Formal fuzzy logic

Formal fuzzy logic: a new chapter of multi-valued logic

Under the name "fuzzy logic" one denotes a series of topics related with the notion of fuzzy subset. Usually these topics are related with the applications. Under the name "formal fuzzy logic" or "fuzzy logic in narrow sense" one denotes a new chapter of formal logic aiming to represent in a formal way the vagueness of the natural language and the reasonings involving notions which are vague in nature. We can consider formal fuzzy also as a chapter of multi-valued logic. In fact from a semantical point of view fuzzy logic is not different from first order multi-valued logic. Neverthless in the deduction apparatus one manifests a basic difference. In fact, in multi-valued logic the deduction operator is a tool to associate every (classical) set of axioms with the related (classical) set of theorems (several authors limite our attention to the generation of the set of valid formulas). From such a point of view the paradigm of the deduction in multi-valued logic is not different in nature from the one of classical logic. Instead in fuzzy logic the notion of approximate reasoning is crucial. This notion enables us to define a deduction operator which associates any fuzzy set of proper axioms with the related fuzzy subset of consequences. Onother basic difference is in the origin and in the agenda. Indeed the origin of fuzzy logic is in control and its aim is to find applications, in general. Instead, multi-valued logic originates from philosophical and theoretical questions.

The semantics

Consider a first order language L whose set of formulas we denote by F'. As in classical logic, in fuzzy logic an interpretation of L is obtained by a domain D and by a function I associating every constant in L with an element of D and every n-ary operation symbol in L with an n-ary function in D. Instead, the interpretation of the predicate names is different since an n-ary predicate symbol is interpreted by an n-ary fuzzy relation in D, i.e. a map r from $D^{n}$ to [0,1]. This is done in order to represent properties which are "vague" are admitted.

Definition. Given a first order language F, a fuzzy interpretation is a pair (D,I) such that D is a nonempty set and I a map associating

- every operation name h with arity n with an n-ary operation $I(h):D^{n}\rightarrow D$ in D,

- every constant c with an element I(c) in D

- every n-ary predicate name r with an n-ary fuzzy relation $I(r):D^{n}\rightarrow [0,1]$ in D.

Every fuzzy interpretation defines a valuation of the set F of formulas. The interpretation of a term t is a function I(t) one defines as in classical logic.

Definition. Given a formula $\alpha \in F$ whose free variables are in $\{x_{1},...x_{n}\}$ , we define the truth degree $Val(I,\alpha ,d_{1},...,d_{n})$ of $\alpha$ by induction on the complexity of $\alpha$ by setting

- $Val(I,r(t_{1},...,t_{n}),d_{1},...,d_{m})=I(r)(I(t_{1})(d_{1},...,d_{n}),...,I(t_{n})(d_{1},...,d_{n}))$

- $Val(I,\alpha \vee \beta ,d_{1},...,d_{n})=Val(I,\alpha ,d_{1},...,d_{n})\oplus Val(I,\beta ,d_{1},...,d_{n}).$

- $Val(I,\alpha \wedge \beta ,d_{1},...,d_{n})=Val(I,\alpha ,d_{1},...,d_{n})\otimes Val(I,\beta ,d_{1},...,d_{n})$

- $Val(I,\neg \alpha ,d_{1},...,d_{n})=$ ~ $Val(I,\alpha ,d_{1},...,d_{n})$

- $Val(I,\exists x_{i}\alpha ,d_{1},...,d_{n})=Sup_{d\in D}Val(I,\alpha ,d_{1},...,d_{i-1},d,d_{n+1},...,d_{n}).$

As usual, if $\alpha$ is a closed formula, then its valuation does not depend on the elements $d_{1},...,d_{n}$ and we write $Val(I,\alpha )$ instead of $Val(I,\alpha ,d_{1},...,d_{n})$ . More in general, given any formula $\alpha$ , we denote by $Val(I,\alpha )$ , the valuation of the universal closure of $\alpha$ .

Definition. Consider a fuzzy set 's' of formulas we interpret as the fuzzy subset of proper axioms. Then we say that a fuzzy interpretation (D,I) is a model of s, in brief $(D,I)\models s$ if $Val(I,\alpha )\geq s(\alpha )$ .

Then the meaning of a fuzzy subset of proper axioms s is that for every sentence $\alpha$ , the value $v(\alpha )$ is a "lower bound constraint" on the unknown truth value of $\alpha$ .

Definition. The logical consequence operator is the map $Lc:[0,1]^{F}\rightarrow [0,1]^{F}$ defined by setting

$Lc(s)(\alpha )=Sup\{Val(I,\alpha ):(D,I)\models s\}$ .

Again, the value $Lc(s)(\alpha )$ is a "lower bound constraint" on the unknown truth value of $\alpha$ . As a matter of fact this is the better constraint we can find given the information s. It is easy to see that Lc is a closure operator, i.e. that

$s\subseteq Lc(s)$

$s\subseteq t\Rightarrow Lc(s)\subseteq Lc(t)$

$Lc(Lc(s))=Lc(s)$ .

The deduction apparatus: approximate reasonings

. . .

Is fuzzy logic a proper extension of classical logic ?

The interpretation of the logical connectives in fuzzy logic is conservative in the sense that its restriction to {0,1} coincides with the classical one. This fact can be interpreted by saying that fuzzy logic is conservative and that it is a proper extension of classical logic. On the other hand it is evident also that fuzzy logic is defined inside classical mathematics and therefore inside classical logic. Then, as a matther of fact fuzzy logic is a (small) chapter of classical mathematics. This means that, differently from intuitionistic logic, fuzzy logic cannot be considered as an alternative philosophy in a trict sense.

Approximate reasonings

In fuzzy logic a deduction apparatus is given by a fuzzy subset of logical axioms and a set of fuzzy inference rules. . . .

Effectiveness for fuzzy subsets

The notions of a "decidable subset" and "recursively enumerable subset" are basic ones for classical mathematics and classical logic. Then, the question of a suitable extension of such concepts to fuzzy set theory arises. A first proposal in such a direction was made by E.S. Santos by the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program. Successively, L. Biacino and G. Gerla proposed the following definition where Ü denotes the set of rational numbers in [0,1].

Definition A fuzzy subset μ : S $\rightarrow$ [0,1] of a set S is recursively enumerable if a recursive map h : S×N $\rightarrow$ Ü exists such that, for every x in S, the function h(x,n) is increasing with respect to n and μ(x) = lim h(x,n). We say that μ is decidable if both μ and its complement –μ are recursively enumerable.

An extension of such a theory to the general case of the L-subsets is proposed in a paper by G. Gerla where one refers to the theory of effective domains. It is an open question to give supports for a Church thesis for fuzzy logic claiming that the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. To this aim, further investigations on the notions of fuzzy grammar and fuzzy Turing machine should be necessary (see for example Wiedermann's paper).

Effectiveness for fuzzy logic

Define the set Val of valid formulas as the set of formulas assuming constantly value equal to 1. Then it is possible to prove that among the usual first order logics only Goedel logic has a recursively enumerable set of valid formulas. In the case of Lukasiewicz and product logic, for example, Val is not recursively enumerable (see B. Scarpellini, Belluce). Such a fact was extensively examined in the book of Hajek. Neverthless, from these results we cannot conclude that these logics are not effective and therefore that an axiomatization is not possible. Indeed, if we refer to the just exposed notion of effectiveness for fuzzy sets, then the following theorem holds true (provided that the deduction apparatus of the fuzzy logic satisfies some obvious effectiveness property).

Theorem. Any axiomatizable fuzzy theory is recursively enumerable. In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general. Moreover, any axiomatizable and complete theory is decidable.

It is an open question to utilize the notion of recursively enumerable fuzzy subset to find an extension of Gödel’s theorems to fuzzy logic.

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Formal fuzzy logic

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