In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which (...) resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus. (shrink)

First - order modal logic is very much under current development, with many different semantics proposed. The use of rigid objects goes back to Saul Kripke. More recently, several semantics based on counterparts have been examined, in a development that goes back to David Lewis. There is yet another line of research, using intensional objects, that traces back to Richard Montague. I have been involved with this line of development for some time. In the present paper, I briefly sketch several (...) of the approaches to first - order modal logic. Then I present one that I call FOIL in the Montague tradition that, I believe, is both expressive and natural. I briefly discuss in what sense it can be made to encompass the other approaches. Finally, I provide tableau rules to go with the FOIL semantics. (shrink)

A new semantics is presented for the logic of proofs (LP), [1, 2], based on the intuition that it is a logic of explicit knowledge. This semantics is used to give new proofs of several basic results concerning LP. In particular, the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator. Finally connections are made with the conventional approach, via soundness and completeness results.

A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel-Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.

We see a systematic set of cut-free axiomatisations for all the basic normal modal logics formed by some combination the axioms d, t, b, 4, 5. They employ a form of deep inference but otherwise stay very close to Gentzen’s sequent calculus, in particular they enjoy a subformula property in the literal sense. No semantic notions are used inside the proof systems, in particular there is no use of labels. All their rules are invertible and the rules cut, weakening and (...) contraction are admissible. All systems admit a straightforward terminating proof search procedure as well as a syntactic cut elimination procedure. (shrink)