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)

We systematically study several versions of the disjunction and the existence properties in modal arithmetic. First, we newly introduce three classes $\mathrm {B}$, $\Delta (\mathrm {B})$, and $\Sigma (\mathrm {B})$ of formulas of modal arithmetic and study basic properties of them. Then, we prove several implications between the properties. In particular, among other things, we prove that for any consistent recursively enumerable extension T of $\mathbf {PA}(\mathbf {K})$ with $T \nvdash \Box \bot $, the $\Sigma (\mathrm {B})$ -disjunction property, the (...) $\Sigma (\mathrm {B})$ -existence property, and the $\mathrm {B}$ -existence property are pairwise equivalent. Moreover, we introduce the notion of the $\Sigma (\mathrm {B})$ -soundness of theories and prove that for any consistent recursively enumerable extension of $\mathbf {PA}(\mathbf {K4})$, the modal disjunction property is equivalent to the $\Sigma (\mathrm {B})$ -soundness. (shrink)

We present a uniform proof-theoretic proof of the Gödel–McKinsey–Tarski embedding for a class of first-order intuitionistic theories. This is achieved by adapting to the case of modal logic the methods of proof analysis in order to convert axioms into rules of inference of a suitable sequent calculus. The soundness and the faithfulness of the embedding are proved by induction on the height of the derivations in the augmented calculi. Finally, we define an extension of the modal system for which the (...) result holds with respect to geometric intuitionistic. (shrink)

Recently, in an ongoing debate about informal provability, non-deterministic logics of informal provability BAT and CABAT were developed to model the notion. CABAT logic is defined as an extension of BAT logics and itself does not have independent and decent semantics. The aim of the paper is to show that, semantically speaking, both logics are rather complex and they can be characterized by neither finitely many valued deterministic semantics nor possible word semantics including neighbourhood semantics.

According to the dialetheist argument from the inconsistency of informal mathematics, the informal version of the Gödelian argument leads us to a true contradiction. On one hand, the dialetheist argues, we can prove that there is a mathematical claim that is neither provable nor refutable in informal mathematics. On the other, the proof of its unprovability is given in informal mathematics and proves that very sentence. We argue that the argument fails, because it relies on the unjustified and unlikely assumption (...) that the informal Gödel sentence is informally provable. (shrink)

We prove that Flagg and Friedman's translation from epistemic to intuitionistic predicate logics is not faithful.

New epistemic principles are formulated in the language of Shapiro's system of Epistemic Arithmetic. It is argued that some plausibility can be attributed to these principles. The relations between these principles and variants of controversial constructivistic principles are investigated. Special attention is given to variants of the intuitionistic version of Church's thesis and to variants of Markov's principle.

This paper presents a defense of Epistemic Arithmetic as used for a formalization of intuitionistic arithmetic and of certain informal mathematical principles. First, objections by Allen Hazen and Craig Smorynski against Epistemic Arithmetic are discussed and found wanting. Second, positive support is given for the research program by showing that Epistemic Arithmetic can give interesting formulations of Church's Thesis.

The subject of this article is Modal-Epistemic Arithmetic (MEA), a theory introduced by Horsten to interpret Epistemic Arithmetic (EA), which in turn was introduced by Shapiro to interpret Heyting Arithmetic. I will show how to interpret MEA in EA such that one can prove that the interpretation of EA is MEA is faithful. Moreover, I will show that one can get rid of a particular Platonist assumption. Then I will discuss models for MEA in light of the problems of logical (...) omniscience and logical competence. Awareness models, impossible worlds models and syntactical models have been introduced to deal with the first problem. Certain conditions on the accessibility relations are needed to deal with the second problem. I go on to argue that those models are subject to the problem of quantifying in, for which I will provide a solution. (shrink)