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Propositional Logic: soundness and completeness revisited

Module by: Ian Barland, John Greiner, Phokion Kolaitis, Moshe Vardi, Matthias Felleisen

Summary: Considering the soundness and completeness of inference rules.

The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us. — Paul Valery, poet and philosopher (1871-1945)

Throughout this discussion, we've implicitly assumed that if we've proven something, it must be true. But we should be careful: What if one of those listed inference rule isn't always valid? What if we introduced a new rule? (Sure, you'd probably balk if we proposed something silly like a∨b⇒a a b a , or even more degenerately false. But what about some more reasonable-sounding rule?) What if our new rule introduces an inconsistency, when combined with the other rules in a some complicated way? In fact, are we absolutely certain that this can't already happen with the inference rules we have?! This brings us back to the questions of soundness and completeness of a proof system. Fortunately, the system presented here is both sound and complete (though proving this is beyond our current scope). However, we can rest assured, that for propositional logic, what we can prove really does correspond entirely to what is true.

Exercise 1

If we omitted the RAA inference rule, would this new system be sound? Would it be complete?

Solution 1

It would be sound: Look at all the possible proofs that can be made in the original system; all those proofs lead to true conclusions (since that original system is sound, as we're claiming). If we just discard all those that include RAA, the remaining proofs are still all true, so the smaller system is sound.

It would not be complete, though: As pointed out, RAA is our only way to prove negations without premises. There are negated formulas that are true (and have no premises) — for example ¬false. Without RAA, we cannot provide a proof of ¬false, so the smaller system is incomplete.

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