DS201504: Sourabh, Sumit (2015) Correspondence and Canonicity in NonClassical Logic. Doctoral thesis, University of Amsterdam.
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Abstract
In this thesis, we study correspondence and canonicity for nonclassical logic using algebraic and ordertopological methods. Correspondence theory is aimed at answering the question of how precisely modal, firstorder, secondorder languages interact and overlap in their shared semantic environment. The line of research in correspondence theory which concerns the present thesis is Sahlqvist correspondence theory  which was originally developed for classical modal logic, and provides a systematic translation between classical modal logic and firstorder logic.
Modal languages are expressive fragments of secondorder logic when interpreted over relational structures. However, the celebrated Sahlqvistvan Benthem theorem, which is the cornerstone of the correspondence theory, states that for every formula in a large, syntactically defined class of modal formulas called Sahlqvist formulas, the correspondent is, in fact, a firstorder sentence. Moreover, this correspondent can be computed effectively. Canonicity is closely related to correspondence, and ensures that logics axiomatized by these formulas are complete with respect to relational semantics. Thus, correspondence and canonicity together establish that Sahlqvist logics are semantically complete with respect to firstorder definable classes of relational structures.
The first part of the thesis focuses on algebraic methods for correspondence and canonicity. In chapter 3, we introduce the algebraic approach to Sahlqvisttype correspondence results by proving the classical Sahlqvist correspondence theorem for basic modal logic in the algebraic setting of complex algebras of frames. In the algebraic setting, the reduction strategies for the elimination of the second order variables can be formulated entirely in ordertheoretic terms. The ordertheoretic conditions that guarantee the applicability of these strategies also lead to a positive characterization of Sahlqvist and inductive formulas across different signatures. Conradie and Palmigiano develop an Ackermann Lemma Based Algorithm (ALBA) for distributive modal logic based on an algebraic analysis of the correspondence theory. We extend the algorithm ALBA to regular modal logic (modal logic with nonnormal modalities) and intuitionistic modal mucalculus in Chapters 4 and 5, respectively. Moreover, we syntactically define the class of inductive inequalities in these languages, and show that the algorithm succeeds on them. In Chapter 6, we develop a version of ALBA for distributive lattice expansions (DLEs), using which we prove the canonicity of certain syntactically defined classes of DLEinequalities (called the metainductive inequalities), relative to the structures in which the formulas asserting the additivity of some given terms are valid.
The second part focuses on ordertopological methods. In Chapter 7, we introduce the concept of a subordination on a Boolean algebra, and develop a full categorical duality between Boolean algebras with a subordination and Stone spaces with a closed relation. We further extend this duality to show that the category of de Vries algebras is dual to the category of Gleason spaces, which are extremely disconnected spaces with a closed irreducible equivalence relation. This provides an alternative JónssonTarski style duality to de Vries duality between de compact Hausdorff spaces and de Vries algebras. It also offers a possibility for developing a topological correspondence theory, and a logical calculus for modal compact Hausdorff spaces. In Chapter 8, we prove a Sahlqvist correspondence and canonicity theorem for topological fixedpoint logic on compact Hausdorff spaces. This generalizes the SambinVaccaro proof of canonicity for the language of positive modal mucalculus interpreted over modal compact Hausdorff spaces.
Item Type:  Thesis (Doctoral) 

Report Nr:  DS201504 
Series Name:  ILLC Dissertation (DS) Series 
Year:  2015 
Subjects:  Language Logic 
Depositing User:  Dr Marco Vervoort 
Date Deposited:  14 Jun 2022 15:17 
Last Modified:  14 Jun 2022 15:17 
URI:  https://eprints.illc.uva.nl/id/eprint/2129 
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