Pretopologies and Completeness Proofs

Pretopologies were introduced in [S] and there shown to give a complete semantics for a propositional sequent calculus BL here called basic linear logic, as well as for its extensions by structural rules, ex falso quodlibet or double negation. Immediately after the Logic Colloquium ’88, conversation with Per Martin-Löf helped me to see how the pretopology semantics should be extended to predicate logic; the result now is a simple and fully constructive completeness proof for first order BL and virtually all its extensions, including usual, or structured, intuitionistic and classical logic. Such a proof clearly illustrates the fact that stronger set-theoretic principles and classical metalogic are necessary only when completeness is sought with respect to a special class of models, such as usual two-valued models. To make the paper self-contained, I briefly review in section 1 the definition of pretopologies; section 2 deals with syntax and section 3 with semantics. The completeness proof in section 4, though similar in structure, is sensibly simpler than that in [S], and this is why it is given in detail. In section 5 it is shown how little is needed to obtain completeness for extensions of BL in the same language. Finally, in section 6 connections with proofs with respect to more traditional semantics are shortly investigated, and some open problems are put forward. The content of this paper, except the last section, was already contained in a lecture given in March 1989 at the Department of Mathematics of the University of Stockholm; I thank Prof. P. Martin-Löf for his kind invitation. Soon after, a first draft of this paper was read by Prof. H. Ono, whose answers [O1] and [O2] in turn influenced the chapter on algebraic semantics in Prof. A. S. Troelstra’s lectures [T]. So by now the completeness proof for BL has partly lost its originality; I will thus stress on the peculiarity of the approach via pretopologies. The main advantage of pretopologies seems to be that of having a middle position: so on one hand little effort is needed to show the completeness of the semantics of pretopologies, as usual with algebraic semantics to which it is closely connected, but