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Distinctness of the approaches (optional)

16 June, 2015 - 11:17
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You might be wondering can we use propositional equivalences as axioms when using inference rules? The short answer is no. First, Boolean equivalences are pairs of formulas, whereas axioms are individual formulas. Second, none of our inference rules mention equivalences.

However, let's reword the question could we use propositional equivalences when using inference rules? It would make sense to add an inference rule to allow this. One possibility would be an inference rule that turns an equivalence into an implication: "if we know φψ, then we know φψ." Another possibility would be an inference rule that allows us to substitute equivalence subterms, as we do in equivalence proofs: "if we know φψ and θ, then we know θ[φψ], i.e., θ, except with instances of φ replaced by ψ." With either, we would also have to allow equivalence proofs as subproofs or lemmas in inference proofs.

Traditionally, and in our presentation, we do not combine equivalences and inference rules in any such way. The disadvantage of combining them is that instead of two relatively simple proof systems, you would have one more complicated proof system. It would be harder to learn all that you could do in such a system, and for theorists, it would be harder to prove things such as soundness and completeness for the combined system. In learning and describing proofs, it is best to keep them separate. However, the advantage would be shorter proofs. When using the combined system, you'd have flexibility to use whichever technique suited the current step best. In practice, people commonly combine these and other proof techniques.