Thursday, August 02, 2007

Hacking on logic

I don't expect that this idea is original to me, but I had thought today about Hacking's view of logic in his "What is logic?" There are some conditions that sequent calculus introduction rules must satisfy in order to be properly logical including being conservative over the base language. There are some conditions that the deducibility relation must satisfy, including admitting cut elimination and elimination of the structural rules. In particular, the system must admit of dilution elimination. This is why modal logic doesn't count as logic to Hacking. It (S4 in particular, since there are lots of modal logics) doesn't admit of general dilution (weakening) elimination. Hacking sees this as a feature, not a bug. What I am now wondering about are substructural logics, like relevance logic. Relevance logic (if I am remembering this rightly) doesn't have weakening in it to begin with, so one can't prove an elimination theorem. Requiring the deducibility relation to admit weakening seems to put a strong constraint on what counts as logic at the outset. What is so special about weakening that it gets picked out? The other rules that get singled out as important are cut and the basic axiom or identity rule. It is hard to argue with those. Granted, Hacking says those are sufficient conditions and makes no effort to give necessary ones, but, I am now puzzled.

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