Section outline

  • 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