) for any generic condition cp(x),cp(a) is true iff'v'icp(i) is true; and similarly for the version (G2)' in terms ofsatisfaction. Given that the predicate 'is an individual number' is not generic, the previous argument then breaks down. This answer to the objection might appear to be excessively weak. Certain counter-examples are provided to the principle of generic attribution. It is then claimed that the principle does not hold for all conditions. But for which? Presumably only those for which it holds. But the status of the revised principle is not as trivial as this caricature of my response would have it. First, it is not as if the principle had no application. Call a language generic if all of the conditions obtainable by its use are themselves generic. Then many languages, of natural and independent interest, will be generic; and so the principle (G3) will have wide application to all such languages. But even when a language is not itself generic, it will correspond to one that is. For we may re-interpret its predicates or other non-logical constants so that they are classically evaluated in their application to individuals but 'generically' evaluated in their application to arbitrary objects. The new language will then be generic by definition and, over the domain of individuals, will be in semantic agreement with the original language. Second, even in those cases in which the principle is not applicable, it will always be clear what the conditions for generic attribution should be-or rather, if there is any lack of clarity here, it will not have its source in the reference to arbitrary objects. Take the conditions that figured in the previous arguments. Then we may suppose that it is just given when an object is to be an individual number or when an individual is to be in the range ofan arbitrary object. In a formal development of the theory, these relations will just be specified as part of the structure ofarbitrary objects. So it is not as ifwe are left in the dark here. There may no longer be any general procedure for evaluating the statements about arbitrary objects;
A DEFENCE OF ARBITRARY OBJECTS
65
but it will always be clear, from case to case, how the evaluation is to proceed. Of course, there may be cases in which a predicate is ambiguous as between a generic and a classical reading. The predicate 'is a number' is a good example. O n a generic reading, it is inclusive of all arbitrary numbers; on a classical reading, it is exclusive of them. So if it is asked whether an arbitrary number is a number, the answer will be 'yes' or 'no', according to whether a generic or a classical reading is intended. It is this ambiguity that may explain why the argument for the ontological parity of individual and arbitrary objects has been seen to be so compelling. For it depends upon the failure to distinguish between the generic and classical readings of the predicate 'is a number'. Attend to the generic reading and it follows that each arbitrary number is a number; switch to the classical reading, and it then appears to follow that each arbitrary number is an individual number. Finally, we may note in support of the distinction between generic and classical conditions, that it corresponds to an intuitive distinction in the way that the names for arbitrary objects can be used. Applied to a generic condition cp(x), the name a plays a merely representative role; it serves to represent the individuals in the range of the arbitrary object. Applied to a classical condition cp(x),the name a plays an essentially referential role; it serves to pick out the arbitrary object itself. So if we wish to know whether a condition cp(x) is generic or classical, we may ask: is the name a in cp(a) being used in a merely representative or an essentially referential capacity? We see therefore that a satisfactory version of the principle of generic attribution can be maintained. In response to the argument from complex predicates, we may insist on a metalinguistic formulation of the intuitive principle, one that makes explicit reference to truth or satisfaction. In response to the argument from special predicates, we may restrict the application of the meta-linguistic formulation to generic conditions. Once these two changes are made, the principle is seen to be impervious to the logical objections that have usually been directed against it. We turn now to the third objection. We have so far concentrated on a single principle, the principle of generic
66
I-KIT
FINE
attribution, and have shown how a coherent formulation of it may be given. But this is a far cry from having a theory of arbitrary objects. A host of questions remain; and unless they can be answered, it is doubtful whether anythingapproachinga satisfactory theory can be obtained. In considering this objection, we can perhaps do no better than to consider some of the detailed points raised by Frege. In arguing against the mathematician Czuber, Frege writes ( [ 5 ] ,p. 160): This [Czuber's account] gives rise to a host of questions. The author obviously distinguishes two classes of numbers: the determinate and the indeterminate. We may then ask, say, to which of these classes the primes belong, or whether maybe some primes are determinate numbers and others indeterminate. We may ask further whether in the case of indeterminate numbers we must distinguish between the rational and the irrational, or whether this distinction can only be applied to determinate numbers. How many indeterminate numbers are there? How are they distinguished from one another? Can you add two indeterminate numbers, and if so, how? How do you find the number that is to be regarded as their sum? The same questions arise for adding a determinate number to an indeterminate one. T o which class does the sum belong? O r maybe it belongs to a third? Let us treat these questions in turn. First, 'to which of these classes [of the determinate and indeterminate numbers] do the primes belong?' Some of the determinate (individual) numbers are prime. But what of the indeterminate (arbitrary) numbers? Suppose 'prime' is taken, in its classical reading, to mean 'individual prime'. Then no arbitrary number is prime. Suppose now that 'prime' is taken in its generic reading. Then the statement 'a is prime', for a an arbitrary number, will not be true, since some individual numbers are not prime. Of course, neither will it be false, since some individual numbers are prime. There is, however, a complication. We have so far taken an arbitrary number to have an unrestricted range, one that includes all individual numbers as values. But it also seems
A DEFENCE OF ARBITRARY OBJECTS
67
reasonable that there should be arbitrary numbers with a restricted range, one that includes only some of the individual numbers as values. Suppose we now ask, of these arbitrary numbers, whether any of them are prime. Then we should say: the statement 'a is prime' is true iff all of the individual numbers in the range of a are prime. So an arbitrary prime is prime, but not an arbitrary (and unrestricted) even number or an arbitrary (and unrestricted) factor of 12. Frege's second question is of the same nature as the first and so need not be considered separately. This takes us to the next question: 'How are they [the indeterminate numbers] to be distinguished from one another?' But why should there be a nontrivial difference between any two arbitrary objects? In Euclidean space, for example, there is, in the absence of a coordinate system, no non-trivial way of distinguishing between any two points. A theory does not require an identity criterion for its objects. But still, the present theory is able to provide one-not as it stands, but upon the introduction of two new elements. T h e first is the simultaneous assignment of values. We have so far explained which individuals can be assigned to arbitrary objects in a very simple manner. Each arbitrary object has been endowed with a range of values. T h e individuals i, j , . . . are then assignable to the objectsa, b, . . . iffeach individual is in the range of its arbitrary object. This simple idea must now be given up; for we must allow an interdependence among the values assigned to the arbitrary objects, so that what individuals are assignable to one object may be constrained by the values assigned to others. We must therefore be told, not merely what ranges the arbitrary objects have, but also what combinations of values from those ranges are admissible. It may be, for example, that i, j are assignable to the two arbitrary numbers a, b iff j = i2. We must now think of the arbitrary objects as working in harness. A single arbitrary object will be representative of a class of individuals, as before. But a pair of arbitrary objects will be representative of a relation between individuals; and similarly for sequences of greater length. T h e principle of generic attribution must accordingly be modified so as to apply to all of the arbitrary objects that might be mentioned.
68
I-KIT
FINE
(G4) If q (x,, x,, . . . , xn) is a generic condition containing no names for arbitrary objects, then cp(al, a,, . . . , a,) is true iff q ( a l , a,, . . . , an) is true for all admissible assignments of individuals i,, i,, . . . , in to the objects a,, a,, . . . , an. We could think of the previous principle (G3) as applying to the objects a,, a,, . . . , an one at a time. But we must think of the present principle (G4) as applying to those objects simultaneously. T h e second component of the new apparatus is the relation of dependence among arbitrary objects or what may be called object dependence. This corresponds pretty well to what mathematicians have in mind when they distinguish between dependent and independent variables. T h e two components are connected; for any value dependence must be sustained, in one way or another, by relations of object dependence. More exactly, when b is a n arbitrary object that depends only upon the arbitrary objects a,, a,, . . . , then the values assignable to b must be determinable upon the basis of the values assigned toa,, a,, . . . . Thus the relation of object dependence provides a principle for the local determination of admissible value assignments. Despite their connection, it is important to keep the two notions of dependence apart. Let a be a n arbitrary real; its values are all the individual reals. Two distinct arbitrary reals that are dependent upon a may be distinguished. There is first the cube a3 of a; this depends upon (stands in the relation of objectdependence to) a and assumes the value j when and only whena assumes the value +j. There is then the cube root +a of a; this likewise depends upon a and assumes the real value j when and only when a assumes the value j3.Now the value dependency between each of the pairs (a, a3) and (+a, a ) is the same; the components can take the real values i, j when and only when . . J = z3. However, the object-dependency between each of the pairs is different; for a3 depends upon a, while +a depends upon a. Mathematicians sometimes talk as if, given a pair ofarbitrary reals (c, d) that stand in an indicated relation of value dependence, we can decide which is to be the dependent and which the independent variable. But on the view being put forward here, we are not free to decide, given the objects c and d,
A DEFENCE OF ARBITRARY OBJECTS
69
which way the relation of dependency goes. Rather we are free to decide, compatibly with the indicated value dependence, what objects we are talking about, whether it is, let us say, the pair (a, a3) or the pair ($a, a). The problem of providing identity criteria for arbitrary objects can now be solved. Say that an arbitrary object is independent if it depends upon no other objects and that otherwise it is dependent. We distinguish two cases, according to whether the objects are independent or dependent. Suppose first thata and b are independent objects. Then we say that a = b iff their ranges are the same. This is pretty much Czuber's account ([5], p. 161 and [4], p. 110). We have taken the ranges to be sets, but we could have equally well taken them in a more intensional sense. Suppose now that a and b are dependent objects. Then we shall say that a = b iff two conditions are satisfied. T h e first is that they should depend upon the same arbitrary objects; their 'dependency range' should be the same. T h e second is that they should depend upon these objects in the same way. Suppose that a and b each depend upon the objects cl, c,, . . . . With a may be associated the relation Ra that holds of i, j,,j,, . . . iff i, j , , j2,. . . may simultaneously be assigned as values to a, c,, c,, . . . ; and similarly for b. The second condition then means that the relations Ra and Rb should be the same. Again, the relation has been taken in extension, but may equally well be construed intensionally. Let us make the reasonable assumption that the relation of dependence is well-founded: any sequence of arbitrary objects a,, a,, a3, . . . , with a, depending upon a,, a, upon as, and so on, must eventually come to an end. T h e above two criteria then enable us to distinguish any two arbitrary objects a and b. First, suppose that a and b are independent. Then they may be distinguished by the first criterion. Now suppose that one is dependent and the other independent. Then they are already distinguished by the fact that the one depends upon another object and the other does not. Finally, suppose that both a and b are dependent. If they depend upon the same objects, they can be distinguished by the second criterion in terms of the way they depend upon those objects (the relations Ra and Rb). SOsuppose that a and b depend upon different arbitrary objects. There is then an arbitrary object c upon which a depends, let us say, but
70
I-KIT
FINE
not b. We can then distinguish a and b if we can distinguish c from all of the objects dl, d,, . . . ,upon which b depends. But the whole of the previous argument may be repeated for the pairs (c, dl), (c, d2), . . . . Of course, this may lead to yet further pairs of objects that stand in need of distinction. But the wellfoundedness of the dependency relation will guarantee that the process eventually comes to an end. In such a way, we may answer Frege's question about identity. But a doubt may remain. Although the theory does not require it, we will want to make an application to the variablesigns of mathematics. In a sentence such as 'Let x andy be two arbitrary reals', we will want to say that the symbols 'x' and 'y' refer to two arbitrary reals. But to which? It is natural to suppose that 'x' and 'y' refer to two unrestricted and independent arbitrary reals. But by the first criterion there is only one such real. So do either of the symbols refer to it and, ifso, to what does the other refer? Frege ([4] p. 109) makes a similar point in arguingagainst the view that these letters designate arbitrary objects: This way of speaking is certainly employed; but these letters are not proper names of variable numbers in the way that '2' and '3' are proper names of constant numbers; for the numbers 2 and 3 differ in a specified way, but what is the difference between the variables that are said to be designated by 'x' and 'y'? We cannot say. We cannot specify what properties x has and what differing properties y has. If we associate anything with these letters at all, it is the same vague image for both of them. When apparent differences do show themselves, it is a matter ofapplication; but we are not here talking about these. Since we cannot conceive of each variable in its individual being, we cannot attach any proper names to variables. There are various ingenious solutions to this problem. But perhaps the most natural is one that makes x and y not be independent at all. There is a unique arbitrary pair of reals,p; it is the independent arbitrary object whose values are all the pairs of reals. We may then take x and y to be the first and second components of this arbitrary pair. More exactly: x a n d y will each depend upon p and upon p alone: when p takes the value
A DEFENCE OF ARBITRARY OBJECTS
71
(i,j),then and only then will x take the value i andy the valuej. By the identity criteria, x and y, as so defined, will be unique. It has to be admitted that on this account the reference of the symbols 'x' and 'y' may vary with the context. If'y' is mentioned first, then 'y' will designate what 'x' would designate if it were mentioned first. Also, we must either suppose that there is implicit an arbitrary a-sequence or we must let the length ofthe sequence vary with the number of arbitrary reals referred to. In this respect the use of the variable-symbols 'x' and 'y' is different from the use of the numerals '2' and '3'. But in the crucial respect of designating a particular object, the use of the two kinds of symbol will be the same. Frege's next question concerns cardinality: 'How many indeterminate numbers are there?' We may sink this into the more general question: what arbitrary objects are there? Consider first the arbitrary objects that take their values from a given set I of individuals. Such objects may be generated in stages, according to their 'degree' of independence. At the first stage are the independent objects. Since there are no essential constraints on the existence of arbitrary objects, we should expect that to each set of individuals from I there will be an arbitrary object with that set as its range. At the second stage are the arbitrary objects that depend upon the objects generated at the first stage, but not on anything else. We should now expect that, for each set X of independent objects and each suitable relation R, there will be an arbitrary object that has Xas the set ofobjects upon which it depends and that has R as the way it depends upon them. At the other stages, both finite and transfinite, the arbitrary objects will be determined in a similar manner. For each set Iof individuals, we thereby obtain a 'system' A,of arbitrary objects. Now the choice of I here is completely free. In particular, it may contain arbitrary objects, so that these may now figure as values to other arbitrary objects. Let us suppose that as the systems A,are generated the base sets I a r e expanded with the objects so obtained. Then with this understanding, we may take the arbitrary objects to be those objects that belong to one of the systems '4,. The question of cardinality may now be considered. The general system A of arbitrary objects and, indeed, each of the individual systems A , is a proper class. Therefore the class of
72
I-KIT
FINE
arbitrary objects has no cardinality or, if one likes, it has the same cardinality as the universe. However, for each I of given cardinality and each ordinal a, we may determine, by a simple combinatorial calculation, how many objects ofA ,are generated by the stage a. Thus we are in as good a position to answer questions concerning cardinality in the case of arbitrary objects as in the case of sets. Frege's final questions concern the application ofarithmetical operations to arbitrary numbers: 'Can you add two indeterminate numbers and, if so, how?' Now the theory does not require that the sum of two arbitrary numbers be defined. Consider the equation 'a t b = b + a'. Then the principle of generic attribution (G4) tells us that the equation is true iff'i tj =j t 2 is true for each i and j that can be assigned to a and b. Thus at no point, in the evaluation of the sentence, need we consider a denotation for the complex terms. However, this argument rests on the identity context 'x = y' being generic. If, as in other parts of mathematics, a complex term is to be applied to classical conditions, then a denotation should be supplied. What then is a suitable denotation c for 'a + b'? First, we would like i,j, k to be admissible values ofa, b, c, iff i, j are admissible values for a, b and k = i +j. Second, we would like c to depend just upon a and b and upon whatever a or b depend upon. Now it follows from our discussion of cardinality, that such an object c exists; and it follows, from our discussion of identity, that c is unique. Therefore the problem of finding a suitable sum is solved. Frege's supplementary queries are readily met: 'The same questions arise for adding a determinate number to an indeterminate one. T o which class does the sum belong? O r maybe it belongs to a third?' The sum of a and i, for a an arbitrary and i an individual number, is the unique arbitrary object b such that (1) k, j are admissible values for b, a iff k = i tj and j is an admissible value for a, (2) b just depends upon a and upon whatever a depends upon. There is no need of a third category of objects here and so no danger of a proliferation of categories. This takes care of all of Frege's questions. Crucial to our answers has been the apparatus of dependence: a class of admissible assignments suitably circumscribed by a relation.
A DEFENCE OF ARBITRARY OBJECTS
73
We may safely say that if it is the meta-linguistic formulation of the principle of generic attribution that makes the theory of arbitrary objects possible, it is the introduction of this apparatus that gives it life. The final objection concerns applications. All theoretical misgivings are put aside: it is granted that a coherent formulation of the principle of generic attribution may be given; it is even granted that a viable theory ofarbitrary objects may be developed. But still the question remains: is the theory of any use, or is it merely a curiosity whose interest lies in the possibility of its existence and not in the value of its applications? It is hard not to be sceptical of the value of our theory. The standard theory of quantification has been so successful in its applications that there appears to be no room for serious rivals. But the very success of the standard theory has blinded us to its weaknesses; the data is now seen through the theory, and difficulties are either minimized or ignored. Once one thinks in terms of an alternative theory, the weaknesses become more apparent and the possibility of removing the difficulties more real. Unfortunately, it will not here be possible to work through any one application of our theory in depth or to establish the detailed respects in which it is superior to an account in terms of standard quantification theory alone. This is something I hope to do elsewhere. But it will be possible to give hints as to where the main applications lie and as to what the superiority might consist in. I conceive of the theory as having application to four areas: (1) the logic of generality; (2) mathematical logic; (3) language; and (4) the history of ideas. Let me give some examples from each. Within the logic of generality, the principal application is to the quantifier rules. In order to show that all individuals cp, we may show that an arbitrary individual cp's; and from the fact that some individual cp's, we may conclude that a n arbitrary or representative such individual cp's. These intuitive methods of argument correspond to the rules of universal generalization (UG) and existential instantiation (EI) in systems of natural deduction. O n the face of it, the intuitive procedures involve reference to arbitrary objects and, indeed, the natural explanation of the rules is in terms of such objects. Such explanations
74
I-KIT
FINE
have usually been dismissed as so much fancy talk. But within our theory, we are able to make them precise and thereby provide a semantics for systems of natural deduction which actually assigns arbitrary objects to the variables involved in the applications of UG and EI. Such a semantics is then able to throw a good deal of light on the different systems of natural deduction: it yields a closer correspondence with ordinary intuitive reasoning; it provides a simple and natural explanation of the various restrictions on the rules; and it is able to explain certain differences in natural deduction systems as differences in the arbitrary objects with which they deal. Within mathematical logic, our theory of arbitrary objects leads to a modest formal development of its own. But it is also able to throw light on more traditional methods and results. It sometimes appears as if a mathematician is making significant use of an arbitrary or 'generic' object. Obvious examplesare the use of generic sets in the independence proofs and the use of arbitrarily small quantities in analysis. Whenever this is so, we may attempt to rework the material within the framework of our theory-and often with interesting results. For example, in the case of analysis, we may treat the arbitrarily small quantities as arbitrary objects whose values tend closer and closer to 0. We are thus led to give an account of analysis in terms of infinitesimals, though one that is closer to the Boolean-valued approach of Takeuti than the 'non-standard' approach of Robinson. Another source of application is Skolem functions. The dependent arbitrary objects correspond loosely to multi-valued Skolem functions. So whenever anything like Skolem functions are used, there is the possibility of using dependent objects in their stead. We are led in this way, for example, to prove some results about branching quantifiers that depend critically upon using multi-valued in place of single-valued Skolem functions. There are two main applications to language; one to mathematical and the other to natural language. As already indicated, the theory of arbitrary objects may be used to explain the role of variables in ordinary mathematical discourse; it may be used to restitute the traditional doctrine that variable signs have the role of referring to variable or arbitrary objects. This is in striking contrast to the standard contemporary account,
A DEFENCE O F ARBITRARY OBJECTS
75
which takes these variables to be 'signs ofgenerality', analogous, in their use, to the bound variables of logic. The superiority of the traditional account can be brought out by considering a typical piece of mathematical discourse: d!;= 2r. Let y = x2. Then a
What will the contemporary logician say? Since we later talk of a derivative, )' must be the sign for the function hxx2. It must therefore be supposed that 'x2' is also a sign for this function and that 'x2' has been improperly used in place of 'kxx2'; similarly for '2x'. The sign for the derivative raises a further problem. Let us ignore the question of whether it should be a ratio and rewrite it as 'DCy, x)' ('the derivative ofy with respect to x'). Since 'DCy, x)' is simply intended to denote the derivativey' ofy = hxx2, it is a mystery why both 'y' and 'x' should make their appearance in the sign. Perhaps this is a mistake arising from the original confusion of 'x2' with 'hxx2'. Correcting all these confusions, we obtain the following 'clean copy' of our original statement: Let y = hxx2. Then DCy) = hx2x. Contrast this now with what the arbitrary object theorist would say. It is understood, either from the immediate context or from general mathematical convention, that 'x' is a sign for an unrestricted and (relatively) independent arbitrary real. The symbol 'y' then denotes the square of that arbitrary real in the sense that has already been defined; and similarly for the sign '2x'. The sign 'DCy, x)' for the derivative denotes an arbitrary real that depends upon x (and perhaps also upony) and that, for a given value i ofx, takes the valuey'(i). It is now perfectly clear why the derivative should be a function of bothx andy; for it will be defined in terms of the values (i,j ) that both can simultaneously receive. The contemporary logician must impute all sorts of confusions and absurdities to the mathematician. The arbitrary object theorist can interpret what he says just as it stands. The application to natural language may be illustrated by the following piece of discourse: Every farmer owns a donkey. He beats it. He feeds it rarely . . .
76
I-KIT FINE
How are we to interpret the pronouns 'he' and 'it'? The simple and natural view of the pronouns is that they are used to refer, in a given discourse, to objects that in some sense are 'in play'. But if we stick to an ontology of individuals, this view can hardly be maintained; for there is no individual farmer or individual donkey to which the pronouns can sensibly be taken to refer. But countenance arbitrary objects and the difficulty disappears. 'He' refers to the arbitrary farmer, 'it' to the arbitrary donkey that he owns. Note that this arbitrary donkey is a dependent object and that for a given farmer as value for the arbitrary farmer, the arbitrary donkey can only take as a value a donkey that the farmer owns. Thus the statement 'He beats it' will be true, just as it should be, iff for all values i and j simultaneously assumed by the arbitrary farmer and donkey, it is true that i beats j. But where do the arbitrary farmer and donkey come from? They must have somehow been introduced into the discourse for reference to them to be possible at all. The only plausible source is the quantifier phrases 'every farmer' and 'some donkey'. We are thus led to give a novel account of quantifier phrases according to which they serve not only to make general assertions but also to introduce arbitrary objects. Very roughly, we may say that universal quantifier phrases introduce unrestricted arbitrary objects and that existential quantifier phrases introduce restricted arbitrary objects, with the role standardly played by scope now performed by the relation of dependence between arbitrary objects. The application to intellectual history rests on the fact that the present theory is a revival of traditional ideas. It may therefore be used to illuminate the way those ideas were presented and applied. There are three main sources here: philosophy, logic and mathematics. Within philosophy, there has been a long line of thought that has taken seriously the idea that certain objects have a distinctively representative capacity. Examples include Plato's Theory of Forms, Locke's Doctrine of General Ideas, and Meinong's theory of Incomplete Objects. Within logic, we have the many cases cited in Barth [l], especially in connection with the doctrine of ekthesis. Within mathematics, there is the traditional explanation of variable signs as designatory of variable objects. It is important to
A DEFENCE O F ARBITRARY OBJECTS
77
emphasize that this was no idle frippery put out by the weak and woolly minded. It was integral to the way mathematical symbolism was conceived and used, and was endorsed by no less an authority than Newton. As an example of how the present theory might provide illumination, we can consider the question of why the introduction of variables into mathematics was so significant. O n the present view, this was not merely a symbolic convenience, but also a theoretical advance; not only were new symbols introduced, but also new objects. Therefore various stages in the implicit theory of these objects might be discerned, from simple algebra-with its independent variables and generic conditions, to calculus-with its complex dependency relations and sophisticated classical conditions. There is more to be said on each particular application; but enough has been said to indicate the value and scope of the theory. It was with irony that Frege wrote: 'Perhaps there is a seminal idea here which we could also find of value outside mathematics'. We now see that the content of his words is justified, and the ironic intent misplaced.* REFERENCES 1 Barth, E. M. T h e Logic of lhe Articles in Tradilional Philosophy, Reidel: DordrechtHolland (1974). 2 Berkeley, G. A Treatise Concerning the Principles of Human Knowledge. 3 Church, A. Introduction lo Mathemalical Logic, Volume I , Princeton University Press: Kew Jersey (1956). 4 Frege, G. Philosophical Writings of Gottlob Frege, (trans. P. Geach and M. Black), Blackwell: Oxford (1970). 5 Frege, G. Posthumous Writings, (ed. H. Hermes et al), Blackwell: Oxford (1979). 6 Frege, G. Philosophical and Mathematical Correspondence, (ed. H. Hermes et al), Blackwell: Oxford (1980). 7 Lewis, D. General Semantics, in Semantics of Natural Language, (eds. D. Davidson and G. Harman), Reidel: Dordrecht-Holland (1972). 8 Luschei, E. L. T h e Logical Sy.rtems of Lesniewski, North Holland: Amsterdam (1962). 9 Menger, K. Selected Papers in Iagic and Foundalions, Didactics, Economics, Reidel: Dordrecht-Holland (1979). 10 Quine, W. V. 0 . M e t h o d of Logic, Routledge Kegan & Paul: London (1952). 11 Rescher, N. Topics in Philosophical Logic, Reidel: Dordrecht-Holland (1968). 12 Russell, B. T h e Principles of Mathematics, Allen & Unwin: London (1903). 13 Tarski, A. Introduction to Logic, Oxford University Press: Oxford (1965).
* I should like to thank Barry Smith, David Over and Mark Sainsbury for their coinments on an earlier draft of this paper.
A DEFENCE OF ARBITRARY OBJECTS Kit Fine and Neil Tennant
II-Neil
Tennant
$0. Fine's theory of arbitrary objects is both intriguing and perplexing. I shall explore the logical structure of difficulties facing it and solutions Fine proposes. In $1 I shall outline the structure of his argument for accepting arbitrary objects, indicating various rejoinders to Fine on certain points. Then I shall take up some of these points in greater detail: in $2, the problematic status of the principle of generic attribution, and in $3 the nature of the commitment Fine would have us make to arbitrary objects. $1. Fine arrives at his theory by successively refining his ideas to take care of certain objections. These are as follows.
Objection I There are no arbitrary objects Fine's reply There are. They are abstractions. But they are not on an ontological par with individuals.
Objection 2 The theory of arbitrary objects is logically incoherent Fine's reply The arguments behind this objection depend upon the failure to distinguish between two basically different formulations of the principle of generic attribution: one in the material mode; the other in the formal mode. Once the distinction is made, the arguments are seen to be without foundation.
Rejoinder In a semantically closed language, the principle of generic attribution, if itself generic, leads to incoherence in a way that makes the principle itself suspect, rather than the fact of semantic closure. I shall develop this point in $2 below.
Objection 2a Semantical rules fail for complex predications on arbitrary objects. Fine's reply The statement Q a ) , regardless of its inner complexity, simply has the same truth conditions as V iQ i).
80
11-NEIL TENNANT
Rejoinder All sorts of differences now emerge between predications on arbitrary objects and predications on individual objects. Moreover they emerge even when the object language is logically perfect, with no vague predicates etc. Fine admits that his proposal for evaluating disjunctions by resorting to lambda abstraction means that the lambda conversion principle fails. He concedes that 'it is impossible to achieve complete logical parity between individual and arbitrary objects'. But this is to play down the importance of the difference: to offer a picture of a progressively diminishing but never disappearing difference in logical behaviour upon successive theoretical adjustments and manoeuvres. It seems to me, however, that the gap between arbitrary and individual objects yawns just as wide as we shunt the difference around from evaluation of complex predications to principles of property abstraction etc; and that one hardly need be an 'adamant logical purist' to be disturbed by this persisting difficulty. Objection 26 Absurdity results from taking the principle of generic attribution within its own scope. Fine's reply Distinguish generic from non-generic (classical) conditions. Rejoinder How? Does the fault lie only in such predicates as 'being an individual number' or 'being in the range o f ? Might it be generated also by certain logical operations, such as unrestricted quantification (especially in the case of set theory)? How do we know when a given condition is generic? Fine nowhere answers this question. O n p. 64 he acknowledges the problem, but offers a circular answer:
. . . it is not as if the principle [of generic attribution] had no application. Call a language generic if all of the conditions obtainable by its use are themselves generic. Then many languages, of natural and independent interest, will be generic; and so the principle (G3)[for any generic condition Q x ) , Q a ) is true iff V i q i ) is true] will have wide application to all such languages. 'Being generic' ought to be a decidable property of conditions expressible in the language. Only then will the principle of generic attribution have application of sure axiomatic status.
A DEFENCE OF ARBITRARY OBJECTS
81
We cannot wait to see whether a given condition, when taken within the scope of the principle, will lead to absurdity. Yet the undecidability of first order logic, and Fine's silence on the matter, would appear to prevent one from doing any better.
$2. Fine distinguishes formulations of the principle of generic attribution in the material and in the formal mode. The reason for this is that if the principle is taken in the material mode then Berkeley's example of odd or even numbers precipitates absurdity. But it is not exactly clear how the distinction is to be drawn. According to Fine, taking the principle in the material mode (as in his (Gl): a Q's iff every individual Q s ) rests on the fallacy of applying the principle internally to only a part of the context in which the name of an arbitrary object appears. For although we may afirm V i Q i ) = ViQ(i), we cannot correctly infer Q ( a )= ViQ(i). But the fallacy involved in the material mode might just as well be taken to be that ofapplying the principle externally to the whole context in which the name of an arbitrary object appears. If we recall Fine's own account of the matter, we ask whether the arbitrary number a satisfies the condition (ExEViEi) of being even iff all individual numbers are even. What the intuitive principle of generic attribution seems to tell us is that an arbitrary number a should satisfy the condition iff all individual numbers do . . . What is this, if not applying the principle to the wider context of the biconditional, rather than 'internally' to the part consisting of its left hand side? Fine's distinction between material and formal modes for his principle of generic attribution can be discarded. He can instead merely prohibit semantic closure. He ought to be calling for something like a type restriction on applications of the principle, especially on the satisfaction and truth predicates occurring in its formulations in the formal mode. (This point will emerge more clearly when formal proofs are examined below.) Tarski's reason for resorting similarly to language levels in the theory of truth was to avoid the semantical paradoxes such as the Liar. Now with the semantical paradoxes, the problem lies
82
11-NEIL
TENNANT
in the natural rules of inference for the truth (or satisfaction) predicate, coupled with inferential moves licenced by the facts of self-reference afforded by semantic closure. Elsewhere' I have analyzed the proofs of absurdity associated with the paradoxes. They all appear to share the feature of not being normalizable. That is, the reduction sequence of a paradoxical proof does not terminate after finitely many steps. Instead it enters a loop whose periodicity depends on the logical structure of the paradoxical statement(s) in question. In the light of this, it seems reasonable to conjecture that the test of looping reduction sequences, applied to an enumeration of proofs in the semantically closed language, would yield an axiomatization of paradox. The relevance of these remarks on paradox is this. If we take Fine's formulation of his principle in the formal mode, but allow the language to be semantically closed, absurdity results from a proof virtually identical to the one he gave for the case of the material mode. Thus semantic closure short-circuits the distinction between modes. In what follows I use these abbreviations:
€52
Q is generic
a19
a satisfies cp
TV
cp is true
PGA (x) Ex Ox
V g Q ( x / Q z T ViQi) x is even x is odd
PGA (x) is the principle of generic attribution for the case where x is an independent arbitrary object. In this case we can simply put TViQi on the right hand side of the biconditional, instead of resorting to talk of satisfaction by each individual in the range of the arbitrary object concerned. For an arbitrary object a, PGA (a) is axiomatic. Let us further assume that we are concerned only with natural numbers. n will range over genuine individual numbers. As an axiom schema we have En vOn. In the absence of explicit criteria allowing us to determine otherwise, let us assume that PGA is a generic property. We have the following proof P of V g Q ( n / Q = T V j Q ) from assumptions PGA (a) and gPGA:
83
A DEFENCE OF ARBITRARY OBJECTS
PGA (a)
i.e. V g Q ( a / Q s TViQi)
gPGA
PGA (a)
We continue now as follows: P
P
gO VgQ(n/Q-VjQ)
gE VgQ(n/Q= TVjQj) (1)-
(1)-
n/E
n/E'--TVjEj
En v On n/EvO n/E vri/O
n/O
n/O
TVjOj
TVjOj
TVjEj v jEj El -
~ j o j
02 A (1)
A A
O u r proof of A is in normal form, by contrast with proofs of semantical paradox, as mentioned above. This leads me strongly to suspect the principle of generic attribution itself, rather than the fact of semantic closure, to be the source of absurdity. T h e full list ofassumptions on which A depends in the proof above is PGA (a),gPGA, gE, g o , En v O n Of these, only PGA (a) could be disputed by Fine. Indeed it is clear from the way he introduced the generic/classical distinction that he would dispute it. But the grounds on which he would do so have not been made clear and justified independently of the reductio provided by our proof. Another would-be reductio is the following:
84
11-NEIL TENNANT I'GA (a):
('1~j (EjvOj) g ( E v 0 ) VgQ(a/Q'--TV,iQj) a/EvO=?'v.j (E,i v Q ~ ) a/EvO (*) a/E v a10
'rvj (Ej vOj)
-
gE a/E
vgQ(a/QsTv,iE,i) a/E=TVjEj
W,jcj v.iEi
similarly for 0 in place ol' E
El A
A
The suspect move here is of course ("). From the fact that an arbitrary object satisfies a disjunction, it does not follow that it satisfies one or other of the disjuncts. Fine notes as much, observing that the semantical rule for evaluating disjunctions fails for statements about arbitrary objects. But I think the point comes out more vividly in a proof theoretic context. The last proof showsjust how abruptly ordinary reasoning about objects can be stopped dead in its tracks by the indeterminacy of the arbitrary. Far from being a theory about arbitrary objects, it is rather a theory of arbitrary obstacles. We lose the distributive laws for satisfaction by objects across logical operators; thus arguably being deprived of a notion of objecthood at all. We have seen from the proofs above that if the language is semantically closed then the principle of generic attribution (in the formal mode) leads to absurdity unless one denies both that the principle itselfexpresses a generic property and that arbitrary objects behave like individuals with respect to satisfaction of predicates. Let us concur with the latter denial for the time being, thereby refusing to admit the last 'proof of absurdity. I want now to make a constructive suggestion which might enable Fine to avoid having to specify what generic properties are, and indeed enable him to formulate the principle of attribution without restriction to generic properties. It appears that merely indexing levels is all that is needed to avoid the first proof of absurdity given above. Let us assign predicates to levels in the obvious way. Let PA,, be the generically unrestricted but type restricted principle Howsoever we now try to write down a version of P observing type restrictions we fail. The type restrictions can be gram-
85
A DEFENCE OF ARBITRARY OBJECTS
matical or inferential. That is, we can either count incorrectly typed formulae as ill-formed, or incorrectly typed inferences as fallacious. T o illustrate the effect of either kind of restriction, let us try to write down a version of P starting with PA1 (a): PA1 (a) PA1 (a)
i.e. V Q ( a / l Q
(I1) a/ 1 PA1
a/~ PA1
TlV i Q i ) TIViPAl i
(1)
etc. Type restrictions are violated repeatedly. (I) instantiates a generalization over zero level predicates with a first level predicate. (11) does not raise the level of the satisfaction predicate from 1 to 2 as it should, given that it has as an argument a predicate of level 1. (111) has T I applying to a formula already of level 1. We thus thoroughly forfeit our rights of passage to A by this route; and this without having to invoke the notion of generic properties and without having to restrict the principle of attribution to generic properties.
93. O n a Carnapian distinction2 between external and internal questions about existence, one might deny (externally) that there are arbitrary objects. This Fine does. But his acharnement about the applications of his theory makes one suspicious of his professed willingness to go along with a programme of reduction that would put his new found crystals into logical solution. The way to a positive answer to the external question about existence was, after all, the utility of applications of the 'linguistic framework' involved. Yet Fine describes himself as answering negatively what he calls the 'ontologically significant' question about existence. He likens himself to the nominalist who denies the ultimate necessity of number talk for our scientific purposes. But if Fine wishes to deny the ultimate necessity of talk about arbitrary objects for our scientific purposes (namely, achieving a
86
11-NEIL
TENNANT
better understanding of truth conditions and valid reasoning in ordinary and mathematical discourse), we can legitimately ask what the point is of developing the theory in the first place. If we already have our reduction in the orthodox view, why go inventing theories about new categories of ill-behaved objects that are to be reduced to it? And if the reduction is to go a different way, why not aim for its terminus straightaway? What is Fine's position qua semantical theorist with regard to the internal question about existence? He calls the question 'ontologically neutral' and answers it affirmatively. He likens himself to the nominalist philosopher of mathematics who is convinced that number talk is dispensable, but useful, and who indulges himself in first order arithmetical assertions. The modern theorist will live with his objects, but not really commit himself to them. I shall not raise here problems that might beset Fine should he wish to use set theory to do the arbitrary object model theory required to treat the set theorist's own sayings of the form 'Let x be an arbitrary set . . .'. Instead I wish to draw attention to a position not yet considered as far as ontological issues are concerned. Is the native speaker (as opposed to the semantic theorist studying his utterances about some subject matter) in any way committed to the existence ofarbitrary objects? I would say not. Assume the native speaks an ordinary first order language with no branching quantifiers. If we are concerned to interpret his utterances by elucidating their truth conditions then it seems we have no way of committing him to arbitrary objects that we might invoke to work out what follows from his general claims about the subject matter in question. It is only if the native reasons by means of certain locutions that can be understood in no other way than as referring to arbitrary objects that it becomes plausible to regard him as 'internally' committed to their existence. Suppose now that I am a first order theorist whose surface linearizations of natural deductions in tree form judiciously eschew all apparent reference to arbitrary objects. For example, instead of saying Let a be an arbitrary object arbitrary, for all x Fx
. . . Then Fa. So, since a was
A DEFENCE OF ARBITRARY OBJECTS
87
I say rather Take a . . . Then Fa. But I assumed nothing about a. So, for all x Fx Each of these manners of speaking adequately represents the inferential move of V-introduction
Fa VxFx
where a does not occur in any assumption on which Fa depends
The latter gloss is deep and robust. The former is ontologically bloating. If I, in the position of the native, persevere with deeply robust renderings of the moves in my language game it is simply wrong to claim I have as much reason to affirm that there are arbitrary objects in this [internal] sense as the nominalist has to affirm that there are numbers. Fine says (p. 57) that the question 'what are they?' may be taken, in an ordinary, non-philosophical way, as a request for an explanation of what objects one is talking about. And he goes on immediately to to say one can do no better than refer to the kind of role that arbitrary objects are intended to play. We may ask here whether he is talking about a role played by arbitrary objects or by parts of language for whose interpretation those objects are (perhaps misguidedly) being invoked. Consider set theory. In his more philosophical mood Fine characterizes arbitrary objects as abstractions like sets or propositions. Let us not pause to ask how, in his more philosophical mood, he would characterize arbitrary sets. Let us instead ask, in the context of set theory, whose intentions about roles to be played are relevant here. Presumably not those of the set theorist. For his intentions might laudably be to describe the universe of sets uncluttered by arbitrary members, sets that do not depend for
88
11-NEIL
TENNANT
their existence on, or derive their character from any role he intends them to play. So the intentions must be those of the semantic theorist. Now is it satisfactory to answer the ordinary non-philosophical questioner who asks 'what are arbitrary objects?' by telling him what role one intends for them as a semantic theorist? Would one be satisfied, upon asking 'what are spirits?' if one's Azande informant told one what role he intended spirits to play in his theory of bad weather and juvenile delinquency? Surely our retort would be that the description of intention is not enough, and that there are further independent methodological constraints to be placed on the theory before a satisfactory answer might be forthcoming. Thus I am interested not so much in the role Fine intends arbitrary objects to play as in the genuinely explanatory role (if any) that they have to be allowed to play if we are to account successfully for the logico-linguistic intentions of native speakers. These intentions are to model and describe reality, and this sometimes in highly schematic fashion, as when they reason about it at first order. I say 'schematic' for good reason. The orthodox construal of Fine's phrase (p. 65) 'names for arbitrary objects' is 'placeholders for names of actual individuals'. We can extend the discussion above of the rule of V-introduction to make this clear. The subproof is schematic in a in the following sense. Any term t may be substituted for appropriate occurrences of a in the subproof so as to yield a proofofFt from the original assumptions. (Similarconsiderations apply to the subproof in the rule of 3 -eIiminati~n.)~ Thus instead of taking the parameter a as a name for an arbitrary object we may consider it as a placeholder for names of actual individuals. Talk 'on the surface' of arbitrary objects a when presenting the proof may thus be construed as remarks about the logically hygienic pattern of occurrences of the parameter a within the proof, given or planned. We thus have a wholly syntactic option for systematically understanding the behaviour of 'names for arbitrary objects'. At the other extreme, arbitrary objects themselves, if admitted, might be better assimilated to the domain of cognitive psychology. The original discussions of the 'general triangle' were highly psychologistic.4 One might regard arbitrary objects as incomplete mental representationsnot so much objects of thought as objects within thought, by
A DEFENCE OF ARBITRARY OBJECTS
89
means of which we reason about the world of actual objects. We may even be able to merge the two accounts. Perhaps the best cognitive account will be one that treats mental representations as clusters of predicates, or 'pigeonholes' within a relational scheme. Fine himself expects that to each set of individuals from I there will be an arbitrary object with that set as its range thereby in effect equating arbitrary objects with subsets ofl. But in all applications of which he has given us any inkling it would appear that only definable subsets matter. So can we not reinterpret his arbitrary objects syntactically as the defining conditions on their range^?^ Is not Fine himself being too orthodox a post-Fregean in devising a semantic or referential theory of arbitrary objects, without offering schemes of arbitrary objects as models of cognitive processes with some smack of psychological-or indeed any kind of-reality? NOTES
'
cf. 'Proof and Paradox', Dialectica 36 (1982) 265-296 'Cf. Carnap, 'Empiricism, semantics and ontology', Revue internationale dephilosophie 4 (1950) 20-40. For more detail see my Natural Logic (Edinburgh, 1978) pp. 42-3, 46, 65-9. 'For references to an extensive literature see Beth, 'Uber Lockes "Allgemeines Dreieck"' Kantstudien 48 (1956/57) 361-380. One is reminded here of Hilbert's terms and Hailperin's restricted variables. See Leisenring, Malhemotical Logic and Hilbert's &-Symbol (MacDonald, London, 19@) and Hailperin, 'A theory of restricted quantification, I and 11', Journal of Symbolic Logic 22 (1957) 19-35 and 113-129.
http://www.jstor.org
LINKED CITATIONS - Page 1 of 1 -
You have printed the following article: A Defence of Arbitrary Objects Kit Fine; Neil Tennant Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 57. (1983), pp. 55-77+79-89. Stable URL: http://links.jstor.org/sici?sici=0309-7013%281983%2957%3C55%3AADOAO%3E2.0.CO%3B2-A
This article references the following linked citations. If you are trying to access articles from an off-campus location, you may be required to first logon via your library web site to access JSTOR. Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR.
Notes 5
A Theory of Restricted Quantification I Theodore Hailperin The Journal of Symbolic Logic, Vol. 22, No. 1. (Mar., 1957), pp. 19-35. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28195703%2922%3A1%3C19%3AATORQI%3E2.0.CO%3B2-C
NOTE: The reference numbering from the original has been maintained in this citation list.
