Dummett's Backward Road to Frege and to Intuitionism - Tripod
Dummett's Backward Road to Frege and to Intuitionism - Tripod Dummett's Backward Road to Frege and to Intuitionism - Tripod
conversely” (1981: 274). This is the forward road from sense to reference. But on Dummett’s program, reference determines sense. This is a problem of circularity. The fifth problem is that the senses of the primitive logical names are underdetermined by their references. Thus even if Dummett’s program magically succeeded in deriving the references of all subsentential names from the truth-conditions, each reference can be presented in indefinitely many ways. Even the identity relation can be presented to us in many ways. Thus it is magical to suppose that we can derive intensional senses from truth-conditions, which consist of extensional references, even if we assign truth-conditions to all statements in the object-language. I call this the problem of sensial underdetermination. 5 Whitehead and Russell distinguish four classic senses of the word “intension.” (1) There are propositional functions that are not truth-functional, e.g., “A believes that p” (1950: 8; see 187 for a derivative sense of “intensional proposition”). (2) There are propositional functions that lack extensional identities—”the same class of objects will have many determining functions” (1950: 23). Such functions are called formally equivalent (1950: 21, 72–73). We may say more generally that different ways of presenting a thing are intensional in this sense. (3) There are intensional functions in the sense that their values need not be specified for them to be specified (1950: 39–40). This sense is inimical to intuitionism. (4) Where extensional classes (“extensions”) are identical just in case their members are identical (1950: *20.31, *20.43), by implication a class is intensional if it is not extensional. (Whitehead and Russell use only extensional classes.) There are four corresponding senses of “extension.” For Frege, senses (2) and (4) are logically tied. That is because for him, all functions are extensional in sense (2) and all classes are extensional in sense (4). Functions correspond one-one with their courses-of-values, where a course-of-values is the class of ordered pairs of values and arguments mapped by the function. For a function and its course- of-values represent each other via the representation function. The representation function is formally 8
well-defined as mutual and is therefore one-one (Frege 1964: 92–94). We may say that functions which have the same course-of-values are representatively identical. Representatively identical functions are not different functions, though due to the “peculiarity” of predicative language we cannot directly say so (Frege 1970d: 46). Since functions are incomplete, they cannot directly stand in the identity relation. A function cannot even be directly said to be identical with itself. For “in view of its predicative nature, it must first be...represented by an object” (Frege 1970d: 46). It is the representing object that stands in the identity relation. We representatively say that functions are identical when we say that their representing objects are identical. Since mutual representation would be impossible if there were not a one-one correspondence between functions and their courses-of-values, the representative identity conditions of functions are exactly as sharp as the identity conditions of the objects that represent them. That this one-one correspondence obtains is the famous extensionality thesis (Furth 1964: xl–xliv), whose name we may honor by saying that for Frege, functions are always extensional in sense (2). A function is definable as any equivalent function (Frege 1970c: 80). Functions are equivalent if and only if their courses-of-values are identical (Frege 1964: 43–44). Thus equivalence is the relation of representative identity. The equivalence relation and the identity relation represent each other, as do their respective relata. But the identity relation itself is indefinable, for a technical reason: since a definition is a stipulated identity, all definitions presuppose identity (Frege, 1970c: 80–81); a definition of a function is a stipulated representative identity. Indeed, the best evidence that representatively identical functions are not to be regarded as different functions is Frege’s explanation (not: definition) of identity as being indiscernibility (Frege 1970c: 80). When Dummett contrasts intensionality with truth-functionality, he has sense (1) implicitly in mind. When he discusses senses as different ways of presenting one object, he has sense (2) implicitly in mind. And when he discusses our inability to traverse infinitely many objects which might fall under a concept, he has sense (3) implicitly in mind. But I cannot recall that he ever expressly distinguishes 9
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conversely” (1981: 274). This is the forward road from sense <strong>to</strong> reference. But on Dummett’s program,<br />
reference determines sense. This is a problem of circularity.<br />
The fifth problem is that the senses of the primitive logical names are underdetermined by their<br />
references. Thus even if Dummett’s program magically succeeded in deriving the references of all<br />
subsentential names from the truth-conditions, each reference can be presented in indefinitely many<br />
ways. Even the identity relation can be presented <strong>to</strong> us in many ways. Thus it is magical <strong>to</strong> suppose that<br />
we can derive intensional senses from truth-conditions, which consist of extensional references, even if<br />
we assign truth-conditions <strong>to</strong> all statements in the object-language. I call this the problem of sensial<br />
underdetermination. 5<br />
Whitehead <strong>and</strong> Russell distinguish four classic senses of the word “intension.” (1) There are<br />
propositional functions that are not truth-functional, e.g., “A believes that p” (1950: 8; see 187 for a<br />
derivative sense of “intensional proposition”). (2) There are propositional functions that lack<br />
extensional identities—”the same class of objects will have many determining functions” (1950: 23).<br />
Such functions are called formally equivalent (1950: 21, 72–73). We may say more generally that<br />
different ways of presenting a thing are intensional in this sense. (3) There are intensional functions in<br />
the sense that their values need not be specified for them <strong>to</strong> be specified (1950: 39–40). This sense is<br />
inimical <strong>to</strong> intuitionism. (4) Where extensional classes (“extensions”) are identical just in case their<br />
members are identical (1950: *20.31, *20.43), by implication a class is intensional if it is not<br />
extensional. (Whitehead <strong>and</strong> Russell use only extensional classes.)<br />
There are four corresponding senses of “extension.” For <strong>Frege</strong>, senses (2) <strong>and</strong> (4) are logically<br />
tied. That is because for him, all functions are extensional in sense (2) <strong>and</strong> all classes are extensional in<br />
sense (4). Functions correspond one-one with their courses-of-values, where a course-of-values is the<br />
class of ordered pairs of values <strong>and</strong> arguments mapped by the function. For a function <strong>and</strong> its course-<br />
of-values represent each other via the representation function. The representation function is formally<br />
8
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