Hilbert’s ∈-Terms in Automated Theorem Proving

Abstract

∈-terms, introduced by David Hilbert [8], have the form ∈x.φ, where x is a variable and φ is a formula. Their syntactical structure is thus similar to that of a quantified formulae, but they are terms, denoting ‘an element for which φ holds, if there is any’.

The topic of this paper is an investigation into the possibilities and limits of using ∈-terms for automated theorem proving. We discuss the relationship between ∈-terms and Skolem terms (which both can be used alternatively for the purpose of ∃-quantifier elimination), in particular with respect to efficiency and intuition. We also discuss the consequences of allowing ∈-terms in theorems (and cuts). This leads to a distinction between (essentially two) semantics and corresponding calculi, one enabling efficient automated proof search, and the other one requiring human guidance but enabling a very intuitive (i.e. semantic) treatment of ∈-terms. We give a theoretical foundation of the usage of both variants in a single framework. Finally, we argue that these two approaches to ∈ are just the extremes of a range of ∈-treatments, corresponding to a range of different possible Skolemization variants.