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The Laws of Excluded Middle and Contradiction in Checklist Paradigm Based Fuzzy Interval Logic

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The Laws of Excluded Middle and Contradiction in Checklist Paradigm Based Fuzzy Interval Logic The Laws of Excluded Middle and Contradiction in Checklist Paradigm Based Fuzzy Interval Logic Eunjin Kim Department of Computer Science University of North Dakota Grand Forks, ND 58202-9015, USA. Email: [email protected] G43 Abstract— This paper continues a study in fuzzy interval logic S-QL implications, where a ! b = min(1; b=a); based on the Checklist Paradigm(CP) semantics of Bandler and a !L b = min(1; 1 − a + b); a KDL! b = min(1; 1 − a + ab); Kohout. The law of excluded middle and the law of contradiction KD EZ are investigated in the fuzzy interval logic system of negation, a ! b = max(1 − a; b); a ! b = max(1 − a; min(a; b)); W [:Bot; :T op], which was defined by the Nicod(NOR) and the a ! b = min(max(1 − a; b); max(a; 1 − a1; 1 − a + ab) Sheffer(NAND) connectives of m1 interval system, respectively. [1],[2]. However, inter-relations of interval systems of Both laws don’t hold in the fuzzy logic with a classic negation, connectives are known much less. :a = 1 − a; however, they do hold both with :Bot and with :T op in the checklist paradigm based fuzzy interval logic. Bandler and Kohout derived five interval systems of fuzzy Index Terms—Fuzzy Interval Logic, Fuzzy Negation, Fuzzy logic, m1; m2; : : : ; m5, based on the Checklist paradigm in Identity, Law of Excluded Middle, Law of Contradiction. 1979 [2]. Since then, the logic systems of connectives that can be generated from the interval of implication by group I. INTRODUCTION transformations have been investigated systematically; in This study continues fuzzy interval logic based on the particular, the m1 interval logic system of 16 connectives Checklist Paradigm semantics of Bandler and Kohout. The has been investigated in depth [3]-[7]. In their m1 logical fuzzy logic default into 2-valued crisp logic for the values system, ten 2-argument connectives such as ^; _; !, ::: 0 and 1, however, the logical operations with the values in ≡(IFF) and ⊕(XOR) yield the interval pairs of connectives the open interval (0; 1), other than 0 and 1, generate different (conbot ≤ m1 ≤ contop) where its implication (!) yields values depending on the definition of logical connective which Łukasiewicz and Kleene-Dienes implication for its TOP-BOT was applied. The question how connectives of the same type pair of interval, in particular. However, a unary connective differ between the boundary points, i.e. within the open such as a negation (:a) and an identity (a) did not yield interval (0; 1), has been well researched. For instance, the interval but just singleton: i.e. :a = 1 − a and a. From this fuzzy logical AND (^), i.e. a t-norm operator, such as minimum question, Kim and Kohout proposed the alternative negations (^m), algebraic product (^a), bounded product (^b) or drastic in [8] defined by the Sheffer(NAND), the Nicod(NOR) and a product (^d) yields a hierarchy of their values as follows: pair of <!; 0 >, which fuzzifies :a, generating the intervals like the fuzzified 2-ary connectives. Since the TOP-BOT pair a ^d b ≤ a ^b b ≤ a ^a b ≤ a ^m b of these generated negation interval lacked of an involutive property on the surface, some pseudo properties such as a where a ^b b = max(0; a + b − 1), a ^a b = ab, a ^m b = nearly involutive property and a convergence of iteration of min(a; b), and a ^d b = a if b = 1; b if a = 1; 0 otherwise. each TOP-BOT pair were also investigated in [9]. Similarly, the fuzzy OR (_a) yields an hierarchy When the fundamental laws in the crisp logic are extended a _m b ≤ a _a b ≤ a _b b ≤ a _d b to a fuzzy logic, most of laws such as De Morgan’s law and where a _m b = max(a; b) ( maximum), a _a b = a + b − ab associativity, etc., do hold in the fuzzy logic; however, both (algebraic sum ), a _b b = min(1; a + b) (bounded sum ) and the law of excluded middle and the law of contradiction fail a^db = a if b = 0; b if a = 0; 1 otherwise. On the other hand, with a classic negation in singleton, :a = 1 − a in the fuzzy the implication connectives such as Łukasiewicz(!L ), Kleene- logic. Thus, it raises a question whether both laws hold with Dienes(KD! ), Reichenbach(KDL! ), Goguen(G!43), Go¨del(!S∗) TOP-BOT pair of the fuzzy interval negation and with those yield an inequality relationship as follows: pair of fuzzy interval identity or not. S∗ G43 L a ! b ≤ a ! b ≤ a ! b : R-implications, In this paper, an interval system of fuzzy identity (a) is defined using the law of idempotence. Then, both the law of a !W b ≤ a EZ! b ≤ a KD! b ≤ a KDL! b ≤ a !L b : excluded middle and the law of contradiction are examined Define: a = r1=n; b = c1=n. with the interval of fuzzy negation (:a) and that of a in m1 logic system based on the Checklist paradigm. No for B Yes for B Row Total II. INTERVAL LOGIC SYSTEM OF CONNECTIVES GENERATED BY THE CHECKLIST PARADIGM Many valued logic interval-based reasoning plays an impor- No for A α00 α01 r0 tant role in fuzzy and other many valued extensions of crisp logic. To be of use in a diversity of application domains, the Yes for A α α r interval-valued inference systems require formal semantics. 10 11 1 The formal semantics that is derived by means of a mathe- matical method, and which also has a sound ontological and Column Total c0 c1 n epistemological base is provided by the so called checklist paradigm developed by Bandler and Kohout [2],[3],[4],[5]. Fig. 1. Checklist Paradigm of the assignment of fuzzy values The checklist paradigm has given interesting theoretical re- sults, shedding light not only on the semantics of various many valued logic connectives, but also on the true method- This assessment operator will be called the contrac- ological importance of fuzzy methods in approximate reason- tion/approximation measure. ing based on the interval methods. The four interior cells α00; α01; α10; α11 of the constraint In its most general form, the checklist paradigm pairs the table constitute its fine structure; the margins r0; r1; c0; c1 distinct connectives of the same logical type to provide the constitute its coarse structure (see Fig. 1). bounds for interval-valued approximate inference. The fine structure gives us the appropriate fuzzy assessments A. The Checklist Paradigm by Bandler and Kohout for all propositional functions of A and B; the coarse structure A checklist template Q is a finite family of properties gives us only the fuzzy assessments of A and B themselves. to what hP1;P2; :::; Pi; :::; Pni; With a template Q, and a given propo- The central question by Bandler and Kohout was, extent can the fine structure be reconstructed from the coarse sition A, one can associate a specific checklist QA = hQ; Ai. ? A valuation f of a checklist Q is a function from Q to A A As shown in Bandler and Kohout’s papers [2],[3],[4],[5] that f0; 1g. the coarse structure imposes bounds upon the fine structure, The value a of the proposition A with respect to a template Q without determining it completely. Hence, associated with the Q (which is the summarized value of the valuation f ) is given A various logical connectives between propositions are their by the formula n extreme values. Thus the inequality restricting the possible X A values of m(F ) is obtained: aQ = pi i=1 conBot ≤ m(F ) ≤ conT op where n =j Q j and pA = f (P ). i A i where con is the name of connective represented by f(i; j). A fine valuation structure of a pair of propositions A, B Q with respect to the template Q is a function fA;B from Q Choosing the implication for the logical type of the connective intof0; 1g assigning to each attribute Pi the ordered pair of its con and making the assessment of the fuzzy value of the truth values hpA; pBi. of a proposition by the formula α10 Let αj;k be the cardinality of the set of all attributes Pi m1(F ) = 1 − n = 1 − u10, it obtains Q such that fA;B(Pi) = hj; ki. Then, there are the following max(1 − a; b) ≤ m1(A ! B) ≤ min(1; 1 − a + b) constraint on the values: α00 + α01 + α10 + α11 = n. Further, which produced the Łukasiewicz implication connective and r0 = α00 + α01, r1 = α10 + α11, c0 = α00 + α10, c1 = the Kleene-Dienes implication connective as the upper and α01 + α11 are defined. lower bound of interval, respectively. This m1 measure is the These entities can be displayed systematically in a constraint classical measure by default. table. In such a table, the inner fine-summarization structure Choosing for con the connective type AND, it yields consists of the four αj;k appropriately arranged, and of mar- max(0; a + b − 1) ≤ m1(AND) ≤ min(a; b). gins c0; c1; r0; r1 (see Fig. 1). Choosing for con the connective type OR, it yields max(a; b) ≤ m (OR) ≤ min(a + b − 1; 1). Now let F be any logical propositional function of propo- 1 sitions A and B. For i; j 2 f0; 1g, let f(i; j) be the clas- It is interesting to note that the last two inequalities are sical truth value of F for the pair i; j of truth values; let identical with those of Schweitzer and Sklar [10] giving the u(i; j) = αi;j=n, the ratio of the number in the ij-cell of bounds on their so called copulas which play an important role the constraint table, to the grand total.
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