BU CLA MA 531 : Computability and Logic

There are formal proof systems other than Hilbert-style for classical logic, whether propositional, first-order, or higher-order. Among these alternatives are tableaux, Gentzen, and natural deduction systems. None of these are discussed in Enderton’s book. In a Hilbert system, finding a derivation for a wff may be a tricky business, as one has to guess which axiom and which inference rule to use without systematic reliance on the syntax of the given wff. In tableaux, Gentzen, and natural deduction systems, there is less of this kind of guesswork, and derivations (or refutations) are largely syntax-directed. In particular, these alternative systems are more suitable for automatic theorem-proving1 and allow for elegant presentations of intuitionistic logic2. One distinctive feature of these systems, not shared by Hilbert systems, is that they systematically and more efficiently produce a formal proof for a given wff — if it is indeed a valid wff, and if it is not, these systems will sometimes (but not always!) produce a counterexample to its validity. (Of course, we can also use a Hilbert system to produce a formal proof for a valid wff, typically by some exhaustive listing of all possible derivations, but this is not efficient at all!) Actually, in some variants of these alternative proof systems, it becomes very clear that a formal proof is basically a record of “an unsuccessful attempt to produce a counterexample”. This is a simple idea, but it seems to be an insight of considerable importance in many applications.3 Not to give the wrong impression, there have been attempts to do automatic theorem-proving as well as presentations of intuitionistic logic based on Hilbert systems (of one form or another), but these are generally ill-suited for a computational task and often seem to obscure it by syntactic details. Of course, there are also subjective considerations (e.g. personal taste, familiarity, etc.) in choosing an appropriate proof system to work with. Moreover, although Hilbert systems are “inefficient and barbarously unintuitive”,4 they have advantages. First, the simplicity of their relatively few inference rules makes them suitable for encoding into arithmetic.5 Second, it is relatively easy to tamper with the axiom schemes of Hilbert systems in order to adapt them to non-classical logics.6 Concerning tableaux systems, Smullyan gives a particularly lucid presentation in [10] (where