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Abstract algebraic logic - NMAL435

Title:	Abstraktní algebraická logika
Guaranteed by:	Department of Algebra (32-KA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2025 to 2025
Semester:	winter
E-Credits:	3
Hours per week, examination:	winter s.:2/0, Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	not taught
Language:	English
Teaching methods:	full-time

Guarantor:	Mgr. Adam Přenosil, Ph.D.
Class:	M Mgr. MSTR > Volitelné

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Annotation -

Advanced course in algebraic logic.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (14.01.2026)

Course completion requirements -

The students will be evaluated by regular homework assignments, or alternatively by a final exam.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (14.01.2026)

Literature -

J.M. Font. Abstract Algebraic Logic: An introductory textbook. College Publications, 2016.

P. Cintula and C. Noguera. Logic and Implication: An introduction to the general algebraic study of non-classical logics. Trends in Logic, vol. 57. Springer, 2021.

J. Czelakowski. Protoalgebraic Logics. Trends in Logic, vol. 10. Kluwer Academic Publishers, 2001.

W.J. Blok and D. Pigozzi. Algebraizable logics. Memoirs of the American Mathematical Society 396, vol. 77. American Mathematical Society, 1989.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (14.01.2026)

Syllabus -

Algebraic logic is the branch of mathematical logic that studies logical systems by giving them an algebraic semantics. Following the application of algebraic methods to particular logics, the field of so-called abstract algebraic logic (AAL) emerged with the aim of understanding the links between logic and algebra in a uniform way. The methodology of AAL has been successfully applied to many families of non-classical logics whose motivations range from mathematics and computer science to philosophy. The central notions of this framework are logical matrices (algebras equipped with a designated subset), the Leibniz operator (which assigns to each logical matrix a natural congruence), and the algebraic counterpart of a logic (which assigns to each logic a class of algebras). The course introduces the key notions of AAL, focusing on the so-called Leibniz hierarchy, which classifies logics accordingly how tightly they are linked to their algebraic counterparts.

Course content:

• Matrix semantics and algebraic semantics. Reduced models and the Leibniz operator. The algebraic counterpart of a logic.

• Protoalgebraic logics. Their characterization via the Leibniz operator and via reduced models. Equivalence formulas with parameters. Protoimplication sets. The correspondence property.

• Equivalential logics. Their characterization via the Leibniz operator and via reduced models. Equivalence formulas without parameters.

• Truth-equational logics. Their characterization via the Leibniz operator and via reduced models. Equational completeness theorems.

• Algebraizable logics. Equivalent algebraic semantics. Blok and Pigozzi’s Isomorphism Theorem.

• Bridge theorems. Deduction theorems and the proof by cases property.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (14.01.2026)

Course registration requirements -

Fundamentals of algebraic logic at the level of lecture NMAL434.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (15.05.2025)