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
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