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Finitary Algebraic Logic

Roger D. Maddux

Mathematical Logic Quarterly 35 (4):321-332 (1989)

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Algebraization of quantifier logics, an introductory overview.István Németi - 1991 - Studia Logica 50 (3):485 - 569.details This paper is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics. The paper has a survey character, too. The most frequently used algebras like cylindric-, relation-, polyadic-, and quasi-polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively comprehensive. In some (...) Download Export citation Bookmark 44 citations
Algebraic Logic, Where Does It Stand Today?Tarek Sayed Ahmed - 2005 - Bulletin of Symbolic Logic 11 (3):465-516.details This is a survey article on algebraic logic. It gives a historical background leading up to a modern perspective. Central problems in algebraic logic (like the representation problem) are discussed in connection to other branches of logic, like modal logic, proof theory, model-theoretic forcing, finite combinatorics, and Gödel’s incompleteness results. We focus on cylindric algebras. Relation algebras and polyadic algebras are mostly covered only insofar as they relate to cylindric algebras, and even there we have not told the whole story. (...) Download Export citation Bookmark 22 citations
On notions of representability for cylindric‐polyadic algebras, and a solution to the finitizability problem for quantifier logics with equality.Tarek Sayed Ahmed - 2015 - Mathematical Logic Quarterly 61 (6):418-477.details We consider countable so‐called rich subsemigroups of ; each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, if and only if is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in polyadic equality set (...) Download Export citation Bookmark 7 citations
Weakly associative relation algebras with projections.Agi Kurucz - 2009 - Mathematical Logic Quarterly 55 (2):138-153.details Built on the foundations laid by Peirce, Schröder, and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong first‐order theories like ZFC, and so their equational theory is undecidable. The less expressive class WA of weakly associative relation algebras was introduced by Maddux [7]. Németi [16] showed that WA's have a decidable universal theory. There has been extensive research on (...) Download Export citation Bookmark 3 citations
(1 other version)Complexity of equational theory of relational algebras with projection elements.Szabolcs Mikulás, Ildikó Sain & Andras Simon - 1992 - Bulletin of the Section of Logic 21 (3):103-111.details The class \ of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of \ nor the first order theory of \ are decidable. Moreover, we show that the set of all equations valid in \ is exactly on the \ level. We consider the class \ of the relation algebra reducts of \ ’s, (...) Download Export citation Bookmark 5 citations
Finite methods in 1-order formalisms.L. Gordeev - 2001 - Annals of Pure and Applied Logic 113 (1-3):121-151.details Familiar proof theoretical and especially automated deduction methods sometimes accept infinity where, in fact, it can be omitted. Our first example deals with the infinite supply of individual variables admitted in 1-order deductions, the second one deals with infinite-branching rules in sequent calculi with number-theoretical induction. The contents of Section 1 summarize and extend basic ideas and results published elsewhere, whereas basic ideas and results of Section 2 are exposed for the first time in the present paper. We consider classical (...) Download Export citation Bookmark
Non-finite-axiomatizability results in algebraic logic.Balázs Biró - 1992 - Journal of Symbolic Logic 57 (3):832 - 843.details Download Export citation Bookmark 15 citations
Equational Reasoning in Non-Classical Logics.Marcelo Frias & Ewa Orlowska - 1998 - Journal of Applied Non-Classical Logics 8 (1-2):27-66.details ABSTRACT In this paper it is shown that a broad class of propositional logics can be interpreted in an equational logic based on fork algebras. This interpetability enables us to develop a fork-algebraic formalization of these logics and, as a consequence, to simulate non-classical means of reasoning with equational theories algebras. Download Export citation Bookmark
Complexity of equations valid in algebras of relations part I: Strong non-finitizability.Hajnal Andréka - 1997 - Annals of Pure and Applied Logic 89 (2):149-209.details We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: (...) Download Export citation Bookmark 23 citations

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