On Algebraization of Classical First Order Logic (original) (raw)

Reconciling first-order logic to algebra

Starting from the algebraic method of theorem-proving based on the translation of logic formulas into polynomials over finite fields, and by adapting the case of first-order formulas by employing certain rings equipped with infinitary operations, this paper defines the notion of M-ring, a kind of polynomial ring defined for each first-order structure, by means of generators and relations. The notion of M-ring allows us to operate with some kind of infinitary version of Boolean sums and products, in this way expressing algebraically first-order logic with a new gist. We then show how this polynomial representation of first-order sentences could be seen as a legitimate algebraic semantics for first-order logic, alternative to cylindric and polyadic algebras an with a higher degree of naturalness. We briefly discuss how the method and their generalizations could be successfully lifted to n-valued logics and to other non-classical logics helping to reconcile some lost ties between algebra and logic.

An overview of generalized basic logic algebras

2003

Classical propositional logic is one of the earliest formal systems of logic, with its origins in the work of Boole and De Morgan. The algebraic semantics of this logic is given by Boolean algebras. Both, the logic and the algebraic semantics have been generalized in many directions over the last 150 years. In this talk we primarily take the algebraic point of view, but we will also use the powerful framework of algebraic logic to clarify the close relationship between algebra and logic. In various applications (such as fuzzy logic) the properties of Boolean conjunction are too stringent, hence a new binary connective •, usually called fusion, is introduced. In Boolean algebra the relationship between conjunction and implication is given by the residuation equivalences x ∧ y ≤ z ⇐⇒ x ≤ y → z ⇐⇒ y ≤ x → z. These equivalences imply many of the properties of ∧ and → (such as commutativity of ∧, distributivity of ∧ over ∨, left-distributivity of → over ∨ and right-distributivity of → over ∧) so one wishes to retain some aspects of these equivalences, but with conjunction replaced by fusion. To avoid imposing commutativity on the fusion operation one has to introduce two implications: x • y ≤ z ⇐⇒ x ≤ y → z ⇐⇒ y ≤ x z.

On the search for a finitizable algebraization of first order logic

Logic Journal of IGPL, 2000

We give an algebraic version of rst order logic without equality in which the class of representable algebras forms a nitely based equational class. Further, the representables are de ned in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of this result is Theorem 1 (a concrete version of which is given by Theorems 1.8 and 3.2), while its logical form is Corollary 4.2. For rst order logic with equality we give a result weaker than the one for rst order logic without equality. Namely, in this case | instead of nitely axiomatizing the corresponding class of all representable algebras | we nitely axiomatize only the equational theory of that class. See subsection 5.1, especially Remark 5.5 there. The proof of Theorem 1 is elaborated in sections 2 and 3. These sections contain theorems which are interesting of their own rights, too, e.g. Theorem 3.2 is a purely semigroup theoretic result. Cf. also \Further main results" in the Introduction.

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...

Polyadic and cylindric algebras of sentences

MLQ, 2006

In this note we give an interpretation of cylindric algebras as algebras of sentences (rather than formulas) of first order logic. We show that the isomorphism types of such algebras of sentences coincide with the class of neat reducts of cylindric algebras. Also we show how this interpretation sheds light on some recent results. This is done by likening Henkin's Neat Embedding Theorem to his celebrated completeness proof.

Curry systems for algebraisation of some non-classical logics

International Journal of Reasoning-based Intelligent Systems, 2011

With the appearance of several non-classical logics notably over the past decades, one natural question arises: What is the algebraic version of these logics? In this paper, we discuss some applications of the concept of Curry algebra for algebraisation of some paraconsistent, paracomplete and non-alethic logics. Such concept is also correlated with some fundamental themes in logic, such as computability, constructability, topology and many other basic branches.

Constructing some logical algebras from EQ-algebras

Filomat, 2021

EQ-algebras were introduced by Nov?ak in [16] as an algebraic structure of truth values for fuzzy type theory (FTT). Nov?k and De Baets in [18] introduced various kinds of EQ-algebras such as good, residuated, and lattice ordered EQ-algebras. In any logical algebraic structures, by using various kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, by means of fantastic filters of EQ-algebras we construct MV-algebras. Also, we study prelinear EQ-algebras and introduce a new kind of filter and named it prelinear filter. Then, we show that the quotient structure which is introduced by a prelinear filter is a distributive lattice-ordered EQ-algebras and under suitable conditions, is a De Morgan algebra, Stone algebra and Boolean algebra.

A Survey of Generalized Basic Logic Algebras

2009

Petr Hájek identified the logic BL, that was later shown to be the logic of continuous t-norms on the unit interval, and defined the corresponding algebraic models, BL-algebras, in the context of residuated lattices. The defining characteristics of BL-algebras are representability and divisibility. In this short note we survey recent developments in the study of divisible residuated lattices and attribute the inspiration for this investigation to Petr Hájek.

Some logical invariants of algebras and logical relations between algebras

St. Petersburg Mathematical Journal, 2008

Let Θ be an arbitrary variety of algebras and H an algebra in Θ. Along with algebraic geometry in Θ over the distinguished algebra H, a logical geometry in Θ over H is considered. This insight leads to a system of notions and stimulates a number of new problems. Some logical invariants of algebras H ∈ Θ are introduced and logical relations between different H 1 and H 2 in Θ are analyzed. The paper contains a brief review of ideas of logical geometry (§1), the necessary material from algebraic logic (§2), and a deeper introduction to the subject (§3). Also, a list of problems is given. 0.1. Introduction. The paper consists of three sections. A reader wishing to get a feeling of the subject and to understand the logic of the main ideas can confine himself to §1. A more advanced look at the topic of the paper is presented in § §2 and 3. In §1 we give a list of the main notions, formulate some results, and specify problems. Not all the notions used in §1 are well formalized and commonly known. In particular, we operate with algebraic logic, referring to §2 for precise definitions. However, §1 is self-contained from the viewpoint of ideas of universal algebraic geometry and logical geometry. Old and new notions from algebraic logic are collected in §2. Here we define the Halmos categories and multisorted Halmos algebras related to a variety Θ of algebras. §3 is a continuation of §1. Here we give necessary proofs and discuss problems. The main problem we are interested in is what are the algebras with the same geometrical logic. The theory described in the paper has deep ties with model theory, and some problems are of a model-theoretic nature. We emphasize once again that §1 gives a complete insight on the subject, while §2 and §3 describe and decode the material of §1. §1. Preliminaries. General view 1.1. Main idea. We fix an arbitrary variety Θ of algebras. Throughout the paper we consider algebras H in Θ. To each algebra H ∈ Θ one can attach an algebraic geometry (AG) in Θ over H and a logical geometry (LG) in Θ over H. In algebraic geometry we consider algebraic sets over H, while in logical geometry we consider logical (elementary) sets over H. These latter sets are related to the elementary logic, i.e., to the first order logic (FOL). Consideration of these sets gives grounds to geometries in an arbitrary variety of algebras. We distinguish algebraic and logical geometries in Θ. However, there is very 2000 Mathematics Subject Classification. Primary 03G25.