On the search for a finitizable algebraization of first order logic (original) (raw)

On the equational theory of representable polyadic equality algebras (extended abstract)

Logic Journal of IGPL, 1998

Among others we will see that the equational theory of ω dimensional representable polyadic equality algebras (RP EAω's) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable (and hence the equational theory of this class is finite schemaaxiomatizable, as well). We will also see that the complexity of the equational theory of RP EAω is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildikó Sain and Viktor Gyuris [10], the following methodological conclusions will be drawn: the negative properties of polyadic (equality) algebras can be removed by switching from what we call the "polyadic algebraic paradigm" to the "cylindric algebraic paradigm". 1

Löwenheim–Skolem theorems for non-classical first-order algebraizable logics: Table 1

Logic Journal of IGPL, 2016

This paper is a contribution to the model theory of non-classical first-order predicate logics. In a wide framework of first-order systems based on algebraizable logics, we study several notions of homomorphisms between models and find suitable definitions of elementary homomorphism, elementary substructure and elementary equivalence. Then we obtain (downward and upward) Löwenheim-Skolem theorems for these non-classical logics, by direct proofs and by describing their models as classical 2-sorted models.

Finite schematizable algebraic logic

Logic Journal of IGPL, 1997

In this work, we attempt to alleviate three (more or less) equivalent negative results. These are (i) non-axiomatizability (by any finite schema) of the valid formula schemas of first order logic, (ii) non-axiomatizability (by finite schema) of any propositional logic equivalent with classical first order logic (i.e., modal logic of quantification and substitution), and (iii) non-axiomatizability (by finite schema) of the class of representable cylindric algebras (i.e., of the algebraic counterpart of first order logic). Here we present two finite schema axiomatizable classes of algebras that contain, as a reduct, the class of representable quasi-polyadic algebras and the class of representable cylindric algebras, respectively. We establish positive results in the direction of finitary algebraization of first order logic without equality as well as that with equality. Finally, we will indicate how these constructions can be applied to turn negative results (i), (ii) above to positive ones. 1

Non-finite-axiomatizability results in algebraic logic

Journal of Symbolic Logic, 1992

This paper deals with relation, cylindric and polyadic equality algebras. First of all it addresses a problem of B. Jónsson. He asked whether relation set algebras can be expanded by finitely many new operations in a “reasonable” way so that the class of these expansions would possess a finite equational base. The present paper gives a negative answer to this problem: Our main theorem states that whenever Rs+ is a class that consists of expansions of relation set algebras such that each operation of Rs+ is logical in Jónsson's sense, i.e., is the algebraic counterpart of some (derived) connective of first-order logic, then the equational theory of Rs+ has no finite axiom systems. Similar results are stated for the other classes mentioned above. As a corollary to this theorem we can solve a problem of Tarski and Givant [87], Namely, we claim that the valid formulas of certain languages cannot be axiomatized by a finite set of logical axiom schemes. At the same time we give a nega...

Gentzen-style axiomatizations in equational logic

Algebra Universalis, 1995

The notion of a Gentzen-style axiomatization of equational theories is presented. In the standard deductive systems for equational logic axioms take the form of equations and the inference rules can be viewed as quasi-equations. In the deductive systems for quasi-equational logic the axioms, which are quasi-equations, can be viewed as sequents and the inference rules as Gentzen-style rules. It is conjectured that every finite algebra of finite type has a finite Gentzen-style axiomatization for its quasi-identities. We verify this conjecture for a class of algebras that includes all finite algebras without nontrivial proper subalgebras, and all finite simple algebras that are embeddable into the free algebra of their variety.

Reducts of Relation Algebras: The Aspects of Axiomatisability and Finite Representability

Logical Foundations of Computer Science, 2021

In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson [11]. We also show that the class of representable join semilattice-ordered semigroups has a recursively enumerable axiomatisation using back-andforth games.

On Algebraization of Classical First Order Logic

2014

Algebraization of first order logic and its deduction are introduced according to Halmos approach. Application to functional polyadic algebra is done. Index Term: Polyadic algebra, Polyadic ideal, Polyadic filter, Functional polyadic algebra.

A de nability theorem for rst order logic

1997

In this paper we will present a de nability theorem for rst order logic This theorem is very easy to state and its proof only uses elementary tools To explain the theorem let us rst observe that if M is a model of a theory T in a language L then clearly any de nable subset S M i e a subset S fa j M j a g de ned by some formula is invariant under all automorphisms of M The same is of course true for subsets of M n de ned by formulas with n free variables Our theorem states that if one allows Boolean valued models the converse holds More precisely for any theory T we will construct a Boolean valued model M in which precisely the T provable formulas hold and in which every Boolean valued subset which is invariant under all automorphisms of M is de nable by a formula of L Our presentation is entirely selfcontained and only requires familiarity with the most elementary properties of model theory In particular we have added a rst section in which we review the basic de nitions concerning ...

An Attempt to Treat Unitarily the Algebras of Logic. New Algebras 1

2007

Since all the algebras connected to logic have, more or less explicitely, an associated order relation, it follows that they have two presentations, dual to each other. We classify these dual presentations in "left" and "right" ones and we consider that, when dealing with several algebras in the same research, it is useful to present them unitarily, either as "left" algebras or as "right" algebras. In some circumstances, this choice is essential, for instance if we want to build the ordinal sum (product) between a BL algebra and an MV algebra. We have chosen the "left" presentation and several algebras of logic have been redefined as particular cases of BCK algebras. We introduce several new properties of algebras of logic, besides those usually existing in the literature, which generate a more refined classification, depending on the pro- perties satisfied. In this work (Parts I-V) we make an exhaustive study of these algebras - wit...