Session 1: The paradigm of algebraizability (from Boole to Blok and Pigozzi, through Lindenbaum and Tarski). Algebraic semantics and equivalent algebraic semantics. What is a logic? What does it mean to algebraize it? Kinds of algebraizability. Can one show that some logic is not algebraizable? What does one get from kowing that some logic is algebraizable? Bridge theorems.

Session 2: Not every logic is algebraizable, so what? Matrix semantics and the protoalgebraic hierarchy. Bridge theorems. Not every logic is protoalgebraic, so what? Generalized matrices, atlases, and their dual view. Full models. Transfer properties and bridge theorems.

Session 3: Algebraization of Gentzen systems. Algebraizability of substructural logics. Models of structural Gentzen systems and full adequacy. Logification of certain varieties through sentential logics and through Gentzen systems. Extended notions of regularity.

References

1) Blok, W. and Pigozzi, D. "Abstract Algebraic Logic and the Deduction Theorem", Bulletin of Symbolic Logic, to appear.
2) Czelakowski, J. "Protoalgebraic logics", Trends in Logic - Studia Logica Library, vol. 10 (Kluwer, Dordrecht, 2001).
3) Font, J. M. "Generalized matrices in abstract algebraic logic", in: V. F. Hendricks and J. Malinowski, eds. "Trends in Logic. 50 years of Studia Logica", Trends in Logic - Studia Logica Library, vol. 21 (Kluwer, Dordrecht, 2003), 57-86.
4) Font, J. M. and Jansana, R. "A General Algebraic Semantics for Sentential Logics", Lecture Notes in Logic, vol. 7 (Springer-Verlag, 1996). Presently distributed by the Association of Symbolic Logic.
5) Font, J. M., Jansana, R. and Pigozzi, D. "A Survey of Abstract Algebraic Logic", Studia Logica 74, 1/2 (2003) 13-97.
6) Wójcicki, R. "Theory of logical calculi", Synthese Library, vol. 199 (Reidel, Dordrecht, 1988).
 

Lecture notes: A and B

Abstract algebraic logic (AAL) deals with the algebraic study of logics in a way that views both terms "algebraic" and "logic" under a general perspective. It studies different abstract processes of assigning a class of algebras to a given logic, and the general properties of these processes, their limitations and their different variants or enhancements, instead of focusing on the resulting class of algebras for a particular logic. AAL crystalized in the eighties and nineties as an evolution from the so-called "Polish"-style general approach to sentential logics with algebraic means, after the fundamental contributions of researchers such as Czelakowski, Blok, Pigozzi, Herrmann, Font, Jansana and Verdú.

The clarification of the different ways in which a logic can be algebraized has thrown light on the behaviour of several logical systems that are not amenable to the standard settings, but it has also clarified the deep roots of the algebraic properties of classical logic. In its recent developments AAL has proven useful for the study of non-monotonic and substructural logics, and of other Gentzen-style systems formulated with hypersequents or many-sided sequents. It does also provide definite criteria to clarify in which sense a logic is said to correspond to a given class of algebras, and it has stimulated a reflection on the issue of what a logic is.

This tutorial is conceived as a reading guide of the available scholarly material. AAL is a mathematical theory, and by now it is deep and technically involved, with several ramifications and its own terminology and notation. A 3-session tutorial can only show the essentials, that is, the "what" and the "why", but not much of the "how". An effort will be made to provide enough examples of logics and of classes of algebras and of their treatment under several of the AAL paradigms.