1. Introduction

Many-valued logic (, 1997; Cignoli et al, 2000; Malinowski, 2001; Miller and Thornton, 2008) differs from classical logic by the fundamental fact that it allows for partial truth. In classical logic, truth takes on values in the set {0, 1}, in other words, only the value 1 or 0, meaning “Yes, it's true,” or “No, it's not,” respectively. The authors propose a t-norm based systems Many-valued logics as their natural extension take on values in the interval [0,1]. Per definition, t-norm based systems are many-valued if the set of valuations is not countable and this set is the interval [0,1].

Let p be the truth value of a proposition or utterance P, P false, P true; P approximated .

We design for a non-countable set F whose algebraic structure is at least that of a semi-ring. We lay the following definitions and fundamental axioms:

Axiom 1: Any proposition P has a truth value p, element of a set E which is a part, not countable and stable for multiplication of the set F.

These systems are basically determined by a strong conjunction connective which has as corresponding truth degree function a t-norm V, i.e. a binary operation V in the unit interval which is associative, commutative, non-decreasing, and has the degree 1 as a neutral element. Let p₁, p₂, p₃ be three truth values.

Then:

Axiom 2: Any proposition P is endowed with a valuation such that, V reciprocal application of E on subject to the following conditions:

1).

2).

Axiom 3:

Axiom 4:

Axiom 5:

For all those t-norms which have the sup-preservation property , there is a standard way to introduce a related implication connective with the truth degree function . This implication connective is connected with the t-norm V by the crucial adjointness condition , which determines uniquely for each V with sup-preservation property.

The language is further enriched with a negation connective, , determined by the truth degree function . We have a conjunction and a disjunction ∨ with truth degree functions.

For t-norms which are continuous functions these additional connectives become even definable. Suitable definitions are

For a t-norm V their sup-preservation property is the left-continuity of this binary function V. And the continuity of such a t-norm V can be characterized through the algebraic divisibility condition .

In this work we develop a many-valued logic: known as quasi-paraconsistent because the contradiction cannot be true, but can be approximated, (Newton da Costa, in Susana Nuccetelli, Ofelia Schutte, and Otávio Bueno, 2010; Bueno, 2010; Carnielli and Marcos, 2001; Fisher, 2007; Priest and Woods, 2007). In 2013, Castiglioni and Ertola Biraben provide some results concerning a logic that results from propositional intuitionistic logic when dual negation is added in certain way, producing a paraconsistent logic that has been called da Costa Logic.

Axiom 6: If truth value denotes the negation or contradiction of P, we must have: .

Let P_i be n propositions, i= 1,2,…, n of p_i andbe the truth values of their contradictories. Then:

Definition 1: A compound proposition (or logical coordination or logical expression) of order n is a proposition whose truth value c is a function of p_i and.

values in F; it determines a truth value if .The condition of existence of a compound proposition defined by is , or what is equivalent, .

Axiom 7: is a polynomial in which each index 1, 2, ...,n must be at least once and that all coefficients are equal to unity.

Condition requires the stability of E for multiplication because possesses the stability and function V is reciprocal.

If e designates the neutral element of the multiplication of truth values, we have and for Axiom 2, . Solution is to reject because for Axiom 2 would lead to, therefore remains.

We apply Axiom 7 and n = 2. Among polynomials f₂ are the monomial p₁p₂ and polynomial.

Definition 2: The conjunction of two propositions P₁ and P₂ is compound proposition, denoted whose truth value is.

For definition 2 and Axiom 2, .

From Axiom 2 the conjunction is commutative. Stability of E and for the multiplication result that.

As in classical logic:

For definition 2, . In general, unless or , the conjunction is not idempotent in many-valued logic.

Definition 3: The complementarity of two propositions P₁ and P₂ is compound proposition, denoted whose truth value is.

For axiom 1, the complementarity is commutative. It exists only if. When and, so that, it is necessary that, so that the addition of the truth values admit opposed.

When n = 1, among the functions f₁, there are polynomial and monomial. For axiom 6 and definition 3, ; complementarity of the two contradictories is always true.

Let be.

Definition 4: u is a denier of the proposition P if the following three conditions are fulfilled:

a)

b) u unitary truth value (from axiom 6)

c) (from axiom 1)

In general, these three conditions can be satisfied by a set of deniers of P, then the contradictory ¬P has a priori, once fixed p, a set of truth values p^*(u); so the choice of a denier who, in a problem of applied logic, will determine the truth value of the contradictory.

Theorem 1: If u is a denier of P, it is also a denier of .

Proof

Indeed,

Definition 5: The conjunction of a proposition and its contradictory is called contradiction.

In paraconsistent logic, a contradiction is not necessarily false. It may be true, then:

2. ALGEBRA OF t-norm based Systems Many-valued FUZZY Logic

Definition 6: A compound proposition of order n is called normal and polynomial that determines its truth value is called normal if is homogeneous polynomial of degree p, if in any of its monomials there is repetitions of index, and no monomial is repeated.

Definition 7: Normal polynomial is said to be complete and denoted if includes all monomials of degree p allowed by combinatorial analysis.

It is easy to see that the complete normal polynomials can be formed from the complementarities of contradictory of truth value . Indeed:

where ij refers to a combination of the two indices; includes monomials of degree 2. Similarly:

where ijk means a combination of three indexes; includes monomials of degree 3. And so on until:

which includes monomials of degree n.

Any normal compound proposition has equal truth value either one of the monomials of a complete normal polynomial, or a combination of several of these monomials.

Definition 8: A family p of normal compound propositions of order n contains all those derived from complete normal polynomial.

Within the same family, the propositions can be classified into groups according to the number of monomials of composing.

Group 1 is that of binary propositions whose truth value is the sum of an odd number of monomials.

The monomials of are the respective truth values of the conjunctions that exist unconditionally.

Definition 9: Polynomial is the truth value of the proposition called incompability of P₁ and P₂ and denoted.

There is incompatibility if admits as denier. Then, after the axiom 6, definition 3 and definition 4, we have:

and the truth value of is fixed, once fixed , by the denier . is commutative.

Definition 10: is the truth value of the compound proposition called disjunction of P₁ and P₂ and denoted.

There is disjunction if admits as denier. Then:

and the truth value of is fixed, once fixed, by the denier .

Definition 11: is the truth value of the proposition called implication of P₂ by P₁ denoted or.

There is implication if admits as denier. Then:

and the truth value of is fixed, once fixed, by the denier .

Condition 1 of existence: Let P₁, P₂ be two propositions, denier of P₁ and denier P₂ such that is a denier of.

Indeed, if E and function V can satisfy this condition, then exists, but as u₁ is also a denier of and u₂ of, other disjunctions may also exist.

The truth value of a proposition of this group is the sum of an even number of monomials of.

Definition 12: is the truth value of the compound proposition called concordance and denoted.

Definition 13: is the truth value of the compound proposition called discordance and denoted.

Condition 2 of existence: exists if denier of P₁ and denier P₂ such that.

Condition 3 of existence: exists if denier of P₁ and denier P₂ such that.

If conditions 2 and 3 are satisfied, then:

We leave aside the coordination of this group whose respective truth values are: , , and whose degrees of truth are determined by the truth value of only one of the propositions .

We have defined the complementarity.

Definition 14: is the truth value of the inverse complementarity of P₁ and P₂ denoted.

Condition 4 of existence: denier of P₁ and denier P₂ such as ifthen.

Note that in Condition 4 the truth values intervene and not just the deniers. Condition 4 satisfied if does not exist, and then exists and vice versa.

Definition 15: is the truth value of the compound proposition called equivalence and denoted.

The denomination equivalence is due to that. Indeed, and , and

Condition 4 satisfied, at least one of two equivalences and exists since and .

We make a summary of the main coordination in the following table (Table 1):

Table 1. Table of principal normal binary propositions

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We consider only those two families and only a few of coordination within them.

The sum of truth values is associative (axiom 1), but the complementarity is not in general. We write in all cases the compound proposition whose veracity is. Iff exists and exists there is associativity, resulting from the addition of the truth values; then have the same truth value.

The complementarity of the n propositions P_i, commutative, is the compound proposition whose truth value is.

It will be even the inverse complementarity of n propositions, associative but not commutative in general, with respect to , denoted and truth value.

If set E and function V satisfy Condition 4, so when the complementarity does not exist, there is the inverse complementarity.

Multiplication of truth values is associative; the conjunction is also because it is not subject to any condition of existence.

always same truth value .

Conjunction of n propositions P_i, commutative and associative, is the compound proposition denoted whose truth value is one of the monomials.

If condition 1 is satisfied by and, the incompatibility exists, then it is the negation by denier of the conjunction, identical to. If satisfies condition 1, the incompatibility exists and it is also the negation by denier of the conjunction.It is seen that if the condition 1 is satisfactory wherever it is necessary, there is an incompatibility of P₁, P₂ and P₃, commutative and associative, which is denoted and will have as truth value .More generally, it may be a commutative and associative incompatibility of n propositions P_i, denoted as and will have as truth value .But we can also, as the complementarity, noted in all cases the compound proposition, non-associative in general, of veracity . Similarly, through a suitable choice of the deniers, it may be a disjunction of n propositions P_i, commutative and associative sometimes, denoted and truth value.

3. t-norm based Systems Many-valued Logic

The set of truth values E is interval; function V is the identity: the degree of truth is equal to the truth value. V does satisfy axiom 2, and; therefore this logic does not know a single denier, number 1.

Axiom 6 is written. Contradiction has a value. It cannot be true, it is false if P is false or if P is true; it is approximated if P is approximated but the maximum degree of truth is 0.25, achieved when p = 0.50.

The four terms of belongs to. Therefore and . Condition 1 is filled by the denier 1. The sum of four terms being 1, and : conditions 2 and 4 are also always met. Finally, if then because : condition 4 is fulfilled too.

Truth values of the main normal binary coordinations are the following:

1. Conjunction:

2. Incompatibility:

3. Disjunction:

4. Implication:

5. Concordance:

6. Discordance:

7. Equivalence^{1}:

8. Complementarity^{2}:

We have, for example, the following degrees of truth:

1. Conjunction:

2. Incompatibility:

3. Disjunction:

4. Complementarities:

4. BOOLEAN REDUCTION OF t-norm Based Systems Many-valued FUZZY Logic

The Boolean reduction of many-valued logic is to reduce the set of valuations to the set . Accordingly, the set E of truth values retains its neutral element 0, since and the set of unitary truth values. The result is a Boolean logic, not many-valued since is countable.

The classical propositional algebra can be deduced from the evaluation of negation and that of conjunction, by defining from it and little by little other compound propositions. We already know that if the valuations of P₁ and P₂ are Boolean that is the same in many-valued logic in classical logic.

Regarding the negation, it follows from, , and that:

If may be selected as denier because, and so that:

The evaluation of many-valued negation may well be made identical to that of classical negation, choosing specific deniers.

Consider some remarkable links between quasi-paraconsistent logic and classical logic.

Concordance, equivalence and reciprocal implication of the quasi-paraconsistent logic just melt in classical logic in a single coordination that is the classic equivalence or equivalence deductive. Indeed, if p₁ and p₂ are Boolean variables

Regarding the reciprocal implication, it has degree of truth in quasi-paraconsistent logic, by defining

When p₁ and p₂ are Boolean, the third term of the last member is always zero (principle of non-contradiction) so that:

Discordance, complementarity and inverse complementarity of quasi-paraconsistent logic blend in classical logic in a single coordination which is the reciprocal exclusion. Adopting the notation of Piaget for reciprocal exclusion we have when p₁ and p₂ are Boolean:

5. Conclusions

The main objective of the authors is to establish a theory of truth-value evaluation for paraconsistent logics, unlike others who are in the literature (Asenjo, 1966; Avron, 2005; Belnap, 1977; Bueno, 1999; Carnielli, Coniglio and Lof D'ottaviano, I.M. 2002; Dunn, 1976: Tanaka et al, 2013), with the goal of using that logic paraconsistent in analyzing ideological, mythical, religious and mystic belief systems (Nescolarde-Selva and Usó-Doménech, 2013^a,b,c,d; Usó-Doménech and Nescolarde-Selva, 2012).

Quasi-Paraconsistent many-valued fuzzy logic includes the special case of classical logic. In fact, our presentation of the propositional many-valued algebra is developed according to the canons of Aristotelian logic, which borrow from the theory of sets. If classical logic does not was a special case of quasi-paraconsistent many-valued logic, it, mined by a fundamental inconsistency, should be rejected, on the spot.

A statement is analytic for pedagogical need that of quasi-paraconsistent many-valued logic remains rational because the latter is subject to conformity with a logic that it encompasses, in definitive with itself.

Note

1. Only exists when, and only exists when. We can define in quasi-paraconsistent logic a unique commutative coordination known as equivalence: which is true iff .

2. Only exists when , and only exists when.

References

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ANNEX A

We will represent in the following table a comparison between two logics: classical (CL), and t-norm based systems many-valued fuzzy logic (MVFL).

Table 2. Truth table of principal normal binary propositions

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