Dummett on Connectives

21 04 2005

There’s an interesting post here on the restrictions Dummett had for logical connectives. In particular, the discussion is about the introduction and elimination rules for a connective being in “harmony” with one another.

I haven’t read that particular chapter yet, but whatever notion he was after, it seems to me that his position that he takes in “The Justification of Deduction” (that deduction is justified in so far as it is a transformation of evidence for the premises into evidence for the conclusion) suggests that the meaning of a connective is just the specification of what counts as evidence for a complex statement built up using it. The standard connectives are such that evidence for a formula is simply related to evidence for the subformulas. For instance, evidence for a conjunction is just the combination of evidence for both conjuncts. This can be used to justify an introduction rule, in that if I have evidence for A and evidence for B, I can easily transform this into evidence for A&B just by concatenating the evidence. Similarly, this justifies the elimination rules, in that anyone who has evidence for the conjunction must have evidence for both conjuncts, and thus can easily generate evidence for either one alone.

For a connective like “tonk”, it seems that no good meaning can be found that yields both the introduction rule and the elimination rule. To justify the introduction rule, one would need evidence for the left contonkt to be sufficient evidence for the whole statement, but to justify the elimination rule, one needs evidence for the whole to be sufficient for the right contonkt. Since the subformulas are general, there doesn’t seem to be any way this can be satisfied.

Thus, on a meaning theory for connectives like Dummett’s, introduction and elimination rules are secondary to the evidentiary relations, and (since he prefers intuitionist logic) it should be clear that truth-tables are largely irrelevant as well.

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