Algebraic logic for the negation fragment of classical logic

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arXiv:2210.01709v2 [math.LO] 13 Mar 2023

Luciano J. Gonz´alez

CONICET (Argentina) – Universidad Nacional de La Pampa. Facultad de Ciencias Exactas y Naturales.

Santa Rosa, Argentina.

Abstract

The general aim of this article is to study the negation fragment of classical logic within

the framework of contemporary (Abstract) Algebraic Logic. More precisely, we shall ﬁnd

the three classes of algebras that are canonically associated with a logic in Algebraic Logic,

that is, we ﬁnd the classes Alg∗, Alg and the intrinsic variety of the negation fragment of

classical logic. In order to achieve this, ﬁrstly we propose a Hilbert-style axiomatization for

this fragment. Then, we characterize the reduced matrix models and the full generalized

matrix models of this logic. Also, we classify the negation fragment in the Leibniz and Frege

hierarchies.

Keywords: Classical logic; Classical negation; Algebraic logic; Reduced matrix models;

Full models

1. Introduction

It is clear that the negation fragment of classical propositional logic (from now on CPL)

is a very inexpressive logic from the syntactic point of view. One can only consider up to

equivalence a proposition pand its negation ¬p, and no other compound sentential can be

built up. However, this fragment has several algebra-based semantics (in the diﬀerent senses

of this term) that are not so trivial. We intend to describe these algebra-based semantics of

the negation fragment of CPL from the point of view of algebraic logic.

In the framework of (abstract) algebraic logic, there are essentially three classes of alge-

bras associated with a propositional logic. These classes are obtained from diﬀerent proce-

dures, which intend to be a kind of generalization or abstraction of the Tarski-Lindenbaum

method applied to classical (intuitionistic) propositional logic. The class Alg∗(S) is the class

of algebras that is canonically associated with a logic Saccording to the theory of logical

matrices. More precisely, Alg∗(S) is the class of the algebraic reducts of the reduced matrix

Email address: lucianogonzalez@exactas.unlpam.edu.ar (Luciano J. Gonz´alez)

models of S. The intrinsic variety of a logic S(denoted by V(S)) is deﬁned as the variety

generated by a quotient algebra on the formula algebra. And the class Alg(S) is determined

from the reduced generalized matrix models, that is, Alg(S) is the class of algebraic reducts

of the reduced g-models of S. We refer the reader to [4,6,7] for the speciﬁc deﬁnitions. In

general, for any logic S, we always have that Alg∗(S)⊆Alg(S)⊆V(S). In many cases, the

three classes coincide, and in many others cases the inclusions are proper. From the point

of view of algebraic logic, the class of algebras which is more representative for a logic Sor

which can be considered as the algebraic counterpart of Sis the class Alg(S). We refer the

reader to [4,6,7] for a deeper explanation of the aims and goals of algebraic logic and the

corresponding algebra-based semantics. We intend to obtain the classes Alg∗, the intrinsic

variety and the class Alg for the negation fragment of CPL.

The article is organized as follows. Section 2is about the signiﬁcance of the negation

fragment of CPL, and we try to justify why it is important to study algebraically this

fragment. In Section 3, we introduce the very basics to start with the work. Throughout

the paper, we will introduce the needed concepts and results of algebraic logic. We assume

that the reader is familiar with the very basics of algebraic logic, for instance, with the

notions of propositional logic (or sentential logic), logical ﬁlter, a theory of a logic, logical

matrices, generalized matrices, etc. We shall provide all those notions that might be less

usual for the reader. We refer the reader to [4,6] for further information on algebraic logic.

Section 4presents a Hilbert-style system for the negation fragment of CPL and shows a

completeness theorem. In Section 5, we characterize the Leibniz-reduced matrix models and

we describe the class Alg∗of the negation fragment. Section 6is devoted to obtaining the

intrinsic variety of the negation fragment. In Section 7, we characterize the full g-models of

the negation fragment. In order to obtain this, we describe the logical ﬁlters generated by a

set on an arbitrary algebra and present several properties of these logical ﬁlters. Then, we

ﬁnd the class Alg of the negation fragment. Finally, in Section 8, we classify the negation

fragment in the Leibniz and Frege hierarchies.

2. Signiﬁcance

As noted in the introduction, the negation fragment of CPL is a very simple logic since

its algebraic language has only one unary connective. In this section, we want to justify

and convince the reader about the importance of having an algebraic description of this

fragment. So, we ask ourselves, what is the algebraic counterpart, under the Algebraic Logic

([7,6,4]) point of view, of the negation fragment of CPL? This question was addressed in

the literature for the others fragments of CPL, see Table 1. It is also important to answer

the above question for the negation fragment beyond its simplicity.

In order to study propositional logics is often important to have an axiomatization,

for instance, a Hilbert-style or Gentzen-style axiomatization. By [13] it is known that

the negation fragment of CPL has a ﬁnite Hilbert-style axiomatization. But there it isn’t

explicitly presented. However, it is known from the folklore that the Hilbert-style rules

x, ¬x⊢y,x⊢ ¬¬xand ¬¬x⊢xare an axiomatization of the negation fragment of CPL.

As far as we know, there isn’t in the literature an argumentation of this. Here we present

one.

As mentioned in the introduction, for the negation fragment of CPL, we described the

three classes of algebras that are naturally associated with a propositional logic in Algebraic

Logic. In spite of the syntactical simplicity of the negation fragment of CPL, we show that

these three classes of algebras are diﬀerent and they are not so simple. In particular, we

characterize the class Alg for this fragment. We notice that this class of algebras was recently

obtained in [11] using the concept of Suszko-reduced matrix. In this paper, we follow an

alternative path to describe the class Alg. We use the Tarski-reduced full g-models. On

the one hand, a logical matrix is a pair hA, F iwhere Ais an algebra (over a corresponding

algebraic language) and F⊆A. On the other hand, a generalized matrix (g-matrix) is a

pair hA, Ci where Ais an algebra and Cis a closure system on A. Both structures serve to

establish algebra-based semantics for propositional logics (see Section 5and 7). These two

algebra-based semantics have their diﬀerences, and regarding the more general diﬀerence

between them, we want to quote Font and Jansana (quoted from [6]):

“Since an abstract logic can be viewed as a“bundle” or family of matrices, one

might think that the new models are essentially equivalent to the old ones; but

we believe, after an overall appreciation of the work done in this area, that it is

precisely the treatment of an abstract logic as a single object what gives rise to a

useful–and beautiful–mathematical theory, able to explain the connections, not

only at the logical level but at the metalogical level, between a sentential logic

and the particular class of models we associate with it, namely the class of its

full models.”

Moreover, to justify why characterize the Tarski congruence and the Tarski-reduced full g-

models rather than the Suszko congruence and the Suszko-reduced models, respectively, we

quote Font et al. (quoted from [7, p. 73]):

“The Suszko congruences appear as particular cases of the Tarski congruence

[. . . ].”

In this article, we obtain a description of the full g-models for the negation fragment

of CPL, showing that they are not simple at all. In order to convince the reader why to

characterize the full g-models of the negation fragment, which seem to be more complex

than the negation fragment itself, we quote Font and Jansana in [6, p. 3]:

“We associate with each sentential logic Sa class of abstract logics called the full

models of S[. . . ] with the conviction that (some of) the interesting metalogical

properties of the sentential logic are precisely those shared by its full models.

[. . . ] And we claim that the notion of full model is a “right” notion of model of

a sentential logic [. . . ].”

Beyond the scope of this article, we want to mention that there is also a great interest

in studying negation from the philosophical, linguistics, artiﬁcial intelligence and logic pro-

gramming point of view. We refer the reader to [10] where there is a compendium of articles

studying negation from diﬀerent perspectives addressed to the question: What is negation?

For instance, in [2] Dunn discusses several properties that a negation can have: ϕ⊢ψonly

if ¬ψ⊢ ¬ϕ(contraposition);ϕ⊢ −¬ψ(Galois double negation); ϕ⊢ ¬¬ϕ(constructive

double negation); ϕ⊢ ¬ψonly if ψ⊢ ¬ϕ(constructive contraposition); ϕ⊢ψand ϕ⊢ ¬ψ

only if ϕ⊢χ(absurdity); ¬¬ϕ⊢ϕ(classical double negation); ¬ϕ⊢ψonly if ¬ψ⊢ϕ

(classical contraposition). These properties are considered in diﬀerent contexts. In [2] Dunn

studies several connections between diﬀerent treatments of the semantics of negation in

non-classical logics: the Kripke deﬁnition of negation for intuitionistic logic, the Goldblatt’s

semantics for negation in orthologic, the deﬁnition of De Morgan negation in relevant logic,

the four-valued semantics of De Morgan negation, and the star semantics. Dunn provides

a detailed correspondence-theoretic classiﬁcation of various notions of negation in terms of

properties of a binary relation interpreted as incompatibility.

3. The {¬}-fragment of classical propositional logic (CPL)

Throughout what follows, we establish the following simple conventions. Given a function

f:X→Yand A∪ {x} ⊆ X, we denote fx instead of f(x) and fA ={f a :a∈A}.

Let L={¬} be an algebraic language of type (1) and let Fm be the algebra of formulas

over the language Land generated by a countably inﬁnite set Var. Unless otherwise stated,

all the algebras considered in the paper are deﬁned over the algebraic language L. Let

us denote by SN=hFm,⊢Nithe {¬}-fragment of CPL, where ⊢Nis the corresponding

consequence relation. Let 2¬=h{0,1},¬i be the {¬}-reduct of the two-element Boolean

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arXiv:2210.01709v2[math.LO]13Mar2023AlgebraiclogicforthenegationfragmentofclassicallogicLucianoJ.Gonz´alezCONICET(Argentina)–UniversidadNacionaldeLaPampa.FacultaddeCienciasExactasyNaturales.SantaRosa,Argentina.AbstractThegeneralaimofthisarticleistostudythenegationfragmentofclassicallogicwithinthefra...

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Algebraic logic for the negation fragment of classical logic

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