Institute of Information Theory and Automation

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Introduction to Algebraic Logic

Matematicko-fyzikální fakulta UK
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Logic is the science of correct reasoning. In its classical form, it gives a standard account of the precise kind of reasoning used in rigorous discourse, such as mathematics. However, in other contexts reasoning is still governed by rules of rationality that can be described by systems of non-classical logic. This course is an introduction to the branch of Mathematical Logic that studies logical systems with the methods and tools of Algebra. The presentation will be based on the study of particular examples that will hint the construction of a general theory. We will focus on propositional logics, discuss their various motivations (from the point of view of Mathematics, Philosophy, Computer science, and Linguistics), explore several families of non-classical logics, and study their algebraic semantics.


Evaluation method: The students will be regularly graded through exercises proposed at each lesson and (if necessary) through a final exam.



1. Classical logic: the standard logic of mathematics. Hilbert-style axiomatization, Boolean algebras, Lindenbaum-Tarski proof of completeness.

2. Elements of Universal Algebra: algebras, terms, (quasi)equations, homomorphisms, subalgebras, congruences, quotients, direct products, subdirect products, subdirect irreducibility, reduced products, ultrafilters, varieties and quasivarieties, Birkhoff and Mal'cev theorems.

3. Intuitionistic logic: the logic of constructive mathematics. Motivation, Hilbert-style axiomatization, Heyting algebras, Lindenbaum-Tarski proof of completeness. Algebraization of finitary extensions of intuitionistic logic. Deduction theorem and selfextensionality.

4. Gentzen calculi LK and LJ for classical and intuitionistic logic. Structural rules. Splitting of connectives in the absence of structural rules.

5. Substructural logics: motivations in Linguistics, Gentzen and Hilbert-style presentations, algebraization with residuated lattice-lattices and their subvarieties, local deduction theorem. Axiomatic extensions. Failure of selfextensionality.

6. Linear logics: motivations in Computer Science, adding exchange to FL, adding involution, expansions with exponentials and their algebraization.

7. Relevant logics: the logic of relevant implication. Philosophical motivations, adding contraction to substructural logics, prominent systems and fragments without unit and their algebraization.

8. Many-valued and fuzzy logics: the logics of vagueness and graded truth. Axiomatization of logics of chains, completeness theorems with respect to real-valued chains, uninorm logic, adding weakening, t-norm-based logics.

9. Modal logics: motivations in Computer Science and Philosophy, axiomatic systems, necessitation, global and local systems. Boolean algebras with operators. Algebraizability and equivalentiality.



A. Chagrov, M. Zakharyaschev. Modal Logic. Oxford Logic Guides, Clarendon Press, 1997.

P. Cintula, P. Hájek, C. Noguera. Handbook of Mathematical Fuzzy Logic – Volume 1, Studies in Logic, Mathematical Logic and Foundations, vol. 37, College Publications, London, 2011. (ISBN: 978-1-84890-039-4)

P. Cintula, C. Noguera. Slabě implikativní logiky: Úvod do abstraktního studia výrokových logik, Univerzita Karlova v Praze, Filozofická fakulta, Prague, 2015 (ISBN: 978-80-7308-576–6).

P. Hájek. Metamathematics of fuzzy logic. Trends in Logic, Springer, 1998.

G. Restall. An introduction to substructural logics. Routledge, 2000.

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