Saturday, September 06, 2008

A note on relevance logic

In the proof theory class I'm taking, Belnap introduced several different axiomatic systems, their natural deduction counterparts, and deduction theorems linking them. We started with the Heyting axiomatization for intuitionistic logic and the Fitch formulation of natural deduction for it.

The neat thing was the explanation of how to turn the intuitionistic natural deduction system into relevance logic. To do this, we add a set of indices to attach to formula. When formulas are assumed, they receive exactly one index (a set containing one index), which is not attached to any other formulas. The rule for →-In still discharges assumptions, but it is changed so that the set of indices attached to A→B is equal to the set attached to B minus the set attached to A, and A's index must be among B's indices. This enforces non-vacuous discharge. It also restricts what things can be discharged. The way it was glossed was that A must be used in the derivation of B.

From what I've said there isn't anyway for a set of indices to get bigger. The rule for →-Elim does just that. When B is obtained from A and A→B, B's indices will be equal to the union of A's and A→B's. This builds up indices on formulas in a derivation, creating a record of what was used to get what. Only the indices of formula used in an instance of →-Elim make it into the set of indices for the conclusion, so superfluous assumptions can't sneak in and appear to be relevant to a conclusion.

This doesn't on the face of it seem like a large change. Just the addition of some indices with a minor change to the assumption rule and the intro and elim rules. The rule for reiterating isn't changed; indices don't change for it. Reiterating a formula into a subproof puts it under the assumption the subproof, in the sense of appearing below it in the fitch proof, but not in the sense of dependence. The indices and the changes in the rules induce new structural restrictions, as others have noted. We haven't gotten to sequent calculi or display logic, so I'm not going to go into what the characterization of relevance logic would look like in those. Given my recent excursion into Howard-Curry land, I do want to mention what must be done to get relevance logic in &lambda- calculus. A restriction has to be placed on the abstraction rule, i.e. no vacuous abstractions are allowed. This is roughly what one would expect. Given the connection between conditionals being functions from proofs of their antecedents to proofs of their consequents and λ-abstraction forming functions, putting a restriction on the former should translate to a restriction on the latter, which it does.

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