Non-classical models of reasoning

Classical propositional logic (CPC) as a point of departure

• logic as algebra - Boolean algebras, algebraic semantics and completeness of classical propositional logic (w.r.t. BA via a Lindenbaum-Tarski algebra construction, w.r.t. powerset algebras via possible worlds, w.r.t the two-element BA). 
• Stone representation theorem and how it connects to the above
• Some distinguishing properties of CPC we usually take for granted: local finiteness (only finitely many formulas in n variables up to provable equivalence), functional completeness (expresses all finite-valued boolean functions), strong completeness and compactness (finitarity), decidability (coNP completeness), normal forms,...

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Non-classical models of reasoning I: Intuitionistic Logic and Mathematics

This part of the course concentrates on intuitionistic logic and its applications in (constructive) metamathematics. The main topics covered by the course are:

• Predicate intuitionistic logic and its main properties (Kripke and algebraic semantics, completeness, disjunction and existence property)
• Intuitionistic axiomatic theories: Heyting arithmetics and its properties (incompleteness, disjunction and existence property, de Jongh's theorem)
• Algebraic semantics of intuitionistic logic (Heyting algebras) and its duality to Kripke semantics
• Decidability of intuitionistic propositional logic

Study materials:

N. Bezhanishvilli, D. de Jongh, Intuitionistic Logic, ESSLLI 2006 Lecture notes.

D. van Dalen, Logic and Structure, Springer 2nd edition 2008 (2013 ebook). available here.

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Non-classical models of reasoning II: Advanced topics in Modal Logics

The second part of the course is devoted to some advanced topics in modal logics (those not covered by the introductory course Modal logics).

• Algebraic semantics of modal logics and its duality to Kripke semantics, applications (Goldblatt-Thomason Theorem - a proof via duality, and a model-theoretic proof)
• van Benthem's theorem - a characterization of the modal fragment of first order logic
• Coalgebraic perspective on modal logics
• Proof theory of modal logics (different formalisms - nested sequents, display calculi, labelled calculi)

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Non-classical models of reasoning III: logics of information

Examples of logics whose semantics is underlined by a concept of information rather than that of a truth value.

• Dunn-Belnap logic of first degree entailment FDE, and its cousins (some of simplest examples of many-valued and/or paraconsistent logics)
• Logics with frame semantics based on information states, including relevant logics
• Substructural logics
• Many-valued (fuzzy) logics
• Two-layered logics for uncertainty