101

On Some Classical Tenets and Fuzzy Logic

Enric Trillas

101.1 Introduction

A fertile seed dropped by Zadeh’s work into the soil of classical logic, refers to thepossibility of reconsidering some of the tenets logicians seems to preserve with al-most no actual debate with thinkers in the ’fuzzy’ community of researchers. Theflourishing potentiality of that seed is partially due to the different perspective fromwhich fuzzy logic looks at what is its object in comparison with what the classicalviews maintain. Such a different perspective comes, in a first place, by consideringimprecise linguistic terms, linguistic connectives, modifiers and quantifiers as theobjects to be represented, instead of the classical formal ones, and commonsensereasoning processes instead of formal ones. That is, for instance, for trying to math-ematically and computationally modeling the Natural Language’s expressions withwhich some dynamical systems are described, or can only be described when no pre-cise mathematical models of them are available. Almost always these expressions arenot representable with classical sets without modifying their meaning as it is givenby their use under some purpose in the corresponding context. Context-sensitiveand purpose-driven meaning are typical characteristics shown by the problems fuzzylogic deals with.

Since the allowed extension for this paper does not permit a large presentation,only the following three topics will be shortly taken into account:

• The validility of some formal fuzzy laws, and the design of fuzzy models.• The universality of the principles of non-contradiction and excluded-middle in

fuzzy logic• The necessity of fuzzy logic to consider non-deductive reasoning.

101.2 On the Validity of Some Formal (Fuzzy) Laws, and theDesign of Fuzzy Models

Since one of the goals of fuzzy logic is the formal representation of natural languagestatements made up with both precise-boolean and imprecise terms, at least involv-ing the linguistic connectives and, or, not, it is important to have formal frames ofrepresentation able to capture as much as possible the different properties these con-nectives show in language. Such frames are known as ’algebras of fuzzy sets’ ,or’fuzzy algebras’, of which no a generally accepted definition is currently known in

R. Seising et al. (Eds.): On Fuzziness: Volume 2, STUDFUZZ 299, pp. 697–705.DOI: 10.1007/978-3-642-35644-5_101 © Springer-Verlag Berlin Heidelberg 2013

698 101 On Some Classical Tenets and Fuzzy Logic

the same form as there are known those of, for instance, ortholattices, and De Mor-gan algebras. Since natural language is very complex and not static at all, a generaldefinition of fuzzy algebra is not an easy task. For instance, neither the linguistic’and’ is always commutative, nor the linguistic ’not’ is always involutive, as they areconsidered in the former lattices.

Additionally, since fuzzy sets are at its turn represented by functions with values inthe unit interval, it is also not obvious that the functions representing the before men-tioned connectives could always be functionally expressible by numerical functions.For instance, if a operation . between fuzzy sets represents the linguistic ’and’, itshould be distinguished if such representation is, or is not, functionally expressible inthe form, μ ·σ = T ◦(μ×σ), with some numerical function T : [0,1]× [0,1]→ [0,1],as it is the case if T is a (commutative!) continuous t-norm.

Once assumed that the set of functions [0,1]X = {μ ; μ : X → [0,1]} is partiallyordered by the pointwise ordering,

μ ≤ σ iff μ(x)≤ σ(x), for all x in X ,

a tentative, but general enough definition of fuzzy algebra for the goal of this paper,is the following [1],

Definition 1. A Basic Formal Fuzzy Algebra (BFFA) is a four-tuple ([0,1]X , ·,+,′ ),where · and + are binary operations [0,1]X × [0,1]X → [0,1]X, and ′ is a unaryoperation [0,1]X → [0,1]X , verifying:

1) If μ ≤ σ , then μ ·λ ≤ σ ·λ , and λ ·μ ≤ λ ·σ ,If μ ≤ σ , then μ +λ ≤ σ +λ , and λ + μ ≤ λ +σ , for all λ in [0,1]X .

2) μ ·μ0 = μ0 ·μ = μ0, μ ·μ1 = μ1 ·μ = μ ,μ + μ0 = μ0 + μ = μ , μ + μ1 = μ1 + μ = μ1,with μ0 and μ1 the constant fuzzy sets respectivelly equal to 0 and 1.

3) μ ′0 = μ1, and μ ′

1 = μ0

If μ ≤ σ , then σ ′ ≤ μ ′4) If μ and σ are in {0,1}X, then also μ ·σ , μ + σ and μ ′, are in {0,1}X, and

μ ·σ = min(μ ,σ), μ +σ = max(μ ,σ), μ ′ = 1− μ .

Axiom (4) is necessary to capture the representation of crisp terms. It is easy to provethat in all BFFA it holds:

a) μ ·σ ≤ min(μ ,σ)≤ max(μ ,σ)≤ μ +σ ;b) The only BFFA that are lattices are the De Morgan algebras ([0,1]X ;min,max,′ )

provided ′ is involutive;c) No BFFA is an ortholattice, thus no one is a Boolean algebra.

Notice that it is not supposed that a BFFA is neither with . or + commutatives, norassociatives, nor that ’ is involutive, nor that distributive or duality laws do hold,etc.A BFFA is functionally expressible (FE) provided there are three numerical functionsT,S : [0,1]× [0,1]→ [0,1], N : [0,1]→ [0,1], such that μ ·σ = T ◦ (μ ×σ), μ +σ =S◦(μ×σ), and μ ′ =N ◦σ , for all μ and σ in [0,1]X . This is the case of the Standard

101.2 On the Validity of Some Formal (Fuzzy) Laws 699

Fuzzy Algebras in which T is a continuous t-norm, S is a continuous t-conorm, and Nis a strong negation (see [2]), that are those currently considered in both the theoreticand applied literature on fuzzy logic.

That some imprecise assertive statements can be satisfactorily represented bymeans of fuzzy sets, does not imply that all the laws of Boolean algebras can beapplied to, for instance, shortening complex statements. This is the case with state-ments of the type “(x is P and y is Q) or (x is P and y is not Q)”, that cannot bealways taken as equivalent to “x is P”, since the logical law of perfect repartition,p ·q+ p ·q′= p, only holds in the setting of Boolean algebras, but neither in proper or-tholattices, nor in proper De Morgan algebras, and less again in most fuzzy algebras.The problem lies in finding fuzzy algebras ([0,1]X , ·,+,′ ) in which the representa-tion of the former statement: (μP · μQ)(x)+ (μP · μ ′

Q)(x), does coincide with μP(x)for all x in X . That is, the fuzzy algebras where the formal law μ ·σ + μ ·σ ′ = μ ,holds. The problem is completely solved [3], and also in the case the algebra isfunctionally expressible by means of a continuous t-norm T, a continuous t.conormS, and a strong negation N, that is, in the particular setting of the Standard FuzzyAlgebras. The only standard algebras of fuzzy sets in which the corresponding lawμ(x) = S(T (μ(x),σ(x)),T (μ(x),N(σ(x)))) holds for all x in X , are those given byT = Prodϕ , S =W ∗

ϕ , N = Nϕ , for any order-automorphism ϕ of the unit interval. Inthese algebras, that are not lattices, neither any law of duality [1], nor several otherlattice’s laws hold.

The law can be used only in the contexts where and can be modeled by T =Prodϕ ,or by S = W ∗

ϕ , and not by Nϕ , with the same ϕ . A situation very different of theclassical-boolean, and with some distant similarity with the quantum-orthomodular.If the involved linguistic terms are imprecise, but representable by fuzzy sets, the lawis only applicable provided the connectives admit the former representations.

This example shows that when imprecise statements in natural language involvinglinguistic connectives do be represented in fuzzy terms, it is strictly necessary tocorrectly choose the corresponding algebra. At its turn the algebra can force thenon validity of some other laws [4], [6] that, if necessary, can be reached by addingnew connectives. For instance, if in the example it were necessary that “(x is P) and(x is not P)” always does not hold, it can be considered the Pexider algebra withtwo ‘intersections’ ([0,1]X ,Prodϕ ,Wϕ ,W ∗

ϕ ,Nϕ ) (see [5]), in which for the secondconjunction holds μ · μ ′ = Wϕ(μ ×N ◦ μ) = μ0,for all μ in [0,1]X . It should benoticed that in large linguistic pieces different uses of the connectives can appearlike it is the case of commutative and non-commutative uses of ‘and’.

Nevertheless, in all case concerning the representation of linguistic pieces the firstproblem is the design or selection of the membership functions of the predicatesappearing in them, that depends on the context and the purpose in which they areused. For instance, the usual representation by piecewise linear membership func-tions could be just erroneous if this kind of functions are not in agreement with whatis known on the use of the corresponding linguistic terms. Concerning the designor selection of the linguistic connectives, their representations in fuzzy terms do bechoosen accordingly with their meanings. For instance, in the case of a rule "If x isP, then y is Q’ in which the negation of the antecedent does not play any role (for

700 101 On Some Classical Tenets and Fuzzy Logic

instance, for not having physical sense), it is possible a conjunctive-type representa-tion of the type T (ϕ(μP(x)),μQ(y)) [16]. Notwithstanding, provided it were knownthat it should always be μP(x)≤ μQ(y), the correct representation could be a residu-ated implication [6], [8] but not a conjunctive-type one.

In any case, before representing anything in terms of fuzzy sets and fuzzy con-nectives, to capture the best than possible knowledge of the system, or the best un-derstanding of the linguistic piece and its parts, is essential. A non correct enoughdesign can conduct to represent in fuzzy terms a problem different from the givenlinguistic one, and at least to eventually finding partially unrealistic solutions of thelinguistically posed problem. A good knowledge on the basic mathematical theoryof fuzzy algebras and fuzzy logic is actually important for the designers, and yet itwill be more important in the path towards Zadeh’s Computing with Words in whichlarger and more complex natural language’s expressions will play a pivotal role.

101.3 On the Universality of the Principles of Non-contradictionand Excluded-Middle in Fuzzy Logic

The linguistic term ’impossible’, Aristotle used to state the principle of Non-Contradiction (NC), was translated by ’false’, by A∩ A′ = /0 in the boolean caseof classical sets, and by μ · μ ′ = μ0 in that of the algebras of fuzzy sets. In thestandard algebras, the principle corresponds with the verification of the equationT (μ(x),N(μ(x))) = 0, for all μ in [0,1]X , and all x in X. Hence, the solutions (T,N)of the functional equation T (a,N(a)) = 0, for all a in [0,1], give the cases in whichNC holds. These solutions are T = Wϕ , and N ≤ Nϕ (see [7]. Hence the principleNC fails in most of the standard algebras of fuzzy sets.

Although Aristotle is very opaque with respect to the principle of Excluded-Middle (EM),it is currently translated by A U A’ = X with classical sets, and byμ + μ ′ = μ1 in the algebras of fuzzy sets. In the standard algebras, this correspondswith the verification of the equation S(μ(x),N(μ(x))) = 1, for all μ and all x. Hencethe solutions (S,N) of the functional equation S(a,N(a)) = 1, give the cases in whichEM holds. These solutions are S =W ∗

ψ , and Nψ ≤ N (see [8], and show that the prin-ciple EM fails in most of the standard algebras.

Hence, both NC and EM principles do hold in the standard algebras of fuzzy setsif and only if T =Wϕ , S =W ∗

ψ , and Nψ ≤ N ≤ Nϕ . There are algebras in which onlyone of the two principles hold, and both jointly fail in many, many cases.

Nevertheless, if ’impossible’ is translated by ’self-contradictory’, both principlesdo hold in all BFFA if posed by:

• NC : μ ·μ ′ ≤ (μ ·μ ′)′, that is, μ ·μ ′ is self-contradictory,• EM : (μ + μ ′)′ ≤ ((μ + μ ′)′)′, that is (μ + μ ′)′ is self-contradictory,

and it is easy to prove that the two inequalities hold in all BFFA (see [9]).

It should be noticed that provided the operations · and + are commutative, the nega-tion is involutive, and the laws of duality hold, the former NC and EM inequalities

101.4 On the Necessity of Formalizing Non-deductive Reasoning 701

reduce to the obviously valid inequality μ ·μ ′ ≤ μ +μ ′. It also should be noticed thatif μ is in {0,1}X , then NC reduces to μ · μ ′ = μ0 and EM reduces to μ + μ ′ = μ1.In [10], and provided the BFFA is FE but without t-norms and t-conorms, the nu-merical functions giving the functional expressibility of · and + verifying NC andEM are characterized. Hence, expressing the principles by understanding ‘impossi-ble’ as ‘self-contradictory’ or ‘absurd’, both NC and EM principles are universallyvalid in all BFFA and, in particular, in the standard algebras of fuzzy sets, somethingthat sheds a different light on what is usually asserted since, in fact, fuzzy sets neverviolate the two Aistotle’s principles. In addition, it does be remarked that the fuzzycase opens the way towards a deductive study of the the general validity of the twoprinciples (see [9]), perhaps against the Aristotle’s view that at least NC cannot besubmitted to proof.

101.4 On the Necessity of Formalizing Non-deductive Reasoning

Most of the commonsense reasonings are not deductive, but conjectural. Perhaps nomore than a 25% of the totality of these reasonings are deductively made step by stepand under well known rules of inference. Hence, the remaining 75% is of relevancefor any methodology trying to represent Commonsense Reasoning (CR) throughoutsome formalization process that, hence, should necessarily take into account the non-deductive ways of reasoning. That is, abductive and inductive types of reasoning inwhich, and contrarily to deduction, are typical the ’jumpings’ to the conclusions.

Of course, deduction does be considered the only ’safe’ form of reasoning in thesense that their conclusions are as valid as premises can be, and for any deductiveconclusion its negation is refused as such. Instead, abduction and induction do notgive ’safe’ conclusions since they are not only doubtful with respect to the givenpremises, but its negations can be also obtained from the same premises. If in de-duction all that is concluded just deploys, or necessarily follows from what is in thepremises, in abduction and induction the situation is different since their conclusionsoften represent something that is ’new’, in the sense of not being directly deployablefrom the premises.

In addition, deduction is monotonic since when new premises are known, no lessconclusions can be deployed. Instead, neither abduction, nor induction can be mono-tonic; experience shows that in these kind of reasonings new premises can easilyconduct to cancel some previously reached conclusions, that is, and cautiously said,no more conclusions can be obtained.

At this respect, a first problem is how to define the concept of conjecture in sucha way that deductive, abductive, and inductive conclusions, can be captured as theironly particular cases. Conjectures are viewed as those elements in the frame of rep-resentation thar are just ’consistent’ with the information conveyed by the premises.

With the goal of not introducing more technical complexities than those that arestrictly necessary, let us try to introduce this new concept (see [11]) in the settingof the De Morgan algebra of fuzzy sets ([0,1]X ,min,max,1− id), that currently ismaybe the most employed one in the applications of fuzzy logic.

702 101 On Some Classical Tenets and Fuzzy Logic

Definition 2. Let P = {μ1, ...,μn} ⊂ [0,1]X , a set of premises such that μ∧ = min(μ1, ...,μn) is not self-contradictory, that is μ∧ �≤ 1−μ∧. The set of conjectures fromP, is Con j(P) = {σ ∈ [0,1]X ; μ∧ �≤ 1−σ}.

Obviously, it is P ⊂Con j(P), and

a) If P ⊂ Q, then Con j(Q) ⊂ Con j(P). That is, the ‘operator’ Con j is anti-monotonic.

b) The operator C(P) = {σ ∈ [0,1]X ; μ∧ ≤ σ}, is extensive: P ⊂C(P), monotonic:P⊂Q→C(P)⊂C(Q), a clausure: C(C(P))=C(P), and consistent: σ ∈C(P)→σ ′ �∈C(P): C is a consistent consequence operator.

c) C(P)⊂Con j(P), and Con j(P) = {σ ∈ [0,1]X ;σ ′ �∈C(P)}.d) It cannot be neither assumed that if σ ∈ Con j(P), then σ ′ �∈ Con j(P), nor that

σ �≤ σ ′.e) It is supposed that μ∧ summarizes the information conveyed by P.

Result (c) could make to think that conjecturing should necessarily come after de-duction. Nevertheless, there are operators that can be called of conjectures and thatdo not ’follow’ from a consequence operator. For instance,

Con j∗(P) = {σ ∈ [0,1]X ; μ∧ ·σ �≤ (μ∧ ·σ)′},which although verifying P ⊂ C(P) ⊂ Con j∗(P), and being anti-monotonic, is notcoming from any conjecture operator since the only possible C∗ with which it can beCon j∗(P) = {σ ;σ �∈ C∗(P)}, is C∗(P) = {σ ; μ∧ ·σ ′ ≤ (μ∧ ·σ ′)′}, that is not a con-sequence operator. Of course, Con j∗ is obtained after understanding the consistencywith μ∧ as “μ∧ ·σ is not self-contradictory", instead that for Conj is understood by‘σ is not contradictory with μ∧’. Yet also the operator

Con j∗∗(P) = {σ ∈ [0,1]X ; μ∧ ·σ �= μ0},obtained by understanding the consistency with μ∧ as ‘non-incompatibility´ with it,only can come from the operator C∗∗(P) = {σ ; μ∧ ·σ ′ = μ0}, that is not also a con-sequence operator. Con j∗∗ is anti-monotonic, and verifies P ⊂ C(P) ⊂ Con j∗∗(P)(see [13]). Hence, it does not seem that conjecturing necessarily comes from a pre-vious form of logical deduction even if always includes deduction as a particularcase.

Which are the other types of conjectures? Since, Con j(P)−C(P) = {σ ; μ0 <σ < μ∧}∪{σ ∈Con j(P);σNCμ∧}, with NC shortening ‘not comparable under theordering ≤’, and μ0 �∈ Con j(P) since μ∧ ≤ μ1 = μ ′

0, it can be defined {σ ; μ0 <σ < μ∧} = Hyp(P), and {σ ;σNCμ∧ �≤ σ ′} = Sp(P), the subsets of hypotheses (orexplanative conjectures), and of speculations (or lucubrative conjectures), respectiv-elly. In this way, since Con j(P) = C(P)∪Hyp(P)∪ Sp(P) is clearly a partition, itcan be said that conjecturing just consists in deducing (obtaining consequences), ab-ducing (obtaining hypotheses), and inducing (obtaining speculations). In addition,the set Re f (P) = Con j(P)′ = {σ ;σ ′ ∈ C(P)}, can be called that of the refutationsof P.

101.5 Conclusion 703

Notice that the partition, [0,1]X = Con j(P) ∪ Re f (P) = (C(P) ∪ Re f (P)) ∪(Hyp(P)∪ Sp(P)), shows that C(P)∪ Re f (P) is the set of C-decidable elements,and Hyp(P) U Sp(P) is that of the C-undecidable elements of [0,1]X .

Remarks

a) Notice that by their own definition in the presented model, to find speculationscould force to make ‘jumps’ in the poset ([0,1]X ,≤).

b) If σ ∈ Sp(P), and provided μ∧ ·σ �= μ0, it is μ∧ ·σ ∈Hyp(P), and μ∧+σ ∈C(P).Hence, once jointly taken with μ∧, speculations can help to obtain hypothesesand consequences [17]. Something that seems in agreement with how peoplesometimes reason.

c) In a slighty different way, also in Con j∗(P) and in Con j∗∗(P), hypotheses andspeculations can be individuated by separating C(P) in them see ( [12] [13]).

d) All that has been presented can be mutatis mutandis translated to the setting ofortholattices [12] by just substituing the condition that in all P, μ∧ is not self-contradictory by the weaker condition that the ’intersection’ of all the premisesin P is not nul (in which case it is not self-contradictory, and it is only equivalentin the particular case of Boolean algebras).

e) With all that has been presented, it is possible to accept that most of reasoningcould be identified with ’deducing+abducing+inducing+refuting’. Of course, ifdeducing and refuting are nothing else than deductive processes, there is not yetclear how the undecidable elements can be systematically obtained in human rea-soning (see [14]). A way that sometimes people use, is by means of an analogywith a previously considered similar case.

101.5 Conclusion

This paper just constitutes a reflection on three topics that, differentiating the method-ologies of classical and fuzzy logics, could be important for the path towards Zadeh’sComputing with Words. Of course, provided this new subject is, also and addition-ally, viewed as an enlargement of fuzzy logic with the goal of representing morecomplex linguistic expressions than those considered in its current applications.

The first topic tries to support the view under which to deal with non-ambiguousexpressions in Natural Language, it is necessary a correct design of all the fuzzyterms representing the linguistic elements, from membership functions and connec-tives to modifiers and quantifiers, since there are no universal ways to represent them.Between lines, it is also pointed out that a more general concept of what is currentlyunderstood by a fuzzy algebra does be introduced for such enterprise. The axiom ofspecification for classical sets (see [15]) has not an immediate translation to fuzzysets since, for instance, imprecise predicates usually neither perfectly classify theuniverse of discourse, nor have the same membership function in all context.

The second topic tries to notice that there are alternative views allowing to proveas theorems what formerly has been taken either as axioms, or as failing properties,

704 101 On Some Classical Tenets and Fuzzy Logic

like are the important cases of the NC and EM principles. These alternatives help,for instance, to see fuzzy logic as founded in more solid grounds, and support the factthat, although its referent is imprecise, fuzzy logic offers precise ways of formallydealing with imprecision.

The third topic tries to show that there are new possibilities to go towards seeinglogic as more than the study of deductive systems, by means of some mathematicalformalizations of the non-deductive forms of reasoning, and once they are based onthe concept of ’conjecture’. Something that seems essential for any useful setting ofrepresentation, in which larger parts of Commonsense Reasoning could be mathe-matically modeled by, for instance, considering most of ’people’s reasoning’ as thesum ’conjecturing+refuting’.

Anyway, this paper is not conclusive but, thoughout some insights it only tries tobe a ’suggestive’ one.

Fig. 101.1. Surrounding Lotfi Zadeh at the Eleventh IEEE International Symposium onMultiple-Valued Logic in Oklahoma City, 1981: Settimo Termini, Ronald Yager, FrancescEsteva, NN, Sergei Ovchinnikov, Teresa Riera, Lorenzo Peña, Enric Trillas.

Acknowledgement. Work partially supported by the Foundation for the Advance-ment of Soft computing (Asturias, Spain), and CICYT (Government of Spain), underproject TIN 2011-29827-C02-01.

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