Formal fuzzy logic

Fuzzy logic is a relatively new chapter of formal logic whose aim is to formalize the reasonings involving predicates that are vague in nature (as an example small, near, similar). An example of such kind of reasoning is

If a tomato is red, then the tomato is ripe. Since this tomato is very red, this tomato is very ripe.

Further examples of reasonings involving vague predicates are in the item Paradoxes and fuzzy logic and in the section Fuzzy logic with no truth-functional semantics. The main tool for fuzzy logic is the notion of a fuzzy subset, since a vague predicate is interpreted by a fuzzy subset. Notice that in literature the name "fuzzy logic" also denotes a large series of topics based on an informal usage of the notion of a fuzzy subset, and which are usually devoted to applications.

As a matter of fact, fuzzy logic is an evolution and an enlargement of multi-valued logic since all the definitions and results in the literature on multi-valued logic are also considered in fuzzy logic. In particular, as in multi-valued logic, the starting point is a fixed valuation structure, i.e. a bounded lattice L equipped with suitable operations to interpret the logical connectives. The minimum 0 means 'False', the maximum 1 means 'True', the remaining elements are interpreted as intermediate truth values. The following is the main class of valuation structures (see Hájek 1998, Novák et al. 1999 and Gottwald 2005) corresponding to the connectives $\wedge$ and $\rightarrow$.

Definition. A standard algebra is an algebraic structure ([0,1], ʘ, →, 0,1) where ʘ is a continuous triangular norm, i.e. a continuous, associative, commutative, order preserving operation such that xʘ1 = 1 and → is the related residuation, i.e. x→y = sup{z | xʘz ≤ y}.

The main examples of standard algebras are obtained by assuming that ʘ is the minimum (Zadeh logic), the usual product (product logic) or that xʘy = Max{x+y-1,0} (Łukasievicz logic). In addition, several authors consider also languages with logical constants to denote rational truth values. Once a valuation structure is fixed, the semantics of the corresponding propositional calculus is defined in a truth-functional way as usual. In first order fuzzy logic the semantics is defined as follows.

Definition. A fuzzy interpretation of a first order language is a pair (D,I) such that D is a nonempty set and I a map associating (as in the classical case) every n-ary operation name h with an n-ary operation in D and every constant c with an element I(c) in D. Moreover, I associates every n-ary predicate name r with an n-ary L-relation I(r) : Dⁿ$\rightarrow$ L in D.

Then the only difference with classical logic is that the interpretation of an n-ary predicate symbol is an n-ary fuzzy relation in D. This enables us to represent properties which are "vague" in nature. Given a fuzzy interpretation we can evaluate the formulas as follows where, given a term t whose variables are in x₁,...,x_n, we denote by $I(t)$ the corresponding n-ary function we define as in classical logic.

Definition. Let (D,I) be a fuzzy interpretation, α a formula whose free variables are in x₁,...,x_n and d₁,...,d_n elements in D. Then we define the truth degree Val(I,α,d₁,...,d_n) by induction as follows :

Val(I, r(t₁,...,t_p), d₁,...,d_n) = I(r)(I(t₁)(d₁,...,d_n), ..., I(t_p)(d₁,...,d_n))

Val(I,α $\wedge$ β, d₁,...,d_n) = Val(I,α,d₁,...,d_n)ʘVal(I,β,d₁,...,d_n)

Val(I,α → β, d₁,...,d_n) = Val(I,α, d₁,...,d_n) → Val(I,β,d₁,...,d_n)

Val(I,$\forall$ x_iα, d₁,...,d_n) = Inf _dєDVal(I,α,d₁,...,d_i-1,d,d_i+1,...,d_n).

In the case there is a propositional constant c^* corresponding to a truth value c, we set

Val(I, c^*,d₁,...,d_n) = c.

Observe that in the case L is not complete it is possible that a quantified formula cannot be evaluated. We call safe an interpretation such that all the formulas are evaluated. As usual, if α is a closed formula, then its valuation does not depend on the elements d₁,...,d_n and we write Val(I,α) instead of Val(I,α,d₁,...,d_n). More in general, given any formula α, we denote by Val(I, α) the valuation of the universal closure of α.

Two approaches

There are two basic approaches to fuzzy logic. The first one, proposed by P. Hajek and followed by Di Nola, Esteva, Gottwald, Godo, Montagna, Mundici and by a large series of students, is very close to the tradition of multi-valued logic. Indeed the deduction apparatus works on a set of hypotheses to give the corresponding set of logical consequences. This is obtained, as it is usual in multi-valued logic, once a set of designed truth values is fixed. We call, ungraded approach such a way to face fuzzy logic. Another approach was proposed by J. A. Goguen, J. Pavelka, V. Novak, G. Gerla and further authors and it is rather out of line with the tradition of multi-valued logic. Indeed, the deduction apparatus works on a given fuzzy subset of hypotheses (the available information) to give the related fuzzy subset of logical consequences. We call graded approach such a way to face fuzzy logic.

The ungraded approach

In the ungraded approach a subset Des of [0,1] is fixed whose elements are called designed truth degrees. The interpretation is that in Des there are the truth degrees which one considers sufficient to claim the validity of a formula. Usually one sets Des = {1}.

Definition. Let ([0,1], ʘ, →, 0, 1) be a fixed standard algebra, and α be a formula. Then we say that a fuzzy interpretation (D,I) satisfies α provided that Val(I,α) is a designed value. Let T be a theory, then (D,I) is a model of T if every formula in T is satisfied in (D,I). We write T $\models$_ʘ α if every model of T satisfies α.

The deduction apparatus in the ungraded approach is defined by adopting the same paradigm of classical logic, i.e. a deduction relation $\vdash$ is defined by a suitable set of logical axioms and suitable inference rules. The fuzzy logic defined by ʘ is axiomatizable provided that a deduction apparatus exists such that $\vdash$ coincides with $\models$_ʘ. Unfortunately, the main fuzzy logics are not axiomatizable.

Theorem. In all the main fuzzy logics (in particular in Łukasievicz logic) the entailment relation $\models$_ʘ is not compact. This entails that these logics are not axiomatizable.

As an attempt to bypass such an obstacle, in the ungraded approach one proposes a different entailment relation related with the variety generated by a given triangular norm.

Definition. Given a standard algebra ([0,1], ʘ, →,0,1), denote by Varl(ʘ) the class of all linearly ordered algebras in the variety generated by ([0,1], ʘ, →, 0, 1). Then a Varl(ʘ)-interpretation is an interpretation in a valuation algebra belonging to Varl(ʘ). Given a set T of formulas and a formula α, we write T $\models$_Varl(ʘ) α provided that every safe Varl(ʘ)-model of T is a safe Varl(ʘ)-model of α.

In such a case, the resulting logic works well. In fact, the following theorem holds true.

Theorem. In all the main fuzzy logics (in particular in Łukasievicz logic) the entailment relation $\models$_Varl(ʘ) is compact. This is in accordance with the fact that these logics are axiomatizable (provided that they are defined by referring to this relation).

Criticisms. A criticism for the ungraded approach, philosophical in nature, concerns its adequateness to represent the daily reasonings in which vague predicates occur. Moreover the structures in Varl(ʘ) look rather unnatural. For example, in Varl(ʘ) there are structures with infinitesimal truth values. Another criticism is that, while the completeness of [0,1] assures that all the formulas are valuated, in the case we refer to the variety Varl(ʘ), we are forced to admit interpretations for which there are unvaluated formulas.

The graded approach: approximate reasonings

The aim of any logic is to elaborate (uncomplete) information to obtain more explicit information. Now, in the case of fuzzy logic it is natural to admit an information like "the truth values of α is between λ and μ", i.e. a constraint on the possible truth value of a formula. Now, observe that if we admit the usual interpretation of the negation, then Val(I,α)≤ λ if and only if Val(I,$\neg$ α)≥ 1-μ. Then we can reduce all the interval constraints to lower bound constraints. In accordance in the graded approach one proposes the following definition.

Definition . Consider a fuzzy theory s, i.e. a fuzzy subset of formulas. Then a fuzzy interpretation (D,I) is a model of s, in brief (D,I) $\models$ s if Val(I,α) ≥ s(α). The logical consequence operator is the map Lc : [0,1]^F → [0,1]^F defined by setting

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

Equivalently, we can refer to a graded entailment relation $\models$^λ by writng s $\models$^λ α where λ = Inf{Val(I,α) : (D,I) $\models$ s}.

These definitions are in accordance with the fact that s represents a system of "lower bound constraints" on the unknown truth value of the formulas. Moreover, Lc(s) is the better lower bound constraint we can find given s. In the graded approach we can obtain a deduction apparatus by extending Hilbert's approach as follows.

Definition. A fuzzy inference rule is a pair r = (syn,sem) where syn, the syntactical part, is a partial n-ary operation in F (i.e. an inference rule in the usual sense) and sem, the semantic part, is an n-ary join-preserving operation in [0,1]. An evaluated syntax is a structure (la,R) where la is a fuzzy set of formulas we call fuzzy subset of logical axioms, and R is a set of fuzzy inference rules.

Usually, n = 2 and sem(λ₁,λ₂) is a product like λ₁ʘ λ₂. As an example, the fuzzy Modus Ponens is defined by assuming that the domain of syn is the set {(α, α→β): α,β are in F}, by setting syn(α, α→β) = β and by assuming that sem(λ,μ) = λʘμ. This rule says that

- if we are able to prove α at degree λ

- and α → β at degree μ

- then we can prove β at degree λʘμ.

Definition. A proof π of a formula α is a sequence α₁,...,α_m of formulas such that α_m = α, together with a sequence of related justifications. This means that, for every formula α_i, we have to specify whether

i) α_i is assumed as a logical axiom or;

ii) α_i is assumed as an hypothesis or;

iii) α_i is obtained by a rule (in this case we have to indicate the rule and the formulas from α₁,...,α_i-1 used to obtain α_i).

The justifications are necessary to valuate the proofs. Indeed, let s be the fuzzy subset of proper axioms and, for every i ≤ m denote by π(i) the proof α₁,...,α_i. Then the information furnished by π given s is the value Val(π,s) is defined by induction on m by setting

Val(π, s) = la(α_m) if α_m is assumed as a logical axiom

Val(π, s) = s(α_m) if α_m is assumed as an hypothesis

Val(π,s) = sem(Val(π(i₁),s),...,Val(π(i_n),s)) if there is a fuzzy rule (syn,sem) such that α_m = syn(α_i1,...,α_in) with i₁ < m,...,i_n < m.

Now, unlike the usual deduction systems, in a fuzzy deduction system, different proofs of a same formula α may give different pieces of information on the truth degree of α. Then, we have to "fuse" all these informations.

Definition. The deduction operator is the operator D defined by setting

D(s)(α)= Sup{Val(π,s)| π is a proof of α}.

A fuzzy logic is axiomatizable if there is a fuzzy deduction system such that Lc = D.

In Novak 2007 one proves the following basic result.

Proposition. In the graded approach Łukasiewicz first order logic is axiomatizable.

Notice also that, under some natural hypotheses, a fuzzy propositional logic is axiomatizable if and only if the logical connectives are interpreted by continuous functions (see Gerla 2001).

Criticisms. A criticism for such an approach is that is not sufficiently flexible. As an example let (D,I) be a fuzzy model of a fuzzy theory s. It is evident that both the assignments (D,I) and s cannot be considered definitive and precise since both depends on a subjective valuation. Assume that either (D,I) or s is subject to a slightly variation as a consequence of a tuning process, an essential component in all the applications in fuzzy mathematics. Then it is possible that (D,I) ceases completely to be a model of s while it should be natural to expect (D,I) is again a model of s at some degree.

Continuous logic

A very important precursor of fuzzy logic is the Continuous logic proposed by Chang and Keisler since 1966. In their book all the model theoretical notions of classical logic are extended to first order multi-valued logic. As an example, the notions of quotient, direct product, ultraproduct are defined and examined. Recently, an interesting reformulation of continuous logic was proposed to give a basis for a model theory for the various kinds of "metric" structures arising in functional analysis and probability theory (see for example Ben I. Y. et al. 2008). Such a reformulation is obtained by referring to the interval [0,1] and by interpreting the logical connectives by a class of continuous functions able to define a complete approximation system. Also one considers models in which a pseudo-metric is defined to interpret the symbol =. One assumes that all the predicates and function symbols are uniformly continuous with respect such a pseudo-metric. As an example, the finitely additive probabilities can be considered models of a continuous logic once we consider a language for Boolean algebras equipped with a vague monadic predicate p such that the interpretation of p(x) is "the event x is probable".

Fuzzy logic with no truth-functional semantics

Fuzzy logic extends beyond the truth-functional tradiction of multi-valued logic. The following are two examples.

Necessity logic

Assume that the deduction apparatus of classical first order logic is presented by a suitable set la of logical axioms, by the MP-rule and the Generalization rule. Then a fuzzy deduction system is obtained by considering as a fuzzy subset of logical axioms the characteristic function of la and as fuzzy inference rules the fuzzy Modus Ponens and the extension of the Generalization Rule obtained by assuming that if we prove α at degree λ then we obtain $\forall$xα(x) at the same degree λ. In suh a case it is easy to see that if s is a fuzzy theory, then

D(s)(α) = 1 i α is a logically true formula,

D(s)(α) = Sup{s(α₁)ʘ ...ʘs(α_n) : α₁,..., α_n $\vdash$α}, otherwise.

By recalling that the existential quantifier is interpreted by the supremum operator, such a formula arises from a multivalued valuation of the (metalogical) claim: "α is a consequence of the fuzzy subset s of axioms provided there are formulas α₁, ...,α_n in s able to prove $\alpha$". In such a case the vagueness originates from s, i.e., from the notion of "hypothesis". Moreover s(α) is not a truth degree but rather a degree of "preference" or "acceptability" for α. For example, let ʘ be the minimum, T be a system of axioms for set theory such that the choice axiom CA does not depend on T. Then we can consider the fuzzy subset of axioms s defined by setting

s(α) = 1 if α є T,

s(α) = 0.8 if α = CA ,

s(α) = 0 otherwise.

A simple calculation shows that:

D(s)(α) = 1 if α is a theorem of T,

D(s)(α) = 0.8 if we cannot prove α from T but α is a theorem of T + CA,

D(s)(α) = 0 otherwise .

Then, despite the fact that no vague predicate is considered in set theory, in the metalanguage we can consider a vague meta-predicate as "is acceptable" and to represent it by a suitable fuzzy subset s.

Similarity logic

In accordance with the ideas of M. S. Ying (1994) we can extend necessity logic by introducing a similarity relation among the predicates (see also Biacino, Gerla, Ying (2002)). As an example, consider an inference like

Since x is a thriller $\Rightarrow$ x good for me +

and b is a detective story +

and "detective story" is synonymous of "thriller"

then "b is good for me".

Now the synonymy is a vague notion we can represent by a suitable similarity in the set W of English worlds, i.e. a fuzzy relation e such that

(a) e(x,x) = 1 (reflexivity),

(b) e(x,z)ʘe(z,y) ≤ e(x,y) (transitivity),

(c) e(x,y) = e(y,x) (symmetry).

Also, as it is usual in fuzzy logic, it is natural to admit that the truth degree of the conclusion "b is good for me" depends on the degree of similarity between the predicates "detective story" and "thriller", obviously. The structure of the corresponding fuzzy inference rule is:

If α was proven at degree λ,

and α’→ β at degree μ,

then β is proven at degree λʘμʘe(α,α’).

Every inference rule can be extended in a similar way, i.e. by relaxing the precise matching of the identity with the approximate matching of a similarity. These ideas are also on the basis for a similarity-based fuzzy logic programming.

Effectiveness

A test to analyze the effectiveness in the ungraded approach to fuzzy logic is to refer to the set of tautologies. Now, since two entailment relations are defined, we have to consider two corresponding notions of tautology.

Definition Given a standard algebra ([0,1], ʘ, →, 0, 1) a formula α is a standard tautology if it is satisfied in every fuzzy interpretation in ([0,1], ʘ, →, 0, 1). The formula α is a general tautology if it is satisfied in every safe Varl(ʘ)-interpretation.

In the first case the following negative result holds true.

Theorem. In the case of Łukasiewicz and product logic the set of standard tautologies is not recursively enumerable (see B. Scarpellini (1962)).

Such a fact gives a further confirm on the impossibility of an axiomatization of the entailment relation $\models$ and it leads to focalize the attention on $\models$_Varl(ʘ). At this regard one proves the following theorem.

Theorem. For each continuous t-norm ʘ, the set of general ʘ-tautologies in first order logic is Σ₁-complete (and therefore recursively enumerable).

In the case of the graded approach to face the question of the effectiveness we have to refer to the notion of effectiveness for fuzzy subset theory. A first proposal in such a direction was made by E.S. Santos by the notions of fuzzy Turing machine. Successively, in Biacino and Gerla 2006 the following definition was proposed where Ü denotes the set of rational numbers in [0,1].

Definition A fuzzy subset s : 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 s(x) = lim h(x,n). We say that s is decidable if both s and its complement –s are recursively enumerable.

An extension of such a theory to the general case of the L-subsets is proposed in Gerla (2006) where one refers to the theory of effective domains. It is an open question to give supports for a Church thesis for fuzzy set theory claiming that the proposed notion of recursive enumerability for fuzzy subsets is the adequate one.

In Gerla (2001) one proves the following theorem where we refer to fuzzy logics whose deduction apparatus satisfies some obvious effectiveness properties.

Theorem. Given an axiomatizable fuzzy logic, its fuzzy subset D(Ø) of tautologies is recursively enumerable. In particular the fuzzy subset of tautologies in Łukasievicz logic is recursively enumerable in spite of the non recursive enumerability of its cut {α : D(Ø)(α) = 1}.

It is an open question to use the notion of recursively enumerable fuzzy subset to extend Gödel’s limitative theorems to fuzzy logic.