Page 291 - Encyclopedia of Philosophy of Language

Now the rule of double negation elimination is a very simple rule which has seemed self-evident to many. The intuitionists explain their rejection of it along these lines: to have a proof of ~ip is to have a proof that p will never be proved. Thus a proof of ~~np is a proof that np will never be proved, that is, a proof that one cannot prove that p is unprovable. But such a proof of ~ n p is entirely distinct from a proof of p itself. Hence since a proof of ~i~lp cannot be transformed into one of p, the latter does not follow from the former.
2. Intuitionism in Contemporary Philosophy of Language
The intuitionist explanation of the meaning of the logical operators in general takes the same overall form as that for negation: it focuses on the ways in which proofs of complex sentences built up by means of the operators are constructed from proofs for their components. Thus the meaning of V for the intuition- ist is given by the rule that a proof of p v q is either a proof of p or else a proof of q.
This proof-theoretic account of the meaning of logical operators has been taken over by Michael Dummett and generalized to take in their application to empirical as well as mathematical sentences, veri- fication and falsification playing the role which proof does in mathematical discourse. Dumniett's motiv- ation is very different from Brouwer's—based on the Wittgensteinian slogan that meaning is use and on an empiricist view of language acquisition. The result is a type of verificationist theory of meaning.
For according to Dummett, one can be credited with full understanding of undecidable sentences (sen- tences for which at present there is no effective method for determining whether they are true or false) only if one has the capacity to recognize verifications and falsifications of the sentences if presented with them. But if one does not have the further capacity to decide the sentence's truth value then one cannot be credited with a grasp of a content for the sentence which deter- mines that it must be either true or false. Since, however, the sentence is undecidable, one cannot rule out coming to have a decision procedure in future, in
which case the sentence would then become deter- minately true or false. Hence intuitionism, with its rejection of excluded middle but also of tertium non datur, is the correct logic.
Against Dummett one might ask why the capacity to recognize verifications and falsifications, if forth- coming, is not sufficient grounds to credit someone with grasp of bivalent propositions. Moreover the Dummettian generalization of intuitionism to ordi- nary language runs into problems coping with the absence of anything like the conclusive verification one has in mathematics.
3. Conclusion
One final, and fundamental, question which arises with intuitionism is whether there really is a sub- stantive debate between intuitionist and classicist. Brouwer, in rejecting bivalence, often presented it not as standardly understood—every proposition is either true or false—but as a modal principle to the effect that every proposition is either correct or impossible. Intuitionists went on to explain the connectives in terms of proof and provability. If one takes these explanations at face value one can translate (albeit with some loss of content) intuitionist language into classical language plus a modal operator Q read as 'it is provable that.' Godel showed that the result is a classical modal logic known as S4 in which the intuitionist excluded middle p v np gets translated as (Dp v D—Dp) Cv' forclassical disjunction '-' for classical negation) read as p is provable or else provably not provable. And the classicist can agree with the intuitionist that this sentence is not provable for all propositions p. So the ever-present danger, in debates about fundamental principles, that the dispute is really just terminological, seems a real one in this case.
Bibliography
Brouwer L 1967 On the significance of the principle of excluded middle. In: van Heijenoort (ed.) From Frege to Gddel. Harvard University Press, Cambridge, MA
Dummett M A E 1977 Elements of Intuitionism. Clarendon Press, Oxford
Heyting A 1956 Intuitionism. North-Holland, Amsterdam
The history of formal logic presents an interesting and initially mysterious phenomenon. Compared to almost any intellectual discipline outside of perhaps philosophy itself (of which it is often taken to be a
part) it has the longest history. But that history is by no means a continuous record of progress and achievement. It is only a slight exaggeration to say that, after a spectacular beginning which culminated
Logic: Historical Survey S. D. Guttenplan
Logic: Historical Survey
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