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Propositional Logic: conclusions

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

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Are we done yet?

These inference rules may seem limited, and you may have some more general ones in mind. Soon, we'll see additional inference rules in the context of first-order logic, which will give us a richer set of proofs. In general, a hard problem is finding a language that is both expressive enough to describe the domain succinctly, but also limited enough to automate reasoning. This is a very practical issue in type checking and other program analysis. While it can be easy to find some program errors automatically, it is very difficult or impossible to guarantee that you can find all errors (of some specific kind, like type errors).

One thing we would like to eliminate is the need (at least technically) to restate structurally identical proofs, as discussed for commutativity. We will be able to add the idea of generalizing such proofs directly into the logic and inference rules.

Despite the desire for more flexible reasoning, we'd also like to consider whether we have more inference rules than are necessary. Are some of them redundant? This is similar to the software rule that we should have a single point of control, or the similar idea that libraries should provide exactly one way of doing something. In general, this is not easy to ensure. We have shown that some potential additional inference rules, like commutativity and associativity, weren't necessary. But we haven't shown our core inference rules to be minimal. What do you think? (See the homework exercise problems on the redundancy of not-elimination, not-introduction, and case-elimination.)

Distinctness of the approaches (optional)

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 φ≡ψφ ψ, then we know φ⇒ψφ ψ.” Another possibility would be an inference rule that allows us to substitute equivalence subterms, as we do in equivalence proofs: “if we know φ≡ψφ ψ and θθ, then we know θ[φ↦ψ]θ[φ↦ψ], i.e., θθ, except with instances of φφ replaced by ψψ.” 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.

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