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)

Reasoning About Knowledge is the first book to provide a general discussion of approaches to reasoning about knowledge and its applications to distributed ...

A well-known problem with Hintikka-style logics of knowledge is that of logical omniscience. One knows too much. This breaks down into two subproblems: one knows all tautologies, and one’s knowledge is closed under consequence. A way of addressing the second of these is to move from knowledge simpliciter, to knowledge for a reason. Then, as consequences become ‘further away’ from one’s basic knowledge, reasons for them become more complex, thus providing a kind of resource measurement. One kind of reason is (...) a formal proof. Sergei Artemov has introduced a logic of explicit proofs, LP. I present a semantics for this, based on the idea that it is a logic of knowledge with explicit reasons. A number of fundamental facts about LP can be established using this semantics. But it is equally important to realize that it provides a natural logic of more general applicability than its original provenance, arithmetic provability. (shrink)

We study here extensions of the Artemov's logic of proofs in the language with quantifiers on proof variables. Since the provability operator □ A could be expressed in this language by the formula u[u]A, the corresponding logic naturally extends the well-known modal provability logic GL. Besides, the presence of quantifiers on proofs allows us to study some properties of provability not covered by the propositional logics.In this paper we study the arithmetical complexity of the provability logic with quantifiers on proofs (...) for a given arithmetical theory T and a class of proof predicates.In the last section we define Kripke style semantics for the logics corresponding to the standard Gödel proof predicate and its multiple conclusion version. (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.