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## Analysis of the MethodIn order to make the method precise, we must define the notion of a canonical argument, for, to repeat, the idea is that an inference is justified if any canonical argument for its premises can be transformed into a canonical argument for its conclusion. The definition should make plausible the following: if a logically complex proposition is provable at all, then it could in principle be proved by a canonical argument. For only if that condition is met will Dummett's procedure have any plausible claim to justificatory force. In line with the underlying idea, it might be tempting to define a canonical argument as one composed only of introduction rules. This does not work, however, because of the nature of the introduction rule for the conditional, to wit: [
But if we are constrained to using only introduction rules, we will not be able to fill in the middle part. Hence the subsidiary arguments, the ones starting from premises that will eventually be discharged, cannot be constrained to contain only introduction rules. All that can be required of such subsidiary arguments is that they themselves be already recognized as justified. The result is a definition, by simultaneous induction on the complexity of the statements in the arguments, of the notion of “valid canonical argument” along with the notion of “valid argument”: A A We have simplified matters slightly by omitting what Dummett calls “boundary rules”, which allow the inference of one atomic sentence from others.2 For example, these may be empirical laws, connecting the primitive notions of the vocabulary. Dummett allows the employment of such rules in valid canonical arguments. For the moment we take there to be no such rules, since the mathematics is clearer without them. In the next section, we shall allow boundary rules and investigate their impact. The validity of an argument depends only on its premises and conclusion, and not on any intervening steps. Hence the second definition is framed as applying to inferences, rather than deductions. The simultaneous induction works because discharge of premises increases logical complexity. Thus, whether a deduction with conclusion These definitions are far from transparent. Applying them involves tracking through the tree structures of deductions in natural deduction systems. Most importantly, the definitions do not readily yield any general information about the range of inferences that are valid or not. However, the definitions can be greatly clarified if we focus not on the prooftheoretic layout but rather on the relation that holds between a set We can now investigate the relation the rule of ∧-introduction. A valid canonical argument for 2These amount to the definitions given by Dummett in [3, p. 261], simplified by the absence of boundary rules and (more importantly) of the need to deal with free variables. valid canonical argument for
A valid canonical argument for
intuitionistic logic, since they are nothing other than rules for the treatment of the connectives in the usual Kripke model semantics, when we take the sets One connective remains to be considered, namely, negation. As Dummett notes, the only way to treat negation that is consonant with his general procedure is to take ¬ following introduction rule: from premises that are all the atomic sentences, it may be inferred. Dummett allows there to be infinitely many atomic sentences; in fact, this treatment of negation fares poorly if there are not. For if If there are infinitely many atomic sentences, then this treatment of negation can most easily be incorporated into our forcing relation by requiring that the domain of sets of atomic sentences
The resulting rule for negation is then: The characterization of the forcing relation will be complete once we give the clause governing atomic sentences themselves. Since we are at the moment allowing no boundary rules, we have: for any atomic sentence As we've just seen, Dummett's notion of valid canonical model yields a relation fthat obeys just the usual semantic rules for models of intuitionism, as given by (1)–(4). However, there is a key difference between fas used in Dummett's method and the ordinary model-theory of intuitionistic logic. In the latter, the validity of an inference would mean that at each node (world) in
¬¬
By the way, since we have shown that, for
The following is easily shown by induction on the construction of sentences: for any sentence Thus, if Thus there are many inferences that turn out valid under Dummett's definition, and yet are logically valid in no plausible sense. The counterexamples show that such inferences exist even in the fragment of the language that does not contain ⊥, and so does not contain negation. As a particularly vivid case, we have the validity of the inference from |

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