Constructive Provability Logic

We present constructive provability logic, an intuitionstic modal logic that validates the L\"ob rule of G\"odel and L\"ob's provability logic by permitting logical reflection over provability. Two distinct variants of this logic, CPL and CPL*, are presented in natural deduction and sequent calculus forms which are then shown to be equivalent. In addition, we discuss the use of constructive provability logic to justify stratified negation in logic programming within an intuitionstic and structural proof theory.Comment: Extended version of IMLA 2011 submission of the same titl

Provability Logic and the Completeness Principle

In this paper, we study the provability logic of intuitionistic theories of arithmetic that prove their own completeness. We prove a completeness theorem for theories equipped with two provability predicates $\Box$ and $\triangle$ that prove the schemes $A\to\triangle A$ and $\Box\triangle S\to\Box S$ for $S\in\Sigma_1$. Using this theorem, we determine the logic of fast provability for a number of intuitionistic theories. Furthermore, we reprove a theorem previously obtained by M. Ardeshir and S. Mojtaba Mojtahedi determining the $\Sigma_1$-provability logic of Heyting Arithmetic

A Short and Readable Proof of Cut Elimination for Two 1st Order Modal Logics

Since 1960s, logicians, philosophers, AI people have cast eyes on modal logic. Among various modal logic systems, propositional provability logic which was established by Godel modeling provability in axiomatic Peano Arithmetic (PA) was the most striking application for mathematicians. After Godel, researchers gradually explored the predicate case in provability logic. However, the most natural application QGL for predicate provability logic is not able to admit cut elimination. Recently, a potential candidate for the predicate provability logic ML3 and its precursors BM and M3 introduced by Toulakis,Kibedi, Schwartz dedicated that A is always closed. Although ML3, BM and M3 are cut free, the cut elimination proof with the unfriendly nested induction of high multiplicity is difficult to understand. In this thesis, I will show a cut elimination proof for all (Gentzenisations) of BM, M3 and ML3, with much more readable inductions of lower multiplicity

Generalized Veltman models with a root

Provability logic is a nonstandard modal logic. Interpretability logic is an extension of provability logic. Generalized Veltman models are Kripke like semantics for interpretability logic. We consider generalized Veltman models with a root, i.e. r-validity, r-satisfiability and a consequence relation. We modify Fine\u27s and Rautenberg\u27s proof and prove non-compactness of interpretability logic

Interpolation and implicit definability in extensions of the provability logic

The provability logic GL was in the field of interest of A.V. Kuznetsov, who had also formulated its intuitionistic analog—the intuitionistic provability logic—and investigated these two logics and their extensions. In the present paper, different versions of interpolation and of the Beth property in normal extensions of the provability logic GL are considered. It is proved that in a large class of extensions of GL (including all finite slice logics over GL) almost all versions of interpolation and of the Beth property are equivalent. It follows that in finite slice logics over GL the three versions CIP, IPD and IPR of the interpolation property are equivalent. Also they are equivalent to the Beth properties B1, PB1 and PB2