What is modal logic

Entailment: The Logic of Relevance and Necessity

Day 5: Applications and Extensions The precise topics for the course will depend in part on the interests and background of the students attending the lectures. This document contains an introduction to the course including some pointers to relevant literature. The focus is on the basics of neighborhood semantics with only pointers to literature relating to the more advanced topics. The website for the course is staff.science.uva.nl/∼epacuit/nbhd esslli.html On this website you will find the lecture slides (and possibly course notes) updated daily. Two other sources will also be useful:

Neighborhood Semantics for Modal Logic

Introduction to Logic and the Methodology of deductive sciences. By Alfred Tarski. (Enlarged and revised edition.) Pp. xviii, 239. 14s. 1941. (Oxford University Press; New York)

Logical Dynamics of Information and Interaction: Preface

The present article aims at establishing formal connections between correspondence phenomena, well known from the area of modal logic, and the theory of display calculi, originated by Belnap. These connections have been seminally observed and exploited by Marcus Kracht, in the context of his characterization of the modal axioms (which he calls primitive formulas) which can be effectively transformed into `analytic' structural rules of display calculi. In this context, a rule is ‘analytic’ if adding it to a display calculus preserves Belnap's cut-elimination theorem. In recent years, the state-of-the-art in correspondence theory has been uniformly extended from classical modal logic to diverse families of non-classical logics, ranging from (bi-)intuitionistic (modal) logics, linear, relevant and other substructural logics, to hybrid logics and mu-calculi. This generalization has given rise to a theory called unified correspondence, the most important technical tools of which are the algorithm ALBA, and the syntactic characterization of Sahlqvist-type classes of formulas and inequalities which is uniform in the setting of normal DLE-logics (logics the algebraic semantics of which is based on bounded distributive lattices). We apply unified correspondence theory, with its tools and insights, to extend Kracht's results and prove his claims in the setting of DLE-logics. The results of the present article characterize the space of properly displayable DLE-logics.

/pdf/unified-correspondence-as-a-proof-theoretic-tool-15b2pc212k.pdf

Unified correspondence as a proof-theoretic tool

https://www.cs.ox.ac.uk/files/4016/LORI.pdf

Algebraic semantics and model completeness for intuitionistic public announcement logic

The present paper aims at establishing formal connections between correspondence phenomena, well known from the area of modal logic, and the theory of display calculi, originated by Belnap. These connections have been seminally observed and exploited by Marcus Kracht, in the context of his characterization of the modal axioms (which he calls primitive formulas) which can be effectively transformed into `analytic' structural rules of display calculi. In this context, a rule is `analytic' if adding it to a display calculus preserves Belnap's cut-elimination theorem. In recent years, the state-of-the-art in correspondence theory has been uniformly extended from classical modal logic to diverse families of nonclassical logics, ranging from (bi-)intuitionistic (modal) logics, linear, relevant and other substructural logics, to hybrid logics and mu-calculi. This generalization has given rise to a theory called unified correspondence, the most important technical tools of which are the algorithm ALBA, and the syntactic characterization of Sahlqvist-type classes of formulas and inequalities which is uniform in the setting of normal DLE-logics (logics the algebraic semantics of which is based on bounded distributive lattices). We apply unified correspondence theory, with its tools and insights, to extend Kracht's results and prove his claims in the setting of DLE-logics. The results of the present paper characterize the space of properly displayable DLE-logics.

Unified Correspondence as a Proof-Theoretic Tool

The unified correspondence theory for distributive lattice expansion logics (DLE-logics) is specialized to strict implication logics. As a consequence of a general semantic consevativity result, a wide range of strict implication logics can be conservatively extended to Lambek Calculi over the bounded distributive full non-associative Lambek calculus (BDFNL). Many strict implication sequents can be transformed into analytic rules employing one of the main tools of unified correspondence theory, namely (a suitably modified version of) the Ackermann lemma based algorithm $\msf{ALBA}$. Gentzen-style cut-free sequent calculi for BDFNL and its extensions with analytic rules which are transformed from strict implication sequents, are developed.

Unified correspondence and proof theory for strict implication

We present a dynamic approach to Peirce’s original construal of abductive logic as a logic of conjecture making, and provide a new decidable, contraction-free and cut-free proof system for the dynamic logic of abductive inferences with neighborhood semantics. Our formulation of the dynamic logic of abduction follows the philosophical and scientific track that led Peirce to his late, post-1903 characterization of abductive conclusions as investigands, namely invitations to investigate propositions conjectured at the level of pre-beliefs.

Minghui Ma

Papers

Unified correspondence as a proof-theoretic tool

Algebraic semantics and model completeness for intuitionistic public announcement logic

Unified Correspondence as a Proof-Theoretic Tool

Unified correspondence and proof theory for strict implication

Let Us Investigate! Dynamic Conjecture-Making as the Formal Logic of Abduction