In Artificial Intelligence, as well as in data base updating or in the design of intelligent agents, it is necessary the use of contradictory information. For that, it is useful to direct our attention to paraconsistent logics. The goal of the present survey is to offer an initial analysis on paraconsistency in some logics like Pac, RM3, Lukasiewicz, G03.
INTRODUCTION TO PARACONSISTENCY This section is devoted to introduce the concepts and basic principles of paraconsistency, as well as its properties.
Definition 1. [
Con1. A ∈ Γ =⇒ Γ ` A Reflexivity Con2. (Γ ` A y A ` B) =⇒ Γ ` B Monotonicity
Con3. (Δ ` A y Γ, A ` B) =⇒ Δ, Γ ` B Transitivity
So, a logic L is defined as a couple hF or, `i containing a consequence relation satisfying, on this paper, Con1, Con2 and Con3 and a set of formulas. We will say that Γ is a theory of L if Γ ⊆ L. We will also say that Γ is closed if it contains all of its consequences (the converse of Con1.)
For our purposes, F or is a numerable set of symbols from the language that
contains ¬ as negation symbol, whether it is defined or native, besides some
other logic symbols proper of each theory. Initially, as we have already said, we
consider our logics with a consequence relation satisfying Con1, Con2 y Con3,
but this does not mean that every consequence relation satisfies them. The
following properties follow directly from Definition 1
Proposition 1. [
Definition 2. 1. We will say that a theory Γ is contradictory, with respect to ¬, if there exists a formula A such that Γ ` A y Γ ` ¬A; 2. We say that a theory Γ is trivial if ∀A : Γ ` A; 3. We say that a theory is explosive if, when adding to it any couple of contradictory formulas, the theory becomes trivial; 4. We say that a logic L is contradictory (trivial, explosive) if all of its theories are contradictory (trivial, explosive).
The empty theory will be required due to the next property.
Proposition 2. [
On Definition 2 they were established the concepts of a logic being contradictory, trivial or explosive; It is natural to ask whether all logics can be classified among these three types of logics. The following principles give an affirmative answer to this question.
Principle of non-contradiction (PNC)
L is not contradictory. ∃Γ : ∀A : ( Γ 6` A or Γ 6` ¬A )
L is not trivial. ∃Γ : ∃B : ( Γ 6` B ) Principle of explosion or Pinciple of pseudo-scotus (PPS)
L is explosive. ∀Γ : ∀A : ∀B : ( Γ, A, ¬A ` B ) 1.1
PARACONSISTENCY
Newton da Costa in [
We say that the logic L is paraconsistent if 1.1
∃Γ : ∃A : ∃B : ( Γ ` A , Γ ` ¬A y Γ 6` B ) The second definition is based on the existence of non-explosive theories. 1. We will say that the formulas A, B are equivalent if one implies the other and viceversa, that is: A ` B and B ` A; 2. In a similar way, the theories Γ and Δ are equivalent if:
( ∀A ∈ Γ : Δ ` A ) y ( ∀B ∈ Δ : Γ ` B ).
From the previous definition, we obtain the following property.
Proposition 4. [
To assume the existence of paraconsistent logics implies assuming the existence of some non-explosive theories. In particular, we do not want the empty theory to be explosive, since it will be convenient to use it in some theoretical proofs.
On the other hand, the triviality is of importance for the study of paraconsistency, since it is of interest to know to what extent a logic can be turned trivial.
Proposition 5. [
Let us take Proposition 5 to the extreme of having a unique formula which trivializes any theory.
Definition 4. We say that the logic L has a bottom particle if there exists a formula C ∈ L such that it, by itself, makes L trivial. That is;
∃C : ∀Γ : ∀B : ( Γ, C ` B )
In the case that such a formula exists, we denote it by ⊥. Evidently, every logic having a bottom particle as one of its thesis, becomes trivial. We have a reformulation of The Principle of explosion .
Definition 5. We say that a logic has a top particle if there exists a formula C, in L, that is consequence of each of its theories, that is:
∃C : ∀Γ : ( Γ ` C ) In case it exists, we denote the top particle by >. Any thesis of a logic is a top particle. It is easy to see that the addition of a top particle to a given theory is pretty innocuous.
Definition 6. We say that the logic L has strong negation (or supplementary negation) if there exists a scheme μ(A), that depends only upon A, such that: a) ∃A : μ(A) is not a bottom particle b) ∀A : ∀Γ : ∀B : ( Γ, A, μ(A) ` B ) We will denote the strong negation by μ(A) = ∼ A. The parallel definition of contradictory theory with respect to ∼ becomes clear, furthermore we have the supplementary version of PPS.
Principle of ex contradiction (sPPS)
L has strong negation 1. If a logic has a bottom particle or it has strong negation, then it is finitely trivializable; 2. If a non-trivial logic has the bottom particle, then it accepts strong negation; 3. If a logic is explosive and non-trivial, then it is supplementarily explosive (it does satisfies ex falso).
But, we would also like to recognize the form taken by ⊥ and ∼, as well as some reciprocal versions of the assertions in 6. In order to do that we need to recognize other connectives that will be very helpful.
Definition 7. [
Such a formula, in case it exists, will be denoted by μ(A, B) = A ∧ B.
In order to characterize the left-adjunctive logics, we have the following proposition.
Proposition 7. [
and Γ, A ∧ B ` B ).
Proposition 8. [
ANALYSIS OF SOME LOGICS
In what follows we will give a diagnostic from paraconsistency in some logics.
We start with an already known logic called Pac, which is treated in [
The Pac logic Pac is determined by Table 1, the designated values are 1 y 12 . Observe that the connectives ∨ and ∧ are determined by the functions max and min, respectively. The consequence relation is determined by:
Γ ` B if and only if for any valuation ν such that ν(A) is a designated value, for every formula A ∈ Γ , ν(B) is also a designated value.
∧ 12 0 1 0 1 1 −→ 0 0 0 – Taking ν, A, B such that ν(A) = 21 , ν(B) = 0. We have that A, ¬A 6` B.
Then, Pac is paraconsistent. – By looking at the tables, we conclude the non-existence of a schema μ(A), that depends only upon A, such that A, μ(A) ` B, for every B. Since taking ν(A) = 12 and ν(B) = 0, then ν(μ(A)) = 12 , therefore Pac does not accept strong negation. – By proposition 6, Pac does not admit a bottom particle – It can be verified, by means of Proposition 7, that Pac is left-adjunctive. – By proposition 8, Pac is not finitely trivializable. 2.2
The Lukasiewicz logic The semantics of Lukasiewicz logic is determined by Table 2.
−→ 0
The conjunction and disjunction connectives are determined by ∨ = max, and ∧ = min. The designated value is 1, and the consequence relation is given by: Γ `Luk A =⇒ ν(A) = 1 ν(B) = 0 ν(B) = 12 = A for some B ∈ Γ.
for some B ∈ Γ – Taking ν such that: ν(A) = 12 and ν(B) = 0, we have A, ¬A 6`Luk B. Hence
Luk is paraconsistent. – Observe, from Table 2, that ν(¬(A → A)) = 0, for all valuation ν. Then we define the bottom particle as ⊥ := ¬(A → A) – By proposition 6, Luk admits strong negation. And for it, we propose ∼ A := ¬((¬A ∨ A) −→ A). – Luk is left-adjunctive.
– By proposition 6, Luk es finitely trivializable. 2.3
The Logic RM3 The semantics of RM3 is determined by the functions min and max, and the next table.
−→ 0 The designated values are 1 and 12 , and the consequence relation is given by: Γ `RM3 B ⇐⇒ ν(B) ν(A) = 0 has the designated value for some A ∈ Γ ν(A) = 12 = ν(B) for some A ∈ Γ. – RM3 is paraconsistent, taking ν(A) = 12 and ν(B) = 0, we conclude that
A, ¬A 6` B. – RM3 is left-adjunctive. – Observe that any scheme μ(A), that depends only on A, is such that if ν(A) = 12 , then ν(μ(A)) = 12 . Then RM3 does not admit strong negation. – By 6, RM3 does not admit a bottom particle.
– By 8 RM3 is not finitely trivializable. 2.4
The ℘-Four logic The semantics of ℘-Four is determined by the following tables:
A
The designated value is 3. In this case, the functions min, max do not coincide with the connectives ∧F and ∨F ; We have, now, two new functions associated to ∧F y ∨F , we will call them minF and maxF , respectively. So, the consequence relation is given by: Γ `F B ⇐⇒ ν(B) = 3 minF {A ∈ Γ } = 0 ν(B) = 2 y minF {A ∈ Γ } = 2 ν(B) = 1 y minF {A ∈ Γ } = 1 – ℘-Four is paraconsistent; taking ν(A) = 1 and ν(B) = 0, we have that
A, ¬F A 6` B. – For every valuation ν, we have ν( A ∧F ¬F A) = 0. Then ℘-Four accepts a bottom particle (⊥ := A ∧F ¬F A). – ℘-Four is left-adjunctive. – Therefore ℘-Four accepts strong negation.
– ℘-Four is finitely trivializable. The G03 semantics is determined by Tables 4, and the functions ∧ := min y ∨ := max.
−→ 0 The designated value is 1, and the consequence relation is determined by: For this logic, there is no n-valuation that defines its semantics, hence we present the system Cω by means of a Hilbert-style axiomatization. The axioms of Cω 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)) Cω1. A ∨ ¬A Cω2. ¬¬A −→ A Pos8. (A → C) −→ ((B → C) → ((A ∨ B) → C))
The only inference rule is Modus Ponens (MP). The consequence relation is given in proof theory, that is to say:
In [
• Cω is not finitely trivializable. – Cω does not have a bottom particle (Proposition 6) – Cω does not have strong negation (Proposition 8) 3
CONCLUSION The present work is just an initial analysis in which, first, we detect the paraconsistent logics, then we recognize their properties like being left-adjunctive, having the bottom particle and strong negation, and we determined whether they are finitely trivializable. All logics in this survey were paraconsistent. For future work, it remains to analyze the concept of a theory being (finitely) gently explosive, partially explosive and controllably explosive.