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## On Dummett's “Proof-Theoretic Justifications of Logical Laws”
Can logical laws be justified? Of course, the question can be answered, trivially, in the affirmative: a logical law can be justified by deriving it from other logical laws. But the question is meant to ask something deeper, something like: can To this question it seems plausible that the answer is negative. Early analytic philosophers might have argued that since the logical laws provide the canons of justification, it does not even make sense to seek to justify them. (This view is, I take it, near to the surface, if not completely explicit, in Frege. It is the cornerstone of Carnap's thought, when he takes the specification of a linguistic framework— including all the logical laws—as a precondition for any rational inquiry or debate at all.) This philosophical view is supported by, or mirrored in, an obvious technical point: any justification would involve a deductive argument; this argument would use logical laws, so that the justification would presuppose what it is supposed to justify. Thus it would be circular, and not a justification at all. This is well illustrated by soundness proofs for deductive systems: ordinarily, in showing soundness of a particular axiom or rule, one uses logical reasoning that is the direct analogue in the metalanguage of that very axiom or rule. Nonetheless, as Michael Dummett has long urged (see, e.g., [1]), a negative answer might be too quick. It might be proposed, for example, that it is the meaning of our words that have, as upshots, the acceptability of the logical laws; might not an account of those meanings therefore be able to play the role of supporting, or even fully justifying logical laws? To put a finer point on it, the suggestion is that logical laws are true by dint of the meanings of the words in them—specifically the meanings of the logical particles; and hence one might be able to find justifications of those laws simply by unfolding what the meanings of the logical particles are. The hope is that this might be done In an odd sense, the idea goes back to Wittgenstein's discovery of truth-functional analysis: for the validity of the truth-functional laws follows at once from the stipulation of the truth-functions that the connectives represent. (I say “in an odd sense”, since for Wittgenstein the logical laws have no content, and it is surely odd to speak of justifying something without content: what is there to justify?) But it should be noted that a strong assumption underlies Wittgenstein's procedure, namely his notion of propositions as bipolar—possibly true, possibly false, and determinately either one or the other. That is a highly suspect assumption, at least to those like Dummett who wish to question classical two-valued logic. So perhaps the question should be rephrased as: can we find noncircular justifications of logical laws by unfolding the meanings of the logical particles, without making strong meaning-theoretic assumptions? Gerhard Gentzen's work in proof theory in the 1930s proved to be suggestive in this regard. Gentzen had developed logical systems in which the role of each connective was isolated, so that each basic inference rule was “about” one and only one connective. Indeed, he showed that two sorts of rules for each connective suffice. One sort allows for the introduction of the connective, and one for its elimination. In the context of a system for natural deduction (rather than in Gentzen's sequent calculus), the rule of ∧-introduction is that which licenses the inference of of premises: if With respect to the project of justifying logical laws on the basis of the meaning of the logical particles, if we accept this view of introduction rules then clearly those rules stand in no further need of justification. As Dummett puts it, they are “selfjustifying”. The question then is whether such self-justifying rules can be used to endow further logical rules with justification, in particular, rules beyond those that amount to iterated use of introduction rules. In Chaps. 11–13 of The clearest illustrative case is an inference by an elimination rule, say, an inference from This method of justifying logical laws is important to Dummett for several reasons. First, it provides a sense in which logical inferences As it turns out, or so Dummett asserts, the method provides justification for intuitionistic logic but not for classical logic, at least not for the classical laws about negation. Thus it gives important support to his position that intuitionistic logic is preferable to classical. Indeed, it exhibits a virtue of intuitionistic logic—justifiability on the basis of laws that merely express the meaning of the connectives—that classical logic fails to have: “[Intuitionistic logic's] logical constants can be understood, and its logical laws acknowledged, without appeal to any semantic theory and with only a very general meaning-theoretical background.” [3, p. 300] The failure of this method for the laws of classical negation thus allows an invidious distinction to be made. In this paper I investigate Dummett's method, as it applies to sentential logic.1 I shall show that, even in this restricted domain, Dummett's method won't do: it provides “justifications” for obviously invalid inferences. I shall consider how to repair the damage, and analyze the question of whether the repair restores confidence in the philosophical framework underlying Dummett's claim that his method does indeed justify. The results are, I think, suggestive of some overlooked, and possibly deep, difficulties in Dummett's overarching project of marrying intuitionism and a verificationistic theory of meaning. |

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