=Paper=
{{Paper
|id=Vol-408/paper-14
|storemode=property
|title=The Pac Logic in the properties of C(w) and C(min)
|pdfUrl=https://ceur-ws.org/Vol-408/Poster2.pdf
|volume=Vol-408
|dblpUrl=https://dblp.org/rec/conf/lanmr/ArrazolaAM08
}}
==The Pac Logic in the properties of C(w) and C(min)==
https://ceur-ws.org/Vol-408/Poster2.pdf
The Pac Logic in the properties of
Cω and Cmin
J. Arrazola1 , E. Ariza1 , and V. Borja2
1
Benemérita Universidad Autónoma de Puebla
Av. San Claudio s/n Puebla, Pue. México
2
Universidad Tecnológica de la Mixteca
Carretera a Acatlima Km 2.5 Huajuapan de León Oaxaca, México
arrazola@fcfm.buap.mx
ariza@hotmail.com
vero0304@mixteco.utm.mx
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 .
Key words: Paraconsistency, C-systems, non-triviality and non-explosiveness
1 Introduction
When proposing the first paraconsistent propositional calculus, Jaskowski ex-
pected it to enjoy the following properties: a) When applied to inconsistent sys-
tems, 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 non-
contradiction must not be a valid schema. This lattice of paraconsistent logics
will be our study object.
2 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.
−→ 0 12 1 ¬
0 1111
1 1 1
2 0 2 1 2
1
1 02 10
Table 1. Pac’s Semantics.
2 J. Arrazola, E. Ariza, and V. Borja
The designated values are 1 and 12 . Pac does not accept strong negation, does
not admit a bottom particle, is left-adjunctive and is not finitely trivializable.
3 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, B o = B 1 = 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))
– Cω 1. A ∨ ¬A
– Cω 2. ¬¬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 ¬n A represents ¬¬ · · · ¬A, n-times). A ≡ ¬m A, ¬2m A ≡
¬2k A, ¬2m A ≡ ¬2k+1 A, ¬2m+1 A ≡ ¬2k+1 A..
It is important to point out that except for C0 , the calculi Cn are not de-
cidable 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 sub-
formulas and bifurcations show up due to the presence of the connective ¬.
The Pac Logic in the properties of Cω and Cmin 3
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 (¬(B n−1 ∧ ¬B n−1 )) = vn (¬(¬B n−1 ∧ B n−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 Paraconsis-
tent 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 matri-
ces, 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.