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EQUIVALENCES IN TWO-VALUED LOGIC
Chapter 7 EQUIVALENCES IN TWO-VALUED LOGIC Two-valued logic (verification) schema can be constructed either on the twovalued set descriptions or on the fuzzy-valued set descriptions. That is we have (a) {D|oi}, V|oi}} (Berztiss, 1971; Copy, 1982), or (b){Dtoj],V(o,'i}} (Turkmen, 1986) as indicated before in Section 2.2. There are a set of equivalences that exits in two-valued set and logic theory which break down in fuzzy-valued set and twovalued logic theory. Next, we treat each one separately. 7.1. Two-Valued Set(Description) and Two-Valued Logic(Verification) Let us review two-valued set and logic formulas for the combination of two linguistic concepts A and B. In this theory, the descriptive membership values A(x)=a and B(x)=b, a, be {0,1}, corresponding to the set symbols A and B and the truth assignments {T,F} of the verity, i.e., verification, of the descriptive membership values, are all mapped to the lattice {0,1}. Let us consider the linguistic combination "W= A OR B", a basic expression, from amongst the 16 possible meta-linguistic expressions, basic expressions, listed in Table 6.3. As an example, we demonstrate the derivation of Conjunctive and Disjunctive Normal Forms, CNF and DNF in general and for W in particular. In accordance with the discussion presented in Chapter 6, let us first write the descriptive set assignments together with the veristic, truth, assignments for the concepts A and B in predicate form. In general for any two linguistic concepts A and B we start with: "XGX isr A, with a=l, is T" "xeX isr A, with a=0, is F" "yeY isr B, with b=l, is T" "y G Y isr B, with b=0, is F"
(7.1) (7.2) (7.3) (7.4)
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Equivalences In Two-Valued Logic
Next we form the linguistic "OR" combination, i.e., W= A OR B and write the predicate expressions for the four possible combinations: ["xeX ["XGX ["xeX ["xeX
isr A, with isr A, with isr A, with isr A, with
a=l, is T" OR "yeY isr B, with b=l, is T"] is T. a=l, is T" OR "y G Y isr B, with b=0, is F"] is T. a=0, is F" OR "yGY isr B, with b=l, is F"] is T. a=0, is F" OR "yeY isr B, with b=0, is F"] is F.
These predicate expressions form the foundation of the "Truth Table" shown in Table 7.1. It is to be emphasized that in this classical "Truth Table" the descriptive membership values for sets A and B are not stated explicitly but are assumed to be {0,1} implicitly. With these and the application of the normal (canonical) form derivation algorithm stated Section 7.2.1 below, one can derive the DNF and CNF expressions for the two-valued set and two-valued logic paradigm. Table 7.1. Classical truth table interpretations of "A OR B" Truth assignments to classical meta-linguistic variables
Truth assignments to the metalinguistic expression
Primary conjunctions
A
B
"A OR B"
T(A)
T(B)
T(A OR B)
AnB
T(A)
F(B)
T(A OR B)
A n c(B)
F(A)
T(B)
T(A OR B)
c(A) n B
F(A)
F(B)
F(A OR B)
c(A) n c(B)
7.2. (Canonical) Normal Form Derivation With the "Truth Table", Table 7.1, one derives the (canonical) normal form expressions in general and in particular for "A OR B" with normal form derivation algorithm as follows: 7.2.1. (Canonical) Normal Form Derivation Algorithm Step 1. To derive the Disjunctive Normal Form Expression, DNF, one does the following:
(Canonical) Normal Form Derivation
173
For each row of the "Truth Table", if a "T" appears in the column for the combined Meta-Linguistic Expression, in our example, "A OR B" column, then: 1(a) Construct the primary conjunction, n , of the meta-linguistic symbols, e.g., "A" and "B" with the requirement that (i) if "T" appears in the truth column of that symbol, the symbol is taken as it appears in the column heading, i.e., in this case, and usually, in its affirmative form; but (ii) if "F" appears in the truth column of that symbol, the complement of the symbol is taken. 1(b) Next, combine all primary conjunctions obtained in (a) above with the disjunction operator, u . For our example "A OR B", we get three phrases in two-variable two dimensional space in STEP 1 (a) as: AnB; Anc(B); c(A)nB. In STEP 1(b), we get one complex clause (AnB)u(Anc(B)u(c(A)nB) in two-variable two dimensional space. Hence, we get: DNF(A OR B)= (AnB)u(Anc(B)u(c(A)nB) As shown in top section of Table 6.4, Table 6.4.(a) row 3. STEP 2. To derive the Conjunctive Normal Form Expression, CNF, one does the following: For each row of the "Truth Table", if an "F" appears in the column for the combined Meta-Linguistic Expression, in our example, "A OR B" column, then: 2(a) construct the primary conjunction, n , of the symbols, e.g., "A" and "B", with the same requirement as stated in STEP 1(a). 2(b) Next, combine all conjunctions obtained in (a) above with the disjunction operator, u . For our example, "A OR B", we get one phrase in two-variable twodimensional space in STEP2(a) as:
174
Equivalences In Two-Valued Logic c(A) n c(B)
In STEP 2(b), we get the same expression, i.e., the phrase c(A) n c(B), since there are no other phrases to combine to obtain a complex clause. 7.2.1.1. Discussion Next note that the expressions obtained in STEP 2(a) and (b) are constructed for the " F " entries under the combined Meta-Linguistic Expression, e.g., "A O R B", thus we must take the compliment of the expression obtained in STEP 2(b) to get the Conjunctive Normal Form, CNF. Thus, for our example, we get CNF(A OR B) = c[c(A) n c(B)] = AuB which is shown in the lower section of Table 6.4, Table 6.4.(b) row 3. The Normal Form Derivation Algorithm stated above can be used to derive all DNF and CNF expression shown in Table 6.4, for all 16 possible combination of concepts shown in Table 6.3. As well, it will be used in the derivation of FDCF and FCCF, Fuzzy Disjunctive and Fuzzy Conjunctive Canonical Forms from the construction of Fuzzy Truth Tables to be discussed later in chapter 8. 7.3. Equivalence of Normal Forms In Section 6.3, it was stated that DNFi(*) = CNFi (•), i=l,2,...,16, for all the possible combination of the linguistic concepts that are shown in Tables 6.3 and 6.4, when the two-valued set and two-valued logic axioms listed in Table 6.2 hold. This is the case of the fourth, boundary-valued, class of fuzzy sets discussed above in (iv) of Section 6.5. It will be shown in this chapter and later in Chapter 8 that these 16 equivalences break down in fuzzy set and two valued logic theory. 7.3.1. Equivalence of DNF and CNF of
'IMPLICATION"
In order to demonstrate the equivalence of the Disjunctive and Conjunctive Normal Forms, DNF and CNF, let us discuss the case of the meta-linguistic expressions: "A IMPLIES B is true, T", i.e., ( A ^ B , T), as an example. Let us rewrite the formulas (6.1) and (6.2). We have:
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175
DNF(A^B, T)=[(AnB) u (c(A)nB) u (c(A)nc(B)),T] CNF(A^B, T)=[(c(A)uB),T] where T designates that the meta-linguistic statements on the left hand side and the propositional statement on the right hand side are "true", T, and A and B are crisp sets, n and u are crisp set conjunction and disjunction symbols, respectively. Next let us apply the idempotency, ID, the distributivity, D, the law of excluded middle, LEM, the identity, I, and the commutativity, C, in that order, to the original expression of DNF(A^'B, T) on the right hand side to see that we derive CNF(A^B, T) as follows: DNF(A^B, T)=[(AnB) u (c(A)nB) u (c(A)nc(B)),T] =[(AnB) u (c(A)nB) u (c(A)nB) u (c(A)nc(B),T] =[(Auc(A)) n B) u (c(A)n(Buc(B)),T] =[I n B u c(A ) n I,T] =[B u c(A),T] =[c(A)uB),T] =CNF(A^B, T)
ID D LEM I C
73.2. Equivalence ofDNF and CNF of "OR " Connective For another example, let us review the equivalence of DNF and CNF expressions of "OR" connective, i.e., "A OR B". We have DNF(A OR B)= (AnB) u (Anc(B)) u (c(A)nB) CNF(AORB)=AuB Applying distributivity and idempotency axioms as shown in Table 6.2 in general and in particular, adding AnB due to idempotency into the DNF expression, we get DNF(A OR B)= (A n (Buc(B))) u (B n (Auc(A))) with additional application of commutativity to the second phrase. Next with the application of the law of "Excluded Middle" we have Auc(A)=I and Buc(B)=I. With the application of the identity axiom, we get DNF(A OR B)= AuB= CNF(A OR B) In a similar manner, with the appropriate applications of the axioms of twovalued set and logic paradigm, it is clear , as is well known, that
176
Equivalences In Two-Valued Logic DNF(-)i-CNF(-)i,
i=l,...,16.
Thus, the following theorem is stated. Theorem 7.1. Tn two-valued set and two-valued logic theory, DNFi(«) = CNFi (•), i=l,2,...,16. A similar proof can be given for all the sixteen possible combination of concepts applying the axioms of Table 6.2. It is for this reason and for this reason alone, in two-valued set and logic studies, one always use the shorter form either the DNF(») or CNF(»). Thus, we use DNF(A AND B) = AnB, But, we use CNF(AORB)=AuB and, we use CNF(A^B)=c(A)uB etc. When we generalize the descriptive set from two-valued to infinite valued fuzzy sets, but keep the verification to be two valued, i.e., when we investigate {D[o,i],V{o,i}}, we have to ask whether these equivalences between DNF and CNF expressions hold. In fact, we find that these equivalences break down as it will be demonstrated next. 7.4. Direct Fuzzification of DNF and CNF Expressions In Section 6.3, it was stated that directly fuzzified Disjunctive and Conjunctive Normal Forms and in turn Fuzzy Disjunctive and Fuzzy Conjunctive Canonical Forms, are not equivalent to each other, i.e., DNFi(«)^CNFi (•), and FDCF(.)^FCCF(.), i=l,2,...,16. We treat this in two parts as indicated before. In part one which is discussed here in this section, we will assume that "Fuzzified DNFi(-)=FDCFi(-) and "Fuzzified CNFi(*)=FCCFi(-), i=l,2,...,16, in form only, without inquiring into the derivation of FDCF and FCCF, from the Fuzzy Truth Tables, which will be treated later (Turkmen, 1986). It should be noted that we use FDCF, Fuzzy Disjunctive Canonical Form, and FCCF, Fuzzy Conjunctive Canonical Form, as a signal that they are not equivalent to each other. Even though, they are equivalent in form only to their two valued descriptive set versions, i.e., DNF and CNF, respectively. Here, "Fuzzified DNFi(*)=FDCFi (•)" in the sense that we take DNFi(«) of the two-valued set expressions and plug in fuzzy-valued descriptive sets in place of
Direct Fuzzification of DNF and CNF Expressions
177
two-valued descriptive sets and try to find out what happens. And we do the same for "Fuzzified CNFi(»)=FCCFi(»)". Note at this time that these are surrogate expressions. In part two, treated in Chapter 8, we will discuss the derivation of FDCF and FCCF expressions directly from Fuzzy Truth Tables and revisit the concern of non-equivalence, i.e., break down of equivalences. The discussion will treat the cases of (i), (ii) and (iii) of section 6.5. Again, we demonstrate the case of the meta-linguistic expression: "A IMPLIES B is True, T", i.e., ( A ^ B , T). But in this demonstration let us rewrite the predicate version of the set expressions of (6.1) and (6.2) in t-norm. A, t-conorm, V, and standard negation, n, in general as:
// {x,y)
=[{A{x)^B{y))V{n{A{x))^b{y)W{n{A{x)^n{B{y))),l]
DNF(A^B,T)
M (x,y)
^[n(A(x))yB(y)j]
CNF(A^B,T)
where A(x) and B(y) designate the membership values of x in the fuzzy sets A and y in the fuzzy set B, and n(A(x))=l-A(x) is the standard negation, i.e., complement operation in the membership domain, i.e., in the computational expression domain. The following predicate expressions are all true, hence, we do not write T after each one of them on the right or left hand sides. But we are indicating the axioms that are used, such as Boundary condition, B, and Monotonicity, M, that all t-norms and t-conorms, including pseudo-t-norms and co-norms, are subject to: 7.4.1. Conjecture To show the non-equivalence, i.e., break down of the equivalence, we conjecture that if V[A(A(x), B(y)), A(n(A(x)), B(y))]
M,B
holds and since in general we have A[n(A(x)), n(B(y)) < A[n(A(x)),l] = n(A(x)),
M,B
Then, we get: V[V[A(A(x), B(y)), A(n(A(x)), B(y))], A[n(A(x)), n(B(y))]]< V(n(A(x)), B(y)).
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Equivalences In Two-Valued Logic
It should be noted that, the proof rests on a conjecture as well as, monotonicity and boundary axioms. The conjecture will be shown to hold for a class of tnorms and t-conorms later. In particular, the conjecture holds for Archimedean t-norms that are strict and nilpotent, as well as special cases of such as, (v, A, n(.)), (0, O, n(.)), (Le, L©, n(.)), etc. Therefore, we have DNF(A^^B, T)c CNF(A^^B, T) whenever, we take two valued normal form expressions and directly substitute fuzzy valued sets to them. Thus, we state the following theorem. This is the theorem that essentially states the break down of the equivalences that hold in two-valued set and two valued logic paradigm. Theorem 7.2. "Fuzzified DNFi(-)"^"Fuzzified DNFi(*)", i= 1,2,.., 16, for all tnorms and t-conorms including pseudo-t-norms and t-conorms for which the conjecture, 7.4.1, holds. A similar proof can be given for all the sixteen possible combination of concepts applying the axioms of t-norms and t-conorms, and pseudo-t-norms and t-conorms, recall that all that was needed monotonicity and boundary axioms as well as the conjecture. A more general treatment will be given later with generating functions of t-norms and conorms. In Figures 7.1.1, 7.1.2; 7.2.1, 7.2.2; 7.3.1, 7.3.2; and 7.4.1, 7.4.2 we have shown the containment of DNF and CNF for two De Morgan Triples, i.e., (v, A, n(.)) and (0, 0,n(.)) for "A OR B" and "A AND B", respectively in Appendix 7, together with the computer programs and graphs. A more general proof the containment of FDCF(.) c FCCF(.) beyond non-equivalence, i.e., FDCF(.)^FCCF(.) will be treated later in Chapter 9. 7.5. Consequences of D|o,i}V(o,i} In two-valued set and logic paradigm mapping the set elements and truth elements both to {0,1} simplifies the knowledge representation but creates certain paradoxes due to the loss of information by imposing an arbitrary cut off to the graded membership values of the elements of the set. In general membership values less than 0.5 are mapped to "0" and those that are greater than or equal to 0.5 are mapped to " 1 " . Naturally other thresholds could and have been used for the purpose of certain applications. At any rate certain paradoxes are created. A well known example is called Russell's paradox or Barber's paradox, etc., under various interpretations in various contests. We state six of these paradoxes this section (Curry, 1963).
Consequences of D{o,i}V{o,i}
179
7.5.7. Russell' s Paradox Our intuition tells us that we can consider classes of objects as forming new objects. Thus, we can consider the class of all chairs in this room, the class of all men, of all houses, of natural numbers. Likewise, we can consider classes of classes, and even such notions as the class of all classes, or the class of all ideas. Among these classes, there will be two sorts, which we shall call proper and improper classes. Proper classes are those, like men, houses, numbers, which are not members of themselves; improper classes are those which, like the class of all classes, or the class of all class of all ideas, are members of themselves. Now, let 7? (the Russell class) be the class of all proper classes. If 7? is a proper class, then, since 7^ is the class of all such classes, 7^ is a member of 7^, and hence 7? is not a proper class. On the other hand, if 7? is not a proper class, then 7? is not a member of 7?, and therefore 7? is a proper class. Either assumption leads to a contradiction. 7.5.1.1. Formal Treatment of Russell's Paradox It is instructive to express the paradox in symbols. Let the statement that x is a member of the class y be symbolized by the notation X e y
'x' and 'y' being variables for which names of arbitrary notions can be substituted, and let -i and <^ be symbols for negation and logical equivalence, respectively. Then, by definition of 7?, we have for arbitrary x, X e 7? <^ —1 (x e x)
and hence, ReR
<^^(ReR)
Thus, the statement that 7? e 7? is equivalent to its own falsehood, and hence, if it is true, it is also false, and vice versa. 7.5.2. Barber'sPseudoparadox The council of a certain village is said to have given orders that the village barber (supposedly unique) was to shave all the men in the village who did not shave themselves, and only those men. Who shaved the barber?
180
Equivalences In Two-Valued Logic
7.5.3. Catalogue Pseudo
paradox
A certain library undertook to compile a bibliographic catalogue listing all bibliographic catalogues, and only those catalogues, which did not list themselves. Did the catalogue list itself? These arguments are here called pseudo paradoxes, because there is no actual contradiction. In the first case, the village barber could not obey the law, which was therefore ridiculous, like that said to have been passed by an American state legislature to the effect that, when two trains approach a crossing at right angles, each one must wait until the other one passes by. Likewise, the library simply could not make a catalogue satisfying the stated requirements. However, we come to realize that how certain orders given by kings, emperors, sultans, dictators, etc., did create ridiculous conflicts and difficulties in the past. But, such explanations as these, do not apply to the Russell Paradox. In terms of logic as it was known in the nineteenth century, the situation is simply inexplicable. This is true in spite of the fact that in the greater sophistication of the present time one may see, or think he sees, wherein the fallacy consists. 7.5.4. Burali-Forti
Paradox
This, the first of the mathematical paradoxes to be published, is of a more technical nature. It involves the theory of transfinite ordinal numbers. For readers acquainted with that theory, the paradox can be stated as follows. It is shown in that theory that (1) every well-ordered set has a (unique) ordinal number; (2) every segment of ordinals (i.e., any set of ordinals arranged in natural order, which contains all predecessors of each of its elements) has an ordinal number, which is greater than any ordinal in the segment; and (3) the set B of all ordinals in natural order is well ordered. Then, by statements (3) and (1), ^ has an ordinal (3; since |3 is in B, we have (3 < |3 by statement (2), which is a contradiction. 7.5.5. Cantor's
paradox
This paradox, although not published until 1932, was known to Cantor as early as 1899; it had a great influence in leading Russell to construct his paradox, in fact rather more than did the earlier paradox of Burali-Forti. It is based on the theory of cardinal numbers. According to that theory the set of all subsets of a set M has a cardinal number higher than that of M. This is a contradiction, if M is the set of all sets.
Consequences of D{o,i}V{o,i}
181
7.5.6. Liar's paradox This paradox takes several forms. The simplest is that of the man, who says "I am lying"; if he lies, he is speaking the truth, and vice versa. Another version is that of Epimenides, the Cretan, who is alleged to have stated that all statements made by Cretans were lies, it being understood that all other statements made by Cretans were certainly false. Modern versions give a statement to the effect that a proposition described in such and such a way is false, the description being constructed so as to apply uniquely to that statement itself. The paradox caused a great commotion in antiquity, and is said to have cause the death of a certain Philites of Cos. The first-mentioned form of the paradox seems to be due to Eubilides of Miletus; any actual connection of Epimenides with any form of it is doubtful. In barber paradox, we run into a paradox of not knowing how the barber gets his shaving. That is if he does not shave himself, he must be shaved by the barber, i.e., himself, and vice versa, if he shaves himself he cannot be shaved by the barber, i.e., himself. Let A be the case that the barber shaves himself, and thus c(A) is the case that the barber does not shave himself. Therefore, in classical paradigm, we have Anc(A)=0 Thus nobody can say how the barber gets shaved. Anc(A)=0 is known as the "Law of Contradiction" which is used in many theorem proving exercises in mathematics that takes its bases as the two-valued set and logic theory. The dual of this is known as the "Law of Excluded Middle", i.e., Auc(A)=I which require an arbitrary threshold to determine the {0,1} assignments of the two-valued paradigm. It will be shown later that in fuzzy theory, we have to replace LEM with "Fuzzy Middle" and contradiction with "Fuzzy Contradiction". They both hold only to a degree. This is yet another example where the classical laws break down but they are upheld in a matter of degree in fuzzy set and two valued logic theory. The difficulty with these paradoxes can in part be resolved by a separation of descriptive and veristic statements as we discussed earlier as an extension of Tarski's explanation of the nature of truth which we had discussed in Chapter 1.
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Equivalences In Two-Valued Logic
7.6. Symbols, Propositions and Predicates A good example of arbitrary cut off can be demonstrated for the "day" and "night" concepts in everyday life vs^hich we had discussed in Chapter 2. Let, for example, the set symbol A stand for our mental state that is linguistically expressed as "the daylight hours of a twenty-four-hour day". And hence, its complement, the set symbol, c(A), stands for our mental state that is linguistically expressed as "not the daylight hours of a twenty-four-hour day". Let also X denote the twenty four hours of a day X = [0:00 a.m., 12:00 p.m. (midnight)], and X' = [6:00 a.m., 6:00 p.m.], be the "daylight hours, and X" = [6:01 p.m., 5:59 a.m.] be not the daylight hours" such that X = X' u X " where U is the set disjunction operator. In studies of logic, we investigate propositions of such linguistic expressions with the symbols of their set representations. It should be recalled that a proposition is "an expression in a language that is either true or false". For the example case stated above, the propositions of the linguistic expressions with their symbols defined above are stated below: (1.1) X' is in the set of "the daylight hours". A, is true, T, i.e., X' isr A is T; (1.2) X' is not in the set of "the daylight hours". A, is false, F, i.e., X' isr not AisF; (2.1) X' is not in the set of "not the daylight hours", c(A), is true, T, i.e., X isr not c(A) is T; (2.2) X' is in the set of "not the daylight hours", c(A), is false, F; i.e., X' isr c(A) is F; where "isr" is a short hand notation that stands for "is in", "belongs to", "compatible with", etc., depending on context in fuzzy theory whereas "is" is used in its usual two-valued sense of the classical set theory. At times, we need to specify a particular time of the day with a predicate. Again it should be recalled that a predicate is "something that is affirmed or denied of the subject in a proposition in logic". For example, the predicates of the two propositions (1.1) and (1.2) stated above are: (1.1.1) the particular time of day, say x=6:30 a.m., or any x e X', in the set of hours of this day, X, is in the set of "the daylight hours", A, with 100% membership value, i.e., |Li(x,A)=a=l, is true, T; i.e., x G X ' isr A, with a=l, is T;
Symbols, Propositions and Predicates
183
(1.2.1) the particular time of day, say x=6:30 a.m., or any x G X', in the set of hours of this day, X, is in the set of "the daylight hours", A, with the 0% membership value, i.e., |i(x,A)=a=0, is false, F, i.e., x e X isr A, with a=0, is F. It should be clear, there is a hidden, but rather arbitrary assumption in these statements. That is, we have defined "the daylight hours", A, crisply, to be equivalent to the time interval of [6:00 a.m., 6:00 p.m.], i.e., X = [6:00 a.m., 6:00 p.m.] and A= {(x,a) | a=l, x e X }, and the set of "not the daylight hours", c(A), to be equivalent to [6:01 p.m., 5:59 a.m.], i.e., X" = [6.01 p.m., 5:59 a.m.], and c(A)={(x,n(a)) | n(a)=l,x e X ' } with the further assumption that our watch or clock is only able to show us the hours and minutes, i.e., a digital clock or watch, of a given day (Figure 2.4). The examples of the predicates of the two propositions (2.1) and (2.2) stated above are: (2.1.1) the particular time of day, say x = 6.30 a.m., or any x e X , in the set of hours of this day, X, is in the set of "not the daylight hours", c(A), with 0% membership value, i.e., |LI(X, C(A)) = n(a) = 0, is true, T; i.e., x G X isr c(A), with n(a) = 0, is T. (2.2.1) the particular time of day, say x = 6.30 a.m., or any x G X , in the set of hours of this day, X, is in the set of "not the daylight hours", c(A), with the 100% membership value, i.e., |i(x,c(A)) = n(a) = 1 is false, F; i.e., x G X isr c(A) with n(a) = 1, is F. (Figure 2.4) It should be noted that in these examples, i.e., (1.1.1), (1.2.1), (2.1.1) and (2.2.1), a > n(a) for the affirmations, T, and a < n(a) for the negations, F, where n(a)=l-a, and a, n(a) G {0,1}. Furthermore, it is clear that for the other case shown in Figure 2.4, i.e., for x = 5:30 a.m., or any x G X " , there are alternate and analogous propositional and predicate expressions. The examples of these alternate and analogous predicate expressions are (in short form): (1.1.1)' (1.2.1)' (2.1.1)' (2.2.1)'
X G X " isrAwitha=OisT. X G X ' isr Awitha=l is F. X G X " isr C(A) with n(a)=l is T. X G X" isr C(A) with n(a)=0 is F.
It should be noted, however, that in these examples, i.e., (1.1.1)', (1.2.1)', (2.1.1)' and (2.2.1)', a < n(a) for the affirmations, T, and a > n(a) for the negations, F.
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Equivalences In Two-Valued Logic
Naturally, in agreement with our arbitrarily assumed two-valued, crisp, and dichotomous partitions of a given day, it was also assumed that membership in these partitions are to be all or none, i.e., 100% or 0% by explicitly stating a, n(a)G{0,l}. But these arbitrary crisp partitions disregard the periods of the sunset and the dawn which have partial darkness, i.e., degrees of light or degrees of darkness. These concern lead us next to discuss fuzzy (graded) membership in describing sets. These will be discussed in Chapter 8. Appendix 7.1. Computer Programs and Graphs. "Figures 7.1. (A OR B) for Max-Min"} Clear [a, b, cnf, dnf, tN, tC]; tN[ a_,b_] = Min[a, b]; tC[a_, b j = Max[a, b]; c[a_] = 1- a; c[b_] = 1- b; cnf = tC[a, b]; dnf = tC[tC[tN[a, b], tN[c[a], b]], tN[a, c[b]]]; Plot3D[dnf, {a, 0, 1}, {b, 0, 1}, AxesLabel -> {"a", "b", "DNF "}]; {"Figure 7.1.1. DNF(A OR B) for Max-Min"} Plot3D[cnf, {a, 0,1}, {b,0,l}, AxesLabel -> {"a", "b", "CNF "}]; {"Figure 7.1.2. CNF(A OR B) for Max-Min"}
Figures 7.1.1. CNF(A OR B) for Max-Min
Figures 7.1.2. DNF(A OR B) for Max-Min
Figures 7.1. (A OR B) for Max-Min
Clear[a, b, cnf, dnf, tN, tC]; tN[a_,bJ = Min[a, b]; tC[a_, b J = Max[a, b]; c [ a j = 1- a; c [ b j = 1- b; cnf = tN[a, b]; dnf = tN[tN[tC[a, b], tC[c[a], b]], tC[a, c[b]]]; Plot3D[dnf, {a, 0, 1}, {b, 0, 1}, AxesLabel -> {"a", "b", "DNF "}]; {"Figure 7.2.1. DNF(A AND B) for Max-Min"} Plot3D[cnf, {a, 0,1}, {b,0,l}, AxesLabel -> {"a", "b", "CNF "}]; {"Figure 7.2.2. CNF(A AND B) for Max-Min"}
Appendix
185
Figure 7.2.1 .DNF(AANDB) for Max-Min Max-Min
Figure 7.2.2. CNF(AANDB) for
Figures 7.2. (A AND B) for Max-Min
{"Figures 7.3. (A AND B) for Algebraic"} Clear[a, b, cnf, dnf, tN, tC]; tN[a_,b_] = ab; tC[a_, b j = a + b - ab; c[a_] = 1- a; c[b_] = 1- b; cnf = tN[tN[tC[a, b], tC[c[a], b]], tC[a, c[b]]]; dnf = tN[a, b]; Plot3D[dnf, {a, 0, 1}, {b, 0, 1}, AxesLabel -> {"a", "b", "DNF "}]; {"Figure 7.3.1. DNF(A AND B) for Algebraic"} Plot3D[cnf, {a, 0,1}, {b,0,l}, AxesLabel -> {"a", "b", "CNF "}]; {"Figure 7.3.2. CNF(A AND B) for Algebraic"}
Figure 7.3.1. DNF(AANDB) for Algebraic
Figure 7.3.2. CNF(AANDB) for Algebraic
Figures 7.3. (A AND B) for Algebraic
Equivalences In Two-Valued Logic
186
{"Figures 7.4. (A OR B) for Algebraic"} Clear[a, b, cnf, dnf, tN, tC]; tN[a_,bJ = ab; tC[a_, b j = a + b - ab; c[a_] = 1- a; c [ b j = l-b; dnf = tC[tC[tN[a, b], tN[c[a], b]], tN[a, c[b]]]; cnf=tC[a,b]; Plot3D[dnf, {a, 0, 1}, {b,0, 1}, AxesLabel -> {"a", "b", "DNF "}]; {"Figure 7.4.1. DNF(A OR B) for Algebraic"} Plot3D[cnf, {a, 0,1}, {b,0,l}, AxesLabel -> {"a", "b", "CNF {"Figure 7.4.2. CNF(A OR B) for Algebraic"}
'}];
Figure 7.4.1. DNF(AORB) for Algebraic Figure 7.4.2. CNF(AORB) for Algebraic Figures 7.4. (A OR B) for Algebraic
References 1. Copy, I.M. (1982), Introduction to Logic, Mac Milland, New York. 2. Lurk§en, LB. (1986), "Interval-Valued Fuzzy Sets Based on Normal Forms", Fuzzy Sets and Systems, 191-210. 3. Curry, H.B. (1963), Foundations of Mathematical Logic, Mc Graw Hill, New York. 4. Berztiss, A.L. (1971), Data Structure, Lheory and Practice, Academic Press, New York.
(7.1) (7.2) (7.3) (7.4)
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Equivalences In Two-Valued Logic
Next we form the linguistic "OR" combination, i.e., W= A OR B and write the predicate expressions for the four possible combinations: ["xeX ["XGX ["xeX ["xeX
isr A, with isr A, with isr A, with isr A, with
a=l, is T" OR "yeY isr B, with b=l, is T"] is T. a=l, is T" OR "y G Y isr B, with b=0, is F"] is T. a=0, is F" OR "yGY isr B, with b=l, is F"] is T. a=0, is F" OR "yeY isr B, with b=0, is F"] is F.
These predicate expressions form the foundation of the "Truth Table" shown in Table 7.1. It is to be emphasized that in this classical "Truth Table" the descriptive membership values for sets A and B are not stated explicitly but are assumed to be {0,1} implicitly. With these and the application of the normal (canonical) form derivation algorithm stated Section 7.2.1 below, one can derive the DNF and CNF expressions for the two-valued set and two-valued logic paradigm. Table 7.1. Classical truth table interpretations of "A OR B" Truth assignments to classical meta-linguistic variables
Truth assignments to the metalinguistic expression
Primary conjunctions
A
B
"A OR B"
T(A)
T(B)
T(A OR B)
AnB
T(A)
F(B)
T(A OR B)
A n c(B)
F(A)
T(B)
T(A OR B)
c(A) n B
F(A)
F(B)
F(A OR B)
c(A) n c(B)
7.2. (Canonical) Normal Form Derivation With the "Truth Table", Table 7.1, one derives the (canonical) normal form expressions in general and in particular for "A OR B" with normal form derivation algorithm as follows: 7.2.1. (Canonical) Normal Form Derivation Algorithm Step 1. To derive the Disjunctive Normal Form Expression, DNF, one does the following:
(Canonical) Normal Form Derivation
173
For each row of the "Truth Table", if a "T" appears in the column for the combined Meta-Linguistic Expression, in our example, "A OR B" column, then: 1(a) Construct the primary conjunction, n , of the meta-linguistic symbols, e.g., "A" and "B" with the requirement that (i) if "T" appears in the truth column of that symbol, the symbol is taken as it appears in the column heading, i.e., in this case, and usually, in its affirmative form; but (ii) if "F" appears in the truth column of that symbol, the complement of the symbol is taken. 1(b) Next, combine all primary conjunctions obtained in (a) above with the disjunction operator, u . For our example "A OR B", we get three phrases in two-variable two dimensional space in STEP 1 (a) as: AnB; Anc(B); c(A)nB. In STEP 1(b), we get one complex clause (AnB)u(Anc(B)u(c(A)nB) in two-variable two dimensional space. Hence, we get: DNF(A OR B)= (AnB)u(Anc(B)u(c(A)nB) As shown in top section of Table 6.4, Table 6.4.(a) row 3. STEP 2. To derive the Conjunctive Normal Form Expression, CNF, one does the following: For each row of the "Truth Table", if an "F" appears in the column for the combined Meta-Linguistic Expression, in our example, "A OR B" column, then: 2(a) construct the primary conjunction, n , of the symbols, e.g., "A" and "B", with the same requirement as stated in STEP 1(a). 2(b) Next, combine all conjunctions obtained in (a) above with the disjunction operator, u . For our example, "A OR B", we get one phrase in two-variable twodimensional space in STEP2(a) as:
174
Equivalences In Two-Valued Logic c(A) n c(B)
In STEP 2(b), we get the same expression, i.e., the phrase c(A) n c(B), since there are no other phrases to combine to obtain a complex clause. 7.2.1.1. Discussion Next note that the expressions obtained in STEP 2(a) and (b) are constructed for the " F " entries under the combined Meta-Linguistic Expression, e.g., "A O R B", thus we must take the compliment of the expression obtained in STEP 2(b) to get the Conjunctive Normal Form, CNF. Thus, for our example, we get CNF(A OR B) = c[c(A) n c(B)] = AuB which is shown in the lower section of Table 6.4, Table 6.4.(b) row 3. The Normal Form Derivation Algorithm stated above can be used to derive all DNF and CNF expression shown in Table 6.4, for all 16 possible combination of concepts shown in Table 6.3. As well, it will be used in the derivation of FDCF and FCCF, Fuzzy Disjunctive and Fuzzy Conjunctive Canonical Forms from the construction of Fuzzy Truth Tables to be discussed later in chapter 8. 7.3. Equivalence of Normal Forms In Section 6.3, it was stated that DNFi(*) = CNFi (•), i=l,2,...,16, for all the possible combination of the linguistic concepts that are shown in Tables 6.3 and 6.4, when the two-valued set and two-valued logic axioms listed in Table 6.2 hold. This is the case of the fourth, boundary-valued, class of fuzzy sets discussed above in (iv) of Section 6.5. It will be shown in this chapter and later in Chapter 8 that these 16 equivalences break down in fuzzy set and two valued logic theory. 7.3.1. Equivalence of DNF and CNF of
'IMPLICATION"
In order to demonstrate the equivalence of the Disjunctive and Conjunctive Normal Forms, DNF and CNF, let us discuss the case of the meta-linguistic expressions: "A IMPLIES B is true, T", i.e., ( A ^ B , T), as an example. Let us rewrite the formulas (6.1) and (6.2). We have:
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175
DNF(A^B, T)=[(AnB) u (c(A)nB) u (c(A)nc(B)),T] CNF(A^B, T)=[(c(A)uB),T] where T designates that the meta-linguistic statements on the left hand side and the propositional statement on the right hand side are "true", T, and A and B are crisp sets, n and u are crisp set conjunction and disjunction symbols, respectively. Next let us apply the idempotency, ID, the distributivity, D, the law of excluded middle, LEM, the identity, I, and the commutativity, C, in that order, to the original expression of DNF(A^'B, T) on the right hand side to see that we derive CNF(A^B, T) as follows: DNF(A^B, T)=[(AnB) u (c(A)nB) u (c(A)nc(B)),T] =[(AnB) u (c(A)nB) u (c(A)nB) u (c(A)nc(B),T] =[(Auc(A)) n B) u (c(A)n(Buc(B)),T] =[I n B u c(A ) n I,T] =[B u c(A),T] =[c(A)uB),T] =CNF(A^B, T)
ID D LEM I C
73.2. Equivalence ofDNF and CNF of "OR " Connective For another example, let us review the equivalence of DNF and CNF expressions of "OR" connective, i.e., "A OR B". We have DNF(A OR B)= (AnB) u (Anc(B)) u (c(A)nB) CNF(AORB)=AuB Applying distributivity and idempotency axioms as shown in Table 6.2 in general and in particular, adding AnB due to idempotency into the DNF expression, we get DNF(A OR B)= (A n (Buc(B))) u (B n (Auc(A))) with additional application of commutativity to the second phrase. Next with the application of the law of "Excluded Middle" we have Auc(A)=I and Buc(B)=I. With the application of the identity axiom, we get DNF(A OR B)= AuB= CNF(A OR B) In a similar manner, with the appropriate applications of the axioms of twovalued set and logic paradigm, it is clear , as is well known, that
176
Equivalences In Two-Valued Logic DNF(-)i-CNF(-)i,
i=l,...,16.
Thus, the following theorem is stated. Theorem 7.1. Tn two-valued set and two-valued logic theory, DNFi(«) = CNFi (•), i=l,2,...,16. A similar proof can be given for all the sixteen possible combination of concepts applying the axioms of Table 6.2. It is for this reason and for this reason alone, in two-valued set and logic studies, one always use the shorter form either the DNF(») or CNF(»). Thus, we use DNF(A AND B) = AnB, But, we use CNF(AORB)=AuB and, we use CNF(A^B)=c(A)uB etc. When we generalize the descriptive set from two-valued to infinite valued fuzzy sets, but keep the verification to be two valued, i.e., when we investigate {D[o,i],V{o,i}}, we have to ask whether these equivalences between DNF and CNF expressions hold. In fact, we find that these equivalences break down as it will be demonstrated next. 7.4. Direct Fuzzification of DNF and CNF Expressions In Section 6.3, it was stated that directly fuzzified Disjunctive and Conjunctive Normal Forms and in turn Fuzzy Disjunctive and Fuzzy Conjunctive Canonical Forms, are not equivalent to each other, i.e., DNFi(«)^CNFi (•), and FDCF(.)^FCCF(.), i=l,2,...,16. We treat this in two parts as indicated before. In part one which is discussed here in this section, we will assume that "Fuzzified DNFi(-)=FDCFi(-) and "Fuzzified CNFi(*)=FCCFi(-), i=l,2,...,16, in form only, without inquiring into the derivation of FDCF and FCCF, from the Fuzzy Truth Tables, which will be treated later (Turkmen, 1986). It should be noted that we use FDCF, Fuzzy Disjunctive Canonical Form, and FCCF, Fuzzy Conjunctive Canonical Form, as a signal that they are not equivalent to each other. Even though, they are equivalent in form only to their two valued descriptive set versions, i.e., DNF and CNF, respectively. Here, "Fuzzified DNFi(*)=FDCFi (•)" in the sense that we take DNFi(«) of the two-valued set expressions and plug in fuzzy-valued descriptive sets in place of
Direct Fuzzification of DNF and CNF Expressions
177
two-valued descriptive sets and try to find out what happens. And we do the same for "Fuzzified CNFi(»)=FCCFi(»)". Note at this time that these are surrogate expressions. In part two, treated in Chapter 8, we will discuss the derivation of FDCF and FCCF expressions directly from Fuzzy Truth Tables and revisit the concern of non-equivalence, i.e., break down of equivalences. The discussion will treat the cases of (i), (ii) and (iii) of section 6.5. Again, we demonstrate the case of the meta-linguistic expression: "A IMPLIES B is True, T", i.e., ( A ^ B , T). But in this demonstration let us rewrite the predicate version of the set expressions of (6.1) and (6.2) in t-norm. A, t-conorm, V, and standard negation, n, in general as:
// {x,y)
=[{A{x)^B{y))V{n{A{x))^b{y)W{n{A{x)^n{B{y))),l]
DNF(A^B,T)
M (x,y)
^[n(A(x))yB(y)j]
CNF(A^B,T)
where A(x) and B(y) designate the membership values of x in the fuzzy sets A and y in the fuzzy set B, and n(A(x))=l-A(x) is the standard negation, i.e., complement operation in the membership domain, i.e., in the computational expression domain. The following predicate expressions are all true, hence, we do not write T after each one of them on the right or left hand sides. But we are indicating the axioms that are used, such as Boundary condition, B, and Monotonicity, M, that all t-norms and t-conorms, including pseudo-t-norms and co-norms, are subject to: 7.4.1. Conjecture To show the non-equivalence, i.e., break down of the equivalence, we conjecture that if V[A(A(x), B(y)), A(n(A(x)), B(y))]
M,B
holds and since in general we have A[n(A(x)), n(B(y)) < A[n(A(x)),l] = n(A(x)),
M,B
Then, we get: V[V[A(A(x), B(y)), A(n(A(x)), B(y))], A[n(A(x)), n(B(y))]]< V(n(A(x)), B(y)).
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Equivalences In Two-Valued Logic
It should be noted that, the proof rests on a conjecture as well as, monotonicity and boundary axioms. The conjecture will be shown to hold for a class of tnorms and t-conorms later. In particular, the conjecture holds for Archimedean t-norms that are strict and nilpotent, as well as special cases of such as, (v, A, n(.)), (0, O, n(.)), (Le, L©, n(.)), etc. Therefore, we have DNF(A^^B, T)c CNF(A^^B, T) whenever, we take two valued normal form expressions and directly substitute fuzzy valued sets to them. Thus, we state the following theorem. This is the theorem that essentially states the break down of the equivalences that hold in two-valued set and two valued logic paradigm. Theorem 7.2. "Fuzzified DNFi(-)"^"Fuzzified DNFi(*)", i= 1,2,.., 16, for all tnorms and t-conorms including pseudo-t-norms and t-conorms for which the conjecture, 7.4.1, holds. A similar proof can be given for all the sixteen possible combination of concepts applying the axioms of t-norms and t-conorms, and pseudo-t-norms and t-conorms, recall that all that was needed monotonicity and boundary axioms as well as the conjecture. A more general treatment will be given later with generating functions of t-norms and conorms. In Figures 7.1.1, 7.1.2; 7.2.1, 7.2.2; 7.3.1, 7.3.2; and 7.4.1, 7.4.2 we have shown the containment of DNF and CNF for two De Morgan Triples, i.e., (v, A, n(.)) and (0, 0,n(.)) for "A OR B" and "A AND B", respectively in Appendix 7, together with the computer programs and graphs. A more general proof the containment of FDCF(.) c FCCF(.) beyond non-equivalence, i.e., FDCF(.)^FCCF(.) will be treated later in Chapter 9. 7.5. Consequences of D|o,i}V(o,i} In two-valued set and logic paradigm mapping the set elements and truth elements both to {0,1} simplifies the knowledge representation but creates certain paradoxes due to the loss of information by imposing an arbitrary cut off to the graded membership values of the elements of the set. In general membership values less than 0.5 are mapped to "0" and those that are greater than or equal to 0.5 are mapped to " 1 " . Naturally other thresholds could and have been used for the purpose of certain applications. At any rate certain paradoxes are created. A well known example is called Russell's paradox or Barber's paradox, etc., under various interpretations in various contests. We state six of these paradoxes this section (Curry, 1963).
Consequences of D{o,i}V{o,i}
179
7.5.7. Russell' s Paradox Our intuition tells us that we can consider classes of objects as forming new objects. Thus, we can consider the class of all chairs in this room, the class of all men, of all houses, of natural numbers. Likewise, we can consider classes of classes, and even such notions as the class of all classes, or the class of all ideas. Among these classes, there will be two sorts, which we shall call proper and improper classes. Proper classes are those, like men, houses, numbers, which are not members of themselves; improper classes are those which, like the class of all classes, or the class of all class of all ideas, are members of themselves. Now, let 7? (the Russell class) be the class of all proper classes. If 7? is a proper class, then, since 7^ is the class of all such classes, 7^ is a member of 7^, and hence 7? is not a proper class. On the other hand, if 7? is not a proper class, then 7? is not a member of 7?, and therefore 7? is a proper class. Either assumption leads to a contradiction. 7.5.1.1. Formal Treatment of Russell's Paradox It is instructive to express the paradox in symbols. Let the statement that x is a member of the class y be symbolized by the notation X e y
'x' and 'y' being variables for which names of arbitrary notions can be substituted, and let -i and <^ be symbols for negation and logical equivalence, respectively. Then, by definition of 7?, we have for arbitrary x, X e 7? <^ —1 (x e x)
and hence, ReR
<^^(ReR)
Thus, the statement that 7? e 7? is equivalent to its own falsehood, and hence, if it is true, it is also false, and vice versa. 7.5.2. Barber'sPseudoparadox The council of a certain village is said to have given orders that the village barber (supposedly unique) was to shave all the men in the village who did not shave themselves, and only those men. Who shaved the barber?
180
Equivalences In Two-Valued Logic
7.5.3. Catalogue Pseudo
paradox
A certain library undertook to compile a bibliographic catalogue listing all bibliographic catalogues, and only those catalogues, which did not list themselves. Did the catalogue list itself? These arguments are here called pseudo paradoxes, because there is no actual contradiction. In the first case, the village barber could not obey the law, which was therefore ridiculous, like that said to have been passed by an American state legislature to the effect that, when two trains approach a crossing at right angles, each one must wait until the other one passes by. Likewise, the library simply could not make a catalogue satisfying the stated requirements. However, we come to realize that how certain orders given by kings, emperors, sultans, dictators, etc., did create ridiculous conflicts and difficulties in the past. But, such explanations as these, do not apply to the Russell Paradox. In terms of logic as it was known in the nineteenth century, the situation is simply inexplicable. This is true in spite of the fact that in the greater sophistication of the present time one may see, or think he sees, wherein the fallacy consists. 7.5.4. Burali-Forti
Paradox
This, the first of the mathematical paradoxes to be published, is of a more technical nature. It involves the theory of transfinite ordinal numbers. For readers acquainted with that theory, the paradox can be stated as follows. It is shown in that theory that (1) every well-ordered set has a (unique) ordinal number; (2) every segment of ordinals (i.e., any set of ordinals arranged in natural order, which contains all predecessors of each of its elements) has an ordinal number, which is greater than any ordinal in the segment; and (3) the set B of all ordinals in natural order is well ordered. Then, by statements (3) and (1), ^ has an ordinal (3; since |3 is in B, we have (3 < |3 by statement (2), which is a contradiction. 7.5.5. Cantor's
paradox
This paradox, although not published until 1932, was known to Cantor as early as 1899; it had a great influence in leading Russell to construct his paradox, in fact rather more than did the earlier paradox of Burali-Forti. It is based on the theory of cardinal numbers. According to that theory the set of all subsets of a set M has a cardinal number higher than that of M. This is a contradiction, if M is the set of all sets.
Consequences of D{o,i}V{o,i}
181
7.5.6. Liar's paradox This paradox takes several forms. The simplest is that of the man, who says "I am lying"; if he lies, he is speaking the truth, and vice versa. Another version is that of Epimenides, the Cretan, who is alleged to have stated that all statements made by Cretans were lies, it being understood that all other statements made by Cretans were certainly false. Modern versions give a statement to the effect that a proposition described in such and such a way is false, the description being constructed so as to apply uniquely to that statement itself. The paradox caused a great commotion in antiquity, and is said to have cause the death of a certain Philites of Cos. The first-mentioned form of the paradox seems to be due to Eubilides of Miletus; any actual connection of Epimenides with any form of it is doubtful. In barber paradox, we run into a paradox of not knowing how the barber gets his shaving. That is if he does not shave himself, he must be shaved by the barber, i.e., himself, and vice versa, if he shaves himself he cannot be shaved by the barber, i.e., himself. Let A be the case that the barber shaves himself, and thus c(A) is the case that the barber does not shave himself. Therefore, in classical paradigm, we have Anc(A)=0 Thus nobody can say how the barber gets shaved. Anc(A)=0 is known as the "Law of Contradiction" which is used in many theorem proving exercises in mathematics that takes its bases as the two-valued set and logic theory. The dual of this is known as the "Law of Excluded Middle", i.e., Auc(A)=I which require an arbitrary threshold to determine the {0,1} assignments of the two-valued paradigm. It will be shown later that in fuzzy theory, we have to replace LEM with "Fuzzy Middle" and contradiction with "Fuzzy Contradiction". They both hold only to a degree. This is yet another example where the classical laws break down but they are upheld in a matter of degree in fuzzy set and two valued logic theory. The difficulty with these paradoxes can in part be resolved by a separation of descriptive and veristic statements as we discussed earlier as an extension of Tarski's explanation of the nature of truth which we had discussed in Chapter 1.
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Equivalences In Two-Valued Logic
7.6. Symbols, Propositions and Predicates A good example of arbitrary cut off can be demonstrated for the "day" and "night" concepts in everyday life vs^hich we had discussed in Chapter 2. Let, for example, the set symbol A stand for our mental state that is linguistically expressed as "the daylight hours of a twenty-four-hour day". And hence, its complement, the set symbol, c(A), stands for our mental state that is linguistically expressed as "not the daylight hours of a twenty-four-hour day". Let also X denote the twenty four hours of a day X = [0:00 a.m., 12:00 p.m. (midnight)], and X' = [6:00 a.m., 6:00 p.m.], be the "daylight hours, and X" = [6:01 p.m., 5:59 a.m.] be not the daylight hours" such that X = X' u X " where U is the set disjunction operator. In studies of logic, we investigate propositions of such linguistic expressions with the symbols of their set representations. It should be recalled that a proposition is "an expression in a language that is either true or false". For the example case stated above, the propositions of the linguistic expressions with their symbols defined above are stated below: (1.1) X' is in the set of "the daylight hours". A, is true, T, i.e., X' isr A is T; (1.2) X' is not in the set of "the daylight hours". A, is false, F, i.e., X' isr not AisF; (2.1) X' is not in the set of "not the daylight hours", c(A), is true, T, i.e., X isr not c(A) is T; (2.2) X' is in the set of "not the daylight hours", c(A), is false, F; i.e., X' isr c(A) is F; where "isr" is a short hand notation that stands for "is in", "belongs to", "compatible with", etc., depending on context in fuzzy theory whereas "is" is used in its usual two-valued sense of the classical set theory. At times, we need to specify a particular time of the day with a predicate. Again it should be recalled that a predicate is "something that is affirmed or denied of the subject in a proposition in logic". For example, the predicates of the two propositions (1.1) and (1.2) stated above are: (1.1.1) the particular time of day, say x=6:30 a.m., or any x e X', in the set of hours of this day, X, is in the set of "the daylight hours", A, with 100% membership value, i.e., |Li(x,A)=a=l, is true, T; i.e., x G X ' isr A, with a=l, is T;
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(1.2.1) the particular time of day, say x=6:30 a.m., or any x G X', in the set of hours of this day, X, is in the set of "the daylight hours", A, with the 0% membership value, i.e., |i(x,A)=a=0, is false, F, i.e., x e X isr A, with a=0, is F. It should be clear, there is a hidden, but rather arbitrary assumption in these statements. That is, we have defined "the daylight hours", A, crisply, to be equivalent to the time interval of [6:00 a.m., 6:00 p.m.], i.e., X = [6:00 a.m., 6:00 p.m.] and A= {(x,a) | a=l, x e X }, and the set of "not the daylight hours", c(A), to be equivalent to [6:01 p.m., 5:59 a.m.], i.e., X" = [6.01 p.m., 5:59 a.m.], and c(A)={(x,n(a)) | n(a)=l,x e X ' } with the further assumption that our watch or clock is only able to show us the hours and minutes, i.e., a digital clock or watch, of a given day (Figure 2.4). The examples of the predicates of the two propositions (2.1) and (2.2) stated above are: (2.1.1) the particular time of day, say x = 6.30 a.m., or any x e X , in the set of hours of this day, X, is in the set of "not the daylight hours", c(A), with 0% membership value, i.e., |LI(X, C(A)) = n(a) = 0, is true, T; i.e., x G X isr c(A), with n(a) = 0, is T. (2.2.1) the particular time of day, say x = 6.30 a.m., or any x G X , in the set of hours of this day, X, is in the set of "not the daylight hours", c(A), with the 100% membership value, i.e., |i(x,c(A)) = n(a) = 1 is false, F; i.e., x G X isr c(A) with n(a) = 1, is F. (Figure 2.4) It should be noted that in these examples, i.e., (1.1.1), (1.2.1), (2.1.1) and (2.2.1), a > n(a) for the affirmations, T, and a < n(a) for the negations, F, where n(a)=l-a, and a, n(a) G {0,1}. Furthermore, it is clear that for the other case shown in Figure 2.4, i.e., for x = 5:30 a.m., or any x G X " , there are alternate and analogous propositional and predicate expressions. The examples of these alternate and analogous predicate expressions are (in short form): (1.1.1)' (1.2.1)' (2.1.1)' (2.2.1)'
X G X " isrAwitha=OisT. X G X ' isr Awitha=l is F. X G X " isr C(A) with n(a)=l is T. X G X" isr C(A) with n(a)=0 is F.
It should be noted, however, that in these examples, i.e., (1.1.1)', (1.2.1)', (2.1.1)' and (2.2.1)', a < n(a) for the affirmations, T, and a > n(a) for the negations, F.
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Naturally, in agreement with our arbitrarily assumed two-valued, crisp, and dichotomous partitions of a given day, it was also assumed that membership in these partitions are to be all or none, i.e., 100% or 0% by explicitly stating a, n(a)G{0,l}. But these arbitrary crisp partitions disregard the periods of the sunset and the dawn which have partial darkness, i.e., degrees of light or degrees of darkness. These concern lead us next to discuss fuzzy (graded) membership in describing sets. These will be discussed in Chapter 8. Appendix 7.1. Computer Programs and Graphs. "Figures 7.1. (A OR B) for Max-Min"} Clear [a, b, cnf, dnf, tN, tC]; tN[ a_,b_] = Min[a, b]; tC[a_, b j = Max[a, b]; c[a_] = 1- a; c[b_] = 1- b; cnf = tC[a, b]; dnf = tC[tC[tN[a, b], tN[c[a], b]], tN[a, c[b]]]; Plot3D[dnf, {a, 0, 1}, {b, 0, 1}, AxesLabel -> {"a", "b", "DNF "}]; {"Figure 7.1.1. DNF(A OR B) for Max-Min"} Plot3D[cnf, {a, 0,1}, {b,0,l}, AxesLabel -> {"a", "b", "CNF "}]; {"Figure 7.1.2. CNF(A OR B) for Max-Min"}
Figures 7.1.1. CNF(A OR B) for Max-Min
Figures 7.1.2. DNF(A OR B) for Max-Min
Figures 7.1. (A OR B) for Max-Min
Clear[a, b, cnf, dnf, tN, tC]; tN[a_,bJ = Min[a, b]; tC[a_, b J = Max[a, b]; c [ a j = 1- a; c [ b j = 1- b; cnf = tN[a, b]; dnf = tN[tN[tC[a, b], tC[c[a], b]], tC[a, c[b]]]; Plot3D[dnf, {a, 0, 1}, {b, 0, 1}, AxesLabel -> {"a", "b", "DNF "}]; {"Figure 7.2.1. DNF(A AND B) for Max-Min"} Plot3D[cnf, {a, 0,1}, {b,0,l}, AxesLabel -> {"a", "b", "CNF "}]; {"Figure 7.2.2. CNF(A AND B) for Max-Min"}
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Figure 7.2.1 .DNF(AANDB) for Max-Min Max-Min
Figure 7.2.2. CNF(AANDB) for
Figures 7.2. (A AND B) for Max-Min
{"Figures 7.3. (A AND B) for Algebraic"} Clear[a, b, cnf, dnf, tN, tC]; tN[a_,b_] = ab; tC[a_, b j = a + b - ab; c[a_] = 1- a; c[b_] = 1- b; cnf = tN[tN[tC[a, b], tC[c[a], b]], tC[a, c[b]]]; dnf = tN[a, b]; Plot3D[dnf, {a, 0, 1}, {b, 0, 1}, AxesLabel -> {"a", "b", "DNF "}]; {"Figure 7.3.1. DNF(A AND B) for Algebraic"} Plot3D[cnf, {a, 0,1}, {b,0,l}, AxesLabel -> {"a", "b", "CNF "}]; {"Figure 7.3.2. CNF(A AND B) for Algebraic"}
Figure 7.3.1. DNF(AANDB) for Algebraic
Figure 7.3.2. CNF(AANDB) for Algebraic
Figures 7.3. (A AND B) for Algebraic
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{"Figures 7.4. (A OR B) for Algebraic"} Clear[a, b, cnf, dnf, tN, tC]; tN[a_,bJ = ab; tC[a_, b j = a + b - ab; c[a_] = 1- a; c [ b j = l-b; dnf = tC[tC[tN[a, b], tN[c[a], b]], tN[a, c[b]]]; cnf=tC[a,b]; Plot3D[dnf, {a, 0, 1}, {b,0, 1}, AxesLabel -> {"a", "b", "DNF "}]; {"Figure 7.4.1. DNF(A OR B) for Algebraic"} Plot3D[cnf, {a, 0,1}, {b,0,l}, AxesLabel -> {"a", "b", "CNF {"Figure 7.4.2. CNF(A OR B) for Algebraic"}
'}];
Figure 7.4.1. DNF(AORB) for Algebraic Figure 7.4.2. CNF(AORB) for Algebraic Figures 7.4. (A OR B) for Algebraic
References 1. Copy, I.M. (1982), Introduction to Logic, Mac Milland, New York. 2. Lurk§en, LB. (1986), "Interval-Valued Fuzzy Sets Based on Normal Forms", Fuzzy Sets and Systems, 191-210. 3. Curry, H.B. (1963), Foundations of Mathematical Logic, Mc Graw Hill, New York. 4. Berztiss, A.L. (1971), Data Structure, Lheory and Practice, Academic Press, New York.