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Glimpsing at guessing
Available online at www.sciencedirect.com
ScienceDirect Fuzzy Sets and Systems 281 (2015) 32–43 www.elsevier.com/locate/fss
Glimpsing at guessing Enric Trillas ∗ European Centre for Soft Computing, Mieres, Asturias, Spain Received 6 October 2014; received in revised form 3 March 2015; accepted 15 June 2015 Available online 17 July 2015 To Professor Lotfi A. Zadeh, whose work opens new and fruitful ways towards glimpsing at Commonsense Reasoning
Abstract This paper just tries to present a new way to look at Commonsense Reasoning, at the end a manifestation of the natural phenomenon ‘thinking’, and, perhaps, the only skill human beings truly share for their survival. Since imprecision is pervasive in Commonsense Reasoning, the development of what is presented is done with fuzzy sets endowed with a very loose structure, is centered on the concepts of conjecture and refutation, allows to define not only informal and formal consequences, but hypotheses and speculations, and, specially, creative speculations. The study of Commonsense Reasoning is one for which researchers cannot be blind by believing that it is just a logical or mathematical subject. It is much more. © 2015 Published by Elsevier B.V. Keywords: Guessing; Refuting; Fuzzy sets; Ordinary reasoning
1. Introduction 1.1. Fifty years after the 1965 seminal paper by Lotfi A. Zadeh, the theoretical continuation of fuzzy logic seems to be partially stagnating, even if good theoretical papers continue to be published, and although most of them are on subjects of a purely mathematical interest. That, perhaps strong qualification, comes up from the author’s own view that beyond Zadeh’s ideas on Computing with Words and Perceptions (CwW), no actually new theoretical challenge is posed to the young researchers in fuzzy logic. For instance and just as a hint, no research project jointly involving CwW and ‘Brain Functioning’ seems to exist, when reasoning under imprecision and non-random uncertainty is the main goal Zadeh endowed fuzzy logic with since its very inception, and reasoning is a natural phenomenon generated in the brain. 1.2. Commonsense, ordinary, everyday, or common reasoning, made and expressed by means of a natural language, cannot be confused with the formal deductive reasoning of mathematical proofs, conducted with the powerful but artificial language of mathematics. People scarcely reason, that is, arrive at conclusions departing from some * Tel.: +34 985 45 65 45; mobile: +34 610 630 813.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.fss.2015.06.026 0165-0114/© 2015 Published by Elsevier B.V.
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premises, by formally deducing, but mainly almost always by guessing with imprecise concepts, doing conjectures of several types and by refuting [1] previously obtained conclusions. At the beginning of his book on Mathematics and Plausible Reasoning [18], George Pólya wrote ‘all our knowledge . . . consists of conjectures. There are, of course, conjectures and conjectures’. In any case, and before trying to prove something new, however in the process of searching for it, also mathematicians conduct reasoning like people do. Contrarily to formal deduction, where when the number of premises grows the number of conclusions cannot decrease, it is typical in common reasoning that new information, that is new premises, conducts to cancel some previously reached conclusions. A process of formal deduction can be represented by a graph of consecutive nodes, in which each node is linked with the former one by a clear rule of inference like it is, for instance, that of ‘Modus Ponens’, connecting them by an elemental, or atomic, deductive step. In any such process there are no jumps from a node to a non-consecutive node, but in common reasoning, jumps are more the rule than the exception. In it there are often nodes between which it lacks some elemental steps linking them. 1.3. Since natural languages are full of imprecisely used words and uncertain statements, their representation by fuzzy sets [2,17] is currently necessary for capturing what everyday reasoning consists in. If both the concepts of conjecture and refutation deserve to be posed by means of fuzzy sets, for this purpose it is crucial to depart with a previous and sufficiently general concept of what is an ‘algebra’, or calculus of fuzzy sets [3]. A character that, for instance, can allow the representation of the conjunction of two statements obtained from the previous representation of its components, as well as to arrive at results with a general enough validity. In such a wide setting, a mathematical modeling of common reasoning posed by means of fuzzy sets, allows to see the importance of ordinary, non-formal, deduction [4], and also to raise a conceptual clarification of guessing. It is not to be forgotten that natural language is not at all static, but dynamic. It is alive, submitted to evolution, and the current meaning on which its words are used is partially endowed with reminiscences coming from their use in previous universes of discourse and, sometimes, in different situations. For instance, very often and until the use of a predicate is currently fixed, it passes through some migrations [20] between different universes of discourse. Typically, usual linguistic terms cannot be defined by an ‘if and only if’ definition, but only described by some rules of use, and it is by this reason that they cannot be represented by crisp sets. Those that are measurable can be specified by fuzzy sets representing nothing else than a state of the predicate given by the information currently available on its use [17]. All this marks a very remarkable difference with what can be done with the precisely defined terms of mathematics and logic. 1.4. If fifty years after Zadeh’s paper [2], the way ahead of Fuzzy Logic is no doubt on the road of ‘Computing with Words’ [5], it seems recommendable to try and follow it in the path of Common Reasoning, and not only in that of Formal Deductive Reasoning. This paper is but a departure in this direction that shows nothing else than how it is possible to pose the study of Common Reasoning by a general enough view on the calculus, or algebra, of fuzzy sets. Finally, fuzzy logic is a way after the one opened by George Boole in 1854, to continue with the old 1650 ‘Calculemus!’ of Gottfried W. Leibniz. A way in which, and perhaps, fifty years after the seminal paper by Lotfi A. Zadeh on fuzzy sets, there could be the right moment for re-thinking and re-posing both fuzzy sets and fuzzy logic. A task that, currently, is waiting to receive the interest of researchers able to adopt a wider point of view. In what follows it will be supposed that common reasoning argues with both precise and imprecise, but not ambiguous, linguistic terms. 1.5. This paper contains nothing allowing to see it as part of the old debate on the justification of induction [19]. In this paper, common reasoning is just considered as an actually existing and observable natural phenomenon, deserving a scientific study for which a mathematical model is presented for trying to start investigations. 2. The mathematical setting 2.1. Reasoning not only consists in the formal deduction synthesized by Alfred Tarski’s consequence operators [6]. For instance, and contrarily to the growing monotonic character of these operators, it is often the case that as more information is reached, less conclusions can be drawn from them. That is, if with consequence operators, C, it holds the law of monotony: P ⊆ Q ⇒ C (P) ⊆ C (Q), in non-deductive ordinary reasoning it often happens that from P ⊆ Q it either follows or not C (Q) ⊆ C (P), that is, it holds either a law of anti-monotony, or there is no law
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of monotony at all. On the other side, consequence operators enjoy the law of closure: C (C (P)) = C (P), for whose validity it is essential that both sets P and C (P) deserve to be accepted as sets conveying ‘good’, ‘safe’, information, something that is not always the case in common reasoning where it cannot be always presumed that the conclusions only convey good, or safe, information. For instance, in searching for the explanation of the information conveyed by P, it is often the case of existing two contradictory hypotheses; that is, consistency cannot be a law for all kinds of reasoning. Most of the varieties of common reasoning are not deductive; at least, are not fully deductive. What are these kinds of ordinary reasoning? Traditionally, scientific reasoning is classified in deductive and inductive reasoning [7], consisting the first in the strict deployment of what is included in the premises, the statements containing all the available information, and the second consisting in the search for what is not included in them but could either explain it, or enlarge its informational content. Anyway, deduction is almost always seen as ‘formal deduction’ that is, what mathematicians do when proving theorems, where no step can be avoided and jumps are not allowed. Something not always occurring with common reasoning in which, when it is said ‘we deduce that’, jumps are more the rule than the exception. Besides reasoning ‘more geometrico’ is always the safest form of reasoning, to end a reasoning with the well known QED (‘quod erat demonstrandum’) is only fully correct when placed at the end of a mathematical proof. Notwithstanding, there is a variety of deductions that, not fully coincidental with the notion of proof, is close to it and constitutes the best form of common reasoning even if it is not possible to use the ending QED. Additionally, when speculating for new ideas, it is not possible to establish if the growing of the information will produce either monotony, or anti-monotony, that is, most of speculation is just non-monotonic since it lacks a law assuring that the number of conclusions will either constantly increase or decrease. Since speculation is also a manifestation of conjecturing or guessing [8], a mathematical model for it that can allow to capture the idea of what can be a ‘creative reasoning’ is, at least, actually convenient. The search for new ideas is crucial in both science and ordinary life. 2.2. To formalize all that, let us place what follows in the setting of a Basic Fuzzy Algebra (BFA) in a universe of discourse X [16] that, defined by a minimal set of laws, can guarantee a good level of generality to what could be deduced. That is (see next Section 2.3), in a five-tuple = ([0, 1]X , ≤; ·, +, ), where, for μ, σ, ρ in [0, 1]X , it holds: 1. μ ≤ σ ⇐⇒ μ(x) ≤ σ (x), for all x in X. 2. With the functions μ0 (x) = 0, and μ1 (x) = 1, for all x in X, the binary operations · and +, the unary , and the partial ordering ≤, jointly verifying, • μ · μ0 = μ0 · μ = μ0 ; μ1 · μ = μ · μ1 = μ • μ + μ0 = μ0 + μ = μ; μ + μ1 = μ1 + μ = μ1 • μ0 = μ1 ; μ1 = μ0 . • μ ≤ σ ⇒ σ ≤ μ . • μ0 ≤ μ ≤ μ 1 , for all μ, σ in [0, 1]X . 3. If μ ≤ σ , for all ρ in [0, 1]X , is: • μ · ρ ≤ σ · ρ; ρ · μ ≤ ρ · σ • μ + ρ ≤ σ + ρ; ρ + μ ≤ ρ + σ . 4. The subset {0, 1}X , endowed with the restriction of the order and the three operations defined in , 0 = ({0, 1}X , min, max, 1-id), is isomorphic to the Boolean algebra P(X), the power set of X. Notice that the pointwise ordering, ≤, between fuzzy sets is a partial, non-total, order, and that 0 is but isomorphic to the power set of X endowed with the structure of Boolean algebra given by the intersection, the union and the complement, of sets. Remarks. 1. It is · ≤ min ≤ max ≤ +, and only with · = min, and + = max, BFAs are lattices, namely De Morgan–Kleene algebras provided the ‘negation’ ( ) is strong, that is, verifies the law (μ ) = μ, for all μ in [0, 1]X .
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2. It is obvious that the Standard Algebras of fuzzy sets, those in which the binary operations · and + are functionally expressible by, respectively, a continuous t-norm and a continuous t-conorm, and the unary operation is functionally expressible by a strong negation function [15], are particular cases of BFA. Of course, the Boolean algebra of the crisp sets P (X), isomorphic to ({0, 1}X , min, max, 1-id), is also a BFA. 3. Notice that BFAs only show a minimal number of properties, and that no functional expressibility, commutative, associative, distributive, duality, etc., laws, are presumed to necessarily hold in them. 2.3. BFAs offer a wide mathematical framework to start dealing with ordinary reasoning, provided it is presumed that – The linguistic ‘and’ is represented by the binary operation · (conjunction), and the linguistic ‘or’ by the binary operation + (disjunction), – The linguistic ‘not’ is represented by the unary operation (negation), – The conditionals if/then are represented by the order ≤, that is, ‘If μ, then σ ’ is represented by μ ≤ σ . With that, two fuzzy sets μ and σ are supposed contradictory provided μ ≤ σ (if μ, then not-σ ), and a fuzzy set σ is self-contradictory if σ ≤ σ . From μ0 ≤ μ1 = μ0 , it follows that μ0 is self-contradictory. If μ represents a predicate P, and σ a predicate Q, it is supposed that μ ≤ σ represents that P is contradictory with Q, and μ ≤ μ represents that P is self-contradictory, that it holds, ‘If x is P, then x is not-P’ for all x in X. Notice that with classical sets there is only the self-contradictory set ø, the empty one [15], but that if, for instance, it is taken μ = 1 − μ, then it is μ ≤ μ ⇐⇒ μ ≤ μ1/2 , with μr the fuzzy sets constantly equal to r in [0, 1]. For each form of representing the negation, there are many self-contradictory fuzzy sets. Notice also that μ ≤ σ only indicates that μ is contradictory with σ , but not reciprocally. Since from μ ≤ σ follows (σ ) ≤ μ , it is sufficient that the negation verify σ ≤ (σ ) = σ (weak negation) for having σ ≤ μ , that also σ is contradictory with μ. In what follows, when saying ‘the fuzzy sets are contradictory’, without quoting their ordering, it will be supposed that the negation is weak. 3. Premises, résumé and conjectures 3.1. Definition. In a BFA , let us consider those sets P = {ρ1 , . . . , ρn } ⊆ [0, 1]X , such that: – For no pair ρi , ρj in P, is ρi ≤ ρj , and – Its résumé, ρ := ρ1 · (ρ2 · (ρ3 · (. . . (ρn−1 · ρn ) . . .), verifies ρ ρ , and ρ ρ. Notice that when ([0, 1]X , ≤, ·, +) is a lattice, that is, when · = min and + = max, the résumé ρ is coincidental with Inf P, always existing in [0, 1]X . In addition, and since the conjunction · = min is associative, in this case and in all those in which the associative law holds, parentheses can be avoided, it is ρ = ρ1 · . . . · ρn . These finite sets P will be called those of admissible premises, as their definition assumes minimal conditions for translating ‘safe’ information. Notice that the definition excludes both ρ = μ0 , and ρ = μ1 . Remark. Provided the conjunction · is not associative, the résumé ρ strictly depends on a previous numbering of the premises in P since, for instance and in general, ρ1 · (ρ2 · ρ3 ) = (ρ1 · ρ2 ) · ρ3 . Hence, in those cases in which the conjunction is not associative, all that follows depends on a numbering that should be previously fixed by attending at some reasonable criteria. Theorem 1. For all ρi in P, it is ρ ≤ ρi .
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Proof. It will be enough to prove the theorem for n = 3. Since it is ρ1 ≤ μ1 , and ρ2 · ρ3 ≤ μ1 , it follows ρ = ρ1 · (ρ2 · ρ3 ) ≤ ρ1 · μ1 = ρ1 , and from ρ2 ≤ μ1 follows ρ2 · ρ3 ≤ μ1 · ρ3 = ρ3 , and hence ρ ≤ ρ1 · ρ3 ≤ ρ3 , and it analogously follows ρ ≤ ρ2 . 2 Corollary 1. In a set of admissible premises, no pair of contradictory premises can exist. Proof. Provided there were in P two premises such that ρi ≤ ρj , from ρ ≤ ρi it will follow ρ ≤ ρj , and from ρ ≤ ρj , or ρj ≤ ρ , and hence the absurd ρ ≤ ρ will follow. 2 3.2. Definitions. 1. Given a set of admissible premises P, the set of its weak consequences [4] is C · (P) = {μ ∈ [0, 1]X ; ρ ≤ μ}, where the symbol · below C refers to the operation ‘and’ with which it is obtained the résumé of P. Provided it were · = min, it will be said that Cmin (P) is a set of strong consequences. 2. Let [μ, σ ] = {γ ; μ ≤ γ ≤ σ }, and (μ, σ ) = [μ, σ ] − {μ, σ }. Hence, C · (P) = [ρ, μ1 ]. Two fuzzy sets μ and σ in a chain of fuzzy sets are consecutive if and only if (μ, σ ) = ∅. It should be pointed out that usually, and in ordinary reasoning, neither the pairs of premises ρj , ρj +1 , nor the pairs ρ, ρi , are consecutive fuzzy sets. Anyway, and in what could be possible, it is better to try to ‘purify’ the set P of admissible premises, like it is always done in the case of mathematics, where it is systematically assumed that, to count with sets of minimal axioms, those pairs be of consecutive (crisp) statements. Of course, in common reasoning this is much more difficult that in the case of mathematics, and even if in it is not always an easy task. Given a set of admissible premises P, – The set of its C · -conjectures is Conj · (P) = {μ; ρ μ }, and – The set of its C · -refutations is Ref · (P) = {σ ; ρ ≤ σ }. Obviously, [0, 1]X = Conj · (P) ∪ Ref · (P), and Conj · (P) ∩ Ref · (P) = ∅, hence, Conj · (P) = Ref · (P)c , both sets are complementary. Then, a conjecture is a fuzzy set whose negation is not weakly deducible from the premises, that is, from the currently available information conveyed by them. Since it is obvious that C · (P) = C · ({ρ}), Conj · (P) = Conj · ({ρ}), and Ref · (P) = Ref · ({ρ}), it is both clear that all these sets are just defined by the résumé of P, and why it receives such name. Notes. 1. If μ and σ are in {0, 1}X , that is both represent crisp sets A and B, respectively, it is ρ σ ⇔ A ∩ B = ∅, ρ ≤ σ ⇔ A ∩ B = ∅, ρ ≤ ρ ⇔ A = ∅, and ρ ≤ ρ ⇔ Ac = ∅ ⇔ A = X. Hence, with crisp sets conjectures are those sets with a non-empty intersection with the premises’ résumé, refutations are those with an empty intersection, and the only self-contradictory crisp set is the empty set. For what concerns the conditions the résumé is submitted to, notice that they are A = ∅, and A = X. 2. Were ρ = μ1 not excluded in the definition of a set of admissible premises through the condition ρ ρ, from: μ1 = μ ⇔ it is not (μ1 ≤ μ ) ⇔ it is not (μ ≤ μ0 ) ⇔ μ0 < μ, it will follow Conj · (P) = [0, 1]X − {μ0 }, and C · (P) = {μ; μ1 ≤ μ} = {μ1 }. The ‘empty’ fuzzy set μ0 is never in Conj · (P) since it is not ρ μ0 = μ1 , but the ‘full’ fuzzy set μ1 is always since it truly holds ρ μ1 = μ0 . 3.3. Theorem 2. No pair of contradictory weak consequences can exist in C · (P ). Proof. Provided σi and σj in C · (P) verify σi ≤ σj , from ρ ≤ σi and ρ ≤ σj , it will follow ρ ≤ σj and σj ≤ ρ , and the absurd ρ ≤ ρ . 2
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Theorem 3. P ⊆ C · (P ), for any set P of admissible premises. It is said that the operators C · are extensive. Proof. It follows immediately from ρ ≤ ρi , for all premise ρi in P.
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Theorem 4. If P and Q are sets of admissible premises, and it is P ⊆ Q, then C · (P ) ⊆ C · (Q). It is said that the operators C · are monotonic. Proof. If it is P = {ρ1 , . . . , ρn }, it is Q = {ρ1 , . . . , ρn , ρn+1 , . . . , ρm }, with résumés ρQ , and ρP , respectively verifying ρQ ≤ ρP . To prove it, let’s just consider the easy case with n = 3, m = 4, in which, ρP = ρ1 · (ρ2 · ρ3 ), and ρQ = ρ1 · (ρ2 · (ρ3 · ρ4 )). From ρ3 · ρ4 ≤ ρ3 , follows ρ2 · (ρ3 · ρ4 ) ≤ ρ2 · ρ3 , and the ρQ ≤ ρ1 · (ρ2 · ρ3 ) = ρP ·. Hence, provided it is ρP ≤ μ, it follows ρQ ≤ μ. 2 Recall that being ≤ a partial order, it enjoys transitivity. Theorem 5. If P and Q are sets of admissible premises, and it is P ⊆ Q, then Conj · (Q) ⊆ Conj · (P ). It is said that the operators Conj · are anti-monotonic. Proof. Recall that it is ρQ ≤ ρP . If σ ∈ Conj · (Q), it is ρQ σ , and if it were ρP ≤ σ , it will follow the absurd ρQ ≤ σ . Hence, it should be ρP σ . 2 Theorem 6. Provided the negation ( ) verifies the law μ ≤ (μ ) , for all μ ∈ [0, 1]X , it is P ⊆ C · (P ) ⊆ Conj · (P ), for all set of admissible premises P. Proof. If it is ρ ≤ μ, and provided it were ρ ≤ μ , it will follow μ ≤ μ ≤ ρ , and ρ ≤ ρ , that is absurd. Hence it should be ρ μ . 2 Were the negation a weak one, that is, submitted to the law μ ≤ (μ ) , and also if it is strong, μ = (μ ) , weak consequences are a particular case of conjectures. Corollary 2. Provided the negation is a weak one, if μ ∈ C · (P ) then μ ∈ / C · (P ). It is said that the operator C · is consistent. Proof. It just follows from Theorem 4, since if it is ρ ≤ μ, and if it were ρ ≤ μ , it will follow ρ ≤ ρ . Hence, it should be ρ μ . 2 Under weak negations, the negation of a weak consequence is never a weak consequence. Theorem 7. Operators Ref · are not extensive, and they are monotonic. Proof. Were P ⊆ Ref · (P), all premises ρi will be refutations, or ρ ≤ ρi , and since ρ ≤ ρi implies ρi ≤ ρ , it will follow the absurd ρ ≤ ρ . If P ⊆ Q, from ρQ ≤ ρP follows that when ρP ≤ σ it is ρQ ≤ σ . 2 Theorem 8. For all conjunction ·, is: 1) Conj · (P ) ⊆ Conjmin (P ); 2) Ref min (P ) ⊆ Ref · (P ); 3) Cmin (P ) ⊆ C · (P ). Proof. Since · ≤ min, a résumé of P obtained with · is always below the one (of the same P) obtained with min. Hence, it immediately follows 3. Since, μ ∈ Ref · (P) ⇔ μ ∈ C · (P), it follows 2. Finally, 1 follows by complementation in 2. 2 Notice that there are no less weak than strong consequences and refutations but that, on the contrary, the largest set of conjectures is obtained with min. Strongly deducing, or refuting, corresponds with the possibility of producing as many conjectures as possible.
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Remarks. 1. The set [ρ, μ1 ] is not necessarily finite and, consequently, it could be not easy to define its résumé by the conjunction of all its elements. Nevertheless, provided the résumé were taken as the infimum, then that of C · (P) will be the same of P, since Inf [ρ, μ1 ] = ρ, and also C · (P) can be taken as a set of admissible premises. This identification of the résumé can be safely done in the particular case in which it is · = min, and then Cmin can be applied to Cmin (P). Thus, since ‘P ⊆ Cmin (P) ⇒ Cmin (P) ⊆ Cmin (Cmin (P))’, and σ ∈ Cmin (Cmin (P)) means Inf Cmin (P) = ρ ≤ σ , it follows σ ∈ Cmin (P), and Cmin (Cmin (P)) ⊆ Cmin (P). Hence, it is Cmin (Cmin (P)) = Cmin (P), the operator Cmin is a closure and, consequently, it is a Tarski’s, or strong, consequence operator. In the case · = min, it can be stated that the consequences are formal, or strong consequences, and that it is possible to go from the résumé ρ up to any consequence σ by following a path ρ = σ0 ≤ σ1 ≤ . . . ≤ σp−1 ≤ σp = σ , of length p, and with consecutive fuzzy sets. That is, by means of a proof. This does not mean, of course, neither that only a single proof conducting from ρ to σ does exist, nor that one can be immediately known, and selecting one with a minimum of steps is a mark of mastering the art of proving. 2. What happens in general when C · (P) is a finite set, and the conjunction · is not coincidental with min? If, for instance, the conjunction · is associative, with P = {ρ1 , . . . , ρn }, and C · (P) = {ρ1 , . . . , ρn , γ1 , . . . , γp }, since P ⊆ C · (P), then it is ρC·(P ) = ρ1 · ρ2 · . . . · ρn · γ1 · γ2 · . . . · γp ≤ ρ1 · ρ2 · . . . · ρn = ρP , an inequality not guaranteeing that ρC·(P ) verifies the conditions required for considering C · (P) an admissible set of premises. Consequently, in general it is not possible to guarantee that the operator C · can be applied to C · (P). Only the operator Cmin is always a mapping A(·) → A(·), with A(·) the set of all sets of admissible premises contained in [0, 1]X with · = min, but, in general, the operators C ·, with · = min, are just mappings A(·) → [0, 1]X , and, hence, the sets C · (P) cannot be usually taken as ‘safe’ information if the conjunction · is not the conjunction min. This is a difference between strong and weak consequences, between formal and common or informal deduction. Since in the classical case (that of {0, 1}X ) only the operator Cmin is used, the other operators C · could correspond to the informal deduction in ordinary reasoning. Analogously, also the operators Conj ·, and Ref · are mappings A(·) → [0, 1]X , the sets Conj · (P) and Ref · (P) cannot be taken as sets of admissible premises. Perhaps, informal or common deduction is done when · = min. 3. Non-monotonic reasoning is not new at all, and it deserved remarkable studies in Artificial Intelligence [23]. This paper does not consider methods to arrive at non-monotonic conclusions, but only which conjectures or refutations show, or not, monotony. In [24] an initial study of the relationships between conditionality and monotony is shown. 4. Deductive paths, hypotheses and guessings 4.1. A deductive path to reach a weak consequence σ from a set of admissible premises P, is a chain ρ = α0 ≤ α1 ≤ . . . ≤ αp−1 ≤ αp = σ, where all αi belong indeed to C · (P). A deductive path is a deductive proof, or simply a proof of σ , provided all pairs (αi , αi+1 ) are of consecutive fuzzy sets. Of course, the number p of steps in a deductive path, or in a proof, is not a constant for it, and proving a consequence by means of a minimal number p of these steps is part of the ‘art of proving’. Notice that in no proof can exist jumps, but that in a deductive path that is not a proof there can lack intermediate steps and that, because of these jumps, it can happen that between some nodes αi and αi+1 , it exists a fuzzy set β such that, for instance, it is αi ≤ β, but β αi+1 , with which the chain is broken and it is not actually deductively conducting from ρ to σ . It does not necessarily mean that it is impossible to prove σ , but that the proof is not yet complete. Since the more restrictive form of deduction is given by the operator Cmin , it is with it that are conducted the safest ways of deducing whenever the linguistic conjunction admits to be represented by min. Remark. What has been presented is degree-free, it is just ‘crisp reasoning’ with fuzzy sets and, hence, not sufficient for a complete covering of common reasoning where everything, and not only elemental statements ‘x is P’, is allowed
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to be graded; it lacks, in particular, a definition of graded conjecture and graded refutation. It lacks to add, at least and between the steps of the deductive paths, degrees up to which each step can hold, that is, a concept of ‘gradual consequence’ [13]. For instance, with ρ ≤r σ ⇔ degree (ρ, σ ) = r ∈ [0, 1], meaning that σ is a consequence of P up to the degree r, that is, changing the crisp relation ≤ to a more general fuzzy relation ≤r reducing to ≤ if r = 1, and to if r = 0 [21]. It still lacks to consider which weight the conjectures or the refutations inherit from the degree up to which the premises’ résumé can hold [14]. With the study of the gradable cases is how it is possible to go further than just crisp conjectures [13] by opening the door towards, for instance, graded consequences [22] that cannot be drawn in the domain of classical logic [21]. 4.2. Since, provided the negation is a weak one, it holds that C · (P) ⊆ Conj · (P), in this case it makes sense to analyze the difference-set Conj · (P) − C · (P) = {σ ; ρ σ & ρ σ } = {σ ; ρ σ & σ < ρ} ∪ {σ ; ρ σ &ρ nc σ }, with nc shortening ‘not comparable under the partial order ≤’. Let us call hypotheses the conjectures in the first of these two sets, and speculations those in the second. The denotation ‘hypothesis’ is suitable since it is from σ that ρ strictly follows in the first case, that is, a deductive chain can exist from σ to the résumé ρ showing that σ ‘explains’ the premises. In the second case, the denotation comes from the fact that no chain can link the conjecture σ with ρ. In principle, a hypothesis is a conjecture that can be weakly deductively reached from ρ by a backwards chain of the type ρ = α0 ≥ α1 ≥ . . . ≥ α0 = σ , although this is not possible in the case of a speculation. Let us designate by Hyp · (P) and by Sp · (P), respectively, these sets that are, obviously, disjoint with the set C · (P), and that give the partition: Conj · (P) = C · (P) ∪ Hyp · (P) ∪ Sp · (P). Theorem 9. Provided the negation is a weak one, and ρ = σ , σ ∈ Hyp · (P ) ⇔ ρ ∈ C · ({σ }). Proof. It is C · (P) ⊆ Conj · (P). If σ ∈ Hyp · (P), it is σ < ρ and ρ σ , hence σ ∈ C · (P). The reciprocal is also obvious. 2 Since it is potentially possible to find ‘backwards’ deductive paths from ρ to σ , it is for why hypotheses deductively ‘explain’ the résumé, that is, the informational content of P. Theorem 10. Operators Hyp · are anti-monotonic. Proof. From P ⊆ Q, and ρQ ≤ ρP , if σ ∈ Hyp · (Q), from σ < ρQ , follows σ < ρP . Since it is ρQ σ , provided it were ρP ≤ σ it will follow the absurd ρQ ≤ σ , and, hence it should be ρP σ . Thus Hyp · (Q) ⊆ Hyp · (P). 2 More premises imply no more hypotheses, that is, as more information is available less hypotheses are possible. With respect to the operators Sp · (P), notice that in the crisp case, a set S is a speculation of a set of admissible premises P with résumé R, provided R ∩ S = ø, and neither R ⊆ S, nor S ⊆ R. In this case, it is really easy to find Venn diagrams with sets of premises P and Q, where neither the monotonic, nor the anti-monotonic property, holds. The operators Sp · are properly non-monotonic. As ρ σ ⇔ Either σ < ρ, or ρ nc σ , let us check what happens in the first case if σ is a hypothesis. In this case, from σ < ρ, is ρ ≤ σ , and then it follows the absurd ρ < ρ, and, finally, the set of hypotheses is reduced to Hyp · (P) = {σ ; ρ nc σ & σ < ρ}. If σ ∈ Hyp · (P), from σ < ρ, it follows ρ ∈ C · ({σ }). Thus, to prove that some conjecture σ is not a hypothesis for P, that is, to falsify σ , it suffices to find a consequence of P that is not among the consequences of {σ }. 4.3. It holds, obviously, Sp · (P) = {σ ; ρ > σ & ρ nc σ } ∪ {σ ; ρ nc σ & ρ nc σ }.
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Calling, respectively, pseudo-guessings (PG), and creative-guessings (CG), those speculations in each of these sets, the partition Sp · (P) = PG · (P) ∪ CG · (P) is obtained. As it is σ < ρ, the negation σ of a pseudo-guessing is a hypothesis, and potentially allows to reach σ , through ‘backwards deductive ways’ starting in ρ. Nevertheless, no creative-guessing can neither be deductively reached from ρ, nor from ρ . Notice that, provided σ is a pseudoguessing, it is ρ ∈ C · ({σ }), but neither ρ ∈ C · ({σ }), nor σ ∈ C · ({ρ}) = C · (P). Theorem 11. Provided the disjunction + is idempotent (μ + μ = μ, for all μ in [0, 1]X ), operators PG · are antimonotonic. Proof. From, P ⊆ Q ⇒ ρQ ≤ ρP , and σ ∈ PG · (Q) ⇔ σ < ρQ & ρQ nc σ , follows σ < ρP . Provided it were ρP ≤ σ , it will follow the absurd ρQ ≤ σ . Provided it were σ < ρP , it will follow ρP ≤ σ + ρP ≤ ρP + ρP = ρP , that is also absurd. Hence, it should be ρP nc σ , and σ ∈ PG · (P), that is, PG · (Q) ⊆ PG · (P). 2 It should be pointed out that this theorem is practically limited to those cases in which + = max. Hence, creativeguessings seem to be the only speculations that, in general, are properly non-monotonic. From the point of view of deductive reasoning, they are ‘wild’ conjectures. Theorem 12. 1) It is ρ + μ ∈ C · (P ), and μ + ρ ∈ C · (P ), for any μ ∈ [0, 1]X . 2) If μ ∈ Conj · (P ) is such that ρ · μ = ρ, and μ · ρ = ρ, then μ · ρ ∈ Hyp · (P ), and ρ · μ ∈ Hyp · (P ). Proof. 1) Obvious, since ρ ≤ ρ + μ, and ρ ≤ μ + ρ. 2 In particular, if μ is either a pseudo-guessing, or a creative guessing, weak consequences are simply reached by ρ + μ and μ + ρ. 2) Obvious, since μ is a conjecture, and it is ρ · μ ≤ ρ, and μ · ρ ≤ ρ, without holding the equalities. 2 In particular, if μ is either a pseudo-guessing or a creative-guessing, whose conjunctions with ρ do not reproduce ρ, hypotheses are simply reached by ρ · μ and μ · ρ. Notice that when · = min, the two conditions in 2 just mean ρ μ, that μ is not a weak consequence. 2 Theorem 12 is actually interesting when μ ∈ Sp · (P), since in this case μ is obtained, and possibly not by deduction, from P and then both weak consequences and hypotheses are reached by directly reasoning on the informational content conveyed by P. Whenever μ ∈ CG · (P), part 2 is especially important since it shows how a true ‘new idea’ can help to ‘explain P’, something well enough corresponding with the intuition of what people sometimes do to answer some questions. It is guessing what often conducts to the key, μ, of the question on which ρ is known. Guessing is actually relevant for reasoning. 5. On the role of analogy in reasoning Although the realm of metaphor is poetry, analogy or similitude between concepts, sometimes in the form of metaphors, it appears very often in common reasoning and, mainly, in either philosophical reasoning, or in the exploratory scientific one for obtaining either explanations, or consequences by speculation. At this respect, the question if analogy is, or is not, a new and different type of reasoning seems to be relevant, and without the pretence of closing the subject, let us show that with a reasonable and simple interpretation of what analogy consists in, it can be proven that it just conducts to either conjectures or refutations. The first problem is to define when it can be asserted that a fuzzy set μ is analogous to another σ . Let us suppose that there is a family A of functions f: [0, 1]X → [0, 1]X , allowing to state that μ and f(μ) are A-analogous fuzzy sets or, less specifically, analogous. Thus, with each fuzzy set μ there is the family {f(μ); f ∈ A} of the fuzzy sets analogous to μ, and with each set of admissible premises P and each f in A, there is the family of analogous fuzzy sets f(ρ1 ), . . . , f(ρn ), as well as the fuzzy set f(ρ) analogous to the résumé ρ = ρ1 · . . . · ρn , and
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possibly not analogous with f(ρ1 ) · . . . · f(ρn ), unless f were an isomorphism for the conjunction (·) in which case it will be f(ρ1 ) · . . . · f(ρn ) = f(ρ1 · . . . · ρn ) = f(ρ). Since the functions f defined by f (μ) = (f(μ)) , for all μ ∈ [0, 1]X , can only verify either id ≤ f (μ ≤ f (μ) for all μ), or id f , there are only the two possibilities: – ρ ≤ f (ρ), under which f(ρ) is a refutation of P, and – ρ f (ρ), under which f(ρ) is a conjecture made from P. That is, based on the given concept of analogy, only new refutations or conjectures can be obtained. Even more, provided it is id f , it only can be id ≤ f, f < id, or the functions id and f are not comparable under the pointwise order ≤, then: – In the first case, it is ρ ≤ f(ρ), and the conjecture f (ρ) is a consequence of P that is analogous to ρ, – In the second case, it is f(ρ) < ρ, and the conjecture f(ρ) is a hypothesis for P that is analogous to ρ, – In the third case, ρ and f(ρ) are not ≤ – comparable, and the conjecture f(ρ) is a speculation made from P. Hence, but depending on the given definition of analogy, it seems that analogy is nothing else but a methodology for conjecturing and refuting, that is, for reasoning. Thus, analogy appears as a methodology for guessing and refuting. For instance, if σ is a creative-guessing with which ρ · σ is a hypothesis, and it is f < id, it is f(ρ · σ ) < ρ · σ ≤ ρ: the fuzzy set f(ρ · σ ), is a hypothesis of P analogous to the previous one ρ · σ . As a metaphor is but a statement reflecting something analogous to something, and even in a fuzzy form, the use of metaphors is not at all avoidable for reasoning. It is an older form of reasoning that, nevertheless, should be kept under control since its actual danger lies in the ‘great distance’ often existing between, for instance, the metaphor, and the persecuted goal. At the end, and as it happened many times in the history of science, the final role of the analogy/metaphor lies in the experience and specific knowledge of who is considering it. Remark. What this section does not, and cannot pretend, is the study of any analogical reasoning’s modality [25]. It just tries to show that it is difficult to believe that conclusions of a different type than conjectures and refutations could be obtained through similarity. Were it the case, similarity will appear as a ‘tool’ for reasoning. 6. Conclusion 6.1. This paper is no more than a glimpse at common reasoning just trying to show that some, at least historically relevant treats of it, can be posed by means of fuzzy sets and under minimal conditions on its calculus. Notwithstanding, there are many things still waiting for clarification. For instance, establishing good, concrete and practically useful mathematical models for representing conjunctions that, in Natural Language where time almost always intervenes, often cannot be commutative as it is the classical associative conjunction. Also for instance, the extent of the use of the associative law is not clear enough in Natural language where it seems to appear, when it is the case, expressed by commas that cannot be moved without constraint in most environments. It should be investigated on which conditions such kind of properties can be stipulated since only in a setting allowing them, a mathematical model with these properties could be realistically taken into account. 6.2. What is important to realize is the possibility of capturing in which type of reasoning actually lies the phenomena of anti-monotony and non-monotony. As the Nobel Laureate Sir Peter B. Medawar wrote in 1984 [9], “No process of logical reasoning can enlarge the information content of the axioms and premises or observation statements from which it proceeds”, to shortly express that the clothes of formal deductive logic are too short and also too large, for a good dressing of creative reasoning, that monotonic deductive logic is not sufficient to mathematically model the creative or, at least
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ampliative, processes of reasoning science requires. And it seems less sufficient again when analogy, imprecision and uncertainty play a crucial role in reasoning. For a true progress of Computing with Words, it is doubtful that its study can be just carried out by logical, formal, methods. It seems plausible to think that such study needs a combination of controlled experimentation and mathematical modeling to be checked against actual common reasoning and natural language [10]. Like it is typical of the scientific reasoning as it was said by the Nobel Laureate Eugene Wigner, with its “unreasonable effectiveness of mathematics in the natural sciences” [11]. At the end, both Natural Language and Commonsense Reasoning are but natural phenomena and deserve to be accordingly studied to discover, for instance, which properties can be stipulated and on which conditions such stipulation is possible. 6.3. All that is presented in this paper is to be kept as a set of theoretical hypotheses until they can be systematically contrasted with language and reasoning. That is, by following a scientific path. The path that allowed, for instance, that the young and then scarcely known scientist Georges Lemaitre, contradicted Albert Einstein referring to the universe’s inflation and that, at the end, Einstein openly recognized that he was mistaken [12]. Without the possibility of contrasting models against reality, the emergence of current science and technology had actually not been possible. In Science there are neither dogmas, nor absolute authorities, and everything is subjected to either falsification, or to some enlargement covering more knowledge; science is a non-stop path for which mathematical models are but a foothold, even if a very important one. Efforts to introduce reasoning in a pre-fabricated mold are purposeless and, on the contrary, the big goal is to find suitable molds in which reasoning, or its most significant parts, can be comfortably included. At the end it should not to be forgotten how difficult is to believe that common reasoning could be submitted to a complete formalization; common reasoning is all people has for surviving and it is often supplied not only by language but by post interpreted memory, blurred imaging, case-based analogy, etc. Complete processes of reasoning are more than what is presented in this paper, and it is perhaps arrived a good moment for a methodological turn in the study of reasoning under imprecision and non-random uncertainty, a turn towards a kind of ‘physics of language and reasoning’. If the usefulness of theoretical concepts in the technological applications is certainly important, and to some extent applications play the role of experimentation in the modern techno-sciences [10], to advance in the study of the natural phenomenon ‘reasoning’, it can be argued that intentional controlled experimentation is suitable for it, and that it should be done under the guide of numerical measurements supported by mathematical models of, at least, some parts of the phenomenon. Acknowledgement This paper is partially funded by the Foundation for the Advancement of Soft Computing, and by the Spanish Government project MICIIN/TIN 2011-29827-C02-01. References [1] K.R. Popper, Conjectures and Refutations: The Growth of Scientific Knowledge, Harper and Row, New York, 1968. [2] P.L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353. [3] I. García-Honrado, E. Trillas, On an attempt to formalize guessing, in: R. Seising, et al. (Eds.), Soft Computing in Humanities and Social Sciences, Springer, Berlin, 2012, pp. 237–255. [4] E. Trillas, A model for ‘crisp reasoning’ with fuzzy sets, Int. J. Intell. Syst. 27 (2012) 859–872. [5] L.A. Zadeh, Computing with Words. Principal Concepts and Ideas, Springer, New York, 2012. [6] A. Tarski, Fundamental concepts of the methodology of deductive sciences, in: Logic, Semantics, Metamathematics, Oxford University Press, 1956. [7] B. Bosanquet, Logic or the Morphology of Knowledge, vol. I, Clarendon Press, London, 1911. [8] S. Watanabe, Knowing & Guessing, John Wiley & Sons, New York, 1969. [9] P.B. Medawar, The Limits of Science, Harper, London, 1984. [10] E.H. Mamdani, E. Trillas, Correspondence between an experimentalist and a theoretician, in: E. Trillas, et al. (Eds.), Combining Experimentation and Theory, Springer, Berlin, 2012, pp. 1–18. [11] E. Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Commun. Pure Appl. Math. 13 (1) (1960) 1–14. [12] C. Rovelli, La realtà non è come ci appare: La struttura elementare delle cose, Raffaello Cortina Editore, Milan, 2014. [13] J.L. Castro, E. Trillas, Tarski’s fuzzy consequences, in: Proceedings International Fuzzy Engineering Symposium, 1991, pp. 70–81.
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[14] E. Trillas, C. Moraga, G. Triviño, Weighting the support conjectures inherit from premises, IEEE Trans. Fuzzy Syst. (2014), http://dx.doi.org/ 10.1109/TFUZZ.2014.2336677, forthcoming. [15] E. Trillas, Non-contradiction, excluded-middle, and fuzzy sets, in: Di Gesù, et al. (Eds.), WILF 2009, in: LNCS, vol. 5571, Springer, Heidelberg, 2009, pp. 1–11. [16] A. Pradera, E. Trillas, E. Renedo, An overview on the construction of fuzzy set theories, New Math. Nat. Comput. 1 (3) (2005) 329–358. [17] E. Trillas, S. Termini, C. Moraga, A naïve way of looking at fuzzy sets, Fuzzy Sets Syst. (2015), http://dx.doi.org/10.1016/j.fss.2014.07.016, forthcoming. [18] G. Pólya, Mathematics and Plausible Reasoning, vol. I: Induction and Analogy in Mathematics, Princeton University Press, Princeton, New Jersey, 1954. [19] H. Reichenbach, Experience and Prediction, University of Chicago Press, 1938. [20] E. Trillas, C. Moraga, A. Sobrino, On ‘family resemblances’ with fuzzy sets, in: Proceedings IFSA-EUSFLAT Conference, Lisbon, 2009, pp. 897–902. [21] E. Trillas, A.R. de Soto, On graded conjectures, ECSC’s report of research (in preparation), 2014. [22] J. Pavelka, On fuzzy logic, I, Z. Math. Log. Grundl. Math. 25 (1979) 45–52. [23] D.M. Gabbay, C.H. Hogger, J.A. Robinson (Eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3, Oxford University Press, 1994. [24] S. Cubillo, E. Trillas, Characterizing non-monotonic fuzzy relations, Soft Comput. 1 (1997) 162–165. [25] H. Prade, G. Richard, From analogical proportion to logical proportions, Log. Univers. 7 (2013) 441–505.
ScienceDirect Fuzzy Sets and Systems 281 (2015) 32–43 www.elsevier.com/locate/fss
Glimpsing at guessing Enric Trillas ∗ European Centre for Soft Computing, Mieres, Asturias, Spain Received 6 October 2014; received in revised form 3 March 2015; accepted 15 June 2015 Available online 17 July 2015 To Professor Lotfi A. Zadeh, whose work opens new and fruitful ways towards glimpsing at Commonsense Reasoning
Abstract This paper just tries to present a new way to look at Commonsense Reasoning, at the end a manifestation of the natural phenomenon ‘thinking’, and, perhaps, the only skill human beings truly share for their survival. Since imprecision is pervasive in Commonsense Reasoning, the development of what is presented is done with fuzzy sets endowed with a very loose structure, is centered on the concepts of conjecture and refutation, allows to define not only informal and formal consequences, but hypotheses and speculations, and, specially, creative speculations. The study of Commonsense Reasoning is one for which researchers cannot be blind by believing that it is just a logical or mathematical subject. It is much more. © 2015 Published by Elsevier B.V. Keywords: Guessing; Refuting; Fuzzy sets; Ordinary reasoning
1. Introduction 1.1. Fifty years after the 1965 seminal paper by Lotfi A. Zadeh, the theoretical continuation of fuzzy logic seems to be partially stagnating, even if good theoretical papers continue to be published, and although most of them are on subjects of a purely mathematical interest. That, perhaps strong qualification, comes up from the author’s own view that beyond Zadeh’s ideas on Computing with Words and Perceptions (CwW), no actually new theoretical challenge is posed to the young researchers in fuzzy logic. For instance and just as a hint, no research project jointly involving CwW and ‘Brain Functioning’ seems to exist, when reasoning under imprecision and non-random uncertainty is the main goal Zadeh endowed fuzzy logic with since its very inception, and reasoning is a natural phenomenon generated in the brain. 1.2. Commonsense, ordinary, everyday, or common reasoning, made and expressed by means of a natural language, cannot be confused with the formal deductive reasoning of mathematical proofs, conducted with the powerful but artificial language of mathematics. People scarcely reason, that is, arrive at conclusions departing from some * Tel.: +34 985 45 65 45; mobile: +34 610 630 813.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.fss.2015.06.026 0165-0114/© 2015 Published by Elsevier B.V.
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premises, by formally deducing, but mainly almost always by guessing with imprecise concepts, doing conjectures of several types and by refuting [1] previously obtained conclusions. At the beginning of his book on Mathematics and Plausible Reasoning [18], George Pólya wrote ‘all our knowledge . . . consists of conjectures. There are, of course, conjectures and conjectures’. In any case, and before trying to prove something new, however in the process of searching for it, also mathematicians conduct reasoning like people do. Contrarily to formal deduction, where when the number of premises grows the number of conclusions cannot decrease, it is typical in common reasoning that new information, that is new premises, conducts to cancel some previously reached conclusions. A process of formal deduction can be represented by a graph of consecutive nodes, in which each node is linked with the former one by a clear rule of inference like it is, for instance, that of ‘Modus Ponens’, connecting them by an elemental, or atomic, deductive step. In any such process there are no jumps from a node to a non-consecutive node, but in common reasoning, jumps are more the rule than the exception. In it there are often nodes between which it lacks some elemental steps linking them. 1.3. Since natural languages are full of imprecisely used words and uncertain statements, their representation by fuzzy sets [2,17] is currently necessary for capturing what everyday reasoning consists in. If both the concepts of conjecture and refutation deserve to be posed by means of fuzzy sets, for this purpose it is crucial to depart with a previous and sufficiently general concept of what is an ‘algebra’, or calculus of fuzzy sets [3]. A character that, for instance, can allow the representation of the conjunction of two statements obtained from the previous representation of its components, as well as to arrive at results with a general enough validity. In such a wide setting, a mathematical modeling of common reasoning posed by means of fuzzy sets, allows to see the importance of ordinary, non-formal, deduction [4], and also to raise a conceptual clarification of guessing. It is not to be forgotten that natural language is not at all static, but dynamic. It is alive, submitted to evolution, and the current meaning on which its words are used is partially endowed with reminiscences coming from their use in previous universes of discourse and, sometimes, in different situations. For instance, very often and until the use of a predicate is currently fixed, it passes through some migrations [20] between different universes of discourse. Typically, usual linguistic terms cannot be defined by an ‘if and only if’ definition, but only described by some rules of use, and it is by this reason that they cannot be represented by crisp sets. Those that are measurable can be specified by fuzzy sets representing nothing else than a state of the predicate given by the information currently available on its use [17]. All this marks a very remarkable difference with what can be done with the precisely defined terms of mathematics and logic. 1.4. If fifty years after Zadeh’s paper [2], the way ahead of Fuzzy Logic is no doubt on the road of ‘Computing with Words’ [5], it seems recommendable to try and follow it in the path of Common Reasoning, and not only in that of Formal Deductive Reasoning. This paper is but a departure in this direction that shows nothing else than how it is possible to pose the study of Common Reasoning by a general enough view on the calculus, or algebra, of fuzzy sets. Finally, fuzzy logic is a way after the one opened by George Boole in 1854, to continue with the old 1650 ‘Calculemus!’ of Gottfried W. Leibniz. A way in which, and perhaps, fifty years after the seminal paper by Lotfi A. Zadeh on fuzzy sets, there could be the right moment for re-thinking and re-posing both fuzzy sets and fuzzy logic. A task that, currently, is waiting to receive the interest of researchers able to adopt a wider point of view. In what follows it will be supposed that common reasoning argues with both precise and imprecise, but not ambiguous, linguistic terms. 1.5. This paper contains nothing allowing to see it as part of the old debate on the justification of induction [19]. In this paper, common reasoning is just considered as an actually existing and observable natural phenomenon, deserving a scientific study for which a mathematical model is presented for trying to start investigations. 2. The mathematical setting 2.1. Reasoning not only consists in the formal deduction synthesized by Alfred Tarski’s consequence operators [6]. For instance, and contrarily to the growing monotonic character of these operators, it is often the case that as more information is reached, less conclusions can be drawn from them. That is, if with consequence operators, C, it holds the law of monotony: P ⊆ Q ⇒ C (P) ⊆ C (Q), in non-deductive ordinary reasoning it often happens that from P ⊆ Q it either follows or not C (Q) ⊆ C (P), that is, it holds either a law of anti-monotony, or there is no law
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of monotony at all. On the other side, consequence operators enjoy the law of closure: C (C (P)) = C (P), for whose validity it is essential that both sets P and C (P) deserve to be accepted as sets conveying ‘good’, ‘safe’, information, something that is not always the case in common reasoning where it cannot be always presumed that the conclusions only convey good, or safe, information. For instance, in searching for the explanation of the information conveyed by P, it is often the case of existing two contradictory hypotheses; that is, consistency cannot be a law for all kinds of reasoning. Most of the varieties of common reasoning are not deductive; at least, are not fully deductive. What are these kinds of ordinary reasoning? Traditionally, scientific reasoning is classified in deductive and inductive reasoning [7], consisting the first in the strict deployment of what is included in the premises, the statements containing all the available information, and the second consisting in the search for what is not included in them but could either explain it, or enlarge its informational content. Anyway, deduction is almost always seen as ‘formal deduction’ that is, what mathematicians do when proving theorems, where no step can be avoided and jumps are not allowed. Something not always occurring with common reasoning in which, when it is said ‘we deduce that’, jumps are more the rule than the exception. Besides reasoning ‘more geometrico’ is always the safest form of reasoning, to end a reasoning with the well known QED (‘quod erat demonstrandum’) is only fully correct when placed at the end of a mathematical proof. Notwithstanding, there is a variety of deductions that, not fully coincidental with the notion of proof, is close to it and constitutes the best form of common reasoning even if it is not possible to use the ending QED. Additionally, when speculating for new ideas, it is not possible to establish if the growing of the information will produce either monotony, or anti-monotony, that is, most of speculation is just non-monotonic since it lacks a law assuring that the number of conclusions will either constantly increase or decrease. Since speculation is also a manifestation of conjecturing or guessing [8], a mathematical model for it that can allow to capture the idea of what can be a ‘creative reasoning’ is, at least, actually convenient. The search for new ideas is crucial in both science and ordinary life. 2.2. To formalize all that, let us place what follows in the setting of a Basic Fuzzy Algebra (BFA) in a universe of discourse X [16] that, defined by a minimal set of laws, can guarantee a good level of generality to what could be deduced. That is (see next Section 2.3), in a five-tuple = ([0, 1]X , ≤; ·, +, ), where, for μ, σ, ρ in [0, 1]X , it holds: 1. μ ≤ σ ⇐⇒ μ(x) ≤ σ (x), for all x in X. 2. With the functions μ0 (x) = 0, and μ1 (x) = 1, for all x in X, the binary operations · and +, the unary , and the partial ordering ≤, jointly verifying, • μ · μ0 = μ0 · μ = μ0 ; μ1 · μ = μ · μ1 = μ • μ + μ0 = μ0 + μ = μ; μ + μ1 = μ1 + μ = μ1 • μ0 = μ1 ; μ1 = μ0 . • μ ≤ σ ⇒ σ ≤ μ . • μ0 ≤ μ ≤ μ 1 , for all μ, σ in [0, 1]X . 3. If μ ≤ σ , for all ρ in [0, 1]X , is: • μ · ρ ≤ σ · ρ; ρ · μ ≤ ρ · σ • μ + ρ ≤ σ + ρ; ρ + μ ≤ ρ + σ . 4. The subset {0, 1}X , endowed with the restriction of the order and the three operations defined in , 0 = ({0, 1}X , min, max, 1-id), is isomorphic to the Boolean algebra P(X), the power set of X. Notice that the pointwise ordering, ≤, between fuzzy sets is a partial, non-total, order, and that 0 is but isomorphic to the power set of X endowed with the structure of Boolean algebra given by the intersection, the union and the complement, of sets. Remarks. 1. It is · ≤ min ≤ max ≤ +, and only with · = min, and + = max, BFAs are lattices, namely De Morgan–Kleene algebras provided the ‘negation’ ( ) is strong, that is, verifies the law (μ ) = μ, for all μ in [0, 1]X .
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2. It is obvious that the Standard Algebras of fuzzy sets, those in which the binary operations · and + are functionally expressible by, respectively, a continuous t-norm and a continuous t-conorm, and the unary operation is functionally expressible by a strong negation function [15], are particular cases of BFA. Of course, the Boolean algebra of the crisp sets P (X), isomorphic to ({0, 1}X , min, max, 1-id), is also a BFA. 3. Notice that BFAs only show a minimal number of properties, and that no functional expressibility, commutative, associative, distributive, duality, etc., laws, are presumed to necessarily hold in them. 2.3. BFAs offer a wide mathematical framework to start dealing with ordinary reasoning, provided it is presumed that – The linguistic ‘and’ is represented by the binary operation · (conjunction), and the linguistic ‘or’ by the binary operation + (disjunction), – The linguistic ‘not’ is represented by the unary operation (negation), – The conditionals if/then are represented by the order ≤, that is, ‘If μ, then σ ’ is represented by μ ≤ σ . With that, two fuzzy sets μ and σ are supposed contradictory provided μ ≤ σ (if μ, then not-σ ), and a fuzzy set σ is self-contradictory if σ ≤ σ . From μ0 ≤ μ1 = μ0 , it follows that μ0 is self-contradictory. If μ represents a predicate P, and σ a predicate Q, it is supposed that μ ≤ σ represents that P is contradictory with Q, and μ ≤ μ represents that P is self-contradictory, that it holds, ‘If x is P, then x is not-P’ for all x in X. Notice that with classical sets there is only the self-contradictory set ø, the empty one [15], but that if, for instance, it is taken μ = 1 − μ, then it is μ ≤ μ ⇐⇒ μ ≤ μ1/2 , with μr the fuzzy sets constantly equal to r in [0, 1]. For each form of representing the negation, there are many self-contradictory fuzzy sets. Notice also that μ ≤ σ only indicates that μ is contradictory with σ , but not reciprocally. Since from μ ≤ σ follows (σ ) ≤ μ , it is sufficient that the negation verify σ ≤ (σ ) = σ (weak negation) for having σ ≤ μ , that also σ is contradictory with μ. In what follows, when saying ‘the fuzzy sets are contradictory’, without quoting their ordering, it will be supposed that the negation is weak. 3. Premises, résumé and conjectures 3.1. Definition. In a BFA , let us consider those sets P = {ρ1 , . . . , ρn } ⊆ [0, 1]X , such that: – For no pair ρi , ρj in P, is ρi ≤ ρj , and – Its résumé, ρ := ρ1 · (ρ2 · (ρ3 · (. . . (ρn−1 · ρn ) . . .), verifies ρ ρ , and ρ ρ. Notice that when ([0, 1]X , ≤, ·, +) is a lattice, that is, when · = min and + = max, the résumé ρ is coincidental with Inf P, always existing in [0, 1]X . In addition, and since the conjunction · = min is associative, in this case and in all those in which the associative law holds, parentheses can be avoided, it is ρ = ρ1 · . . . · ρn . These finite sets P will be called those of admissible premises, as their definition assumes minimal conditions for translating ‘safe’ information. Notice that the definition excludes both ρ = μ0 , and ρ = μ1 . Remark. Provided the conjunction · is not associative, the résumé ρ strictly depends on a previous numbering of the premises in P since, for instance and in general, ρ1 · (ρ2 · ρ3 ) = (ρ1 · ρ2 ) · ρ3 . Hence, in those cases in which the conjunction is not associative, all that follows depends on a numbering that should be previously fixed by attending at some reasonable criteria. Theorem 1. For all ρi in P, it is ρ ≤ ρi .
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Proof. It will be enough to prove the theorem for n = 3. Since it is ρ1 ≤ μ1 , and ρ2 · ρ3 ≤ μ1 , it follows ρ = ρ1 · (ρ2 · ρ3 ) ≤ ρ1 · μ1 = ρ1 , and from ρ2 ≤ μ1 follows ρ2 · ρ3 ≤ μ1 · ρ3 = ρ3 , and hence ρ ≤ ρ1 · ρ3 ≤ ρ3 , and it analogously follows ρ ≤ ρ2 . 2 Corollary 1. In a set of admissible premises, no pair of contradictory premises can exist. Proof. Provided there were in P two premises such that ρi ≤ ρj , from ρ ≤ ρi it will follow ρ ≤ ρj , and from ρ ≤ ρj , or ρj ≤ ρ , and hence the absurd ρ ≤ ρ will follow. 2 3.2. Definitions. 1. Given a set of admissible premises P, the set of its weak consequences [4] is C · (P) = {μ ∈ [0, 1]X ; ρ ≤ μ}, where the symbol · below C refers to the operation ‘and’ with which it is obtained the résumé of P. Provided it were · = min, it will be said that Cmin (P) is a set of strong consequences. 2. Let [μ, σ ] = {γ ; μ ≤ γ ≤ σ }, and (μ, σ ) = [μ, σ ] − {μ, σ }. Hence, C · (P) = [ρ, μ1 ]. Two fuzzy sets μ and σ in a chain of fuzzy sets are consecutive if and only if (μ, σ ) = ∅. It should be pointed out that usually, and in ordinary reasoning, neither the pairs of premises ρj , ρj +1 , nor the pairs ρ, ρi , are consecutive fuzzy sets. Anyway, and in what could be possible, it is better to try to ‘purify’ the set P of admissible premises, like it is always done in the case of mathematics, where it is systematically assumed that, to count with sets of minimal axioms, those pairs be of consecutive (crisp) statements. Of course, in common reasoning this is much more difficult that in the case of mathematics, and even if in it is not always an easy task. Given a set of admissible premises P, – The set of its C · -conjectures is Conj · (P) = {μ; ρ μ }, and – The set of its C · -refutations is Ref · (P) = {σ ; ρ ≤ σ }. Obviously, [0, 1]X = Conj · (P) ∪ Ref · (P), and Conj · (P) ∩ Ref · (P) = ∅, hence, Conj · (P) = Ref · (P)c , both sets are complementary. Then, a conjecture is a fuzzy set whose negation is not weakly deducible from the premises, that is, from the currently available information conveyed by them. Since it is obvious that C · (P) = C · ({ρ}), Conj · (P) = Conj · ({ρ}), and Ref · (P) = Ref · ({ρ}), it is both clear that all these sets are just defined by the résumé of P, and why it receives such name. Notes. 1. If μ and σ are in {0, 1}X , that is both represent crisp sets A and B, respectively, it is ρ σ ⇔ A ∩ B = ∅, ρ ≤ σ ⇔ A ∩ B = ∅, ρ ≤ ρ ⇔ A = ∅, and ρ ≤ ρ ⇔ Ac = ∅ ⇔ A = X. Hence, with crisp sets conjectures are those sets with a non-empty intersection with the premises’ résumé, refutations are those with an empty intersection, and the only self-contradictory crisp set is the empty set. For what concerns the conditions the résumé is submitted to, notice that they are A = ∅, and A = X. 2. Were ρ = μ1 not excluded in the definition of a set of admissible premises through the condition ρ ρ, from: μ1 = μ ⇔ it is not (μ1 ≤ μ ) ⇔ it is not (μ ≤ μ0 ) ⇔ μ0 < μ, it will follow Conj · (P) = [0, 1]X − {μ0 }, and C · (P) = {μ; μ1 ≤ μ} = {μ1 }. The ‘empty’ fuzzy set μ0 is never in Conj · (P) since it is not ρ μ0 = μ1 , but the ‘full’ fuzzy set μ1 is always since it truly holds ρ μ1 = μ0 . 3.3. Theorem 2. No pair of contradictory weak consequences can exist in C · (P ). Proof. Provided σi and σj in C · (P) verify σi ≤ σj , from ρ ≤ σi and ρ ≤ σj , it will follow ρ ≤ σj and σj ≤ ρ , and the absurd ρ ≤ ρ . 2
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Theorem 3. P ⊆ C · (P ), for any set P of admissible premises. It is said that the operators C · are extensive. Proof. It follows immediately from ρ ≤ ρi , for all premise ρi in P.
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Theorem 4. If P and Q are sets of admissible premises, and it is P ⊆ Q, then C · (P ) ⊆ C · (Q). It is said that the operators C · are monotonic. Proof. If it is P = {ρ1 , . . . , ρn }, it is Q = {ρ1 , . . . , ρn , ρn+1 , . . . , ρm }, with résumés ρQ , and ρP , respectively verifying ρQ ≤ ρP . To prove it, let’s just consider the easy case with n = 3, m = 4, in which, ρP = ρ1 · (ρ2 · ρ3 ), and ρQ = ρ1 · (ρ2 · (ρ3 · ρ4 )). From ρ3 · ρ4 ≤ ρ3 , follows ρ2 · (ρ3 · ρ4 ) ≤ ρ2 · ρ3 , and the ρQ ≤ ρ1 · (ρ2 · ρ3 ) = ρP ·. Hence, provided it is ρP ≤ μ, it follows ρQ ≤ μ. 2 Recall that being ≤ a partial order, it enjoys transitivity. Theorem 5. If P and Q are sets of admissible premises, and it is P ⊆ Q, then Conj · (Q) ⊆ Conj · (P ). It is said that the operators Conj · are anti-monotonic. Proof. Recall that it is ρQ ≤ ρP . If σ ∈ Conj · (Q), it is ρQ σ , and if it were ρP ≤ σ , it will follow the absurd ρQ ≤ σ . Hence, it should be ρP σ . 2 Theorem 6. Provided the negation ( ) verifies the law μ ≤ (μ ) , for all μ ∈ [0, 1]X , it is P ⊆ C · (P ) ⊆ Conj · (P ), for all set of admissible premises P. Proof. If it is ρ ≤ μ, and provided it were ρ ≤ μ , it will follow μ ≤ μ ≤ ρ , and ρ ≤ ρ , that is absurd. Hence it should be ρ μ . 2 Were the negation a weak one, that is, submitted to the law μ ≤ (μ ) , and also if it is strong, μ = (μ ) , weak consequences are a particular case of conjectures. Corollary 2. Provided the negation is a weak one, if μ ∈ C · (P ) then μ ∈ / C · (P ). It is said that the operator C · is consistent. Proof. It just follows from Theorem 4, since if it is ρ ≤ μ, and if it were ρ ≤ μ , it will follow ρ ≤ ρ . Hence, it should be ρ μ . 2 Under weak negations, the negation of a weak consequence is never a weak consequence. Theorem 7. Operators Ref · are not extensive, and they are monotonic. Proof. Were P ⊆ Ref · (P), all premises ρi will be refutations, or ρ ≤ ρi , and since ρ ≤ ρi implies ρi ≤ ρ , it will follow the absurd ρ ≤ ρ . If P ⊆ Q, from ρQ ≤ ρP follows that when ρP ≤ σ it is ρQ ≤ σ . 2 Theorem 8. For all conjunction ·, is: 1) Conj · (P ) ⊆ Conjmin (P ); 2) Ref min (P ) ⊆ Ref · (P ); 3) Cmin (P ) ⊆ C · (P ). Proof. Since · ≤ min, a résumé of P obtained with · is always below the one (of the same P) obtained with min. Hence, it immediately follows 3. Since, μ ∈ Ref · (P) ⇔ μ ∈ C · (P), it follows 2. Finally, 1 follows by complementation in 2. 2 Notice that there are no less weak than strong consequences and refutations but that, on the contrary, the largest set of conjectures is obtained with min. Strongly deducing, or refuting, corresponds with the possibility of producing as many conjectures as possible.
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Remarks. 1. The set [ρ, μ1 ] is not necessarily finite and, consequently, it could be not easy to define its résumé by the conjunction of all its elements. Nevertheless, provided the résumé were taken as the infimum, then that of C · (P) will be the same of P, since Inf [ρ, μ1 ] = ρ, and also C · (P) can be taken as a set of admissible premises. This identification of the résumé can be safely done in the particular case in which it is · = min, and then Cmin can be applied to Cmin (P). Thus, since ‘P ⊆ Cmin (P) ⇒ Cmin (P) ⊆ Cmin (Cmin (P))’, and σ ∈ Cmin (Cmin (P)) means Inf Cmin (P) = ρ ≤ σ , it follows σ ∈ Cmin (P), and Cmin (Cmin (P)) ⊆ Cmin (P). Hence, it is Cmin (Cmin (P)) = Cmin (P), the operator Cmin is a closure and, consequently, it is a Tarski’s, or strong, consequence operator. In the case · = min, it can be stated that the consequences are formal, or strong consequences, and that it is possible to go from the résumé ρ up to any consequence σ by following a path ρ = σ0 ≤ σ1 ≤ . . . ≤ σp−1 ≤ σp = σ , of length p, and with consecutive fuzzy sets. That is, by means of a proof. This does not mean, of course, neither that only a single proof conducting from ρ to σ does exist, nor that one can be immediately known, and selecting one with a minimum of steps is a mark of mastering the art of proving. 2. What happens in general when C · (P) is a finite set, and the conjunction · is not coincidental with min? If, for instance, the conjunction · is associative, with P = {ρ1 , . . . , ρn }, and C · (P) = {ρ1 , . . . , ρn , γ1 , . . . , γp }, since P ⊆ C · (P), then it is ρC·(P ) = ρ1 · ρ2 · . . . · ρn · γ1 · γ2 · . . . · γp ≤ ρ1 · ρ2 · . . . · ρn = ρP , an inequality not guaranteeing that ρC·(P ) verifies the conditions required for considering C · (P) an admissible set of premises. Consequently, in general it is not possible to guarantee that the operator C · can be applied to C · (P). Only the operator Cmin is always a mapping A(·) → A(·), with A(·) the set of all sets of admissible premises contained in [0, 1]X with · = min, but, in general, the operators C ·, with · = min, are just mappings A(·) → [0, 1]X , and, hence, the sets C · (P) cannot be usually taken as ‘safe’ information if the conjunction · is not the conjunction min. This is a difference between strong and weak consequences, between formal and common or informal deduction. Since in the classical case (that of {0, 1}X ) only the operator Cmin is used, the other operators C · could correspond to the informal deduction in ordinary reasoning. Analogously, also the operators Conj ·, and Ref · are mappings A(·) → [0, 1]X , the sets Conj · (P) and Ref · (P) cannot be taken as sets of admissible premises. Perhaps, informal or common deduction is done when · = min. 3. Non-monotonic reasoning is not new at all, and it deserved remarkable studies in Artificial Intelligence [23]. This paper does not consider methods to arrive at non-monotonic conclusions, but only which conjectures or refutations show, or not, monotony. In [24] an initial study of the relationships between conditionality and monotony is shown. 4. Deductive paths, hypotheses and guessings 4.1. A deductive path to reach a weak consequence σ from a set of admissible premises P, is a chain ρ = α0 ≤ α1 ≤ . . . ≤ αp−1 ≤ αp = σ, where all αi belong indeed to C · (P). A deductive path is a deductive proof, or simply a proof of σ , provided all pairs (αi , αi+1 ) are of consecutive fuzzy sets. Of course, the number p of steps in a deductive path, or in a proof, is not a constant for it, and proving a consequence by means of a minimal number p of these steps is part of the ‘art of proving’. Notice that in no proof can exist jumps, but that in a deductive path that is not a proof there can lack intermediate steps and that, because of these jumps, it can happen that between some nodes αi and αi+1 , it exists a fuzzy set β such that, for instance, it is αi ≤ β, but β αi+1 , with which the chain is broken and it is not actually deductively conducting from ρ to σ . It does not necessarily mean that it is impossible to prove σ , but that the proof is not yet complete. Since the more restrictive form of deduction is given by the operator Cmin , it is with it that are conducted the safest ways of deducing whenever the linguistic conjunction admits to be represented by min. Remark. What has been presented is degree-free, it is just ‘crisp reasoning’ with fuzzy sets and, hence, not sufficient for a complete covering of common reasoning where everything, and not only elemental statements ‘x is P’, is allowed
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to be graded; it lacks, in particular, a definition of graded conjecture and graded refutation. It lacks to add, at least and between the steps of the deductive paths, degrees up to which each step can hold, that is, a concept of ‘gradual consequence’ [13]. For instance, with ρ ≤r σ ⇔ degree (ρ, σ ) = r ∈ [0, 1], meaning that σ is a consequence of P up to the degree r, that is, changing the crisp relation ≤ to a more general fuzzy relation ≤r reducing to ≤ if r = 1, and to if r = 0 [21]. It still lacks to consider which weight the conjectures or the refutations inherit from the degree up to which the premises’ résumé can hold [14]. With the study of the gradable cases is how it is possible to go further than just crisp conjectures [13] by opening the door towards, for instance, graded consequences [22] that cannot be drawn in the domain of classical logic [21]. 4.2. Since, provided the negation is a weak one, it holds that C · (P) ⊆ Conj · (P), in this case it makes sense to analyze the difference-set Conj · (P) − C · (P) = {σ ; ρ σ & ρ σ } = {σ ; ρ σ & σ < ρ} ∪ {σ ; ρ σ &ρ nc σ }, with nc shortening ‘not comparable under the partial order ≤’. Let us call hypotheses the conjectures in the first of these two sets, and speculations those in the second. The denotation ‘hypothesis’ is suitable since it is from σ that ρ strictly follows in the first case, that is, a deductive chain can exist from σ to the résumé ρ showing that σ ‘explains’ the premises. In the second case, the denotation comes from the fact that no chain can link the conjecture σ with ρ. In principle, a hypothesis is a conjecture that can be weakly deductively reached from ρ by a backwards chain of the type ρ = α0 ≥ α1 ≥ . . . ≥ α0 = σ , although this is not possible in the case of a speculation. Let us designate by Hyp · (P) and by Sp · (P), respectively, these sets that are, obviously, disjoint with the set C · (P), and that give the partition: Conj · (P) = C · (P) ∪ Hyp · (P) ∪ Sp · (P). Theorem 9. Provided the negation is a weak one, and ρ = σ , σ ∈ Hyp · (P ) ⇔ ρ ∈ C · ({σ }). Proof. It is C · (P) ⊆ Conj · (P). If σ ∈ Hyp · (P), it is σ < ρ and ρ σ , hence σ ∈ C · (P). The reciprocal is also obvious. 2 Since it is potentially possible to find ‘backwards’ deductive paths from ρ to σ , it is for why hypotheses deductively ‘explain’ the résumé, that is, the informational content of P. Theorem 10. Operators Hyp · are anti-monotonic. Proof. From P ⊆ Q, and ρQ ≤ ρP , if σ ∈ Hyp · (Q), from σ < ρQ , follows σ < ρP . Since it is ρQ σ , provided it were ρP ≤ σ it will follow the absurd ρQ ≤ σ , and, hence it should be ρP σ . Thus Hyp · (Q) ⊆ Hyp · (P). 2 More premises imply no more hypotheses, that is, as more information is available less hypotheses are possible. With respect to the operators Sp · (P), notice that in the crisp case, a set S is a speculation of a set of admissible premises P with résumé R, provided R ∩ S = ø, and neither R ⊆ S, nor S ⊆ R. In this case, it is really easy to find Venn diagrams with sets of premises P and Q, where neither the monotonic, nor the anti-monotonic property, holds. The operators Sp · are properly non-monotonic. As ρ σ ⇔ Either σ < ρ, or ρ nc σ , let us check what happens in the first case if σ is a hypothesis. In this case, from σ < ρ, is ρ ≤ σ , and then it follows the absurd ρ < ρ, and, finally, the set of hypotheses is reduced to Hyp · (P) = {σ ; ρ nc σ & σ < ρ}. If σ ∈ Hyp · (P), from σ < ρ, it follows ρ ∈ C · ({σ }). Thus, to prove that some conjecture σ is not a hypothesis for P, that is, to falsify σ , it suffices to find a consequence of P that is not among the consequences of {σ }. 4.3. It holds, obviously, Sp · (P) = {σ ; ρ > σ & ρ nc σ } ∪ {σ ; ρ nc σ & ρ nc σ }.
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Calling, respectively, pseudo-guessings (PG), and creative-guessings (CG), those speculations in each of these sets, the partition Sp · (P) = PG · (P) ∪ CG · (P) is obtained. As it is σ < ρ, the negation σ of a pseudo-guessing is a hypothesis, and potentially allows to reach σ , through ‘backwards deductive ways’ starting in ρ. Nevertheless, no creative-guessing can neither be deductively reached from ρ, nor from ρ . Notice that, provided σ is a pseudoguessing, it is ρ ∈ C · ({σ }), but neither ρ ∈ C · ({σ }), nor σ ∈ C · ({ρ}) = C · (P). Theorem 11. Provided the disjunction + is idempotent (μ + μ = μ, for all μ in [0, 1]X ), operators PG · are antimonotonic. Proof. From, P ⊆ Q ⇒ ρQ ≤ ρP , and σ ∈ PG · (Q) ⇔ σ < ρQ & ρQ nc σ , follows σ < ρP . Provided it were ρP ≤ σ , it will follow the absurd ρQ ≤ σ . Provided it were σ < ρP , it will follow ρP ≤ σ + ρP ≤ ρP + ρP = ρP , that is also absurd. Hence, it should be ρP nc σ , and σ ∈ PG · (P), that is, PG · (Q) ⊆ PG · (P). 2 It should be pointed out that this theorem is practically limited to those cases in which + = max. Hence, creativeguessings seem to be the only speculations that, in general, are properly non-monotonic. From the point of view of deductive reasoning, they are ‘wild’ conjectures. Theorem 12. 1) It is ρ + μ ∈ C · (P ), and μ + ρ ∈ C · (P ), for any μ ∈ [0, 1]X . 2) If μ ∈ Conj · (P ) is such that ρ · μ = ρ, and μ · ρ = ρ, then μ · ρ ∈ Hyp · (P ), and ρ · μ ∈ Hyp · (P ). Proof. 1) Obvious, since ρ ≤ ρ + μ, and ρ ≤ μ + ρ. 2 In particular, if μ is either a pseudo-guessing, or a creative guessing, weak consequences are simply reached by ρ + μ and μ + ρ. 2) Obvious, since μ is a conjecture, and it is ρ · μ ≤ ρ, and μ · ρ ≤ ρ, without holding the equalities. 2 In particular, if μ is either a pseudo-guessing or a creative-guessing, whose conjunctions with ρ do not reproduce ρ, hypotheses are simply reached by ρ · μ and μ · ρ. Notice that when · = min, the two conditions in 2 just mean ρ μ, that μ is not a weak consequence. 2 Theorem 12 is actually interesting when μ ∈ Sp · (P), since in this case μ is obtained, and possibly not by deduction, from P and then both weak consequences and hypotheses are reached by directly reasoning on the informational content conveyed by P. Whenever μ ∈ CG · (P), part 2 is especially important since it shows how a true ‘new idea’ can help to ‘explain P’, something well enough corresponding with the intuition of what people sometimes do to answer some questions. It is guessing what often conducts to the key, μ, of the question on which ρ is known. Guessing is actually relevant for reasoning. 5. On the role of analogy in reasoning Although the realm of metaphor is poetry, analogy or similitude between concepts, sometimes in the form of metaphors, it appears very often in common reasoning and, mainly, in either philosophical reasoning, or in the exploratory scientific one for obtaining either explanations, or consequences by speculation. At this respect, the question if analogy is, or is not, a new and different type of reasoning seems to be relevant, and without the pretence of closing the subject, let us show that with a reasonable and simple interpretation of what analogy consists in, it can be proven that it just conducts to either conjectures or refutations. The first problem is to define when it can be asserted that a fuzzy set μ is analogous to another σ . Let us suppose that there is a family A of functions f: [0, 1]X → [0, 1]X , allowing to state that μ and f(μ) are A-analogous fuzzy sets or, less specifically, analogous. Thus, with each fuzzy set μ there is the family {f(μ); f ∈ A} of the fuzzy sets analogous to μ, and with each set of admissible premises P and each f in A, there is the family of analogous fuzzy sets f(ρ1 ), . . . , f(ρn ), as well as the fuzzy set f(ρ) analogous to the résumé ρ = ρ1 · . . . · ρn , and
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possibly not analogous with f(ρ1 ) · . . . · f(ρn ), unless f were an isomorphism for the conjunction (·) in which case it will be f(ρ1 ) · . . . · f(ρn ) = f(ρ1 · . . . · ρn ) = f(ρ). Since the functions f defined by f (μ) = (f(μ)) , for all μ ∈ [0, 1]X , can only verify either id ≤ f (μ ≤ f (μ) for all μ), or id f , there are only the two possibilities: – ρ ≤ f (ρ), under which f(ρ) is a refutation of P, and – ρ f (ρ), under which f(ρ) is a conjecture made from P. That is, based on the given concept of analogy, only new refutations or conjectures can be obtained. Even more, provided it is id f , it only can be id ≤ f, f < id, or the functions id and f are not comparable under the pointwise order ≤, then: – In the first case, it is ρ ≤ f(ρ), and the conjecture f (ρ) is a consequence of P that is analogous to ρ, – In the second case, it is f(ρ) < ρ, and the conjecture f(ρ) is a hypothesis for P that is analogous to ρ, – In the third case, ρ and f(ρ) are not ≤ – comparable, and the conjecture f(ρ) is a speculation made from P. Hence, but depending on the given definition of analogy, it seems that analogy is nothing else but a methodology for conjecturing and refuting, that is, for reasoning. Thus, analogy appears as a methodology for guessing and refuting. For instance, if σ is a creative-guessing with which ρ · σ is a hypothesis, and it is f < id, it is f(ρ · σ ) < ρ · σ ≤ ρ: the fuzzy set f(ρ · σ ), is a hypothesis of P analogous to the previous one ρ · σ . As a metaphor is but a statement reflecting something analogous to something, and even in a fuzzy form, the use of metaphors is not at all avoidable for reasoning. It is an older form of reasoning that, nevertheless, should be kept under control since its actual danger lies in the ‘great distance’ often existing between, for instance, the metaphor, and the persecuted goal. At the end, and as it happened many times in the history of science, the final role of the analogy/metaphor lies in the experience and specific knowledge of who is considering it. Remark. What this section does not, and cannot pretend, is the study of any analogical reasoning’s modality [25]. It just tries to show that it is difficult to believe that conclusions of a different type than conjectures and refutations could be obtained through similarity. Were it the case, similarity will appear as a ‘tool’ for reasoning. 6. Conclusion 6.1. This paper is no more than a glimpse at common reasoning just trying to show that some, at least historically relevant treats of it, can be posed by means of fuzzy sets and under minimal conditions on its calculus. Notwithstanding, there are many things still waiting for clarification. For instance, establishing good, concrete and practically useful mathematical models for representing conjunctions that, in Natural Language where time almost always intervenes, often cannot be commutative as it is the classical associative conjunction. Also for instance, the extent of the use of the associative law is not clear enough in Natural language where it seems to appear, when it is the case, expressed by commas that cannot be moved without constraint in most environments. It should be investigated on which conditions such kind of properties can be stipulated since only in a setting allowing them, a mathematical model with these properties could be realistically taken into account. 6.2. What is important to realize is the possibility of capturing in which type of reasoning actually lies the phenomena of anti-monotony and non-monotony. As the Nobel Laureate Sir Peter B. Medawar wrote in 1984 [9], “No process of logical reasoning can enlarge the information content of the axioms and premises or observation statements from which it proceeds”, to shortly express that the clothes of formal deductive logic are too short and also too large, for a good dressing of creative reasoning, that monotonic deductive logic is not sufficient to mathematically model the creative or, at least
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ampliative, processes of reasoning science requires. And it seems less sufficient again when analogy, imprecision and uncertainty play a crucial role in reasoning. For a true progress of Computing with Words, it is doubtful that its study can be just carried out by logical, formal, methods. It seems plausible to think that such study needs a combination of controlled experimentation and mathematical modeling to be checked against actual common reasoning and natural language [10]. Like it is typical of the scientific reasoning as it was said by the Nobel Laureate Eugene Wigner, with its “unreasonable effectiveness of mathematics in the natural sciences” [11]. At the end, both Natural Language and Commonsense Reasoning are but natural phenomena and deserve to be accordingly studied to discover, for instance, which properties can be stipulated and on which conditions such stipulation is possible. 6.3. All that is presented in this paper is to be kept as a set of theoretical hypotheses until they can be systematically contrasted with language and reasoning. That is, by following a scientific path. The path that allowed, for instance, that the young and then scarcely known scientist Georges Lemaitre, contradicted Albert Einstein referring to the universe’s inflation and that, at the end, Einstein openly recognized that he was mistaken [12]. Without the possibility of contrasting models against reality, the emergence of current science and technology had actually not been possible. In Science there are neither dogmas, nor absolute authorities, and everything is subjected to either falsification, or to some enlargement covering more knowledge; science is a non-stop path for which mathematical models are but a foothold, even if a very important one. Efforts to introduce reasoning in a pre-fabricated mold are purposeless and, on the contrary, the big goal is to find suitable molds in which reasoning, or its most significant parts, can be comfortably included. At the end it should not to be forgotten how difficult is to believe that common reasoning could be submitted to a complete formalization; common reasoning is all people has for surviving and it is often supplied not only by language but by post interpreted memory, blurred imaging, case-based analogy, etc. Complete processes of reasoning are more than what is presented in this paper, and it is perhaps arrived a good moment for a methodological turn in the study of reasoning under imprecision and non-random uncertainty, a turn towards a kind of ‘physics of language and reasoning’. If the usefulness of theoretical concepts in the technological applications is certainly important, and to some extent applications play the role of experimentation in the modern techno-sciences [10], to advance in the study of the natural phenomenon ‘reasoning’, it can be argued that intentional controlled experimentation is suitable for it, and that it should be done under the guide of numerical measurements supported by mathematical models of, at least, some parts of the phenomenon. Acknowledgement This paper is partially funded by the Foundation for the Advancement of Soft Computing, and by the Spanish Government project MICIIN/TIN 2011-29827-C02-01. References [1] K.R. Popper, Conjectures and Refutations: The Growth of Scientific Knowledge, Harper and Row, New York, 1968. [2] P.L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353. [3] I. García-Honrado, E. Trillas, On an attempt to formalize guessing, in: R. Seising, et al. (Eds.), Soft Computing in Humanities and Social Sciences, Springer, Berlin, 2012, pp. 237–255. [4] E. Trillas, A model for ‘crisp reasoning’ with fuzzy sets, Int. J. Intell. Syst. 27 (2012) 859–872. [5] L.A. Zadeh, Computing with Words. Principal Concepts and Ideas, Springer, New York, 2012. [6] A. Tarski, Fundamental concepts of the methodology of deductive sciences, in: Logic, Semantics, Metamathematics, Oxford University Press, 1956. [7] B. Bosanquet, Logic or the Morphology of Knowledge, vol. I, Clarendon Press, London, 1911. [8] S. Watanabe, Knowing & Guessing, John Wiley & Sons, New York, 1969. [9] P.B. Medawar, The Limits of Science, Harper, London, 1984. [10] E.H. Mamdani, E. Trillas, Correspondence between an experimentalist and a theoretician, in: E. Trillas, et al. (Eds.), Combining Experimentation and Theory, Springer, Berlin, 2012, pp. 1–18. [11] E. Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Commun. Pure Appl. Math. 13 (1) (1960) 1–14. [12] C. Rovelli, La realtà non è come ci appare: La struttura elementare delle cose, Raffaello Cortina Editore, Milan, 2014. [13] J.L. Castro, E. Trillas, Tarski’s fuzzy consequences, in: Proceedings International Fuzzy Engineering Symposium, 1991, pp. 70–81.
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