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## Schematic InferencesThe counterexamples I have presented are not The question naturally arises as to how Dummett's definitions fare on schematic inferences. Let us call an inference Now any inference that is schematically valid is classically valid, since if replaced with “⊥”. The resulting instances ∅ f However, schematic validity outstrips intuitionistic logic.
However, we can obtain a positive result if we combine the notion of schematic inference with that of validity-in-the-revised-sense, that is, validity given any collection of boundary rules. That is, it is possible to prove the following:
For the proof, see the Appendix. ## AssessmentDummett can't take too much comfort in this positive result. Dummett is careful to point out, in framing his method, that the inferences treated are actual inferences, involving particular meaningful sentences, the atomic components of which are actual atomic sentences, not schematic parts [3, p. 254]. That is, he treats the language as fully interpreted. On the view he is propounding, an inference is justified by its validity; the justification of a schematic inference (an inference rule) can lie only in the fact that each of its instances is justified. There is simply no room, on his view, for a position to the effect that validity does not justify an inference unless all inferences like it in being subsumable under a particular rule are also justified. Let us return to Counterexample 3. Intuitionistically, the inference from
demonstration of I believe the basic views that lead to Dummett's difficulties are well exhibited in the following remark (he is speaking here only of mathematics, but presumably he would maintain the same for a language with empirical vocabulary as well): If the intuitionistic explanations of the logical constants and, more generally, of the meanings of mathematical statements are to be considered as constituting a coherent theory of meaning for the language of mathematics, then the notion of proof which is appealed to must be such that we can fully grasp the concept of a proof of any constituent of a given sentence in advance of grasping that of a proof of that sentence. It cannot, therefore, be identified with the notion of the sort of proof that we may, at some future time, come to consider valid . . . [2, p. 402] This remark expresses a fundamental view of Dummett's; and from it we can see three sources of the problems with his program for “proof-theoretic justification”. Of course, most generally, his underlying concern to meld intuitionistic logic with theory of meaning impels him to differ with the Brouwerian tradition of the openendedness of the notion of demonstration, and as we saw that was key to the anomalies exemplified by Counterexamples 1 and 2. But the remark also expresses Dummett's commitment to molecularity: that what it is to prove a sentence is explained in terms of what it is to prove each constituent of the sentence. That, of course, is a denial of the impredicative nature of intuitionistic conditionals; and it signals his commitment to just the view of the proofs of atomic sentences as, if you like, logically unanalyzable, that engenders Counterexample 3. The difference between Dummett and the treatments of intuitionism standard in mathematical logic on these two points has not been sufficiently explored. In classical truth-functional semantics, atomic sentences are the basic building blocks, and it is clear why. Dummett takes atomic sentences to be the basic building blocks of a proof-theoretic semantics—and on both the “basic” aspect and the “building block” aspect he differs from classical intuitionism. It is not clear why one should believe this, except perhaps for the conflation of the notion of verification and a mathematical notion of proof. The third factor expressed in Dummett's remark is that there will be no definite meaning ascribed to a sentence unless it is fixed what a demonstration is for that sentence. That will presuppose that not just the logical rules but also the boundary rules are fixed. This tells us that, from Dummett's viewpoint, the revised notion of validity is not acceptable. For, in considering all possible boundary rules, it should be clear, the revised notion of validity treats the atomic components of sentences in abstraction from their actual content. It takes them to be schematic, in that their connections to one another (and hence also to complex sentences) are varied at will, but this is precisely what Dummett's insistence that he seeks to justify actual inferences, not schematic ones, would rule out.6 Finally, I suppose the following line might be taken. The claim that Dummett's method provides justifications of logical laws might be abandoned or weakened, while still it be pressed that the method does show |

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