On the Church-Frege Solution of the Paradox of Analysis Morton G. White Philosophy and Phenomenological Research, Vol. 9, No. 2. (Dec., 1948), pp. 305-308. Stable URL: http://links.jstor.org/sici?sici=0031-8205%28194812%299%3A2%3C305%3AOTCSOT%3E2.0.CO%3B2-P Philosophy and Phenomenological Research is currently published by International Phenomenological Society.
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DISCUSSION ON T H E CHURCH-FREGE SOLUTION OF T H E PARADOX OF ANALYSIS Church has recently1 proposed a solution of the paradox of analysis as propounded by Langford2 in which Church makes use of Frege's distinction ~ between the sense (Sinn) of a name and its denotation ( B e d e ~ t u n g ) .The main purpose of the present note is to show that a version of the paradox may be presented which is not directly solved by Church in his review but which, in turn, may be solved by using another distinction of Frege-that between the ordinary (gewohnlich)and the oblique (ungerade) use of a name. The second part of the note will be concerned with the relative merits of this second distinction of Frege and Quine's related distinction between purely designative occurrences of a name and occurrences which are not they will be compared merely with respect to their purely de~ignative;~ effectiveness in removing or solving the paradox of analysis.
I h analysis as here understood is a statement of identity of attributes of the following kind: (1) The attribute of being a brother = the attribute of being a male sibling. The paradox as it was formulated earlier arises as a result of putting the expression 'the attribute of being a brother' for the expression 'theattribute of being a male sibling' in ( 1 ) and concluding that ( I ) expresses the same proposition as that expressed by: (2) The attribute of being a brother = the attribute of being a brother. According to Church this puzzle is a special case of a puzzle of Frege insofar as the entities said to be identical in ( I ) are attributes. He therefore uses essentially Frege's method in solving it. Church concludes from the fact that all names have a sense and a denotation that the names of attributes do. His solution depends on assuming that the sense of 'the attribute of being a brother' is different from the sense of 'the attribute In a review in The Journal of Symbolic Logic, V o l . XI (1946), pp. 132-133. "The Notion o f Analysis in Moore's Philosophy," The Philosophy of G . E . Mooye, e d . P . A . Schilpp, pp. 321-342. ''Uber Sinn und Bedeutung," Zeitschrift fur Philosophie und philosophische Kritik," V o l . C (1892),pp. 25-50. iiNotes on Existence and Necessity," The Journal of Philosophy, V o l . XL (1943), pp. 113-127. 305 2
of being a male sibling' although the denotations of these names are identical. I t follows that we cannot conclude that (1) and (2) have the same sense (express the same proposition). I t seems to me that this is conclusive as against the mode of reasoning used above, but now I wish to present a closely related puzzle which can only be solved along these lines, so far as I can see, by using Frege's other distinction. This puzzle begins with.the assumption that the name 'brother' expresses the attribute of being a brother but does not denote it. A similar assumption is made about the name 'male sibling.' I t follows from (1) that these two names have the same sense. Church says el~ewhere:~ "If a name forming part of a longer name is replaced by another having the same sense, the sense of the whole is not altered." Accordingly, the sense of (1) is the same as the sense of (2); since sentences on Frege's view are also names whose senses are the propositions they express; therefore me have a puzzle again. The solution of this puzzle follows merely upon introducing Frege's distinction between the two uses of a name. The point is that the name 'brother' in (1) is used obliquely and therefore does not denote the class of all brothers, but rather the attribute of being a brother. Therefore it does not express that attribute. The case of 'male sibling' in (1) is analogous. I t follows that 'brother' as used in (1) does not have the same sense as 'male sibling' in (1) because both are being used obliquely in that context. I t follows that sentences (I) and (2) have not been shown synonymous by this new argument and that the second version of the puzzle is solved. Church has noted the need for a similar distinction in connection with predicates occurring in clauses like 'that all men are mortal,' pointing out that 'man' and 'mortal' occur obliquely therein.6 My remarks are merely an extension of this point to the case of predicates within a context governed by the phrase 'the attribute of,' and hence to the situation which has been most discussed in connection with the paradox of analysis.
The proposal to regard 'brother' and 'male sibling' as used obliquely in (1) is related to a problem of Quine's'. Since the number of planets I: 9 we might expect by substitutivity of identity, according to Quine, to deduce from the truth: (3) The attribute of exceeding 9 = the attribute of exceeding 9. the falsehood : 6
Review of Carnap's Introduction to Semantics, Philosophical Review, Vol. LII
(1943),p. 301. 6 Zbid., p. 302. Op. cit., p. 126.
(-1) The attribute of exceeding 9 = the attribute of exceeding the number of planets. Quine's coriclusion is that names like '9' and 'the number of planets' do not occur purely designatively in these contexts. The Church-Frege approach, judging from Church's review of the article of Quine in which this lead us to say that these two names are used distinction is d r a ~ v n would ,~ obliquely in these contexts and hence that they respectively denote what mould be their respective senses in ordinary usage. Since the two distinctions are so closely related one might propose the Quinemethod as a way of solving the paradox of analysis as restated in part I of this note. Since 'brother' and 'male sibling' are predicates which occur in a context governed by the phrase 'the attribute of' the Quine method suggests that we conclude that 'brother' and 'male sibling' do not occur in (1) in a purely designative manner. The possible effect of this on situations in which predicates appear within the scope of 'the attribute of' may be seen better by choosing another illustration. Let us suppose the following statements to be true: (5) The attribute of being a man = the attribute of being a rational animal (6) rational animal = featherless bipedg And let us suppose that the follo\ving is false: (7) The attribute of being a man = the attribute of being a featherless biped. Quine's method will prevent us from inferring the falsehood (7) from ( 5 ) via (6) and substitutivity of identity. I t will do so merely by arguing that 'rational animal' does not occur purely designatively in (5). But nothing that Quine says would exclude the claim that the meaning (sense) of 'rational animal' is identical in contexts (5) and (6). I gather that on Quine's view the predicate has the same meaning in all contexts, and if we ever construe it as denoting it will never denote what would be its sense in ordinary use. The important point here is that Quine's theory does not allow us to change the sense of the predicate 'brother' when we move it into a context governed by the phrase 'the attribute of.'1° The Journal of Symbolic Logic, Vol. V I I I (1943), p . 46. The identity here intended would be more clearly rendered by 'The class of all rational animals is identical with the class of all featherless bipeds'. lo I mean here that Quine does not deal with this question explicitly in the paper referred to. I t should also be observed t h a t Quine is generally inclined not t o think of predicates themselves as designating. This is in conformity with his practice of restricting designation t o terms, i.e., expressions which are substituends of variables that are bindable. See his "On the Logic of Quantification," The Journal of Symbolzc Logic, Vol. X (19451, pp. 1-3 especially. Contrary t o the Church-Frege view, then, Quine does not construe predicates as names. But he does construe them as having meanings; indeed they have meaning onlv The Church-Frege theory 1s forced t o 8
9
The conclusion that may be drawn is that while Quine's method permits a solution of his own puzzle in connection with the inference from (3) to (4)) it does not present any obvious solution of the paradox of analysis as restated in the first part of these remarks. The Church-Frege method, on the other hand, would appear adequate to both puzzles. The virtue of the latter is the fact that it permits the sense of a name to vary with context. Frege's ontology is, of course, well-populated, but it should be noted that the paradox of analysis starts by assuming the existence of attributes, and it would appear arbitrary, as Prof. Nelson Goodman has pointed out to me, to assume enough intensions to generate the paradox but not enough to solve it. I t should also be pointed out that Quine's own puzzle in connection with (3) and (4) also assumes attributes provisionally. The chief difference between the purpose of the two distinctions must be noted, however. Quine is led to make his in this context mainly to point out the limitations which we must impose upon inference from statements involving 'the attribute of'. Indeed, if one should maintain that (1) and (2) are not synonymous and yet reject Frege's solution of the puzzle, one will be forced to limit the application of intensional concepts even more severely than Quine does; one will conclude that putting synonyms for synonyms does not always yield synonyms. On the other hand, Frege's distinction would seem to preserve this rule in the manner illustrated above, albeit with restrictions of the sort indicated; whether two expressions are synonymous will d e ~ e n don context." MORTON G. WHITE.
change the meaning (sense) of a name when that name shifts from a context in which it is ordinarily used t o one,in which i t is obliquely used. But since Quine's predicates, strictly speaking, are not names, they cannot change their denotation, and hence cannot change their meaning or sense in the Fregean manner. I t would be worthwhile t o consider whether Quine's predicates might change their meanings in spite of lacking denotation, but the writer has not thought about this sufficiently t o offer any proposals. " Since the above was written Rudolf Carnap has proposed another solution of the paradox in his Meaning and Necessity, pp. 63-64, and Max Black has translated Frege's ''Uber Sinn und Bedentung" in T h Philosophical Review, Vol. LVII (1948)' pp. 207-230. I have used Church's translation of 'Sinn' as 'sense' and 'Bedentung' as 'denotation.'
