Dummett's Backward Road to Frege and to Intuitionism - Tripod

these three Principia senses of “intension.” I doubt even more that he distinguishes the corresponding
three Principia senses of “extension.” In his Frege books, he introduces a single notion of extension in
terms of both truth-functionality and opacity, blurring senses (1) and (2). The slip from sense (1) to
sense (2) is subtle:
...are...extensional: that is, provided that the truth-value of a complex sentence depends
only on the truth-values of the constituents....What happens, then, in a case in which a
method of sentence-transformation is used which is not extensional: that is, when a
sentence is formed which contains an opaque context? (Dummett 1981: 189–90, my
emphasis; see 1981a: 155)
But he does clearly identify intension in sense (2) in discussing intuitionism (2000: 16–17).
For Frege, all sentential contexts, even opaque contexts, are extensional in sense (1). To
preserve truth-functionality in opaque contexts, Frege introduces a systematic reference shift from
customary references to senses. Thus senses are extensional in sense (1). But senses are intensional in
sense (2). That is the whole point of Frege’s explanation of informative identity statements. Namely,
different names expressing different senses can refer to the same reference.
All senses are intensional in sense (2), while all customary, i.e., direct, references are
extensional in sense (2). For any customary reference, there are indefinitely many senses containing
modes of presentation of that reference. Insofar as a reference is informatively identifiable indefinitely
many times, there are indefinitely many senses through which it can be identified. Thus in general, the
relation of a reference to the senses which are ways of identifying it is one-many. Thus it is magical to
suppose that starting from any given set of references, i.e., objects and functions, we can work back so
as to determine any one sense in particular as the sense expressed by a name of any one of those
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