The Pac Logic in the properties of Cω and Cmin

Abstract

In this work we try to answer some questions related to the theory of paraconsistent logics. We study a chain of paraconsistent logics stronger than Cω. −→ 0 12 1 ¬ 0 1 1 1 1 1 0 12 1 12 2 1 0 12 1 0 Table 1. Pac's Semantics.

Keywords

• Paraconsistency
• C-systems
• non-triviality and non-explosiveness

Introduction

When proposing the first paraconsistent propositional calculus, Jaskowski expected it to enjoy the following properties: a) When applied to inconsistent systems, it should not always entail their trivialization; b)It should be rich enough to enable practical inference and c)It should have an intuitive justification. In 1963, da Costa [ 1 ] proposed a whole hierarchy of paraconsistent propositional calculi, known as Cn, with 1 ≤ n < ω: in this calculi, the principle of noncontradiction must not be a valid schema. This lattice of paraconsistent logics will be our study object.

Paraconsistent logics

In [ 1 ], da Costa suggests the study of non-trivial contradictory logics, which he called paraconsistent logics. A logic L is paraconsistent if ∃Γ : ∃A : ∃B : (Γ ` A, Γ ` ¬A y Γ 6` B).

The Pac logic. This logic is of particular importance in our study, it is determined by Table 1, and the connectives ∨ and ∧ are determined by the functions max and min, respectively. The designated values are 1 and 21 . Pac does not accept strong negation, does not admit a bottom particle, is left-adjunctive and is not finitely trivializable.

The hierarchy Cn

Let 1 ≤ n ≤ ω. To define Cn, we start with Ao = ¬(A ∧ ¬A) and we write An instead of Ao··· o (n-times). We also write A(n) for A1 ∧ A2 ∧ · · · ∧ An. It is necessary to clarify that for n = 1, Bo = B1 = B(1). The only inference rule is Modus Ponens (MP), and the axioms for each Cn are: – Pos1. A → (B → A) – Pos2. (A → (B → C)) → ((A → B) → (A → C)) – Pos3. (A ∧ B) → A – Pos4. (A ∧ B) → B – Pos5. A → (B → A ∧ B) – Pos6. A → (A ∨ B)) – Pos7. B → (A ∨ B)) – Pos8. (A → C) → ((B → C) → ((A ∨ B) → C)) – Cω1. A ∨ ¬A – Cω2. ¬¬A → A – 12-n. B(n) → ( (A → B) → ((A → ¬B) → ¬A) ) – 13-n. A(n) ∧ B(n) → (A ∧ B)(n) – 14-n. A(n) ∧ B(n) → (A ∨ B)(n) – 15-n. A(n) ∧ B(n) → (A → B)(n) The Cω logic. It is built with the same axioms of Cn, except 12-n to 15-n. Let us denote by C0 the classical propositional calculus. Then Cn, with 0 ≤ n < ω, is finitely trivializable. Cω is not finitely trivializable.

One important results of Arruda is that for 1 ≤ n ≤ ω, it is impossible to reduce the negation. In other words, for m 6= k, the following schemes are not valid in Cn (where ¬nA represents ¬¬ · · · ¬A, n-times). A ≡ ¬mA, ¬2mA ≡ ¬2kA, ¬2mA ≡ ¬2k+1A, ¬2m+1A ≡ ¬2k+1A..

It is important to point out that except for C0, the calculi Cn are not decidable by using finite matrices. In fact there are valuations (not satisfying the principle of functional truth) that let us prove the soundness of each Cn. 3.1

Decidability of Cn In [ 4 ] da Costa defines valuations for each Cn those valuations do not satisfy the principle of functional truth, then we can not determine thorough valuations whether a formula is valid or not. It is not possible in general to associate a matrix to a formula so we will use the concept of quasi-matrix to refer to an array that differs from a matrix in the following way: A quasi-matrix can show bifurcations in a row starting at some column, the last column is reserved to represent the principal formula, the remaining columns represent proper subformulas and bifurcations show up due to the presence of the connective ¬.

The Pac Logic in the properties of Cω and Cmin

Some important results are that: For every line k in the quasi-matrix M, there exists a valuation v such that vΓ corresponds to k, where Γ is the set of formulas in M and that C1 is decidable through the valuation v. In order to construct quasi-matrices, it is necessary the following observation, which is characteristic of them: vn(¬(Bn−1 ∧ ¬Bn−1)) = vn(¬(¬Bn−1 ∧ Bn−1)) = 0. Also for 1 ≤ n < ω. Then Cn is deducible through quasi-matrices. and for 0 ≤ n ≤ ω each of the calculi in the hierarchy Cn is strictly stronger than its successor.

Due to the previous result, we have a family of strictly decreasing Paraconsistent logics which are finitely trivializable due to the fact that they accept strong negation; therefore, they have the bottom particle. Also these results motivates to consider Cω was a syntactic limit of Cn. Let us remember that Cω is not finitely trivializable, and can not be finitely gently explosive. We will keep exploding the idea of regarding Cω as a syntactic limit in order to get more properties.

The Cmin logic. It is the logic defined when adding the formula A ∨ (A → B) as an axiom to Cω. Using a Similar valuation to those of Cn we have soundness and completeness for Cmin.

Theorem 1. The calculus Cmin is not decidable through finite matrices, does not have a bottom particle, it is not finitely trivializable and it does not accept strong negation.

The proof of this result is a consequence of the semantics proposed by Arruda and the fact that Cmin is sound under the matrices of Pac logic, and Pac does not accept a bottom particle. With the previous theorem we can realize that Cmin is not the syntactic limit of Cn, however is seems to be the syntactic closure of Cω . 4

Conclusions

Pacs semantic makes the Cω and Cmin logics sound, as a consequence these logics do not have a bottom particle. It is important to notice that the Arruda’s proposal to attack the same problem is much more complicated. The logic Cmin has come to substitute Cω as the syntactic limit of the hierarchy Cn.

References

1. N. C. da Costa , Presente par M. Ren Garnier , Calculs Propositionnels pour les Systmes Formels Inconsistants , 1963 .

2. Newton C. A. da Costa , On the Theory of Inconsistent Formal Systems , 1974 .

3. Aida I. Arruda , Presente par M. Ren Garnier , Remarques sur les Systmes Cn , 1975 .

4. N. C. da Costa and E. H. Alves , A Semantical anlisis of the Calculi Cn , 1977 .