Axioms for a theory of semantic equivalence

Fuzzy Sets and Systems 21(1987) 319-349 North-Holland

AXIOMS

FOR A THEORY

319

OF SEMANTIC

EQUIVALENCE

Daniel G. SCHWARTZ Depar!ment

of Computer

Science,

The Florida

State University,

Tallahassee,

FL 32306,

USA

Received July 1985 Revised November 1985 GSdel-type semantic completeness theorems are established for a formal theory of semantic equivalence based on L.A. Zadeh’s concept of a linguistic variable. The linguistics that is employed allows for the expression of propositions such as “it is not the case that ‘young’ is semantically equivalent with ‘not old”‘, or, in symbols l(young(x) = -old(x)). The result is a two-leveled semantics which, at the lower level, is a multivalent interpretation of fuzzy assertions (e.g., -old(x)) in terms of fuzzy subsets of a given universe and, at the upper level, is a two-valued (classical) interpretation based on the fact that two closed fuzzy assertions either do or do not have the same truth value. The main proof is of the Henkin variety, employing adaptations of the algebraic methods found in Rasiowa (91 and Rasiowa and Sikorski [lo]. Keywords:

Axiomatization,

Semantic equivalence, Linguistic variables, Approximate Formalization, Semantic completeness.

reasoning, Fuzzy logic,

1. Introduction The subject of fuzzy sets as an approach to the mathematical representation of vagueness in everyday language was introduced by L.A. Zadeh in 1965 [14]. Although it was slow to gain momentum, the subject has subsequently undergone extensive development in both its theory and its applications. Up to the present time, however, little attempt has been made toward axiomatizing the associated modes of reasoning. The incentive for axiomatizing the theory of approximate reasoning is of course the same as for any other mathematical theory. First, an axiomatization lays down a more or less minimal set of assumptions from which all the true, or ‘semantically valid’, propositions of the theory may be derived. This in turn provides a clear representation of the underlying logic, considered as a system of formal deduction. Second, for the purpose of stating the axioms in a sufficiently precise manner that their semantic completeness can be proved, one inadvertently develops a complete and rigorously defined linguistics adequate for expressing all the relevant concepts of the theory. Hence an axiomatization provides an overall clarification of both the logic and the language associated with the theory in question and thereby serves as a useful guide in its various applications. For the theory of semantic equivalence, which comprises a fragment of approximate reasoning, an axiomatization has potential value especially in the areas of expert systems and fuzzy-logic programming. 01650114/87/$3.50

@ 1987, Elsevier Science Publishers B.V. (North-Holland)

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Prior to the present writing, the only work explicitly concerned with establishing a semantically complete axiomatization of approximate reasoning appears to be that of Gaines [4]. That paper showed that Zadeh’s original multivalent interpretations of the ‘not’, ‘or’, and ‘and’ connectives can be represented by the known axioms for Lukasiewicz’ logic L,. This was significant inasmuch as Bellman and Giertz [l] had previously shown that these interpretations were the only ones which are consistent with a certain simple and intuitively appealing set of arithmetical postulates. On the other hand, [4] additionally showed that the L, version of the logical ‘implies’ is but one of several potentially acceptable modes of fuzzy inference. Thus, while the Lukasiewicz axioms capture an important aspect of approximate reasoning, they are also somewhat overly restrictive. Aside from issues pertaining to the L, axioms, however, there clearly is a much more crucial restriction inherent in the tukasiewicz system. Namely, the language of L, is essentially only that of the propositional calculus and, therefore, a priori forbids expressing many of approximate reasoning’s other more salient ideas. From the seminal paper [14], the subject of fuzzy sets evolved into the study of linguistic hedges [15]; subsequently, it went into linguistic variables [16] and fuzzy logic [2, 171; and, more recently, it has developed into the study of possibilistic reasoning [18, 191. Evidently, in order to encompass this larger collection of ideas, much more sophisticated types of formal systems need to be devised. The work presented in this paper proceeds beyond the currently existing formalisms to what may be regarded as the most natural next stage - specifically, a formalization of Zadeh’s concept of semantic equivalence and its implicit role in the theory of linguistic variables. The end product is a class of formal logical systems that is characterized by a linguistics which operates at two distinct levels. First, a lower level, akin to a first-order language (but without quantifiers), allows for the representation of n-ary fuzzy relations and their use in expressing fuzzy propositions. Then, an upper level formalizes part of the metalanguage that is used for expressing assertions of semantic equivalence between fuzzy propositions. Section 2 gives a cursory overview of this ‘dual-leveled’ linguistics, in process of laying down the specific concepts from approximate reasoning that are to be axiomatized. Section 3 gives the rigorous, fully detailed, definition of the desired class of formal logical systems, including (i) the requisite class of formal languages, (ii) a semantics for those languages, and (iii) the proposed axioms and inference rules. Section 4 (i) undertakes a brief discussion of the problem of semantic completeness, (ii) presents a series of definitions and results leading up to the formulation of a ‘canonical interpretation’ for an arbitrary system in the given class, and (iii) employs this concept in a proof that these systems are semantically complete. Establishing the latter result is, in effect, the primary goal of this paper. The property of semantic completeness ensures that the proposed axiomatization faithfully represents the logic that is inherent in the chosen semantics - where this semantics, in turn, is a well-defined formulation of the chosen ideas from the theory of approximate reasoning. Although one may easily verify, via the

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semantics, that the desired ideas are indeed expressible in the given languages, without a proof of semantic completeness, there is no guarantee that the proposed axioms and inference rules are adequate to derive all of the semantically true (or valid) linguistic expressions. 2. Preliminaries

What follows is essentially self-contained in its use of concepts from the theory of formal logical systems, with the various notations and definitions being a mixture of those found in Mendelson [8] and Shoenfield [12]. The concepts from the theory of approximate reasoning that are elaborated here have been extracted from the above-mentioned works by Zadeh. A fuzzy subset A of a universe V is represented as a mapping pA : U--, [0, 11, where [0, l] denotes the unit interval. The function pA is the membership function for A, and the value pA(x), x E V, is the degree of membership of x in A. For A and B fuzzy subsets of V, fuzzy complement, fuzzy union, and fuzzy intersection are defined by

for all x E V, and equality (notation:

A = B) is defined by

for all x E V. An n-ary fuzzy relation on V is a fuzzy subset of the Cartesian product of n copies of V. By taking x as an n-tuple, the above definitions apply also to n-ary fuzzy relations. Note that a fuzzy subset of V may be regarded equivalently as a unary fuzzy relation on V. A linguistic variable is a variable whose values are expressions in a natural or artificial language [16]. In this paper, such expressions will be referred to as linguistic attributes, and the attribute set A(T) for a linguistic variable ‘V will be generated from a smaller set of terminals by means of a contexf-free grammar. To illustrate, a specific attribute set for the linguistic variable V = Age may be generated by the grammar G = (V,, V,, P), where (i) VT is the set of terminals, consisting of the atomic attributes ‘young’, ‘old’, and ‘middle-aged’, the logical connectives ‘not’, ‘or’, and ‘and’, and the linguistic hedge ‘very’, (ii) V, is a set of nonferminals, consisting of the root R, together with the letters A, B, C, D, and E, and (iii) P is a production system given by R+A, R+R

C+ D,

orA,

C+E,

and B,

C + middle-aged, D + very D,

A+B, A+A

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D + young,

B-+C,

B-+not

C,

C-* CR),

E + very E, E+

old.

A composite linguistic attribute (YE A(v) is produced in G by any sequence of productions in P which starts with the nonterminal R and ends with LY. For example, a production of the composite attribute ‘not very young and not very old’ is R+A*A and B+B and B+not C and B-not D and B+not very D and B+ not very young and B+ not very young and not C-t not very young and not E + not very young and not very E-, not very young and not very old.

Properly chosen, a grammar effectively allows for the production of all the desired linguistic attributes while forbidding any nonsense expressions. For example, the above G disallows such expressions as ‘very middle-aged’ and ‘very not young’. A semantic interpretation I for a linguistic variable 7f consists of a universe V,, together with an assignment of a meaning I(&) in U, to each linguistic attribute (YE A(T). In practice the meaning assignments are established inductively. First, one defines I(&) for each atomic (Y and, as well, selects a specific meaning for each linguistic hedge. Then, one extends I to the composite attributes in A(T) in accordance with some previously established interpretations for the logical connectives. Under a multivalent interpretation, as is the subject of the present paper, the I(&) are given as fuzzy relations on V,, the linguistic hedges are interpreted as operators on fuzzy relations, and the logical connectives are taken as the fuzzy complement, union, and intersection operators defined above. To illustrate, an interpretation I for the linguistic variable “I’ = Age, as defined above, may be developed as follows. Let the universe r/, consist of the ages (integers) from 0 to 100. Following Zadeh [16, Part II, pp. 313-3231 let the fuzzy subsets Y and 0 of VI be defined by f1 py”)=[[l+(~~]m’

f-0 In addition,

if x S 25, ifx>25, if x < 50,

let M be defined by

h(x) or equivalently

= min(l - cL&),

M=-Yfl-0.

1 - Mx))~

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Then the atomic linguistic attributes for Age may be given the assignments Z(young) = Y, Z(old) = 0, and Z(middle-aged) = M, and it remains only to provide an interpretation for the linguistic hedge ‘very’. For the present purpose, we may use the operator that is most frequently suggested in the works of Zadeh. This version of the ‘very’ hedge is of course only exemplary, and other interpretations may equally suffice. For A a fuzzy subset of U,, let the fuzzy subset vA be defined by, for all x E U,, PIA = bh(-#. Then, for each cyE A(Y) we may assign Z(very (u) = r~Z(cu). These assignments, together with the above-mentioned interpretations of ‘not’, ‘or’, and ‘and’, provide all the machinery that is needed for giving meaning assignments to any composite attribute in A(V). For example, the attribute ‘not very young and not middle-aged’ would be interpreted as Z(not very young and not middle-aged) = Z(not very young) n Z (not middle-aged) = Z(very young) n -Z(middle-aged) = -(vZ(young)) fl -Z(middle-aged) = -(VU)

n -M.

Two linguistic attributes

(Y and /3 of a linguistic variable V will be semanticaZZy Z for ‘V if

equivalent with respect to a given interpretation

Z(a) = wa where here = is equality of fuzzy subsets of U,. Note that, with respect to the foregoing interpretation Z for the variable Age, the attribute ‘middle-aged’ is semantically equivalent with the attribute ‘not young and not old’, by the definition of M. On the other hand, it can be seen that ‘young’ is not semantically equivalent with ‘not old’ (and ‘old’ is not semantically equivalent with ‘not young’) so that, under this particular interpretation of Age, ‘young’ and ‘old’ are not antonyms. If (Y is a linguistic attribute of a linguistic variable “Ir and Z is an interpretation for M, then, for each individual a E U,, the degree of membership of a in Z(a), plCa,(a), is the degree of computability of a with (Y. Transition from the concept of a linguistic variable to that of a formal logic begins by writing linguistic attributes as mathematical relations, so as to express propositions, and by interpreting degrees of compatability as degrees of truth. For example, to express the proposition “a is young”

where a E U,, one may write yowd4 and assign this expression the truth value ~l(young)(a). Continuing

with this line of

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development, it is natural to next let -, v, and A be written for ‘not’, ‘or’, and ‘and’, and to treat composite linguistic attributes as logical combinations of elementary relational expressions. For example, the proposition “a is not young and not old” may be written as -young(u)

A -old(a),

which would be assigned the truth value min(1 - ~,~youn,,~(u), 1 - ~~c,,i&a)). Thus we have, in brief outline, the lower part of the dual-leveled linguistics mentioned in Section 1. Development of the upper linguistic level begins by agreeing to take uninstantiated relational expressions as representing linguistic attributes per se and by introducing a new symbol = to express semantic equivalence. To illustrate, the proposition that ‘middle-aged’ is semantically equivalent with ‘not young and not old’ may be expressed by middle-aged(x)

= -young(x)

A -old(x),

where x is understood to range over all individuals in U,. Such a proposition is said to be valid in an interpretation I if all of its instantiations with individuals from U, are true, where an instantiated expression of semantic equivalence is true if the lower-level propositions on either side of the = sign wind up with the same numeric truth value. Evidently, the above proposition is valid in the foregoing interpretation I for Age simply by definition of the fuzzy subset M. It is natural to also allow instantiated propositions at the upper level, in which case = expresses a more restricted sense of ‘meaning the same thing as’ - i.e., that the two lower-level propositions are ‘equally true’. In this case, one would say that the overall proposition is valid if it is true. Accordingly, if a E U, was such that PI(yo”ng)w = 1 - k(old)(Q)9 then the proposition young(u) = -old(u) would be valid, even though the expression young(x) = -old(x) is invalid (since there is a b E U, such that young(b) z -old(b) is not true). Development of the upper linguistic level is completed by introducing some additional connectives, denoted by 1, V, and &, for expressing logical combinations of propositions of semantic equivalence. For example, to express the proposition “it is not the case that u is young to the same degree that a is not old”, one may write l(young(u)

= -old(u)),

which will be true if k(youn&)

=+ 1 -

/-b(old)(~>*

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Evidently, since two instantiated lower-level propositions either do or do not have the same numeric truth value, the upper-level logic is automatically two-valued. For this reason it is natural to interpret the upper-level connectives in accordance with classical propositional calculus. For the purposes of the formal linguistics developed in the following sections it has turned out to be convenient to additionally provide a means for formally representing degrees of truth (or compatibility) at the lower linguistic level. This is accomplished by introducing, for each r E [0, 11, a 0-ary relation symbol K~ which is to always be interpreted by PI&) = rt for all x E V,. Then, to express “a is 0.7 young”,

one writes

young(a) = ~~~~~ and one writes (young(a) =

~0.5

v young(a)) & (young(a) = ~~~~A

young(a))

to express “a is between 0.5 and 0.7 young”. While all the foregoing has concerned only unary relations, it should be noted that the same considerations extend easily to n-ary relations for arbitrary rz. For example, to express the transitivity of a fuzzy similarity relation, one may write, for r, s E [0, 11, (simlr(x, y) =

K,)

& (simlr(y, 2) = K,) 2 (simlr(x, 2) = K, A K,).

In words, “if x is r similar to y and y is s similar to z, then x is min(r, S) similar to 2”. Here, of course, the 3 connective is interpreted classically: P 2 Q stands for

+VQ. 3. Linguistic

theories

3.1. Languages

We shall begin with a complete and rigorous definition of the requisite formal languages. These languages collectively provide the capacity for formalizing the foregoing concepts of interest for the purposes of an arbitrary set of linguistic variables over a single universe. 3.1.1. Symbols Let there be given any proper class of known as symbols. The use of proper below, ensure that there will always be for any particular formal language. The mutually disjoint subclasses of the given 1. Zndividuul variables: a countably alinguistically by x, y, z, etc.

objects (e.g., the ordinal numbers) to be classes, here and in the various places as many symbols available as necessary following collections are assumed to be class: infinite set of symbols, denoted met-

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2. Individual consfants: a proper class of symbols, denoted metalinguistically by a, b, c, etc. This collection is assumed to strictly include as a subclass a proper class of individual names, which may receive the alternate denotations i, j, etc. Individual names are individual constants that will be used to formally represent the specific individuals in the universe of a given semantic interpretation (cf. Section 3.2.1). 3. Equality symbol: one symbol, denoted metalinguistically by =. 4. Fuzzy relation symbols: for each n 2 0 an uncountably infinite set of n-ary relation symbols, denoted metalinguistically by CX,p, y, etc. The 0-ary relation symbols are assumed to include a special symbol, K,, for each number r E [0, 11. The K, will be used as formal representatives of individual truth values from [0, l] (cf. Sections 3.2.1, 3.2.2 and 3.4.3). 5. Multivalent (or fuzzy) connectives: three symbols denoted metalinguistically by - (not; negation), v (or; disjunction), and A (and; conjunction). 6. Hedge (or operator) symbols: for each n 2 1, an uncountably infinite set of n-ary operator symbols, denoted metalinguistically by #, ~JJ,etc. 7. Equivalence symbol: one symbol, denoted metalinguistically by z. 8. Punctuation marks: three symbols, denoted metalinguistically by the comma and the left and right parentheses. 9. Bivalent (or classical) connectives: two symbols, denoted metalinguistically by 1 (not; negation) and V (or; disjunction). Individual variables and individual constants (hence also individual names) are collectively called individual ferms and will have the common notations t, t’, etc. The logical symbols are the individual variables, the equality symbol, the multivalent connectives, the equivalence symbol, the punctuation marks, and the bivalent connectives. These will be common to all languages as defined in Section 3.1.3. The other symbols are nonlogical symbols, any of which might or might not be used in a specific language. 3.1.2. Expressions An expression is any finite sequence of symbols. The class of expressions known as fuzzy assertions will consist of: 1. Atomic fuzzy assertions: expressions of the form c~(t,, . . . , t,), where (Y is

an n-at-y fuzzy relation symbol and tr, . . . , f, are individual terms (note that this includes expressions of the form K where K is 0-ary). 2. Composite fuzzy assertions: all expressions which can be generated from the atomic fuzzy assertions by means of a context-free grammar (Section 2) subject only to the requirement that every such assertion has one of the following four h-m:

-p,

Cp v q), (p A q), and @(PI, . . . , PA

where p, q, pl, . . . , pn are

fuzzy assertions and $ is a hedge symbol. The class of expressions known as formulas will consist of: 1. Atomic formulas: all equations of the form (t = t’), where r and t’ are individual terms, and all equivalences of the form (p = q), where p and q are fuzzy assertions. 2. Composite formulas: all expressions of the forms -IP and (P V Q) where P and Q are formulas.

Axioms

Some abbreviations P& Q PxQ P= Q

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to be used are: -~(lPvlQ) 1PVQ (P =I Q) & (Q 3 P)

(bivalent conjunction), (bivalent implication), (bivalent logical equivalence).

Some notational conventions will be: (i) parentheses are left unwritten when not needed for readability, e.g., (P V Q) may be shortened to P V Q, and (ii) as in the above definitions of & and 2, the connective -I has a higher precedence than any of the others, so that 1P V Q means the same as (1P) V Q, rather than l(P V Q). Similar conventions will apply for composite fuzzy assertions. An expression is closed if it contains no occurrences of individual variables; otherwise it is open. The notation e(e,, . . . , e,/xr, . . . , x,), where e, and el, . . . , e, are expressions, denotes the uniform substitution of an occurrence of ei for each occurrence of Xi in the expression e, with the understanding that no substitution is performed in case Xi does not actually occur in e. Typically, e is either a fuzzy assertion or a formula, and the ei are individual terms. A notation of the form e(e,, . . . , err) indicates only that el, . . . , e, do occur in e. 3.1.3. Formal languages A language L shall be comprised of: 1. Symbols for L: the logical symbols, together with an empty or nonempty

set of each kind of nonlogical symbol. 2. Grammars for L: zero, one, or more context-free grammars which meet the requirement mentioned in Section 3.1.2. 3. Fuzzy assertions of L: all atomic fuzzy assertions which can be made from symbols of L, together with all composite fuzzy assertions that can then be generated by the grammars of L. 4. Formulas of L: all atomic and composite formulas that can be made up from the individual terms and fuzzy assertions of L. The minimal language will be the (unique) language that contains no nonlogical symbols and no grammars. Thus the minimal language contains no fuzzy assertions, but it does contain formulas - e.g., there would be a formula of the form 1(x = y). A specific language for one or more linguistic variables is obtained from the minimal language by adjoining an appropriate set of nonlogical symbols, together with one or more grammars. Typically, one grammar will suffice, but there is no technical difficulty with having several. 3.1.4. Example We may illustrate the foregoing definition by representing the linguistic variable Age, discussed in Section 2, in a formal language L. The nonlogical symbols of L shall consist of four individual constants, denoted by a, b, c, and d, three unary fuzzy relation symbols, denoted by CY,/3, and y, and one unary hedge symbol, denoted by $. The individual constants are taken as representing four distinct ages from a given universe - e.g., from the ages between 0 and 100, inclusive. For the sake of the illustration, let us say that a, b, c, and d are,

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respectively, the ages of Ken, Jeni, Linda, and Russell. The fuzzy relation symbols LY, /3, and y will represent the fuzzy relations ‘young’, ‘old’, and ‘middle aged’, and the symbol @ will represent the hedge ‘very’. It follows that atomic fuzzy assertions can be only of the forms a(t), /3(t), and y(t), where t is either an individual variable or one of the constants II, b, c, and d. When t is a constant, e.g., as in c(b), then the expression represents an actual assertion in the usual sense -in this case, the assertion ‘Jeni is young’. On the other hand, when t is a variable, e.g., as in a(x), then the expression is taken as representing a specific fuzzy subset of the given universe - i.e., the fuzzy set of ‘young’ ages. As a grammar for L, we may use essentially the same grammar as described in Section 2: (i) let the terminals be the atomic fuzzy assertions of L, the multivalent connectives -, v , and A, the hedge symbol @, and the left and right parentheses, (ii) let the nonterminals be the letters R, A, B, C, D, and E (as before) together with the letter 1, (iii) rewrite the production system of Section 2 by everywhere replacing ‘young’ with cu(_t), ‘old’ with p(j), and ‘middle-aged’ with y@), replacing ‘not’, ‘or’, and ‘and’ with -, v, and A, replacing ‘very’ with Cp, adding the production rules j-a,

J+b,

_t+c,

j-d,

and adding the rule -t-)x, for each individual variable X. This completely specifies a language of the kind defined in Section 3.1.3. The composite fuzzy assertions of L are those which can be introduced in accordance with the production rules of the given grammar. The atomic and composite formulas of L are those which can be made up from the fuzzy assertions in accordance with the definition appearing in Section 3.1.2. The composite fuzzy assertion -444)

A --W(d))

expresses the proposition the assertion

“Russell is neither very young nor very old”, whereas

-4x) A -B(x) represents the fuzzy set of ages which are neither young nor old. It is worth noting that L also admits such assertions as -a(u)

v y(c)

and

-(u(x)

v y(y).

The former has the natural interpretation “either Ken is not young or Linda is middle-aged”. The latter, on the other hand, effectively describes a binary fuzzy relation on the universe of ages - to wit, the pairs of ages in which the first age is ‘not young’ or the second is ‘middle-aged’. The manner in which such composite expressions are realized in terms of fuzzy set membership functions will be taken up in Section 3.2.3.

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The atomic formulas of L express propositions between fuzzy assertions. For example, y(x) = -a(x)

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equivalence

A -P(x)

expresses the proposition “middle-aged is semantically equivalent with not young and not old”. Composite formulas accordingly represent expressions of logical relationships between propositions of semantic equivalence. For example, l(~(X)

= -P(x))

3 l(P(X) = -a(x))

expresses: “if ‘young’ is not semantically equivalent with the antonym of ‘old’, then ‘old’ is not semantically equivalent with the antonym of ‘young”‘. The special 0-ary relation symbols K, may be used to express such assertions as “the individual a is 0.7 young” in an equivalence of the form a(a) =

K0.7.

This is based on the interpretations

of the

K~

given formally below.

3.2. Semantic interpretations

The semantics to be used in this paper shall consist of all possible interpretations of the following kind, for all languages of the kind described in Section 3.1. 3.2.1. Definition An interpretation I for a language L has: 1. A universe, U,, comprising a (finite or infinite) set of objects to be known as individuals. For each individual in U,, there is assumed to be a unique individual

name (cf. Section 3.1.1, item 2) individual. The notation i will corresponding denotation is i. The all the names of the individuals in

which serves as the name of that particular denote the name of the individual whose language that is obtained from L by adjoining U, is denoted by L(I).

2. Meaning assignments:

a. To each individual constant a of L, assignment of an individual Z(a) in CJ,. For each individual name i^ of L(I), it is understood that 1(f) = i -that is, I assigns to each name the unique individual of which it is the name. b. To each n-ary fuzzy relation symbol CYof L, assignment of an n-ary fuzzy relation r(a) in U,. This is equivalent to specifying a membership function p+): Uy+ [0, 11. In particular, for each special 0-ary fuzzy relation symbol K~ (if such appear among the nonlogical symbols of L), it is understood that for some r’ E [0, 11, p+,(i) = r’ for all individuals i E U,. Typically, but not necessarily, r’=r.

c. To each n-ary hedge symbol on fuzzy relations in U, such that, in U,, then Z($)(A,, . . . , A,) is equivalent to specifying a function

@ in L, assignment of an n-ary operation I(@) if Al, . . . , A,, are well-defined fuzzy relations a well-defined fuzzy relation in U,. This is F&,, : UT+ [0, l] such that

cLt(+jbt ,,...,A.)(ib . . . , i,) = ~t&b&i.l~ where {il, . . . , i,,} =

ULl {ik.l, . . . , kmt).

. . . , id,

. . . , cLA,(in,l, . . . , in.,.)),

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3. Valuations:

the mapping

I : {closed fuzzy assertions and closed formulas of L(Z)} + [0, l] defined by a. b.

Z(cu(tl, . . * 3 0) = P,(a)(~(f*)~ . - . ) q&J), I(-p) = 1 -Z(p),

c.

z(p v 4) = mW(p),

d. e.

g.

z(p A q) = min[z(p), z(q)], z(Nzh, . . * , PA) = F,(,,wPd~ 1 if Z(t) = Z(t’), Z(t = t’) = i 0 if not, 1 if Z(p) = Z(q), GP = 4) = (o if not,

h.

Z(V)

f.

z(q)], * . . 9 GL)),

= 1 -Z(P),

i. z(P V Q> = mW(P), Z(Q)]. An Z-instance of an expression e of L(Z) is a closed expression of the form e(l,, . . . , t/x,, . . . , x,,), where il, . . . , i,, are individuals in U,. A formula P of L(Z) is valid in Z (notation: Z k P) if Z(P’) = 1 for every Z-instance P’ of P. Note that an interpretation Z for a language L is uniquely determined by specifying a universe U, and assigning a specific meaning in that universe to each nonlogical symbol of L. 3.2.2. Example Let us now develop an interpretation Z for the language L of Section 3.1.4. The universe U, will be the set of integers from 0 to 100. For the sake of the illustration, we shall have that, for each integer n in U,, the individual name for n is denoted by i,,. Thus L(Z) is the language that is obtained from L by adjoining all the i,, for 0 < n G 100; and, for each n, I(;,,) = n. As meaning assignments for the individual constants of L, let us select Z(a) = 40 (age of Ken), Z(b) = 26 (age of Jeni), Z(c) = 34 (age of Linda), and Z(d) = 40 (age of Russell). It follows that, for example, I&) = Z(b), so that the individual 26 has two distinct representatives in L(Z). As meaning assignments for the fuzzy relation symbols a, /3, and y of L, let us specify that ,ul(,) is the membership function py given in Section 2, that p,(@) is the membership function ,uo of Section 2, and that plCv) is defined by CL~(~) = mint1 -

PY,

1 - PO).

Last, we shall specify a meaning for the hedge symbol @ in accordance with the definition of the ‘very’ operator given in Section 2. Let the function F,(+) of Section 3.2.1, item 2.c, be defined by F,(+)(i) = i2, for all i E U,.

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This completely determines an interpretation I for L as defined in Section 3.2.1. It is a simple matter to check that I interprets L as being a formal linguistics for the linguistic variable Age discussed in Section 2. Note, however, that there is nothing intrinsic to L which demands this particular interpretation. An alternative interpretation might be obtained, for example, by expanding the universe to include the ages from 0 to 120, or by changing any of the above meaning assignments, e.g., selecting alternative definitions of py and ,uo. Thus L can be seen as representing several different renditions of Age. On the other hand-and perhaps even more important-there is nothing intrinsic to L which says that a particular interpretation need have anything at all to do with ages. For example, an interpretation I’ for L can be given in which U,, is a set of heights, the four individual constants are assigned four specific heights, the fuzzy relation symbols are interpreted as meaning ‘short’, ‘tall’, and ‘medium’, and the hedge symbol is assigned some rendition of the linguistic ‘more-or-less’. This shows that, rather than representing any specific linguistic variable, L characterizes a particular class of linguistic variables, all of which have the same basic structure - i.e., those which have four individual constants, three unary fuzzy relation symbols, one unary hedge symbol, and whose composite fuzzy assertions are generated by the same grammar. As an illustration of the action of valuation mappings in Z, consider the following: Z(44)

= PdW))

= PI(a~(40)

= PY(40)

=[1+(40-25)2]-1=o.10.

using the definition have

in Section 2 for p y. Similarly, using the definition of po, we

Z@(4) = 0. Hence, I( - n(d) A -B(4)

= mint1 - z(44), 1 -W(4)) = min( 1 - 0.10, 1 - 0.00) = 0.90.

Since, by the foregoing definition of ,ulo,), Z(Y@)) = ~I(,~N4) = mint1 - ~YW)), 1 - P~UG-W) = min(1 - py(40), 1 - ~~(40)) = 0.90, it follows that Z(y(d) = -a(d)

A -/3(d))

= 1.

Thus, this particular interpretation is such that the atomic formula which asserts “‘Russell is middle-aged’ is semantically equivalent with ‘Russell is not young and Russell is not old”’ is ‘true’. In addition, it is easy to see from the definition of ,u~(,,)that the valuation Z will be 1 (true) for every Z-instance of the open formula y(x) = -(u(x)

A -/al(x).

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D. G. Schwartz

Hence this formula is valid in I. On the other hand, the formula cr(X) = -P(x) is not valid in I. For consider the Z-instance where x is replaced by i,,,,. The previous calculations show that I( c&J)

= 0.09,

whereas q-+3(&J)

= 1 - Z(/3&rJ)) = 1 - 0 = 1,

so that k(~40))

and

# P+(L3))

Z(CI&)

The action of Z on @ is straightforward, Z(qb(d))

= F,&Z(a(d)))

= -p(&))

= 0.

e.g.,

= Z&,(0.90)

= (0.90)‘=

0.81.

We may also note that Z(a = d) = 1, since Z(u) = Z(d) = 40. Similarly, in L(Z), since I&,)

= 40,

Z(a = &,) = 1. On the other hand, since Z(b) = 26, we have that Z(b = d) = 0.

From these considerations, the form

it is easy to see that any atomic formula of L having

x=x is valid in I, whereas one of the form x=y is not valid. For the latter, consider the Z-instance (x = y)(L

i^,,lx, y).

This reduces to &)=&, which formula is clearly false in I. Last, observe that, if the symbol K O.10was present in L, then the formula 44

=

KJ. 10

would be true in Z, if in fact Z is such that Z(Q.J

= 0.10.

3.2.3. Remark

The assignments and valuations of an interpretation Z for a language L induce the assignment of a unique fuzzy relation in U, to each open fuzzy assertion of L,

Axioms

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a rheory

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333

according to: 1. for m>O, p I(n(a ,,..., n,,.x,,...,x.))(ilt * . . , i,,) is a restriction of prCn), considered as a function of jr, . . . , j ,,,, j,,,+,, . . * *lm+n, * to the variables jnr+,, . . . , j,,,+,, (and hence is a uniquely defined membership function), 2.

PI(-,,)

= 1 - Pl@),

5. Pl(l$@,,...,P”)) = ~I(~)(P1@,)~ *. . 9Clld,

where it is understood that, for example, if p,(P) is a function of two variables ir, i2 and if plCp) is a function of iz, ig, then pIbVs) will be a function of the three variables iI, iz, ix. 3.3. Axioms and inference rules

This section describes the basic proof-theoretic theories to be defined in Section 3.4.

mechanism for the linguistic

3.3.1. Logical axioms and logical inference rules

These shall consist of the following categories of formulas and mappings of formulas, where ‘formula’ here refers to the entire class as defined in Section 3.1: 1. Hilbert and Ackermann’s axiomatization of the propositional calculus [5]: all formulas having the forms a. (PVP)>P,

b. PI(PVQ>, c. (f’ V Q) = = ((R V PI = CRV Q>>, together with modus ponens: from P and P ZI Q infer Q. 2. Axioms for equality of individuals: all formulas having the forms a. x=x, b. x1 =y, 2 (x2 =y2 3 (x1 =x2 =~yt =y2)), c. x1 = y, 3 (- * * 3 (x,, = y,, 3 (Y(Xl, . . . ) X”) = &I, . . . ) y,)) * * -). 3. Substitution rule: from P infer P(t,, . . . , t,,/x,, . . . , x,,) for any sequence of terms tl, . . . , t,,.

4. Axioms for semantic equivalence of fuzzy assertions: all formulas having the forms a. pzp, b.

~~~~~~~~~~~~~~~~~~~~~~~~~~

c. p=qzI-ps-q, d. p1=q1~072=qq2

=)@1vP2=q*vq2)), e. p1=q41~(P2=q2~031Ap2=41

Aq2)),

f. p,=q411(’ * * =I (p,, = q,, 1 $Q,, * * * , Pn) = @(q1, * * * ,qn)) - - *I. 5. Axioms for a De Morgan lattice, cf. Rasiowa [9. p. 381: all equivalences having the forms a.pvq=qvp,

b. pv(qvr)=(pvq)vr,

PA4=:4AP pA(qAr)=(pAq)vr

C. P V 0, A 4) =‘p,

PA(P”cI)=P

(commutativity) (associativity) (absorption)

D. G. Schwartz

334

d. p v (q A r) = (p v q) A (p v r),p A (q v r) z (p A q) v (JJ A r) (distributivity) e. --p zp, (involution) f. -(pVq)=-PA--q, -@Aq)=‘-pV-4 (De Morgan laws) 6. Linear-ordering

axioms: all formulas having the forms

(Pvq=p)V(Pvq=q)

and

(pAq=p)V(p

Aq=q).

The latter axioms are so named because they are used in Proposition 4.3-3 to establish that a certain lattice ordering is linear. Here it may be noted that these axioms do impose a linear ordering on the closed fuzzy assertions of L, but not on the open ones, since it is possible that p and q are such that, for some substitution instances p’ and q’, we have Z(p’ v q’ =p’) = 1, while, for other substitution instances p” and q”, we have I@” v q” = q”) = 0. We may note also that the right-hand forms for distributivity and the De Morgan laws could be omitted, since they can be derived from those that would remain. 3.3.2. Inference rules in general

In the foregoing items 1 and 4, an inference rule is regarded as a mapping from the class of all formulas (as defined in Section 3.1.2) into itself. Accordingly, an instance of an n-ary inference rule may be denoted by an (n + 1)-tuple (Z-4,. . . , H,,, C), where HI, . . . , H,, are the hypotheses of the instance and C is the conclusion. Such an instance is in a language L if the Hi and C are formulas of L. Let L be a language which contains instances of a rule p, and let I be an interpretation for L. Then p is validity preserving in I if, for every instance (H,, . . . , H,,C)ofpinL,wehavethatI~Hi,fori=1,...,n,onlyifZ~C.An inference rule p will be regular if it is validity preserving in every interpretation for every language which contains instances of p. It is easily verified that modus ponens and the substitution rule are regular. The primary use of the concept of regularity appears in Proposition 3.4.2-3. 3.4. Theories and models

We are now in a position to define the desired class of formal logical systems. 3.4.1. Linguistic theories A linguistic theory - or simply a theory - T of zero, one, or more linguistic

variables is comprised of: 1. A language, L(T): a formal language of the kind described in Section 3.1.3, subject only to the restriction that it be countable - i.e., none of its sets of nonlogical symbols may be uncountably infinite, 2. Axioms: (i) the set of all logical axioms (Section 3.3.1), that are formulas of L(T), together with (ii) an empty or nonempty set of specially chosen formulas of L(T), to serve as nonlogical axioms of T,

Axioms

for

a theory

of semantic

equivalence

335

3. Inference rules: (i) the logical inference rules-modus ponens and the substitution rule - together with (ii) an empty or nonempty set of specially chosen nonlogical rules of T, 4. Theorems: the axioms of T, together with all formulas of L(T) that can be derived from those axioms by means of the inference rules of T. It follows that a theory T may be specified by selecting a language L(T), some nonlogical axioms, and some nonlogical rules. Examples appear in Section 3.4.3. It will be seen in Section 4.3 that the semantic completeness theorems depend crucially on the assumption that L(T) be countable. The minimal theory will be the theory whose language is the minimal language, and which has no nonlogical axioms and no nonlogical rules. A proof of a formula P in a theory T is a finite sequence PI, . . . , P, of formulas of L(T) such that P is P,, and, for each index i = 1, . . . , n, either: 1. fi is an axiom of T, or 2. e is the conclusion in an instance of an inference rule of T in which the hypotheses of that instance are all among the 4 for j < i. Such a proof is said to be of length n. The notation T l- P is used to assert the condition that there exists a proof of P in T, and T 3 P is used to assert the contrary. Proposition

3.4.1-1. A formula

P of L(T)

is a theorem of T if and only if T k P.

Proof. To show that, if P is a theorem of T, then T t P, one uses induction on the theorems of T. This involves two cases: (i) P is an axiom of T, and (ii) P is the conclusion in some instance in L(T) of an inference rule of T. To show that, if T I- P, then P is a theorem of T, one uses induction on the length of proofs in T, with two cases: (i) length n = 1, and (ii) length n > 1. Each of the above parts (ii) employ straightforward applications of the appropriate induction hypothesis. q Observe that a formula of L(T) becomes a formula of the ordinary propositional calculus if we treat atomic formulas as if they were propositional variables. It follows that we may refer to a formula P as being, or not being, a tautology in the usual sense, i.e., P is a tautology if, under the classical interpretation of the logical connectives, P receives the value 1 (or true) under all possible assignments of values (O’s or l’s) to its propositional variables. Proposition

3.4.1-2 (Tautology

of the propositional P,, . . . , P,,andifTkc,

Theorem). Zf a formula P of L(T) is a tautology calculus, then T !- P. If P is a tautological consequence of foralli=l,. . . ,n, then TI-P.

Proof. The first assertion follows from the fact that the axioms and inference rules of T include those for the propositional calculus. The second assertion is a corollary. Details of the analogous theorem for first-order theories may be obtained from Shoenfield [12, pp. 27ff]. 0

D. G. Schwartz

336

A contradiction in T is a formula of L(T) having the form P & 1P. A theory T is consistent if its theorems do not include a contradiction. It happens that, for any formulas P and Q of L(T), Q is a tautological consequence of P & 1P. Thus, by the tautology theorem, together with Proposition 3.4.1-1, this implies that a theory T is consistent if and only if there is at least one formula Q of L(T) such that T 3 Q. In the following, all further applications of Proposition 3.4.1-1 will go without explicit mention - i.e., the assertions “P is a theorem of T” and “T t P” will be used interchangeably. 3.4.2. Semantic models The semantics Z(T)

for a theory T shall consist of all possible interpretations, as defined in Section 3.2, for L(T). An interpretation I a model of T if every theorem of T is valid in 1. This property may be by the notation I k T. A formula P of L(T) is valid in Z(T) (notation: if P is valid in every model of T. Proposition

3.4.2-l (Validity Theorem).

Proof. Immediate, Proposition

semantic in Z(T) is expressed Z(T)

k P)

If T t P, then Z(T) k P.

by the definition of model.

0

3.4.2-2. A theory T is consistent if it has a model.

Proof. Suppose that T is inconsistent, say T t P & 1P. Since a formula of the form P & 1P cannot be valid in any interpretation of L(T) (a consequence of the definition of & in Section 3.1.2), Proposition 3.4.2-l imples that T has no models. Thus if T has a model, it must be consistent. 0 3.4.2-3. If the nonlogical inference rules of a theory T are regular rules, then an interpretation I for L(T) is a model of T if and only if every nonlogical axiom of T is valid in I.

Proposition

Proof. Since axioms of T are theorems of T, it is obvious that, if I k T, then I k P for every nonlogical axiom P. Suppose that an interpretation I is such that I k P for every nonlogical axiom P. It is easily verified that I !=P, for every logical axiom P of T. And it has been noted that the logical rules are regular (Section 3.3.2). Thus, all axioms of T are valid in I, and all inference rules of T always act so that validity is preserved. It follows by the definition of ‘theorem’ that, if P is a theorem of T, then I !=P. q Proposition

3.4.2-4.

The minimal theory is consistent.

Proof. Let T be the minimal theory. By Proposition 3.4.2-3, any interpretation of L(T) is a model of T. Hence T is consistent by Proposition 3.4.2-2. Cl

I

Axioms for a theory of semantic equivalence

337

3.4.3. Example A formal theory T of the linguistic variable Age discussed in the prior examples may be specified as follows. Let L(T) be the language L of Section 3.1.4, and for the time being, let T have as its only nonlogical axiom the formula y(x) = -(Y(x) A -P(x). Then this formula serves as a defining axiom for y in terms of CYand /I, and the interpretation I of Section 3.2.2 is a model of T. For, by previous considerations, the above axiom is valid in I, and the inference rules of T are regular. Hence I k T, by Proposition 3.4.2-3. Next observe that, since the squaring function, which was used previously to define the ‘very’ operator, is not definable in terms of l-, max, and min, there is no formula of L(T) which explicitly defines @ as such. However, an ad hoc version of ‘very’ can be introduced, if one first extends T to a system which effectively delimits the truth values to be a finite set. For example, to limit the set to just the five values 0.0, 0.25, 0.5, 0.75, 1.0, include the 0-ary fuzzy relation symbols ~0.00, ~0.25, ~0.50, ~0.75, and K,.~ as nonlogical symbols of L(T), and add as nonlogical axioms the formulas Ko.ooV K0.z =

K0.251

Ko.z.5

V Ko.so=

Ko.50

Ko.75,

K0.75

V KLOO

V Ko.75

=

K0.50, =

KI.OO>

which rank order the five values, together with the formulas Ko.oo= -KI.oo, K0.25 z -K0.75, K0.50 = -K0.50, which describe their interrelations. In addition, for every n-ary relation symbol (Y of L(T) add the formula 4.h

* * * >X,)

=

KO.O v

* * *v

CY(X,,

. . . , X,,)

= K1,o.

Then, for the purposes of the theory T, every model of Twill treat each K, as if it were the number r in [0, l] and will effectively use only these five numbers as truth values. Having accomplished this, then the general effect of the ‘very’ operator is achieved by adding the following nonlogical rules: from p =

K~.~

infer $~p = K1.OO,

from p = from p z from p z

~~~~~

K0.25

infer @p = K0.50, infer @p G K0.25, infer @p E KO,OO,

from p =

Ko.~

infer (bp = K,,&o.

K0.50

For this particular version of T, the interpretation I develo+. .n Section 3.2.2 becomes a model of T if we agree that the squaring function additionally ‘rounds’ its value to the nearest member of the new value set. At the same time, however, there is nothing inherent in T which requires that this I be its only semantic model. Alternate models may be obtained, for example, by selecting alternate

D. G. Schwartz

338

interpretations for (Y and /3 (perhaps reinterpreting them in such a way that they become antonyms), or by shrinking or enlarging the universe of discourse. On the other hand, it is possible to extend T in such a way that it explicitly requires that all its models have, say, exactly 100 elements. This may be accomplished by adjoining some individual constants ai, u2, . . . . , ulM) to L(T), and adjoining as nonlogical axioms a formula of the form

together will all formulas l(Ui =

aj),

for i #j. The models of this extension of T would then differ essentially only in their variations on the interpretations of cx and 6. Last, let us note that the Kleene-Dienes mode of fuzzy inference can be introduced by adding a binary operator symbol denoted by + to L(T) and adjoining the formulas of the form p+q=-pvq as defining axioms. A future work will demonstrate that many of the more sophisticated modes of fuzzy inference can similarly be defined in this type of formalism. These include, in particular, the well-known Lukasiewicz connective, the concept of semantic entailment [18], and the generalized modus ponens [16].

4. Semantic completeness The results established in this section show that the logical axioms and inference rules of Section 3.3 exactly capture the logic which is inherent in the interpretations defined in Section 3.2. Following Shoenfield [12, p. 41ff], the concept of semantic complefeness may be expressed in two forms: First Form. For any theory T and formula P of L(T), if Z(T) k P then T I- P. Second Form. For any theory T, if T is consistent then T has a model. The former is the converse of the validity theorem, Proposition 3.4.2-1, and the latter is the converse of Proposition 3.4.2-2. Both forms will be established here for linguistic theories, using an adaptation of an algebraic method which was developed by the author and his master’s thesis advisor, S.K. Thomason, for establishing semantic completeness theorems for a class of ‘free-variable theories’ [ll]. That work, in turn, was an adaptation of some techniques of Rasiowa and Sikorsky [lo]. The general strategy may be outlined as follows. We invoke the theory of Boolean algebras (Section 4.1) and consider the ‘Lindenbaum-Tarski algebra’ l+ of equivalence classes of formulas in the language of a given theory T (Section 4.2). If T is consistent, then the equivalence classes of theorems of T are contained in one or more ultrafilters V in r+. Any such V can be used to define some equivalence classes of fuzzy assertions, which turn out to form a De Morgan

Axioms

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a theory

of semantic

equivalence

339

lattice (Section 4.3). Then - the main step - one can define a ‘canonical embedding’ Qv which embeds this lattice in [0, l] in such a way that the lattice ordering is preserved (Section 4.3). This leads to the definition of some ‘canonical interpretations’ Iv for the language L(T) of any consistent theory T (Section 4.4). The basic properties of canonical interpretations are established in Section 4.5, and the two completeness results are proved in Section 4.6. The second form is established by showing that any canonical interpretation of a consistent theory T is a model of T. The first form is established by using algebraic principles to show that, if T f P, then V can be chosen in such a way that Zp is a model of T in which P is not valid. From this it follows that T b P if P is valid in every model of T. 4.1. Boolean algebras

The contents of this section are adapted from [lo]. All bracketed page numbers indicate places in that book where the relevant items are discussed. Let r be a Boolean algebra with respect to some operations U, fl, and - [p. 681. A non-empty subset V of r is a filter in r if, for all a, b E r, (i) a, b E V implies a II b E V, and (ii) a E V implies a U b E V [p. 44, bottom]. A filter V in r is proper in Z if Vf r. A proper filter V in I’ is an ultrafilter in r if it is maximal - i.e., for every proper filter V’ in r, if V’c P, then V= P [p. 461. We have the following propositions: Pl. r contains a unique unit element i defined by a U i = i and a fl i = a, for all a E r, and a unique zero element 8 defined by a n 6 = 8 and a U 6 = a for all a E Z. It follows that I= a U -a and 6 = a rl -a, for all a E r [pp. 37 and 681. P2. Every subset Vof rgenerates a unique filter in r - namely, the intersection of all filters V’ in rsuch that Vc V’ [p. 451. P3. A filter V is proper in r if and only if 6 4 V [p. 461. P4. If V is a proper filter in r and a 4 V, then the filter generated by the set VU {-a} is a proper filter in r [p. 791. P5. Every proper filter is contained in an ultrafilter - i.e., if V is a proper filter in r, then there exists an ultrafilter V’ in r such that Vc P [p. 461. P6. If V is an ultrafilter in r, then, for all a E r, either a E V, or -a E V, but not both [pp. 66-671. P7. Let a + b denote the complement of a relative to b, defined by a j b = -a U b. Then a subset V of r is a filter in r if and only if: (i) 1 E V, and (ii) for all a,bEI’,ifaeVandajbEV,thenbeV[pp.54and56]. P8. A nonempty subset V of r is a filter in r if and only if, for all a, b E r, we have a, b E V iff a fl b E V [p. 441. and -(anb)= P9. De Morgan laws: for all a, b E r, -(aUb)=-an-b -a U -b [p. 691. PlO. If Vis an ultrafilter in I+, then, for all a, b E r, a U b E Viff either a E Vor b E V.

Proposition PlO may be established as follows. If either a E V or b E V, then a U b E V by the definition of filter, part (i). To show that a U b E V implies either a E V or b E V, suppose otherwise, i.e., suppose that both a 4 V and b 4 V. Then, since V is an ultrafilter, both -a E V and -b E V, by P6. This implies that

340

D.G. Schwurrz

-a n -b E V, by P8; so -(a U b) E V, by P9. But then a U b 4 V, by P6 - which is a contradiction. 4.2. Lindenbaum-Tarski

algebras

Let T be any linguistic theory. Define: P - Q iff T I- P = Q, where P and Q are formulas of L(T). Proposition

4.2-l.

P - Q is an equivalence

relation

on the set of formulas

of

L(T).

Proof. Each of reflexivity, symmetry, and transitivity may be established by a straightforward application of the tautology theorem, Proposition 3.4.1-2. Cl Let [P] = {Q 1P - Q}, VT = {[PI 1T t- P}. Proposition

let r’ = {[PI 1P is a formula

of L(T)},

and let

4.2-2. P, is a Boolean algebra with respect to the operations U, n,

and - defined by

[PIU[Ql= P’ V Ql,

PI n [Ql= [P& 121,

-[PI

= [TP].

Proof. All the axioms for a Boolean algebra may be verified by means of the tautology theorem. 0 Proposition

4.2-3.

VT is a filter in r,.

Proof. This uses P7. Part (i) of P7 is established as follows. T F P V lP, by the tautology theorem. So [P VlP] E Vr, which means that [P] U -[PI E VT. Hence 1 E VT, by Pl. For part (ii), observe that [P 1 Q] = [P] j [Q], by the definition of 3 (Section 3.1.2) and + (W). It follows by modus ponens and the tautology theorem that, if [P] E VT and if [P] 3 [Q] E VT, then [Q] E Vr. 0 Proposition

4.2-4.

VT is a proper filter in r, if and only if T is consistent.

Proof.

VT is a filter in r,, by Proposition 4.2-3. By P3, Vr is proper in rr iff By Pl, 6=[P]n-[P]=[P&lP] for all P. Hence OE Vriff Tl-P&iP for any P. It follows that Vr is proper in rr iff T f P & 1P for any P. Cl 8$V,.

The Boolean algebra r, is commonly referred to as the Lindenbaum-Tarski algebra for T. Discussion of such algebras may be found in [9, p. 172ff] and [lo, p. 209fq. 4.3. Canonical embedding

Throughout this section, let T be any consistent theory; let [PI, r,, and Vr be as in Section 4.2; and, in accordance with Proposition 4.2-4 and P5, let V be any

Axioms

for

a theory

of semantic

ultrafilter in r, such that VT c V. Define: p -q fuzzy assertions of L(T). Proposition

equivalence

341

iff [p = q] E V, where p and q are

4.3-l. p - q is an equivalence relation on the set of fuzzy assertions of

L(T).

Proof. Reflexivity. The formula p =p is an axiom of the form 3.3.1-4.a, so Tkp=p. Hence [p=p]~ V=TC V, andp-p. Symmetry. p = q =I @ =p 2 (p =p ZI q sp)) is an axiom of the form 4.b, so the associated equivalence class is in VP It follows, by the definitions of 3 and 3, that [P=q]+(([P=p]+([Psp]$[q=p]))~ VTc V. That [p=p]~ Vwas demonstrated above. Thus, if [p = q] E V, then, by three applications of P7, part (ii), we have that [q =p] E V It follows that, if p -4, then q -p. Transitivity. Similar to the above, using the formula p =p 3 (q = r 3 (p = q I p=r)).

cl

A set A is a De Morgan lattice [9, p. 38 and p. 441 with respect to operations U, tl, and - if it is a distributive lattice for U and fl and if - satisfies involution (- - a = a) and the De Morgan laws. Let 6 denote the lattice ordering of A, defined by a G 6 iff a U b = b. Then G is a partial ordering. It is easily established that, for all a, b E A. Dl. aSb iff -bS-a, D2. a
4.3-2. A, is a De Morgan lattice with respect to U, n, and - defined

by

[PIu kI1= b ” 419 [PIr-l141 = b A419 --[PI = I--PI* Proof. By the axioms listed in Section 3.3.1-5. Proposition

Cl

4.3-3. The lattice ordering of A, is a linear ordering.

Proof. It is required to show that, for all [p] and [q], either [p] G [q] or [q] G [p]. Suppose that [p] C [q]. Then [p] U [q] #[q], by definition of G; so that [p v q = q] 4 V. Then, since V is an ultrafilter, -[p v q = q] E V, by P6. By the linear ordering axioms (3.3.1-6) and the tautology theorem, we have that T kl(pvq=q)x(pvq=p). It follows that -[pvq=q]+[pvq=p]EV, from which one application of P7, part (ii), yields [p v q sp] E V. This implies that [p] U [q] = [p]; hence [q] U [p] = [p], so that [q] % [p]. Similarly, if

M C [PI, then[PI C [ql. 0 It is desired to establish that A, under U,

tl, and -

can be embedded

342

isomorphically in the interval [0, l] under max, min, and l-. can prove the following.

More generally, we

Theorem 4.3-I. Let A be any countable De Morga,l lattice with operations U, fl, and -, and assume that the lattice orderiq of A is linear. Thert there is a one-to-one mapping Q, : A+ [0, l] such that, for all a, b E A, (i) @(a U 6) = max(@(a), 0(b)), (ii) @(a n b) = min(@(a), @p(b)>, (iii) @(-a) = 1 - @(a). Proof.

Let < be the strict ordering of A defined by a
iff

aGbanda#b.

Then, for each a E A, exactly one of the following must hold: (a) -a
Let A,, AZ, and A, be the a in A satisfying, respectively, (a), (b), and (c). It will be shown that the desired 0 can be given in three well-defined pieces by: (1) @: A,+ (4, l] is any S-preserving embedding, where the image ordering is the natural ordering of the reals, (2) @ : AZ-, [tl, (3) @: A3+ [0, 4) is such that @(a) = 1 - @(-a). Clearly, if a, b E AZ, then @(N) = 3 = @i(b), so that @(A,) is well defined. Moreover, @(A,) is well defined if @(A,) is, since, by Dl, N E A3 iff -a E A,.

Hence it remains only to show that there is a well-defined mapping of the kind @(A,). From classical set theory it is known that any countable linearly ordered set can be embedded in any set of type n, where 11 is the ordinal type of the rationals, cf. [13, p. 2111 or [7, p. 2141. It is easily established that the set of rationals in the open interval (4, 1) is of type q. Hence there exists an embedding @: A,+ (4, l] as desired. To show that @ is G-preserving on all of A, one must examine all possible choices of a, b from among A,, AZ, or A3. Except for the case of both N and b in A,, all cases are immediate consequences of the foregoing. For the former, one can use Dl and the fact that Qi is C-preserving on A,. With this settled, we can now establish the three parts of the theorem. (i) Let a, b E A. By D2, we have a
and

bsaUb.

(1)

Since < is linear, we have either a=aUb

or

Since @ is s-preserving imply

b=aUb.

and one-to-one

@(a U 6) = max( @(a), Q(b)).

(2)

(i.e., an embedding),

(1) and (2) together

Axiomsfor

a theory

of semantic

equivalence

343

(ii) Similar to (i), using D3. (iii) Consider the three cases of a in A. If a E A,, then -a E A3; and @(-a) = 1 - @(- - a), by definition of @(A,). Hence @(-a) = 1 - @(a), by involution. If a E AZ, then -a E AZ, and @(-a) = 4 = 1 - f = 1 - @(a), by definition of @(A,). If a E A3, then @(a) = 1 - @(-a), by definition of @(A,), and solving gives @(-a) = 1 - @(a). Cl Now consider the lattice Av of Propositions 4.3-2 and 4.3-3. The set A, is countable, since it has been required that L(T) employ at most countably many symbols (Section 3.4.1). Hence, Theorem 4.3-l applies. This means that there exists a distinct embedding @: A v-* [0, 11 for each ultrafilter V 2 0,. This mapping will be denoted by @v, and it will be referred to as the canonical embedding bused on V, 4.4. Canonical interpretations

Let T, rr, VT, and V be as in the preceding section. [t = t’] E V, where t and t’ are individual terms of L(T). Proposition

4.4-l.

Define:

t - t’ is un equivalence relation on the set of individual

t-t’

iff

terms of

W“).

Proof. Similar to Proposition 4.3-1, using the axioms for equality of individual terms (forms 2.a and 2.b), the substitution rule, and P7. The appropriate theorems of T are obtained by replacing p, q, and r, respectively, by t, t’, and t I’. cl Let [t] = {t’ 1t’ - t}. The canonical interpretation for Iv, is defined as follows: 1. U, = {[t] 1t is an (open or closed) individual term 2. For each individual constant a of L(T), let Zdu) 3. For each n-ary fuzzy relation symbol (Yof L(T),

L(T)

bused on V, denoted

of L(T)}. = [a]. define I,,(a) by

P&>([tIl9. ** r [t,,l)= @v([4L** *3Oh where Qzv is the canonical embedding based on V, and where each t,f is any representative from the equivalence class [ti]. 4. For each n-ary hedge symbol $, define Zlp(r$) by . . ,p,,)]), where pl,. . . ,p,, are fuzzy assertions of L(T) such that Qv([pi]) = ri, for all i = fi~(@)(rl’ * ’ * ’ rn) = 1, . . . , n, assuming that such pi exist, 0 if, for some i, no such pi exists. i In the following, for individuals in U,, denoted by [t], [a], and [xl, their respective individual names will be denoted by ;, B, and f. @d[@(pl,.

Proposition

4.4-2. The universe UI, is well defined.

D. G. Schwartz

344

Proof.

[t] is well defined, by Proposition

Proposition Proof.

4.4-3. For each individual

4.4-l.

constant a of L(T),

[a] is well defined, by Proposition

Proposition

Cl

4.4-l.

&(a) is well defined.

q

4.4-4. For each fuzzy relation symbol a, IV(o) is well defined.

Proof. It is required to show that, if [t,] = [ti], . . . , [t,,] = [t:], then

P,,w([tIl, *- *JM>= PI&)(M . . . >KID. The proof is based on the fact that G+, is well defined (Theorem 4.3-l) similar to that of Propositions 4.3-l and 4.4-l. Here one uses the formula

and is

Cl= t; 3 (* * * 3 (t, = t:, 3 cu(t,, . . . , tJ = aft;, . . . ) tA)> * * *), which can be inferred from an axiom of the form 2.c by means of the substitution rule. 0 Proposition

4.4-S. For each n-ary special operator

symbol

$I, IV($)

is well

defined.

Proof. Similar to Proposition Pl

4.4-4, using the formula

=P; = to -*~CPn=PA~@@l,.

which is an axiom of the form 3.f.

. . ,p”)=@;,.

. . ,p:)>-*-),

•i

Note that, if T is not consistent, then no such interpretation Z, can exist. For, if , then VT = r,, and there is no ultrafilter V in r, such that VT c V (recall that an ultrafilter must be proper). Hence there is no canonical embedding Q5, as required in the definition of Iv. T is inconsistent

4.5. Preliminary

Proposition

results

4.51.

interpretation

Let T be a consistent theory, and let IV be a canonical for L(T). For all linguistic assertions p and q of L(T), we have:

1. @d-PI) = 1 - @dPl)~ @v(b ” 41)= mW@&4), 3. @v(b A 41) = miWMP1)~ 2.

@d91)lJ @&l)l,

where Gv is the canonical embedding associated with Iv.

Proof. By Theorem 4.3-l and the definitions Proposition

of U, fl, and - for A,.

Cl

4.52. Let T and Iv be, as in Proposition 4.5-1, let p be a fuzzy assertion of L(T), and let p(tl, . . . , t,,) be any I,-instance of p (where the notation ii is as in Section 4.4). Then Mp(L

. . . > k) = @v@(t1, . * * , t”)]).

Axioms for a theory of semantic equivalence

345

Proof. We shall use mathematical induction on the length of fuzzy assertions in L(T). IH denotes the induction hypothesis. Case 1: p atomic, of the form cu(t;, . . . , t;). We may assume without loss of generality that all ofi the tf are individual variables so that p(il, . . . , i,,) is in this case just a(?,, . . . , f,). Then we have the following: Z&r(i,,

. . * 9 i, = p&)(qQ,

. . . , Z(L))

= PIv(4([hl, * . . P[4x1) = @vt[~Ol, * * * . , &)I)

(3.2.1-3.a) (3.2.1-2.a) (def. Z,, item 3).

Case 2: p of the form -4. Then * * . ) in)) = 1 - Z,(q(&, . . . ) F,)

M-4,

(3.2.1-3.b)

= 1 - @v(kl(hr . . . , fn)]) = @d--&l9 . * * 5 fn)l> Care 3: p of form q v r. Without loss of generality, same variables appear in both q and r. Then we have Mq6,

W-0 (Prop. 451.1). we may assume that the

* * . ) in) v r(i1, . . . ) i”)) = m=(Mq(L

. . . , id),

W(h,

. . . , L)>>

(3.2.1-3.~)

= m=(%([q(fl, . . . , t,)]), @d[r(tl, . . . , &)I)) W-U (Prop. 451.2). = @&(fl, * . * , t,) v a, . * * 3fn)l) Case 4: p of the form q A r. Similar to Case 3. Care 5: p of the form (p(pi, . . . ,p,), $I m-ary. Here assume that the same variables appear in all of p,, . . . , Pm. Then ~vMpl(L

. . . 9 L),

= F,,(#)(Mp,(L

* * * 9 P,(&>

. . . 3 k)))

* . . 1 EJ), . . . 9 MPm(L

f * * 9 tJ>>

(3.2.1-3.e)

=~v(~)(@v([Pdt~,. . . atn>l>> . . . , @&-dt~~ . . . t cdl)> W-0 . , m), where ri = @d([pi(tlp . . . , t,,)]) (notation) = F,($J)(hl. . = @V([gdpI(C,,

. . . , 0,

. . . , pm(tl,

This is what we are required to show.

. . . , tn)>])

q

Proposition L(T),

(def. Zv, item 4).

4.5-3. Let T-and Z, be as in Proposition 4.5-1, let P be a formula and let P(i,, . . . , t,,) be any Z&stance of P. Then ~v(P(L

* . . , in) = 1 if

[P(tl,

of

. . . , tn)] E V.

Proof. This uses mathematical induction on the length of formulas of L(T). Case 1: P atomic of the form t = t’. Then t and t’ are either individual constants or individual variables of L(T). This means that p(il, . . . , i,) is of the form a = b, where a and b are either individual constants of L(T) or names of individuals in U,. Assume first that both are names, say that a is i, and b is i2.

346

D. G. Schwartz

Then we have Z&, = i,) = 1 iff

Z&r) = I&

(3.2.1-3.f)

iff M = Lb1 iff iff

(3.2.1-2.a) (def. [r], Sec. 4.4) (def. t-r’, Sec. 4.4).

tl-rC2 [I, = r2] E v

Now assume that a is an individual constant of L(T). Then the second step becomes ‘iff [a] = [r2]’ and additionally employs item 2 in the definition of Z,, together with Proposition 4.4-3. The same considerations apply in case b is an individual constant of L(T). &se 2: P atomic of the form p = q. Assume without loss of generality that the same variables appear in both p and q. Then b@(L *. . , in) = q(i*, . . . ) i”)) = 1 iff Z&(ir, . . . , tn)) = &(q(tll, . . . , ?J) iff @&I([,, . . * 901) = @vML . . * 1Ql)

iff [P(h,. - *9fn)]= [4(L . * . 9L)l iff

p(h, . . . , r,) -q(h,

Case 3. P of the form lQ.

. . . ) r”)] E v

Then

Z+Q(&, . . . , in)) = 1 iff ZdQ(;,, . . . , i,,)) = 0 iff iff iff

[Q
(Prop. 4.5-2) ( Gppis 1-1, Thm. 4.3-l) (def. [p], Sec. 4.3) (def. p -9, Sec. 4.3).

. . . , t,>

iiT [P(r,, . . . ) rn) zq(r,,

(3.2.1-3.g)

. . . , r,)] E V . . . , fn)] E V

(3.2.1-3. h) W) (P6, Sec. 4.1) (def. -[PI, Prop. 4.2-2).

Case 4: P of the form Q V R. Assume that the same variables appear in both Q and R. Then

MQ(b

. . . >rn) V NO,, . . . , tJ> = 1 iffeitherZdQ(r,, . . . ,f,,))=lorZ~(R(rI,. . . ,r,,))=l iff either [Q(tr, . . . , tn)] E Vor [R(t,, . . . , rn)] E V

[Qh . . . , fn)] u [W,, . . . , fn)] E V iff [Q(L . . . , 4,) V WI, . . . , Ql E V

iff

(3.2.1-3.i) V-=0 (PlO, Sec. 4.1)

W. PI U[Ql, Prop. 4.2-2).

This completes the proof of Proposition

4.5-3.

Cl

4.6. Completeness theorems Theorem 4.6-I.

Let T be a consistent theory, and let V be any ultrafilter in I-, such

Axioms

for

that VT c V. Then the canonical of T.

a rheory

of semantic

interpretation

equivalence

for L(T)

347

based on V is a model

Proof. Lei T and {v be as specified, let P be any formula of L(T) such that T I- P, and let P(tl, . . . , t,) be any &instance of P. By the definition of model (Section 3.4.2), it is sufficient to show that ZdP(t^,, . . . , I,,)) = 1. Since T l- P, we have that T I- P(tl, . . . , t,,), by the substitution rule. Then [WI, * . . 9 tn)] E VT c V, by definition of VP Hence Zv(P(il, . . . , t,,)) = i, by Proposition 4.5-3. 0 Corollary

4.6-l

(Second Form of the Completeness Theorem).

Every consistent

theory has a model.

Proof. If T is consistent, then there is at least one canonical interpretation for L(T), by Proposition 4.2-4 and the discussion at the end of Section 4.4. By the theorem, any canonical interpretation for L(T) is a model of T. Theorem

4.6-2 (First Form of the Completeness Theorem).

of L(T).

If

P is valid in E(T),

then P is a theorem of T.

Let P be a formula

Cl

Proof. We shall prove the contrapositive. Suppose that P is a formula of L(T) such that T)C P. Then T is consistent, by the discussion at the end of Section 3.4.1. Consider the filter VT in r,. Since T Y P, we have that [P] 4 VT. It follows by P4 (Section 4.1) that the filter generated by the set V, U {-[PI} is a proper filter in r,. In accordance with P5, let V be any ultrafilter in Zr such that VT U {-[PI} c V. Assume that xi, . . . , x, are all the distinct individual variables that occur in P, so that P is P(xl, . . . , x,). Then [lP(x,, . . . , x,)] E V, by the definition of -[PI. Let ZV be the canonical interpretation for L(T) based on V, and consider the ITinstance TP(~~, . . . , a,.,) of lP(xI, . . . , x,). We have that Zv(lP(&, . . . , a,)) = 1, by Proposition 4.5-3. So Z,(P(i,, . . . ,a,)) = 0, by 3.2.13.h. Therefore, Z,tiP. But ZV is a model of T, by Theorem 4.6-l. Hence 2(T)

tip.

Cl

5. Concluding

remarks

Thus we have established the semantically complete axioms for a sizeable fragment of the theory of approximate reasoning. At the same time, however, the present work clearly falls short of being a full axiomatization of the entire theory. Several points deserve mentioning in this regard. First, conspicuously absent in this study is any treatment of the logical quantifiers, either classical or fuzzy. This has here been avoided primarily because of the added complexity that would thereby be introduced into the problem of establishing semantic completeness. Given the foregoing results, however, it now seems reasonable that the existing formulation could be extended to include quantification at a later time. Second, while one here has the ability to explicitly represent the linguistic

348

D. G. Schwartz

attributes associated with any particular linguistic variable, one cannot similarly portray the variable itself. To illustrate, the present systems provide the linguistic machinery necessary to express a proposition such as young(John)

= -old(John),

but not one such as Age(John) = young, as would be required in a full axiomatization of possibility theory. Such linguistic constructs evidently require a somewhat more sophisticated semantics, perhaps based of fuzzy sets of type 2. Third, while the version of approximate reasoning that is considered in the present work is consistent with that originally proposed by Zadeh [14, 15, and 161, it must also be acknowledged that subsequent studies of approximate reasoning have evolved a large assortment of variants. Most of these are based on alternate interpretations of the fuzzy-logical connectives, and each adds a new richness to the expressive power of the original scheme. Different logics will of course lead to different sets of axioms. Yet many are sufficiently similar to the present logic that their axioms could likely be discovered through the same means. Last, it is reasonable to question whether the present notion of semantic equivalence itself might be replaced with a notion of ‘approximate equivalence’, perhaps based on a version of Zadeh’s concept of linguistic approximation [16 and 171. This would lead to a somewhat different class of systems, ones which are multivalent - rather than classical - at the second linguistic level.’ Acknowledgements This work was undertaken as part of the author’s doctoral dissertation, completed at the Portland State University Systems Science Ph.D. Program, Portland, Oregon, U.S.A., in 1981. Here I would like to thank especially Professors George Lendaris and Robert Stanley, Portland State University, and Professor Lotfi Zadeh, University of California at Berkeley, for their guidance and inspiration. Thanks also are due to Professors Hilbert Levitz and Warren Nichols, Florida State University, for several discussions leading to the correction of an important oversight in the earlier proofs. References [l] R. Beihnan and M. Giertz, On the semantic formalism of the theory of fuzzy sets, Inform. Sci. 5 (1973) 149-156. [2] R. Bellman and L.A. Zadeh, Local and fuzzy log&, in: J.M. Dunn and G. Epstein, Eds., Modem Uses of Multiple-Valued Logic (Reidel, Dordrecht, 1977) 103-165. ‘The need to emphasize the last two points was made clear to me from a review of this paper by Professor Dimiter Driankov, University of Linkiiping, Sweden.

Axioms

for

a theory

of semantic

equivalence

349

[3] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1979). [4] B.R. Gaines, Foundations of fuzzy reasoning, Internat. J. Man-Machine Stud. 8 (1976) 623-668. [5] D. Hilbert and W. Ackermann, Grundzuge der Theoretischen Logik (Springer, Berlin, 1928). In English as: R.E. Lute, Ed., L.M. Hammond et al., Transl., Principles of Mathematical Logic (Chelsea, New York, 1950). [6] T.J. Jech, The Axiom of Choice (Elsevier, New York, 1973). [7] K. Kuratowski and A. Mostowski, Set Theory: With an Introduction to Descriptive Set Theory (North-Holland, Amsterdam-New York, 1976). [8] E. Mendelson, Introduction to Mathematical Logic (Van Nostrand, New York, 1964). [9] H. Rasiowa, An Algebraic Approach to Nonclassical Logics (North-Holland, Amsterdam, 1974). [IO] H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics (PNW (Polish Scientific Publishers), Warsaw, 1963). [ll] D.G. Schwartz, Free-variable theories, Masters Thesis, Simon Fraser University, Burnaby, B.C. (1973). [12] J.R. Shoenfield, Mathematical Logic (Addison-Weseley, Reading, MA, 1967). [13] W. Sierpinski, Cardinal and Ordinal Numbers (PWN (Polish Scientific Publishers), Warsaw, 1958). [14] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [15] L.A. Zadeh, A fuzzy-set-theoretic interpretation of linguistic hedges, J. Cybernet. 2 (1972) 4-34. [16] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci. Part I, 8 (1975) 119-249; Part II, 8 (1975) 301-357; Part III, 9 (1976) 43-80. [17] L.A. Zadeh, Fuzzy logic and approximate reasoning, Synthese 30 (1975) 407-428. [18] L.A. Zadeh, PRUF- a meaning representation language for natural languages, Internat. J. Man-Machine Stud. 10 (1978) 395-460. [19] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28.