Skip to content Skip to navigation

Connexions

You are here: Home » Content » Propositional Logic: soundness and completeness revisited

Navigation

Content Actions

  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the authors

Recently Viewed

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 aba 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.

Comments, questions, feedback, criticisms?

Send feedback