1 Introduction
In computer science, it is well known the successful application of logics as a foundation of programming languages, that is, programs can be characterized as proofs in logical inference systems (Curry-Howard isomorphism) [11]. In development Artificial Intelligence (AI), logical languages have also played a key role: in the burgeoning of reasoning systems and even as a tool for proving algorithm correctness. There are even cases in which some logical theories have served for the advanced programming paradigms, such as logic programming [25].
Paraconsistent logics form a family of languages designed to analyze and reason from inconsistencies (from the point of view of classical logic), as is often useful in many AI contexts, such as signal and image processing and expert systems [26]. Within the family of paraconsistent logics, Annotated logics, which encompass fuzzy set theory, are the most widely applied in AI [1]. Another scope of paraconsistent logic is non-monotonic reasoning, a fundamental notion in the development of intelligent systems.
In [2, 3], it is introduced a standard semantics for non-monotonic reasoning in the setting of annotated logics and annotated logic programs. Multi-valued logics are non-classical logics [5]. Like in logic classical, multi-valued logics also enjoy the principle of truth functionality: the truth value of a compound sentence is determined through the truth values of its component sentences and remains the same when one of the component sentences is replacing by another sentence with the same truth value. However, in contrast to the classical case, multi-valued logics do not restrict the number of truth values to just two. A larger set of degrees of truth is the distinctive feature in the context of many-valued logics.
In [15], it is reported a detailed summary of multi-valued logics. Some multi-valued logic systems are presented as families of systems of uniformly defined finite and infinite values, for example, Łukasiewicz logic, Gödel’s logic, systems based on the t-norm, 3-valued systems, Dunn-Belnap’s 4-value system. Most common inference systems for multi-valued logics are Hilbert and Gentzen (sequent) calculus, and Tableaux [15]. A broad class of infinitely valued logics is described by [20].
Classical logic, as well as intuitionistic logic, suffer a disadvantage when reasoning with inconsistent information. According to the principle of explosion, also known as “ex contradictione sequitur quodlibet”, all theory or inconsistent knowledge base is trivial. Classical logic is then useless to reason with inconsistencies. As a result, alternatives to classical logic that do not have this drawback have been developed, called “paraconsistent” approaches. In 1954 F. Asenjo, in his doctoral dissertation, proposes for the first time to use multi-valued logic as a form of paraconsistent logic (logics whose logical consequence relationship semantics or proof theory is not explosive [16]). The focus of many truth values is to abandon the classical assumption and allow more than two values. The most common strategy is to use three truth values: true, false, and both (true and false) for evaluation of formulas.
George Boole introduced the algebra of logic or algebraic logic in [7] as an explicit algebraic system showing the underlying mathematical structure of logic. The methodology started by Boole was continued in the 19th century for the work of A. De Morgan, W. S. Jevons, C. S. Peirce, and E. Schröder. A summary of these works can be found in [8]. The relationship between logic and algebra from the contemporary perspective goes back to the ideas of Lindenbaum and Tarski, as follows: formulas of a given logic are interpreted in algebras with operations associated with the logical connectives. In [6], Blok and Pigozzi proposed a generalization of the techniques of original algebra to encompass a broader range of logics. Generalization of the Blok and Pigozzi method was suggested in the literature [12, 13, 14]. Algebraic foundations for logic have been shown useful in the development of reasoning systems [10, 17]. In this paper, we show that CG′3 and G′3 are Blok-Pigozzi algebrizable. We believe this result may help in the development of paraconsistent reasoning systems.
This article is organized as follows: in Section 2, we present some known definitions and results according to the setting of the present manuscript; in Section 3, we study the CG′3 and G′3 logics which are defined in terms of four connectives ∧, ∨, → and ¬ where the implication is deductive to CG′3. The main result of the paper is also described in this Section, that is, it is shown that CG′3 and G′3 are algebraizable logics with the Blok-Pigozzi method. Finally, in the last Section, we give a summary of the paper and we describe a list of open problems to be studied in the future.
2 Background
We first introduce the syntax of the logical formulas considered in this paper. We follow standard notation and basic definitions as W. Carnielli and M. Coniglio in [9].
Definition 1 (Propositional signatures). A proposition signature is a set Θ of symbols called connectives, together with the information concerning the arity of each connective.
The following symbols will be used for logical connectives: ∧ (conjunction, binary); (disjunction, binary); → (implication, binary); ¬ (weak negation, unary); • (inconsistency operator, unary); ∼ (strong negation, unary); ⊥ (bottom formula, 0-ary).
Definition 2 (Propositional language). Let Var={p1,p2,…} be a denumerable set of propositional variables, and let Θ be any propositional signature. The propositional language generated by Θ from Var will be denoted by ℒΘ.
Definition 3 (Standard logic). A logic L defined over a language L which has a consequence relation ⊢, is Tarskian if it satisfies the following properties, for every Γ∪Δ∪{α}⊆ℒ:
(i) If α∈Γ then Γ⊢α;
(ii) If Γ⊢α and Γ⊆Δ then Δ⊢α;
(iii) If Δ⊢α and Γ⊢β for every β∈Δ, then Γ⊢α.
A logic satisfying item (ii) above is called monotonic. A logic L is said to be finitary if it satisfies the following:
(iv) If Γ⊢α, then there exists a finite subset Γ0 of Γ such that Γ0⊢α.
A logic L defined over a propositional language ℒ generated by a signature from a set of propositional variables is namely structural, if it satisfies the following property:
(v) Is Γ⊢α then σ[Γ]⊢σ[α], for every substitution σ of formulas for variables.
Propositional logic is standard if it is Tarskian, finitary, and structural.
From now on, a logic L will be represented by a pair L=〈ℒ,⊢〉, where ℒ and ⊢ denote the language and the consequence relation of ℒ, respectively. ℒ is generated by a propositional signature Θ from Var, this is, ℒ=ℒΘ then we will write L=〈Θ,⊢〉.
Let L=〈ℒ,⊢〉 be a logic. Let α be a formula in ℒ and let X1…Xn be a finite sequence (for n≥1) such that each Xi is either a set for formulas in ℒ or formula in ℒ. Then, as usual, X1,…,Xn⊢α will stand for X′1∪⋯∪X′n⊢α, where, for each i, X′i is Xi, if Xi is a set of formulas, or X′i is {Xi} if Xi is a formula.
Definition 4 (Paraconsistent logic). A Tarskian logic L is paraconsistent if it has a (primitive or defined) negation ¬ such that α, ¬α⊬Lβ for some formulas α and β in the language of L.
Remark 1. If L has a deductive implication →, in the sense that it satisfies the Deduction meta-theorem DMT, then L is paraconsistent if and only if the schema formula φ→(¬φ→ψ) is not valid, i.e., the explosion law is not valid in L with respect to the negation ¬. That is, the negation ¬ is not explosive.
Now, we present the notion of Logic of Formal Inconsistency.
Definition 5 (Logic of Formal Inconsistency). Let L=〈Θ,⊢〉 be a standard logic. Assume that the signature Θ of L contains a negation ¬, and let ○(p) be a nonempty set of formulas depending exactly on the propositional variable p. Accordingly, L is a Logic of Formal Inconsistency, (LFI), with respect to ¬ and ○(p) if the following holds:
(i) φ, ¬φ⊬ψ for some φ and ψ;
(ii) There are two formulas α and β such that:
(a) ○(α), α⊬β;
(b) ○(α), ¬α⊬β;
(iii) ○(φ), φ, ¬φ⊢ψ for every φ and ψ.
Remark 2.
— When ○ is a singleton, its elements are denoted by ∘p, where ∘ is the consistency operator.
— A logic that satisfies the property (iii) is called gently explosive.
Finally, we define a stronger notion of LFIs for more reference, see [9].
Definition 6 (Strong Logic of Formal Inconsistency). Let L=〈Θ,⊢〉 be a standard logic. Assume that the signature Θ of L contains a negation ¬, and let ○(p) be a nonempty set of formulas depending exactly on the propositional variable p. Then L is a strong LFI with respect to ¬ and ○(p) if the following holds:
-
(i) there are two formulas α and β such that:
(a) α, ¬α⊬β;
(b) ○(α), α⊬β;
(c) ○(α), ¬α⊬β; and
(ii) ○(φ), φ, ¬φ⊢ψ for every φ and ψ.
Remark 3.
Definition 7 (Blok and Pigozzi algebraizability). Let Θ be a propositional signature, and let L be a standard propositional logic defined over the language ℒΘ, with a consequence relation ⊢ℒ. Then L is algebraizable in the sense of Blok and Pigozzi if there exists a nonempty set Δ(p1,p2)⊆ℒΘ of formulas depending on variables p1 and p2, and a nonempty set E(p1)⊆ℒΘ×ℒΘ of pairs of formulas depending on variable p1 satisfying the following properties:
(i) ⊢ℒδ(p1,p1), for every δ(p1,p2)∈Δ(p1,p2);
(ii) Δ(p1,p2)⊢ℒδ(p2,p1), for every δ(p1,p2)∈Δ(p1,p2);
(iii) Δ(p1,p2), Δ(p2,p3)⊢ℒδ(p1,p3), for every δ(p1,p2)∈Δ(p1,p2);
(iv) Δ(p1,pn+1),…,Δ(pn,p2n)⊢ℒδ(#(p1,…,pn),#(pn+1,…,p2n)), for every δ(p1,p2)∈Δ(p1,p2), every n-ary connective # of Θ and every n≥1;
(v) p1⊢ℒδ(γ(p1),ϵ(p1)), for every δ(p1,p2)∈Δ(p1,p2) and every 〈γ(p1),ϵ(p1)〉∈E(p1);
(vi) {δ(γ(p1),ϵ(p1)):δ(p1,p2)∈Δ(p1,p2),〈γ(p1),ϵ(p1)〉∈E(p1)}⊢Lp1.
The sets Δ(p1,p2) and E(p1) are called systems of equivalence formulas and defining equations, respectively.
Definition 8 (Relation). Let Θ be a propositional signature, and let θ⊆ℒΘ×ℒΘ be a relation defined over the algebra of formulas ℒΘ if it satisfies the following properties:
(i) αθα for every α∈ℒΘ (reflexivity);
(ii) αθβ implies βθα for every α,β∈ℒΘ (symmetry);
(iii) αθβ and βθγ implies αθγ for every α,βγ∈ℒΘ (transitivity);
(iv) Given αi and βi in ℒΘ (for 1≤i≤n) such that α1θβ1,…,αnθβn, then #(α1,…,αn)θ#(β1,…,βn) for every n-ary connective # of Θ and every n≥1.
A congruence θ in ℒΘ is trivial if either θ=ℒΘ×ℒΘ or θ={(α,α):α∈ℒΘ}.
Definition 9 (Logical congruence). Let L be a standard logic defined over the language ℒΘ.
αθβ and Γ⊢Lα implies that Γ⊢Lβ.
αθβ implies that α⊢Lβ and β⊢Lα.
The usual mode to define the many-valued semantics of logic is through a matrix. We introduce the definition of the deterministic matrix, also known as the logical matrix or just as a matrix. In [19], we can find and an exhaustive discussion about many-valued logic and some examples.
Definition 10 (Matrix). Given a logic L in the language ℒ, the matrix of L is a structure M=〈D,D*,F〉M = hD, D∗, F i, where:
(i) D is a non-empty set of truth values (domain),
(ii) D* is a subset of D (set of designated values),
(iii) F={fc|c∈C} is a set of truth functions, with one function for each logical connective c of ℒ.
Definition 11 (Interpretation). Given a logic L in the language ℒ, an interpretation t, is a function t:Var→D that maps propositional variables to elements in the domain.
Any interpretation t can extend to a function on all formulas in ℒΣ as usual, i.e., applying recursively the truth functions of logical connectives in F. If t is a valuation in the logic L, we will say that t is an L-valuation. Interpretations allow us to define the notion of validity in this type of semantics as follows:
Definition 12 (Valid formula). Given a formula φ and an interpretation t in a logic L, we say that the formula φ is valid under t in L, if t(φ)∈D*, and we denote it as t⊨Lφ.
Let us note that validity depends on the interpretation, but if we want to talk about “logical truths” in the system, then the validity should be absolute, as stated in the following definition:
Definition 13 (Tautology). Given a formula φ in the language of a logic L, we say φ is a tautology in L, if for every possible interpretation, the formula φ is valid, and we denote it as ⊨Lφ.
If φ is a tautology in the logic L, we say that φ is an L-tautology. When logic defined via a many-valued semantics, it is common to define the set of theorems of L as the set of tautologies obtained from the many-valued semantics, i.e., φ∈L if and only if ⊨Lφ.
Definition 14 (Translation between Logics). Let L1 and L2 be logics with sets of formulas ℒ1 and ℒ2, respectively. A mapping ∗:ℒ1→ℒ2 is said to be a translation from L1 to L2 if, for every Γ∪{α}⊆ℒ1:
Γ⊨L1α then Γ*⊨L1α*.
And it is said to be a conservative translation if if satisfies the stronger property:
Γ⊨L1α if and only if Γ*⊨L1α*.
If * is a mapping defined on formulas and Γ is a set of formulas, then Γ*={γ*:γ∈Γ}.
3 Blok-Pigozzi Algebrization
In this section, we study the CG′3; and G′3 logics, which are defined in terms of four connectives ∧, ∨, →, and ¬ where the implication is deductive to CG′3. We establish that CG′3 and G′3 are algebraizable logics with the Blok-Pigozzi method.
3.1 The CG′3 logic
This section aims to analyze the algebraizability of CG′3 in the sense of Blok and Pigozzi, and we see the CG′3 logic as a Logic of Formal Inconsistency (LFI).
Osorio et al. defined CG′3 logic as a three-valued logic in [21], where the matrix is giving by the structure ℳ=〈D,D*,F〉 over Σ={∨,∧,→,¬}, where D={0,1,2}, the set D* of designated values is {1,2}, and ℱ is the set of truth functions defined in Table 1.
Table 1 Truth functions of the connectives in CG′3 and G′3
| ∨ |
|
0 |
1 |
2 |
|
∧ |
|
0 |
1 |
2 |
| |
|
|
|
|
|
|
|
|
|
|
| 0 |
|
0 |
1 |
2 |
|
0 |
|
0 |
0 |
0 |
| 1 |
|
1 |
1 |
2 |
|
1 |
|
0 |
1 |
1 |
| 2 |
|
2 |
2 |
2 |
|
2 |
|
0 |
1 |
2 |
| |
|
|
|
|
|
|
|
|
|
|
| → |
|
0 |
1 |
2 |
|
|
|
¬ |
|
|
| 0 |
|
2 |
2 |
2 |
|
0 |
|
2 |
|
|
| 1 |
|
0 |
2 |
2 |
|
1 |
|
2 |
|
|
| 2 |
|
0 |
1 |
2 |
|
2 |
|
0 |
|
|
Remark 4.
— Observe that → is a deductive implication: Γ,α⊨CG′3β if and only if Γ⊨CG′3α→β.
— Considering the natural order 0≤1≤2 in D, the ∨ corresponds to the supremum, ∧ corresponds to the infimum and → is the residuum of ∧:
z∧x≤y if and only if z≤x→y,for every x,y,z∈D.
Table 2 Truth functions of the connectives ∼, •, and ∘ in CG′3
| |
|
∼ |
|
|
|
• |
|
|
|
∘ |
| |
|
|
|
|
|
|
|
|
|
|
| 0 |
|
2 |
|
0 |
|
0 |
|
0 |
|
2 |
| 1 |
|
0 |
|
1 |
|
2 |
|
1 |
|
0 |
| 2 |
|
0 |
|
2 |
|
0 |
|
2 |
|
2 |
In [6], Blok and Pigozzi gave a mathematical concept of algebraizable logic. The main idea of this definition is the following:
A logic is algebraizable if there exists a class of algebras associated with the system of reasoning. In the same way as the class of Boolean algebras is related to classical propositional logic.
Proposition 1. The logic CG′3 is a strong LFI with consistency operator ∘ defined as above.
Proof. Assume that p and q are two different propositional variables. By considering the valuation v1 such that v1(p)=1, v1(¬p)=2, and v1(q)=0, it follows that p, ¬p⊬CG′3q and clause (i.a) of Remark 3 is verified. Consider the valuation v2 such that v2(p)=2, v2(∘p)=2, and v2(q)=0, it follows that p, ∘p⊬CG′3q and clause (i.b) of Remark 3 is satisfied. Now, considering the valuation v3 such that v3(¬p)=1, v3(∘p)=2, and v3(q)=0, it follows that ¬p, ∘p⊬CG′3q and clause (i.c) of Remark 3 is verified. Finally, there is no valuation that makes formulas φ, ¬φ, and ∘φ simultaneously true. Thus, item (ii) of Remark 3 is satisfied. Thus CG′3 is a strong LFI w.r.t ¬ and ∘.
Proposition 2. Let h be a valuation for CG′3. Then:
(i) h(p1→p2)∈D* if and only if h(p1)=0 or h(p2)∈D*;
(ii) h(p1∧p2)∈D* if and only if h(p1)∈D* and h(p2)∈D*;
(iii) h(p1∨p2)∈D* if and only if h(p1)∈D* or h(p2)∈D*;
(iv) h(p1↔p2)∈D* if and only if either h(p1)∈D* and h(p2)∈D*, or h(p1)=h(p2)=0.
Proof. Immediate from the truth-tables.
Definition 15. Let δ(p1,p2) be the following formula of ℒΣ:
δ(p1,p2)=(p1↔p2)∧(∘p1↔∘p2).
Proposition 3. Let α and β be formulas in ℒΣ. Then for every valuation h for CG′3 it holds that:
-
1. h(δ(α,β))∈D* if and only if we have:
Thus, h(δ(α,β))∈D* if and only if h(α)=h(β).
2. h(•α→α)∈D*.
3. h(∘α)=h(∘(•α→α)).
4. h(δ(α,•α→α))∈D* if and only if h(α)∈D*.
Proof.
“Only if” part. h(δ(α,β))∈D* if and only if in accordance with Proposition 2(ii), h(α↔β)∈D* and h(∘α↔∘β)∈D*. By Proposition 2(iv), h(α↔β)∈D* is equivalent to either h(α)∈D* and h(β)∈D*, or h(α)=h(β)=0, while h(∘α↔∘β)∈D* if and only if either h(∘α)∈D* and h(∘β)∈D*, or h(∘α)=h(∘β)=0. By the definition of ∘, h(∘α↔∘β)∈D* is equivalent to h(∘α)=h(∘β). Now, suppose that h(δ(α,β))∈D*. If h(α)=2 and h(β)=1, then h(∘α)=2 and h(∘β)=0, violating that h(∘α)=h(∘β). Analogously, it is impossible to have h(α)=1 and h(β)=2. This shows that h(α)=h(β). “If” part is obvious, in light of the clauses.
The proof is straightforward.
If h(∘α)=2 then h(α)∈{0,2} and h(•α)=0 and so ; hence, h(∘(•α→α))=2. If h(∘α)=0 then h(α)=1 and so h(•α)=2; hence, h(•α→α)=1; therefore, h(∘(•α→α))=0.
“Only if” part. By item 1, h(δ(α,•α→α))∈D* implies that h(α)=h(•α→α). So, by item 2, h(α)∈D*. “If” part. Suppose that h(α)∈D*. By item 2, h(•α→α)∈D* and, by item 3, h(∘α)=h(∘(•α→α)). Finally, applying item 1, h(δ(α,•α→α))∈D*.
Theorem 1. The logic CG′3 is algebraizable in the sense of Blok and Pigozzi with a system of equivalence formulas given by Δ(p1,p2)={δ(p1,p2)} and a system of defining equations given by E(p1)={p1,•p1→p1}.
Proof. It is easy to prove that the system Δ(p1,p2) satisfies conditions (i)-(iv) by item 1 of Proposition 3. By item 4 of the same proposition, conditions (v)-(vi) follow easily.
3.2 The G′3 Logic
The logic G′3 is defining as a three-valued logic in [22]. The matrix is giving by the structure ℳ=〈D,D*,F〉 over Σ={∨,∧,→,¬}, where D={0,1,2}, the set D* of designated values is {2}, and ℱ is the set of truth functions defined in Table 1. Note that G′3 is defined in the signature of CG′3.
It is easy to check from truth functions that ⊨CG′3((α→β)→α)→α but ⊭G′3((α→β)→α)→α. So the two deductive systems are different.
We now define a mapping CG′3 into G′3.
Definition 16. Let the mapping *:ℒΣ→ℒΣ given by α*=∼α→α.
We now show the mapping defined above is a conservative translation.
Proposition 4. For every Γ∪{α}⊆ℒΣ:
Γ⊨CG′3α if and only if Γ*⊨G′3α*.
Proof. “Only if” part. Suppose otherwise, that is, Γ⊨CG′3α and Γ*⊭G′3α*. Then there are a valuation h such that h(Γ*)∈D* and h(α*)=0, then h(∼α)∈(1,2) and h(α)=0, but ⊨CG′3α then for that valuation h(α)∈D*, is a contradiction. “If” part. Γ*⊨G′3α* implies Γ⊨CG′3α*, hence, using the following valid formula (∼α→α)→α in CG′3, we obtain Γ⊨CG′3α.
Corollary 1. G′3 is algebraizable in the sense of Blok and Pigozzi.
The systems CG′3 and G′3 despite, being different, are algebraizable in the sense of Blok and Pigozzi.
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