American Journal of Systems and Software, 2014, Vol. 2, No. 6, 139-145 Available online at http://pubs.sciepub.com/ajss/2/6/1 © Science and Education Publishing DOI:10.12691/ajss-2-6-1

Truth Values in t-norm based Systems Many-valued FUZZY Logic

Usó-Doménech J.L., Nescolarde-Selva J.*, Perez-Gonzaga S. Department of Applied Mathematics, University of Alicante, Alicante, Spain *Corresponding author: [email protected] Received November 28, 2014; Revised December 04, 2014; Accepted December 08, 2014 Abstract In t-norm based systems many-valued logic, valuations of propositions form a non-countable set: interval [0,1]. In addition, we are given a set E of truth values p, subject to certain conditions, the valuation v is v=V(p), V reciprocal application of E on [0,1]. The general propositional algebra of t-norm based many-valued logic is then constructed from seven axioms. It contains classical logic (not many-valued) as a special case. It is first applied to the case where E=[0,1] and V is the identity. The result is a t-norm based many-valued logic in which contradiction can have a nonzero degree of truth but cannot be true; for this reason, this logic is called quasi- paraconsistent. Keywords: contradiction, denier, logic coordinations, propositions, truth value Cite This Article: Usó-Doménech J.L., Nescolarde-Selva J., and Perez-Gonzaga S., “Truth Values in t-norm based Systems Many-valued FUZZY Logic.” American Journal of Systems and Software, vol. 2, no. 6 (2014): 139-145. doi: 10.12691/ajss-2-6-1.

1). V −1 (00) =

1. Introduction 2). Vpp( 12, ) = VpVp( 1) ( 2)

Axiom 3: V(p1,V(p2, p3)) = V(V(p1, p2), p3), Many-valued logic (Béziau, 1997; Cignoli et al, 2000; Axiom 4: V(p1, p2) = V(p2, p1), p1≤ p2 V(p1, p3) ≤ V(p2, p3), Malinowski, 2001; Miller and Thornton, 2008) differs Axiom 5: V(p1,1) = p1. from classical logic by the fundamental fact that it allows For all those t-norms which have⇒ the sup-preservation for partial truth. In classical logic, truth takes on values in property V(p, supi pi) = supi V(p,pi), there is a standard the set {0, 1}, in other words, only the value 1 or 0, way to introduce a related implication connective meaning “Yes, it's true,” or “No, it's not,” respectively. → V The authors propose a t-norm based systems Many-valued logics as their natural extension take on values in the with the truth degree function p1 → p2 = sup {p | V(p1, p) interval [0,1]. Per definition, t-norm based systems are V many-valued if the set of valuations is not countable and ≤ p2}. This implication connective is connected with the t- this set is the interval [0,1]. norm V by the crucial adjointness condition V(p1, p2) ≤ p3 Let p be the truth value of a proposition or utterance P, p1 ≤ (p2 → p3), which determines →V uniquely for P false p = 0 , P true p = 1; P approximated E = [0,1] = {p V ⇔each V with sup-preservation property. R | 0 ≤ p ≤ 1}. The language is further enriched with a negation We design for a non-countable set F whose algebraic connective, ¬ , determined by the truth degree function ¬ structure∈ is at least that of a semi-ring. We lay the V V following definitions and fundamental axioms: p = p → 0. We have a conjunction and a disjunction Axiom 1: Any proposition P has a truth value p, element V ∧ ∨ of a set E which is a part, not countable and stable for with truth degree functions. multiplication of the set F. p1∧= p 2 min { pp 12 ,} , These systems are basically determined by a strong p1∨= p 2 max { pp 12 ,} . conjunction connective ΛV which has as corresponding For t-norms which are continuous functions these truth degree function a t-norm V, i.e. a binary operation V additional connectives become even definable. Suitable in the unit interval which is associative, commutative, definitions are non-decreasing, and has the degree 1 as a neutral element. = Let p1, p2, p3 be three truth values. min { pp12 ,} V( p 1 , ( p 1→ p 2 )) , Then: V Axiom 2: Any proposition P is endowed with a valuation  = v ∈[0,1] such that v= Vp( ) , V reciprocal application of max{pp12 ,} min  p1→→ p 2 p 2, p 2 →→ p 1 p 1. V V VV E on [0,1] subject to the following conditions: 140 American Journal of Systems and Software

For a t-norm V their sup-preservation property is the vP( 1) = vP( 2) =⇔∧=11 vP( 12 P) left-continuity of this binary function V. And the continuity =⇒∧== of such a t-norm V can be characterized through the vP( i ) 0 vP( 12 P) 0, i 1, 2 algebraic divisibility condition Λ (p1 p2) = p1 p2. 2 V → For definition 2, vPP∧= VppVp =2 = Vp . V ( ) ( ) ( ) ( ) ∧ In this work we develop a many-valued logic: known as In general, unless vP( ) = 0 or vP( ) = 1, the conjunction quasi-paraconsistent because the contradiction cannot be is not idempotent in many-valued logic. true, but can be approximated, E = [0,1] (Newton da Definition 3: The complementarity of two propositions P1 Costa, in Susana Nuccetelli, Ofelia Schutte, and Otávio and P2 is compound proposition, denoted PPℑ whose Bueno, 2010; Bueno, 2010; Carnielli and Marcos, 2001; 12 Fisher, 2007; Priest and Woods, 2007). In 2013, truth value is pp12+ . Castiglioni and Ertola Biraben provide some results For axiom 1, the complementarity is commutative. It concerning a logic that results from propositional exists only if pp12+∈ E. When p1 ≠ 0 and p2 ≠ 0 , so intuitionistic logic when dual negation is added in certain that vP( ℑ= P) 0 , it is necessary that pp+=0 , so that way, producing a paraconsistent logic that has been called 12 12 da Costa Logic. the addition of the truth values admit opposed. When n = 1, among the functions f1, there are Axiom 6: If ¬P truth value p* denotes the negation or polynomial pp+ * and monomial pp* . For axiom 6 and * contradiction of P, we must have: Vp( += p) 1 . * definition 3, vP( ℑ¬ P) = V( p + p) = 1; complementarity * Let Pi be n propositions, i= 1,2,…, n of pi and pi be the of the two contradictories is always true. truth values of their contradictories. Then: * Let pp+= ube. Definition 1: A compound proposition (or logical coordination or logical expression) of order n is a Definition 4: u is a denier of the proposition P if the proposition whose truth value c is a function f of p following three conditions are fulfilled: n i a) uE∈ * and pi . b) Vu( )= 1; u unitary truth value (from axiom 6) ** * c) up−= p* ∈ E (from axiom 1) c= fn( ppp11, , 2 , p 2 ,... pnn , p) In general, these three conditions can be satisfied by a fn values in F; it determines a truth value if cE∈ .The set of deniers of P, then the contradictory ¬P has a priori, * condition of existence of a compound proposition defined 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 by fn is cE∈ , or what is equivalent, Vc( )∈[0,1] . the truth value of the contradictory. Axiom 7: fn is a polynomial in which each index 1, Theorem 1: If u is a denier of P, it is also a denier of ¬P. 2, ...,n must be at least once and that all coefficients are Proof equal to unity. Indeed, u∈ E;1 Vu( ) = u −=∈ p* p E Condition Vpp( 12, ) = VpVp( 1) ( 2) requires the Definition 5: The conjunction of a proposition and its stability of E for multiplication because [0,1] possesses contradictory is called contradiction. the stability and function V is reciprocal. In paraconsistent logic, a contradiction is not If e designates the neutral element of the multiplication necessarily false. It may be true, then: = of truth values, we have Vee( , ) Ve( ) and for Axiom 2, vP( ∧¬=⇔ P) 11 vP( ) =¬= v( P) VeVe( ) ( ) = Ve( ) . Solution Ve( ) = 0 is to reject because for Axiom 2 would lead to e = 0 , therefore remainsVe( ) = 1. 2. ALGEBRA OF t-norm based Systems Many-valued FUZZY Logic We apply Axiom 7 and n = 2. Among polynomials f2 are the monomial p p and pp+ polynomial. 1 2 12 Definition 6: A compound proposition of order n is called Definition 2: The conjunction of two propositions P1 and P normal and polynomial Pn that determines its truth value P2 is compound proposition, denoted PP12∧ whose truth is called normal if is homogeneous polynomial of degree p, value is . pp12 if in any of its monomials there is repetitions of index, and For definition 2 and Axiom 2, no monomial is repeated. vP( 121212∧= P) Vpp( ) = VpVp( ) ( ) . Definition 7: Normal polynomial is said to be complete From Axiom 2 the conjunction is commutative. _ P Stability of E and [0,1] for the multiplication result and denoted Pn if includes all monomials of degree p allowed by combinatorial analysis. ∀∀ ∃ ∧ that p12,, p( PP 12) . It is easy to see that the complete normal polynomials As in classical logic: can be formed from the complementarities of American Journal of Systems and Software 141 contradictory PPiiℑ¬ of truth value and the truth value of ¬PP12 ∨¬ is fixed, once fixed pp12, by the denier uu12. ¬PP12 ∨¬ is commutative. pp+=* ui, = 1,2,..., n. Indeed: ii i ** ** Definition 10: ppppppuupp12++ 12 1 2 =− 12 1 2is the _ 1* truth value of the compound proposition called disjunction Pn=∑∑ pp ii += u i ( ) of P1 and P2 and denoted PP12∨ . ii There is disjunction if ¬PP12 ∧¬ admits uu12 as denier. _ 2 ** Then: Pn=∑∑( p i + p i)( p j += p j) uu ij ij ij vP( 12∨ P) = v ¬( ¬ P 1 ∧¬ P 2) where ij refers to a combination of the two indices; ∧ ** _ and the truth value of PP12is fixed, once fixed pp12, by 2 22 the denier uu . Pn includes 2 Cn monomials of degree 2. Similarly: 12 * ** * _ Definition 11: ppppppuupp12++ 12 1 2 =− 12 1 2is the 3 *** truth value of the proposition called implication of P by Pn=∑∑( p i + p i)( p j + p j)( p k += p k) uu i jk u 2 ijk ijk P1 denoted ¬∨PP12or PP12⇒ . ¬∧ _ There is implication if PP12admits uu12 as denier. 3 Then: where ijk means a combination of three indexes; Pn 33 vP( 1⇒ P 2) = v ¬¬( P 12 ∧ P) includes 2 Cn monomials of degree 3. And so on until: * _ and the truth value of PP12⇒ is fixed, once fixed pp12, n ** * Pn=+( p1 p 1)( p 2 + p 2)....( pnn += p) uu12... u n by the denier uu12. Condition 1 of existence: Let P1, P2 be two propositions, n which includes 2 monomials of degree n. ∃u1 denier of P1 and ∃u2 denier P2 such that uu12 is a Any normal compound proposition has equal truth denier of PP∧ . value either one of the monomials of a complete normal 12 Indeed, if E and function V can satisfy this condition, polynomial, or a combination of several of these ¬ ∨¬ ¬ monomials. then PP12exists, but as u1 is also a denier of P1 Definition 8: A family p of normal compound and u2 of ¬P2 , other disjunctions may also exist. propositions of order n contains all those derived from _ p 2.2. Normal Binary Propositions: Family 2, complete normal polynomial Pn . Group 2 Within the same family, the propositions can be classified into groups according to the number of The truth value of a proposition of this group is the sum _ _ 2 p P of an even number of monomials of P2 . monomials of Pn composing Pn . ** Definition 12: pp12+ pp 1 2is the truth value of the 2.1. Normal Binary Propositions: Family 2, compound proposition called concordance and Group 1 denoted PP12Ξ . ** Group 1 is that of binary propositions whose truth value Definition 13: pp12+ pp 1 2 is the truth value of the _ compound proposition called discordance and is the sum of an odd number of monomials P2 . 2 denoted PP12Χ . _ Condition 2 of existence: PP12Ξ exists if ∃u1 denier of P1 2 * * ** P2=+++ ppppppppuu 1212121212 =. ** and ∃u2 denier P2 such that pp12+∈ pp 1 2 E. _ Condition 3 of existence: PP12Χ exists if ∃u1 denier of 2 The monomials of P2 are the respective truth values of ** P1 and ∃u2 denier P2 such that pp12+∈ pp 1 2 E. the conjunctions PPP∧,,, ∧¬ P ¬ PP ∧ ¬ P ∧¬ P 121 2121 2 If conditions 2 and 3 are satisfied, then: that exist unconditionally. Ξ =¬Χ * * ** vP( 12 P) v( P 12 P) Definition 9: Polynomial ppppppuupp12++=− 1 2 1 2 12 12 Χ=¬⇔ is the truth value of the proposition called incompability vP( 12 P) v( P 1 P 2) of P1 and P2 and denoted ¬PP12 ∨¬ . We leave aside the coordination of this group whose ∧ * There is incompatibility if PP12 admits uu12 as respective truth values are: pp12+= pp 12 u 21 p , denier. Then, after the axiom 6, definition 3 and definition pp+= pp* up , pp*+= pp ** up * and 4, we have: 12 1 2 12 12 1 2 12 * ** * pp12+= pp 12 up 21 whose degrees of truth are vP(¬1 ∨¬ P 2) = v ¬( PP 12 ∧ ) 142 American Journal of Systems and Software determined by the truth value of only one of the _ 1 ** * propositions P1,¬¬ PP 12 ,, P 2. Pppppppnnn=12 + ++... + 1 + 2 ++... The sum of truth values is associative (axiom 1), but the 2.3. Normal Binary Propositions: Family 1 complementarity is not in general. We write in all cases _ PP123ℑℑ P the compound proposition whose veracity 1 ** P2=+++ pp 121 pp 2 is rppprE=++, ∈ . Iff PPℑ exists We have defined the complementarity. 123 12 pp+∈ Eand PPℑ exists ppE+∈ there is Definition 14: pp+ * is the truth value of the inverse 12 23 13 12 associativity, resulting from the addition of the truth complementarity of P1 and P2 denoted ¬PP12 ℑ¬ . values; PPℑ ℑ P,, P ℑ P ℑ P PP ℑℑ Pthen have Condition 4 of existence: ∃u1 denier of P1 and ( 12) 3 1( 23) 123 ** the same truth value pp123++ p. ∃u2 denier P2 such as if pp12+∉ Ethen ppE12+∈. Note that in Condition 4 the truth values intervene and The complementarity of the n propositions Pi, commutative, is the compound proposition not just the deniers. Condition 4 satisfied if PP12ℑ does PP12ℑ ℑℑ... P 3whose truth value is pp12+ ++... pn . not exist, and then ¬PP ℑ¬ exists and vice versa. 12 It will be even the inverse complementarity of n * Definition 15: pp12+ is the truth value of the compound propositions, associative but not commutative in general, with respect to ¬Pi, denoted ¬PP ℑ¬ ℑ... ℑ¬ P and truth proposition called equivalence and denoted PP12℘ . 12 3 ** * The denomination equivalence is due to value pp12+ ++... pn . that p1= p 2 ⇒ vP( 21 ℘= P) vP( 12 ℘ P) =1 . Indeed, If set E and function V satisfy Condition 4, so when the complementarity does not exist, there is the inverse +=−+* and * + = −+ , pp12 ppu 122 p1 p 2 p 2 pu 11 complementarity. p= p ⇒ vP ℘= P Vu =1 and 1 2( 21) ( 2) _ ℘= = n ** * vP( 12 P) Vu( 1) 1. Pn=+( p1 p 1)( p 2 + p 2)....( pnn += p) uu12... u n Condition 4 satisfied, at least one of two equivalences PP℘ and PP℘ exists since PP℘ ≡ P ℑ¬ Pand Multiplication of truth values is associative; the 21 12 21 1 2 conjunction is also because it is not subject to any P1℘ P 2 ≡¬ PP 12 ℑ . condition of existence. We make a summary of the main coordination in the following table (Table 1): P1∧∧( P 23 P),,( PP 12 ∧∧) PPP 3123 ∧∧ P Table 1. Table of principal normal binary propositions always same truth value pp p . Notation Name Truth value 123 Conjunction of n propositions P , commutative and PP∧ Conjunction pp i 12 12 associative, is the compound proposition denoted ¬PP ∨¬ Incompatibility uu− p p 12 12 1 2 PP12∧ ∧∧... Pn whose truth value pp12... pn is one of the ** uu12− p 1 p 2 monomials Pn . PP12∨ Disjunction n =+−u p up pp 21 1 2 12 If condition 1 is satisfied by uu23 and uu123 u , the − * uu12 p 1 p 2 incompatibility ¬P1 ∨( ¬ PP 23 ∨¬ ) exists, then it is the PP12⇒ Implication =−− uupup12 1( 2 2) negation by denier u1() uu 23= uuu 123 of the ** conjunction P∧∧ PP, identical to PP∧∧ P. If pp12+ pp 1 2 1( 23) 123 PPΞ Concordance 12 uu satisfies condition 1, the incompatibility =−++uu12( u 21 p u 12 p) 2 p 12 p 12 ** (¬PP12 ∨¬) ∨¬ P 3 exists and it is also the negation by pp12+ pp 1 2 PP12Χ Discordance denier uu123 u of the conjunction PP123∧∧ P.It is seen that =+−u21 p up 122 pp 12 if the condition 1 is satisfactory wherever it is necessary, PP12ℑ Complementarity pp12+ there is an incompatibility of P1, P2 and P3, commutative Inverse ¬ ℑ¬ ** ¬ ∨¬ ∨¬ PP12complementarity pp12+ and associative, which is denoted PP123 Pand ℘ Equivalence * will have as truth value uu123 u− p 1 p 2 p 3.More generally, PP21 p1+ p 2 =+− pu 12 p 2 it may be a commutative and associative incompatibility Inverse * PP12℘ ppupp+ =−+ equivalence 1 2112 of n propositions Pi, denoted as ¬PP12 ∨¬ ∨... ∨¬ Pn and will have as truth value uu12... unn− p 1 p 2 ... p .But we can 2.4. Normal propositions of order n: 1 and n also, as the complementarity, noted in all cases families ¬PP12 ∨¬ ∨... ∨¬ Pn the compound proposition, non- We consider only those two families and only a few of associative in general, of veracity uu12... unn− p 1 p 2 ... p . coordination within them. Similarly, through a suitable choice of the deniers, it may be a disjunction of n propositions Pi, commutative and American Journal of Systems and Software 143 associative sometimes, denoted PP12∨ ∨∨... Pn and truth vP()12ℑ=+ P p 1 p 2 ** * 2 ** value uu12... unn− p 1 p 2 ... p . 8. Complementarity : vP()¬1 ℑ¬ P 2 = p 12 + p =−+2 ( pp12) 3. t-norm based Systems Many-valued 3.2. Normal Propositions of n Order Logic We have, for example, the following degrees of truth: The set of truth values E is interval [0,1] .; function V is 1. Conjunction: vP(1∧ P 2 ∧∧ ... Pnn ) = pp12 ... p the identity: the degree of truth is equal to the truth value. vP(¬12 ∨¬ P ∨... ∨¬ Pn ) V does satisfy axiom 2, Vu()= 1 and Vu()=⇒= u u 1; =1 − pp12 ... pn 2. Incompatibility: therefore this logic does not know a single denier, number ¬PP12 ∨¬ ∨... ∨¬ Pn 1. =¬∧(PP12 ∧∧... Pn ) Axiom 6 is written pp+=* 1 . Contradiction has a vP(∨ P ∨∨ ... P ) =− 1 p** p ... p * value vP( ∧¬ P) = p(1 − p) .It cannot be true, it is false if 3. Disjunction: 1 2 nn12 PP∨ ∨... ∨ P =¬ ¬ P ∧¬ P ∧... ¬ P P is false or if P is true; it is approximated if P is 12 nn( 1 2 ) approximated but the maximum degree of truth is 0.25, 4. Complementarities: achieved when p = 0.50. vP(12ℑ P ℑℑ ... Pnn ) = p 1 + p 2 + ... + p 3.1. Normal Binary Propositions. Conditions: 1-4 iff( p12+ p ++... pn ) ∈[ 0,1]

_ vP(¬1 ℑ¬ P 2 ℑ ... ℑ Pnn ) = n −( p12 + p +... + p) 2 * * ** P2=+++ pppppppp 12121212 =1 iff( p12+ p ++... pn ) ≥− n 1 _ 2 The four terms of P2 belongs to [0,1] . Therefore −∈ −** − −∈** 4. BOOLEAN REDUCTION OF t-norm (1pp12) [ 0,1] and (1pp12) ,1( pp 12) ,1( pp 12) [ 0,1] . Condition 1 is filled by the denier 1. The sum of four Based Systems Many-valued FUZZY ** Logic terms being 1, ( pp12+∈ pp 1 2) [0,1] and The Boolean reduction of many-valued logic is to pp**+∈ pp [0,1] : conditions 2 and 4 are also always ( 12 1 2) reduce the set of valuations [0,1] to the set {0,1} . ** met. Finally, if pp12+≥1 then pp1+=−+ 221( pp 12) ≤ Accordingly, the set E of truth values retains its neutral −1 because pp12+≤2 : condition 4 is fulfilled too. element 0, since V (00) = and the set of unitary truth Truth values of the main normal binary coordinations values. The result is a Boolean logic, not many-valued are the following: since {0,1} is countable. 1. Conjunction: vP()∧= P pp 1 2 12 The classical propositional algebra can be deduced v(¬ P1 ∨¬ P 2 )1 = − pp 12 from the evaluation of negation and that of conjunction, 2. Incompatibility: by defining from it and little by little other compound ¬P1 ∨¬ P 2 =¬( PP 12 ∧ ) propositions. We already know that if the valuations of P1 ** vP(∨ P )1 =− p p =+− p p pp and P are Boolean that PP∧ is the same in many- 3. Disjunction: 1 2 1 2 1 2 12 2 12 PP12∨ =¬( ¬ P 1 ∧¬ P 2) valued logic in classical logic. + * vP(⇒=−=−− P )1 pp* 1 p( 1 p ) Regarding the negation, it follows from pp, 4. Implication: 1 2 12 1 2 V −1 (00) = , V (00) = and Vu( ) = 1 that: P1⇒ P 2 =¬( P 1 ∧¬ P 2) =¬ PP 12 ∨ vP()Ξ= P pp + p** p ∀uv,0( P) =⇒¬= v( P) 1 5. Concordance: 1 2 12 1 2 =−+12( p1 p 2) + pp 12 ∀uv,0( ¬=⇒ P) v( P) = 1 ** vP()Χ= P pp +=+− p p p2 pp = = 6. Discordance: 1 2 12 1 1 2 12 If vP( ) 1 may be selected up as denier PP12Χ=¬Ξ( PP 12) because pE∈ , vp( ) = 1and up−=∈0 E so that: * 1 vP()℘ P = p + p = p +− 1 p 7. Equivalence : 21 1 2 1 2 * vP()12℘ P = p 1 + p 2 =−+ 1 p 1 p 2 commutative coordination known as equivalence: ℘= ℘ =−− vP()()121 P vP 12 P p 1 p 2 which is true iff pp12= . 1 2 Only vP()21℘ P exists when pp21≥ , and only vP()12℘ P exists Only vP()12ℑ P exists when ( pp12+∈) [0,1] , and only ¬ ℑ¬ +≥ when pp12≥ . We can define in quasi-paraconsistent logic a unique vP()12 Pexists when pp121 . 144 American Journal of Systems and Software

vP( ) =⇒¬=1 v( P) 0 provided that u= p Quasi-Paraconsistent many-valued fuzzy logic includes the special case of classical logic. In fact, our presentation ¬=⇒ = = v( P) 1 vP( ) 0 provided thatu p . of the propositional many-valued algebra is developed The evaluation of many-valued negation may well be according to the canons of Aristotelian logic, which made identical to that of classical negation, choosing borrow from the theory of sets. If classical logic does not specific deniers. was a special case of quasi-paraconsistent many-valued Consider some remarkable links between quasi- logic, it, mined by a fundamental inconsistency, should be paraconsistent logic and classical logic. rejected, on the spot. A statement is analytic for pedagogical need that of 4.1. Deductive Equivalence quasi-paraconsistent many-valued logic remains rational because the latter is subject to conformity with a logic that Concordance, equivalence and reciprocal implication of it encompasses, in definitive with itself. the quasi-paraconsistent logic just melt in classical logic in a single coordination that is the classic equivalence or equivalence deductive PP12⇔ . Indeed, if p1 and p2 are References Boolean variables [1] Arruda A. I., Chuaqui R., da Costa N. C. A. (eds.) 1980 . Mathematical Logic in Latin America. North-Holland Publishing v( P1⇔ P 2) =−+1( p 1 p 2) + 2 pp 12 = vP () 1 Ξ P 2 Company, Amsterdam, New York, Oxford. vP( 1⇔ P 2) =−−=1 p 1 p 2 vP( 12 ℘= P) vP( 21 ℘ P) [2] Asenjo, F.G. 1966. A calculus of antinomies. Notre Dame Journal of Formal Logic 7. Pp 103-105. Regarding the reciprocal implication, it has degree of [3] Avron, A. 2005. Combining classical logic, paraconsistency and truth in quasi-paraconsistent logic, by defining relevance. 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Ex contradictione non sequitur 4.2. Mutual Exclusion quodlibet. Proc. 2nd Conf. on Reasoning and Logic (Bucharest, July 2000). Discordance, complementarity and inverse [10] Castiglioni, J. L. and Ertola Biraben, R. C. 2013. Strict complementarity of quasi-paraconsistent logic blend in paraconsistency of truth-degree preserving intuitionistic logic with classical logic in a single coordination which is the dual negation. Logic Journal of the IGPL. Published online reciprocal exclusion. Adopting the notation of Piaget August 11, 2013. [11] Cignoli, R. L. O., D'Ottaviano, I, M. L. and Mundici, D., 2000. P12 WP for reciprocal exclusion we have when p1 and p2 Algebraic Foundations of Many-valued Reasoning. Kluwer. are Boolean: [12] Da Costa, N., Nuccetelli, S., Schutte, O. and Bueno, O. 2010. “Paraconsistent Logic” (with (eds.), A Companion to Latin American Philosophy (Oxford: Wiley-Blackwell), pp. 217-229. vPWP( 1 2) =+− p 1 p 22 pp 12 = vPXP( 1 2) [13] Dunn, J.M. 1976. Intuitive semantics for first-degree entailments and coupled trees. Philosophical Studies 29. pp 149-168. vPWP=+= p p vP ℑ Pwhen p +≤ p 1 ( 1212) ( 12) 12 [14] Fisher, J. 2007. On the Philosophy of Logic. Cengage Learning. pp. 132-134. v( P WP) =2( − p + p ) = v( ¬ P ℑ¬ P ) [15] Malinowski, G. 2001. Many-Valued Logics, in Goble, Lou. Ed., 12 1 2 1 2 whenpp+≥ 1. The Blackwell Guide to Philosophical Logic. Blackwell. 12 [16] Miller, D, M. and Thornton, M. A. 2008. Multiple valued logic: concepts and representations. Synthesis lectures on digital circuits and systems 12. Morgan & Claypool Publishers. 5. Conclusions [17] Nescolarde-Selva, J. and Usó-Domènech, J. L. 2013a. Semiotic vision of ideologies. Foundations of Science. [18] Nescolarde-Selva, J. and Usó-Domènech, J. L. 2013b. Reality, The main objective of the authors is to establish a Systems and Impure Systems. Foundations of Science. theory of truth-value evaluation for paraconsistent logics, [19] Nescolarde-Selva, J. and Usó-Doménech, J. 2013c. Topological unlike others who are in the literature (Asenjo, 1966; Structures of Complex Belief Systems. Complexity. pp 46-62. Avron, 2005; Belnap, 1977; Bueno, 1999; Carnielli, [20] Nescolarde-Selva, J. and Usó-Doménech, J. 2013d. Topological Coniglio and Lof D'ottaviano, I.M. 2002; Dunn, 1976: Structures of Complex Belief Systems (II): Textual materialization. [21] Priest G. and Woods, J. 2007. Paraconsistency and Dialetheism. Tanaka et al, 2013), with the goal of using that logic The Many Valued and Nonmonotonic Turn in Logic. Elsevier. paraconsistent in analyzing ideological, mythical, [22] Usó-Domènech, J.L. and Nescolarde-Selva, J.A. 2012. religious and mystic belief systems (Nescolarde-Selva and Mathematic and Semiotic Theory of Ideological Systems. A Usó-Doménech, 2013a,b,c,d; Usó-Doménech and systemic vision of the Beliefs. LAP LAMBERT Academic Nescolarde-Selva, 2012). Publishing. Saarbrücken. Germany. American Journal of Systems and Software 145

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 Notation Name MVFL truth values pp12,∈{ 0,1} QPL truth values pp12,∈[ 0,1]

PP12∧ Conjunction pp12 pp12 ¬PP12 ∨¬ Incompatibility 1− pp12 uu12− p 1 p 2 PP12∨ Disjunction p1+− p 2 pp 12 u21 p+− up 1 2 pp 12

PP12⇒ Implication 1−+p1 pp 12 uupup12−− 1( 2 2)

PP12⇔ Concordance 12−−p1 p 2 + pp 12 uu12−++( u 21 p u 12 p) 2 p 12 p

PP12 Discordance p1+− p 22 pp 12 u21 p+− up 122 pp 12 PP12ℑ Complementarity pp12+ pp12+ ¬ ℑ¬ Inverse complementarity −− ** PP12 2 pp12 pp12+ PP21℘ Equivalence 1+−pp12 pu12+− p 2 PP12℘ Inverse equivalence 1−+pp12 upp112−+