Formal fuzzy logic

Formal fuzzy logic

"Fuzzy logic" comprises topics based on the notion of a fuzzy subset and usually is devoted to applications. "Formal fuzzy logic" or "fuzzy logic in narrow sense" comprises a relatively new chapter of formal logic. Its aim is to represent in a formal way the vagueness of the natural language and to formalize the reasonings involving notions which are vague in nature.

Formal fuzzy logic is an evolution and an enlargement of multi-valued logic. The fuzzy logics usually proposed have a semantics which is not different from the usual truth-functional semantics of a first order multi-valued logic. In addition there are fuzzy logic whose semantics is not truth-functional (as an example, see necessity logic and probability logic) and also fuzzy logics whith no semantics (see similarity logic) since are obtained by a fuzzyfication of the metalogic we use for classical logic.

The main difference with traditional multi-valued logic is in the deduction apparatus. In multi-valued logic the deduction operator is a tool to associate every (classical) set of axioms with the related (classical) set of theorems. 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 and this leads to define a deduction operator working from a fuzzy set of proper axioms (the available information) to give the related fuzzy subset of consequences.

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 enables us to represent properties which are "vague" in nature.

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. Given a term t its interpretation 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:

$Val(I,r(t_{1},...,t_{n}),d_{1},...,d_{m})=I(r)(I(t_{1})(d_{1},...,d_{n}),...,I(t_{n})(d_{1},...,d_{n}))$
$Val(I,\alpha \vee \beta ,d_{1},...,d_{n})$ = $Val(I,\alpha ,d_{1},...,d_{n})\oplus Val(I,\beta ,d_{1},...,d_{n}).$
$Val(I,\alpha \wedge \beta ,d_{1},...,d_{n})=Val(I,\alpha ,d_{1},...,d_{n})\otimes Val(I,\beta ,d_{1},...,d_{n})$
$Val(I,\neg \alpha ,d_{1},...,d_{n})=n(Val(I,\alpha ,d_{1},...,d_{n}))$
$Val(I,\exists x_{i}\alpha ,d_{1},...,d_{n})=Sup_{d\in D}Val(I,\alpha ,d_{1},...,d_{i-1},d,d_{n+1},...,d_{n}).$

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 $Lc(s)$ 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

Once we have defined the logical consequence operator Lc, we have to search for a "deduction apparatus" able to calculate Lc(s) in some way. As an example, by extending the Hilbert's aproach, we can define a deduction apparatus by a fuzzy subset of formulas la we call fuzzy subset of logical axioms and by a set R of fuzzy inference rules. In turn, and inference rule is a pair (r,s) where r is a partial n-ary operation in F (i.e. an inference rule in the usual sense) and s is an n-ary operation in [0,1]. The meaning of an inference rule is:

- if we are able to prove $\alpha _{1},...,\alpha _{n}$ at degree $\lambda _{1},...,\lambda _{n}$, respectively

- and we can apply r to $\alpha _{1},...,\alpha _{n}$

- then we can prove $r(\alpha _{1},...,\alpha _{n})$ at degree $s(\lambda _{1},...,\lambda _{n})$.

As an example, let $\otimes$ be an operation in [0,1] able to interpret the conjunction. Then the fuzzy Modus Ponens is defined as the pair $(r,s)$ in which the domain of r is the set $\{(\alpha ,\alpha \rightarrow \beta :\alpha ,\beta \in F\}$, $r(\alpha ,\alpha \rightarrow \beta )=\beta$ and $s(\lambda _{1},\lambda _{2})=\lambda _{1}\otimes \lambda _{2}$. This rule says that if we are able to prove $\alpha$ and $\alpha \rightarrow \beta$ at degree $\lambda _{1}$ and $\lambda _{2}$, respectively, then we can prove $\beta$ at degree $\lambda _{1}\otimes \lambda _{2}$.

The fuzzy $\land$-introduction rule is a totally defined rule such that r($\alpha _{1},\alpha _{2})=\alpha _{1}\land \alpha _{2}$ and again $s(\lambda _{1},\lambda _{2})=\lambda _{1}\otimes \lambda _{2}$. This rule says that if we are able to prove $\alpha _{1}$ and $\alpha _{2}$ at degree $\lambda _{1}$ and $\lambda _{2}$, respectively, then we can prove $\alpha _{1}\land \alpha _{2}$ at degree $\lambda _{1}\otimes \lambda _{2}$.

A proof $\pi$ of a formula $\alpha$ is a sequence $\alpha _{1},...,\alpha _{m}$ of formulas such that $\alpha _{m}=\alpha$, together with a sequence of related justifications. This means that, for every formula $\alpha _{i}$, we have to specify whether

i) $\alpha _{i}$ is assumed as a logical axiom or;

ii) $\alpha _{i}$ is assumed as a proper axiom or;

iii) $\alpha _{i}$ is obtained by a rule (in this case we have to indicate also the rule and the formulas from $\alpha _{1},...,\alpha _{i-1}$ used to obtain $\alpha _{i}$).

The justifications are necessary to valuate the proofs. Let v be any fuzzy set of formulas and, for every $i\leq m$ denote by $\pi (i)$ the proof $\alpha _{1},...,\alpha _{i}$. Then the valuation $Val(\pi ,v)$ of $\pi$ with respect to v is defined by induction on m by setting

$Val(\pi ,v)=a(\alpha _{m})$ if $\alpha _{m}$ is assumed as a logical axiom

$Val(\pi ,v)=v(\alpha _{m})$ if $\alpha _{m}$ is assumed as a proper axiom

$Val(\pi ,v)=s(Val(\pi (i_{1}),v),...,Val(\pi (i_{n}),v))$ if there is fuzzy rule $(r,s)$ such that $\alpha _{m}=r(\alpha _{i(1)},...,\alpha _{i(n)})$ with $i_{1}<m,...,i_{n}<m$.

Now, unlike the crisp deduction systems, in a fuzzy deduction system different proofs of a same formula $\alpha$ may give different contributions to the degree of validity of $\alpha$. This suggests setting

$D(v)(\alpha )=Sup\{Val(\pi ,v)|\pi$ is a proof of $\alpha \}$.

This formula defines, for every initial valuation v, a fuzzy subset D(v) we call the fuzzy set of formulas deduced from v. Also, we call deduction operator the function D so defined, i.e., the operator associating any fuzzy subset v of hypotheses with the fuzzy subset D(v) of its consequences. ( (to be continued) ...

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.

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.

See also

• Fuzzy subalgebra
• Fuzzy associative matrix
• Fuzzy programing logic
• Fuzzy set
• Paradoxes
• Rough set

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