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 {0, 1}. 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 =| Q | 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 into{0, 1} 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 ∈ {0, 1}, 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. Then what we define in their theory of norms and conorms. It also coincides with the (non-truth-functional) fuzzy assessment of the truth of the Novak’s´ derivation of bounds on fuzzy sets approximating proposition F (A, B) to be classes of Vopenka’sˇ Alternative Set Theory [11]. X m(F (A, B)) = f(i, j) · uij. i,j Other four inequalities of mi(F ) were also retrieved by the distinct measures below: From this implicational interval pair [ m1(→bot), m1(→top)],

α11 α10 all 16 pairs of connectives of corresponding interval systems • m2(F ) = = 1 − : performance measure. α10+α11 r1 could be generated by group transformation of logic. Among • m3(F ) = u11 ∨ (u00 + u01) 16 pairs of logical connectives, 10 of them are genuine interval : by performance and by default. pairs: {→, ←, 6←, 6→, ∨, ↓, |, ∧, ≡ (IFF), ⊕ (XOR)}. The rest of • m4(F ) = m3 ∧ ¬m3 6 connectives, however, collapse into a single point, not an : lower contrapositivization of m3. interval: {true, false, identity (a, b), negation (¬a, ¬b)}; r0 • m5(F ) = m2 ∨ n = m2 ∨ (u00 + u01) see [5] for the details of intervals [conbot, conbot] of 16 con- : the less conservative performance. nectives. It has been investigated systematically over the years In each of four measures, the interval of logical connective in the series of papers [6]-[7],[13]. These 16 connectives of of implication (→) are generated and further build five distinct measure m1 −m5 could be generated by group transformation. 8 of these interval pairs of connectives of m1 system are shown interval systems of connectives m1 −−m5 in the next section. at Table 1 in section III. In the following section, the operations B. Five Implication Operator Based Interval Systems of Ban- of group transformation is described. dler and Kohout III.CHARACTERIZATIONOFLOGICSBYGROUP The structure of five fuzzy interval systems m1 – m5, based on the Checklist paradigm by Bandler and Kohout in [2] is TRANSFORMATION generated by a distinct measure that performs the summa- The m1 interval system of logic can be generated by group rization of the information contained in certain well-defined transformations of logical connectives. binary structures called fine structures as it was described in Definition 1: Group Transformations in Logic: the above section. The interval produced by a measure mi pair Let f be one of the 10 two-argument propositional connectives of connectives of one type can be generically characterized by of a logic system ({→, ←,..., ≡, ⊕}) and ¬ be an involutive the following inequality: negation. Then, the following transformations over f are con ≤ m ≤ con , i ∈ {1, 2,..., 5} defined: Bot i T op 1. I(f) = f(x, y) : Identity transformation For the more details of the checklist paradigm and its various 2. D(f) = ¬f(¬x, ¬y) : Dual transformation uses, refer to the papers [2] -[7],[12]. 3. C(f) = f(¬x, ¬y) : Contradual transformation 4. N(f) = ¬f(x, y) : Negation transformation The following five inequalities linking the interval bounds for implication operators [ →bot, →top ] with corresponding It is well known that for the crisp (2-valued) logic these measures, mi, i = {1, 2, 3, 4, 5} have been generated from T = {I,D,C,N} transformations determine the Piaget group five different measures described in Sec.II-A [2]. which is a realization of the abstract Klein 4-element group. 1) The Kleene-Dienes implication(KD) and Łukasiewicz The new 4 non-symmetrical transformations below were implication (Ł) respectively, are attainable lower and added to the above 4 basic transformations by Bandler and upper bounds of m1: Kohout in 1979 [14],[15]. These non-symmetrical transforma- max(1 − a, b) ≤ m1(→) ≤ min(1, 1 − a + b). tions enrich the algebraic structure of logical transformations. 2) A certain new function of (a, b) and the Goguen-Gaines Definition 2: 8-element Group Transformations: Bandler- (G43) implication (the left-hand side) are respectively Kohout attainable lower and upper bounds of m2: 5. LC(f) = f(¬x, y) : Left Contradual max(0, (a + b − 1)/a) ≤ m2(→) ≤ min(1, b/a). 6. RC(f) = f(x, ¬y) : Right Contradual 3) Another function of (a, b) and the Early Zadeh impli- 7. LD(f) = ¬f(¬x, y) : Left Dual cation (EZ) are respectively attainable lower and upper 8. RD(f) = ¬f(x, ¬y) : Right Dual bounds of m3: This enlarged set of transformations T = {I, D, C, N, LC, max(a+b−1, 1−a) ≤ m3(→) ≤ max[min(a, b), 1−a] RC, LD, RD} forms 8-element group {T, ◦} called S2×2×2 4) Still another function of (a, b) and the Wilmott impli- group. The equation, N 2 = C2 = D2 = LC2 = LD2 = cation (W) respectively, are attainable lower and upper RC2 = RD2 = I, provides sufficient information to identify bounds of m4: this group. Its group operations are shown in [17] in detail. min[max(a + b − 1, 1 − a), max(b, 1 − a − b)] ≤ m4(→) ≤ When T = {I, D, C, N, LC, RC, LD, RD} are applied min[max(1 − a, b), (max(a, (1 − b), min(b, 1 − a)))]. to Łukasiewicz implication (→top) and to Kleene-Dienese 5) Yet another function of (a, b) and one of G43 respec- implication (→bot) of m1 system of logic, respectively, they tively, are attainable lower and upper bounds of m5: yield the closed set of connectives in Table I [6],[17]. This max[(a + b − 1)/a, 1 − a] ≤ m5(→) ≤ result of m1 interval system of 8 connectives is further max[min(1, b/a), 1 − a]. summarized in Table II. As Table II shows, neither negation In m1 system, a Kleene-Dienes logic system forms a a lower nor identity belonged to the interval system m1. bound of the interval (m1(→bot)) wile a Łukasiewicz logic The graph of transformations showing a more detailed system forms a upper bound of the interval (m1(→top)). structure of the transformation of connectives that realize TABLE I GROUP TRANSFORMATION OF m1 SYSTEM: <→top, →bot>. Transformation Type of of Connective Interval Bound g1t = I(→T op) = min(1, 1 − a + b) →T op g2t = C(→T op) = min(1, 1 + a − b) ←T op g3t = D(→T op)= max(0, b − a) 6←Bot g4t = N(→T op)= max(0, a − b) 6→Bot g5t = LC(→T op)= min(1, a + b) ∨T op g6t = LD(→T op)= max(0, 1 − a − b) ↓Bot g7t = RC(→T op) = min(1, 2 − a − b) |T op g8t = RD(→T op) = max(0, a + b − 1) ∧Bot

g1b = I(→Bot) = max(1 − a, b) →Bot g2b = C(→Bot) = max(a, 1 − b) ←Bot g3b = D(→Bot) = min(1 − a, b) 6←T op g4b = N(→Bot) = min(a, 1 − b) 6→T op g5b = LC(→Bot) = max(a, b) ∨Bot g6b = LD(→Bot) = min(1 − a, 1 − b) ↓T op g7b = RC(→Bot) = max(1 − a, 1 − b) |Bot g8b = RD(→Bot) = min(a, b). ∧T op

Fig. 2. Symmetric 8 element abstract group S2×2×2

S2×2×2 group is shown in Fig. 2. The graph of the Piaget group (cf. [16] p.160) is a subgraph of the graph in Fig. 2 [13] Thus, ¬S a defined by Sheffer generates the interval of negation

TABLE II [ |Bot, |T op ] = [ 1 − a, min(1, 2(1 − a)) ]. TOP/BOTCONNECTIVESIN m1 SYSTEM

Logical Type Valuation Similarly, the negation by Nicod (¬N ) could be defined by No. Proposition/Connective Conbot ≤ Contop applying ↓T op and ↓Bot which are duals of |Bot and |T op, 1 Conjunction(AND) max(0, a + b − 1) a&b ≤ min(a, b) respectively. 2 Non-Disjunction(Nicod) max(0, 1 − a − b)  a ↓ b ≤ min(1 − a, 1 − b) a ↓T op a = 1 − a = D(a |Bot a) 3 Non-Conjunction(Sheffer) max(1 − a, 1 − b) ¬N a = a ↓Bot a = max(0, 1 − 2a) = D(a |T op a). a | b ≤ min(1, 2 − a − b) 4 Disjunction(OR) max(a, b) a ∨ b ≤ min(1, a + b) Thus, the interval of ¬N a generated by Nicod is: 5 Non-Inverse Implication max(0, b − a) a ←| b ≤ min(1 − a, b) 6 Non-Implication max(0, a − b) [↓Bot, ↓T op] = [max(0, 1 − 2a), 1 − a]. a →| b ≤ min(a, 1 − b) 7 Inverse Implication max(a, 1 − b) Hence, the interval of negation is generated by combining both a ← b ≤ min(1, 1 + a − b) 8 Implication max(1 − a, b) intervals of ¬S and ¬N as below: a → b ≤ min(1, 1 − a + b) Definition 3: Interval of Negation 9 Equivalence (IFF) max(1 − a − b, a + b − 1) a ≡ b ≤ min(1 − a + b, 1 + a − b) [¬Bot, ¬T op] = [max(0, 1 − 2a), min(1, 2(1 − a)] 10 Exclusion(XOR) max(a − b, b − a)  [ 0, 2(1 − a)], a ∈ [.5, 1] a ⊕ b ≤ min(2 − a − b, a + b) = [ 1 − 2a, 1 ] a ∈ [0,.5)

IV. INTERVAL NEGATIONS IN MANY-VALUED LOGICS where the median value of the interval negation is As Table II shows, 8 connectives could be ¬ = ¬PLY a = a ↓ a = a | a = 1 − a. intervalized through a group transformation; however, MID T op Bot the rest of 6 unary connectives were not yet: Since both ¬ and ¬ are a non-involutive TOP-BOT pair {true, false, identity (a, b), negation (¬a, ¬b)}. In T op Bot of fuzzy interval negations, we have examined the convergence [8], Kim and Kohout defined the interval of negation in m 1 of their iteration and their nearly involutive property in [9]. logical system using the interval of Sheffer (|) and Nicod (↓) connectives. First, the negation by Sheffer (¬S) is defined by employing V. INTERVAL OF a BYIDEMPOTENCE |T op and |Bot, respectively. The unary connective identity of a can be similarly inter-  a |T op a = min(1, 2(1 − a)) valized to an interval [aBot, aT op] by extending the law of ¬S a = a |Bot a = 1 − a idempotence in the crisp logic: a ∧ a = a and a ∨ a = a. A. Intervalization of a on m1 Defined by the AND Connective

Since the AND connective (∧) of the interval system m1 yields a TOP-BOT pair by means of right dual (RD) transfor- mation of <→Bot, →T op> as in Table I,

 a ∧ b = min(a, b) a ∧ b = T op a ∧ b = max(0, a + b − 1), Bot Fig. 3. The intervals of [aB , aT ] and [¬B , ¬T ] the identity of a generated by the logical AND, aA, is yielded in two forms:  Theorem 5: Gap Theorem 1. (Bandler and Kohout [5]) a ∧T op a = a aA = a AND b − a AND b = a OR b − a OR b a ∧Bot a = max(0, 2a − 1) T op Bot T op Bot = a PLYT op b − a PLYBot b Thus, the interval of aA is = min(ϕa, ϕb). a IFF b − a IFF b = a XOR b − a XOR b [ aA , aA ] = [ max(0, 2a − 1), a ]. T op Bot T op Bot Bot T op = 2min(ϕa, ϕb). B. Intervalization of a in m Defined by the OR Connective 1 Similarly, the margins of imprecision can be formulated for a Since the OR connective (∨) of the interval system m1 TOP-BOT pair of negation and that of identity below. similarly yields a TOP-BOT pair of connectives by means of Theorem 6: Gap Theorem 2. left contra dual (LC) transformation of <→T op, →Bot> as below, ¬T op a − ¬Bot a = aT − aB = min(2a, 2(1 − a))  a ∨ b = min(1, a + b) = 2ϕa a ∨ b = T op a ∨Bot b = max(a, b), Hence, the margins of imprecision can be directly measured by the degree of fuzziness ϕ where ϕa = min(a, 1 − a). the identity of a generated by the logical OR, aO, also appears in two forms:  a ∨T op a = min(1, 2a) = D(a ∧Bot a) VI.LAWS OF EXCLUDED MIDDLEAND CONTRADICTION aO = a ∨Bot a = a = D(a ∧T op a). IN FUZZY INTERVAL LOGIC A. Law of Excluded Middle Thus, the interval of aO by the logical OR connective is: This section investigates the law of excluded middle with the [a , a ] = [a, min(1, 2a)]. OBot OT op interval of negation and that of identity which were defined in Section V where C. Interval of a in m1 logic system

From two intervals of identity idA and idO, an interval of [aBot, aT op] = [max(0, 2a − 1), min(1, 2a)], and identity ([aBot, aT op]) may be generated by combining both [¬Bot, ¬T op] = [max(0, 1 − 2a), min(1, 2(1 − a))]. intervals since a is both the lower bound of aO and the upper bound of aA, respectively. Thus, the interval of identity is Since each of aT , aB, ¬T a and ¬B a was generated using defined as follows. ∨T op, ∧Bot, |T op and ↓Bot, respectively, which belongs to the Definition 4: Interval of Identity closed set of m1 logic system of Łukasiewicz connective, i.e. {→ , ← , 6← , 6→ , ∨ , ↓ , | , ∧ }, as it [a , a ] = [ a , a ] ∪ [ a , a ] T op T op Bot Bot T op Bot T op Bot Bot T op ABot AT op OBot OT op is shown in Table I, ∨ is applied for ∨ to investigate = [ max(0, 2a − 1), min(1, 2a)] T op the law of excluded middle, a ∨ ¬a in each case where  [ 2a − 1, 1 ], a ∈ [.5, 1] = a ∨ b = min(1, a + b). [ 0, 2a ] a ∈ [0,.5) T op In addition, since ¬B a = 1 − aT a and ¬T a = 1 − aB a, where aMID = a ∧T op a = a ∨Bot a = a. an excluded middle is computed with a pair of < aT , ¬B > and that of < aB, ¬T > with ∨T op. Thus, OR(∨T op) and AND(∧Bot) form the TOP and BOT system of identity, respectively, while a, is a median value of aT ∨T op (¬B a) = min(1, 2a) ∨T op max(0, 1 − 2a) the interval of fuzzy identity, aMid. = min(1, min(1, a) + max(0, 1 − 2a)) In addition: = 1 [¬B, ¬T ] = 1 − [aB, aT ]. (1) aB ∨T op (¬T a) = max(0, 2a − 1) ∨T op min(1, 2(1 − a)) Both interval of fuzzy negation and the interval of fuzzy = min(1, max(0, 2a − 1) identity are depicted in Fig. 3. + min(1, 2(1 − a))) The margins of imprecision in the interval of connectives were = 1 formulated in Gap Theorem. (2) Thus, the law of excluded middle holds with TOP-BOT pair [3] W. Bandler and L.J. Kohout. 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Interval Negation in Fuzzy Logics. 1 − aB a, we also compute a degree of contradiction with a In Proc. of NAFIPS (North American Fuzzy Information Processing pair of < aT , ¬B > and that of < aB, ¬T > with ∧Bot. Society), pp. 537-542, San Diego, CA, June 24-27 2007. IEEE. [9] E. Kim and L.J. Kohout. Multiple Negation in Fuzzy Interval Logic. In Proc. World Congress of IFSA (International Fuzzy Systems Associ- a ∧ (¬ a) = min(1, 2a) ∧ max(0, 1 − 2a) ation), pp. 1773-1778, Lisbon, Portugal, July 20-24 2009. T Bot B Bot [10] B. Schweizer and A. Sklar. Probabilistic metric spaces. North Holland, = max(0, min(1, a) (3) New York, 1983. + max(0, 1 − 2a) − 1) [11] V. Novak.´ On the position of fuzzy sets in modeling of vague = 0 phenomena. In R. Lowen and M. Roubens, editors, IFSA ’91 Brussels, vol. Artificial Intelligence, pages 165–167. International Fuzzy Systems Association, 1991. aB ∧Bot (¬T a) = max(0, 2a − 1) ∧Bot min(1, 2(1 − a)) [12] P. Ha´jek and L.J. Kohout. Fuzzy implications and generalized quanti- = max(0, max(0, 2a − 1) fiers. Internat. Journal of Uncertainty, Fuzziness and Knowledge Based + min(1, 2(1 − a) − 1)) Systems, 4(3):225-233, 1996. [13] L.J. Kohout and E. Kim. Group transformations of systems of logic = 0 connectives. In Proc. of IEEE-FUZZ’97, 1:157–162, IEEE, New York, (4) July 1997. Thus, the law of contradiction holds with with TOP-BOT pair [14] L.J. Kohout and W. Bandler. Checklist paradigm and group transfor- mations. Technical Note EES-MMS-ckl91.2, 1979. Dept. of Electrical of negation and identity, vice versa, in the fuzzy interval logic Engineering, University of Essex. U.K. of m1 system: < aT , ¬B a, ∧Bot > and < aB, ¬T a, ∨T op >. [15] L.J. Kohout and W. Bandler. How the checklist paradigm elucidates the semantics of fuzzy inference. In Proc. of the IEEE Internat. Conference VII.CONCLUSIONS on Fuzzy Systems 1992, pp. 571–578. IEEE, New York, 1992. [16] D. Dubois and H. Prade. Fuzzy Sets and Systems: Theory and Both the fuzzy identity (a or b) and the fuzzy negation Applications. Academic Press, New York, 1980. [17] L.J. Kohout and E. Kim. Global characterization of fuzzy logic systems ¬a = 1 − a which were collapsed into a single point are now with para-consistent and gray set features. In P. Wang, ed., Proc. 3rd extended to their intervals of [aBot, aT op] and [¬Bot, ¬T op] Joint Conf. on Information Sciences JCIS’97 (5th Int. Conf. on Fuzzy in the m1 logic system of intervals based on the semantics Theory and Technology), pp. 238–241, Research Triangle Park, NC, March 1997. Duke University. In volume 1: Fuzzy Logic, Intelligent of Checklist paradigm by Bandler and kohout. With both Control and Genetic Algorithms. interval of fuzzy negation and that of fuzzy identity, the law of excluded middle and the law of contradiction were investigated at TOP-BOT pair of both intervals. Both laws fail with the classic negation and an identity of a single point, i.e. ¬a = 1−a and a, which is also a median value of the interval of negation [¬Bot, ¬T op], and a median of [aBot, aT op], respectively. However, they do hold with at TOP-BOT pair of both intervals [aBot, aT op] and [¬Bot, ¬T op] when they were computed with the above interval and a corresponding ORT op (i.e. ∨T ) and ANDBot (i.e. ∧B).

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