A Seligman-Style Tableau System

Proof systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labeling approach lies at the heart of most resolution, natural deduction, and tableau systems for hybrid logic. But there is another, less well-known approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was first introduced and explored by Jerry Seligman in the setting of natural deduction and sequent calculus in the late 1990s. The purpose of this paper is to introduce a Seligman-style tableau system. The most obvious feature of Seligman-style systems is that they work with arbitrary formulas, not just formulas prefixed by @-operators. To achieve this in a tableau system, we introduce a rule called GoTo which allows us to “jump to a named world” on a tableau branch, thereby creating a local proof context (which we call a block) on that branch. To the surprise of some of the authors (who have worked extensively on developing the labeling approach) Seligman-style inference is often clearer: not only is the approach more modular, individual proofs can be more direct. We briefly discuss termination and extensions to richer logics, and relate our system to Seligman’s original sequent calculus.