What is logic? What is a logic? (Two different questions, cf. [1],[2]
for answers.) Why algebraize? Algebraization of logics as a method for
putting all kinds of logics into a unifying, mathematically streamlined
perspective [3],[4]. Duality theory between the world of logics and the
world of (classes of) algebras. Characterization of domains of validity
of fundamental theorems of logic, like interpolation (Craig),
definability (Beth), deduction, compactness,..., undecidability. The
approach of Tarski’s school (still active, presently expanding) to these
and related issues [5],[6]. Algebraizing the semantic/model theoretic
aspects leads to algebras of sets of sequences, i.e. of relations. Hence
a central unifying tool for all the above is provided by theories of
algebras of relations of various ranks: cylindric algebras, polyadic
algebras of relations and their variants. Relativization is used as a
tool for turning negative results to positive [7],[8]. This leads to
positive results for the guarded fragment and bounded fragment of FOL as
well as for logics of the dynamic (or arrow) trend, for relational
semantics in general, and for making the finite variable hierarchy of
FOL behave well. Definitional equivalence of many-sorted first-order
theories as a step toward defining equivalence of logics [9]. Broadening
the scope of applicability of logic: logic foundation of spacetime
theories (two-way connections). Cosmologic. The G¨odel-Einstein
collaboration. The unity of Tarski’s approaches to (i) logic, (ii)
universal algebra, (iii) algebraic logic, and (iv) geometry
(space-time). Convergence of major schools: Tarski, Quine, van Benthem &
Goldblatt.
References:
[1] Andreka-Nemeti-Sain: Algebraic Logic, In: Handbook of
Philosophical Logic Vol.2, Kluwer 2002.
[2] Nemeti-Andr´eka: General Algebraic Logic: a perspective on What
is logic, In: What is a logical system, Clarendron, 1994.
[3] Andreka: Universal Algebraic Logic, PhD Diss., 1977.
[4] Andreka-Gergely-Nemeti: Universal algebraic construction of
logics, Studia Logica 1977.
[5] Henkin-Monk-Tarski: Cylindric Algebras, Chap.5.3, North-Holland
1981, 1985.
[6] Andreka-Madarasz-Nemeti: Algebraic Logic, J. Rel. Methods in
Comp.Sci. 2004.
[7] Simon: Nonrepresentable algebras of relations, PhD Diss., 1977.
[8] Andreka-van Benthem-Nemeti: Modal languages and bounded
fragments of predicate logic, J. Phil. Logic 1998.
[9] Madarasz: Logic and Relativity, section 4.3, 2002. Many of the
above works are available
here