Interpolation properties for the bimodal provability logic GR\mathbf{GR}

We study interpolation properties for Shavrukov's bimodal logic $\mathbf{GR}$ of usual and Rosser provability predicates. For this purpose, we introduce a new sublogic $\mathbf{GR}^\circ$ of $\mathbf{GR}$ and its relational semantics. Based on our new semantics, we prove that $\mathbf{GR}^\circ$ and $\mathbf{GR}$ enjoy Lyndon interpolation property and uniform interpolation property.Comment: 22 page

The provability logic of all provability predicates

We prove that the provability logic of all provability predicates is exactly Fitting, Marek, and Truszczy\'nski's pure logic of necessitation $\mathsf{N}$. Moreover, we introduce three extensions $\mathsf{N4}$, $\mathsf{NR}$, and $\mathsf{NR4}$ of $\mathsf{N}$ and investigate the arithmetical semantics of these logics. In fact, we prove that $\mathsf{N4}$, $\mathsf{NR}$, and $\mathsf{NR4}$ are the provability logics of all provability predicates satisfying the third condition $\mathbf{D3}$ of the derivabiity conditions, all Rosser's provability predicates, and all Rosser's provability predicates satisfying $\mathbf{D3}$, respectively.Comment: 34 page

Reasoning with Inconsistent Information

In this thesis we are concerned with developing formal and representational mechanisms for reasoning with inconsistent information. Strictly speaking there are two conceptually distinct senses in which we are interested in reasoning with inconsistent information. In one sense, we are interested in using logical deduction to draw inferences in a symbolic system. More specifically, we are interested in mechanisms that can continue to perform deduction in a reasonable manner despite the threat of inconsistencies as a direct result of errors or misrepresentations. So in this sense we are interested in inconsistency-tolerant or paraconsistent deduction. … ¶ In this thesis we adopt a novel framework to unify both logic-as-deduction and logic-as-representation approaches to reasoning with inconsistent information. …