Formal fuzzy logic

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Introduction

Fuzzy logic is a relatively new chapter of formal logic. Its aim is to represent predicates which are vague in nature as big, near, or similar (for example), and to formalize the reasonings involving these predicates. As an example, the aim of fuzzy logic is to manage pieces of information like

IF temperature IS very cold THEN stop fan

IF temperature IS cold THEN turn down fan

IF temperature IS normal THEN maintain level

IF temperature IS hot THEN speed up fan

where "very cold", "cold" ... "speed up" are all vague notions. The main tool for fuzzy logic is the notion of a fuzzy subset, proposed by L. A. Zadeh since 1965. Indeed, 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.

We can consider fuzzy logic as 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. Indeed, as in multi-valued logic, the starting point is a 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 intermediate truth values. The following is the main class of valuation structures (see Hájek 1998, Novák et al. 1999 and Gottwald 2005).

Definition A standard algebra is an algebraic structure ([0,1], ʘ, $\rightarrow$, 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 L-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, we denote by $I(t)$ the corresponding function we define as in classical logic.

Definition. Let (D,I) be a fuzzy interpretation, then for every formula α whose free variables are in x₁,...,x_n and d₁,...,d_n in D, 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_i α,d₁,...,d_n) = Inf d є D Val(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 by a large series of students, is strictly closed to the tradition of multi-valued logic. Indeed the entailment relation is a crisp one, equivalently, the logical consequence operator works on a given classical subset of hypotheses to give the related classical 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, in brief U-approach, such a way to face fuzzy logic. Another approach was proposed by J. A. Goguen, J. Pavelka and many authors and it is rather out of line with the tradition of multi-valued logic. Indeed, the entailment relation is a fuzzy relation. Equivalently, the logical consequence operator works on a given fuzzy subset of hypotheses (the available information) to give the related fuzzy subset of logical consequences. We call graded approach, in brief G-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 (L, ʘ, →, 0, 1) be a fixed standard algebra. Then we say that a fuzzy interpretation (D,I) is a model of a formula 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 is a model of 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 almost all the fuzzy logics (for example Łukasievicz logic) the entailment relation $\models$_ʘ is not compact. This entails that these logic 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 almost all the fuzzy logics (for example Łukasievicz logic) the entailment relation $\models$_Varl(ʘ) is compact. This is in accordance with the fact that, once we refer to this relation, these logics are axiomatizable.

A criticism for such a solution is that in Varl(ʘ) there are unnatural valuation structures. For example, structures with infinitesimal truth values. This is rather far from the uman intuition. Moreover, 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 unsafe interpretations.

The graded approach: approximate reasonings

The graded approach is perhaps closer to the spirit of fuzzy logic. In fact the aim of any logic is to βelaborate (uncomplete) information and, in the case of fuzzy logic, should be 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. Taking in account that for a large class of fuzzy semantics we can split such an interval constraint into the two lower bound constraints "the truth values of α is greater or equal to λ" and "the truth value of $\neg$α is greater or equal to 1-μ", the following definitions are proposed.

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 define a graded entailment relation by setting s $\models$^λ α provided that λ = Inf{Val(I,α) : (D,I) $\models$ s}. Such a definition is in accordance with the fact that the information carried on by s is that, for every sentence α, the value s(α) is a "constraint" on the unknown truth value of α. More precisely s(α) is a lower bound for such a value. Again, the value Lc(s)(α) is a "constraint" on the unknown truth value of α. As a matter of fact it is the better constraint we can find given the information s.

In the graded approach we can obtain a deduction apparatus by extending the 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 semantical part, is an n-ary joing-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.

The meaning of an inference rule r is:

- IF we are able to prove α₁,...,α_n at degree λ₁,...,λ_n, respectively

- AND we can apply syn to α₁,...,α_n

- THEN we can prove syn(α₁,...,α_n) at degree sem(λ₁,...,λ_n).

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 and α → β at degree λ and μ, respectively, 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

also 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 contributions to the degree of validity of α. This suggests the following definition.

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.

Notice 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). As was shown in Novak 2007, the following axiomatizability theorem holds true.

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

Effectiveness

A test to analize the effectiveness in the ungraded approach 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 give a suitable notion of effectiveness for fuzzy sets. 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.

Paradoxes

The heap paradox

To show an example of approximate reasoning in fuzzy logic we refer to the famous "heap paradox". Let n be a natural number and denote by Small(n) a sentence whose intended meaning is "a heap with n stones is small" (n is a numeral to denote n). Then it is natural to assume the validity of the atomic formula Small(1) and, for every n, the validity of Small(n) → Small(n+1).

On the other hand from these formulas given any natural number n, by applying MP (Modus Ponens) rule several times we can prove that a heap with n stones is small. Indeed,

from Small(1) and Small(1)→ Small(2) by MP we may state Small(2);

from Small(2) and Small(2)→ Small(3) by MP we may state Small(3),

…

from Small(n-1) and Small(n-1)→ Small(n) by MP we may state Small(n).

Obviously, a conclusion like Small(20.000) is contrary to our intuition in spite of the fact that the reasoning is correct and the premises appear very reasonable. Clearly, the core of such a paradox lies in the vagueness of the predicate " small" and therefore, as proposed by Goguen (1968/69), we can refer to the notion of approximate reasoning to face it. Indeed it is a fact that everyone is convinced that the implications Small(n) → Small(n+1) are very close to the truth but not completely true, in general. We can try to "respect" this conviction by assigning to these formulas a truth value λ very close to 1 but different from 1. Then, for example, we can express the hypothesis of the heap paradox by the following fuzzy theory

Small(1) [to degree 1]

Small(2) [to degree 1]

...

Small(10.000) [to degree 1]

Small(10.000)→ Small(10.001) [to degree λ]

Small(10.001)→ Small(10.002) [to degree λ]

...

In accordance, the Heap Paradox argument can be restated as follows where we denote by λ⁽ⁿ⁾ the n-power of λ with respect to ʘ.

Since Small(10.000) [to degree 1]

and Small(10.000)→ Small(10.001) [to degree λ]

we state Small(10.001) [to degree 1ʘλ = λ⁽¹⁾]

since Small(10.001) [to degree λ]

and Small(10.001)→ Small(10.002) [to degree λ]

we state Small(10.002) [to degree λʘλ = λ⁽²⁾ ]

. . .

since Small(10.000+n-1) [to degree λ^(n-1)]

and Small(10.000+n-1) → Small(10.000+n) [to degree λ]

we state Small(10.000+n) [to degree λ^(n-1)ʘλ = λ⁽ⁿ⁾].

In particular, we can prove Small(10.000+10.000) at degree λ^(10.000) . Now, this is not paradoxical. Indeed if $\otimes$ is Łukasievicz triangular norm, then λ⁽ⁿ⁾ = max {nλ-n+1, 0}. As a consequence, we have that λ⁽ⁿ⁾ = 0 for every n ≥ 1/(1-λ). Assume that λ = 1-10^-4 then λ^(10.000) = 0. In this way we get a formal representation of heap argument preserving its intuitive content but avoiding its paradoxical character.

The argument on the basis of heap paradox enables us to show an interesting fact:

"the induction principle is not valid in fuzzy logic, i.e. we cannot extend such a principle to vague properties".

In fact, assume that the formula

Small(1) → (($\forall$n(Small(n) → Small(n+1)) → $\forall$n Small(n))

is satisfied at degree μ ≠ 0 and let λ ≠ 1 such that λʘμ ≠ 0. Then, by two applications of MP we can prove $\forall$n Small(n) to degree λʘμ ≠ 0. This contradicts the fact that $\forall$n Small(n) is false.

The Poincaré paradox

The so called “paradox” of Poincaré refers to indistinguishability by emphasizing that, in spite of common intuition, this relation is not transitive. In fact, let d₁,…, d_m be a sequence of objects such that we are not able to distinguish d_i from d_i+1 and that, nevertheless, that we have no difficulty in distinguishing d₁ from d_m. Also, consider a first order language with a predicate symbol E to denote the indistinguishability relation and, for every i in N, with a constant c_i to denote d_i. Then it is natural to consider the theory defined by the following formulas:

E(c₁,c₂),…, E(c_i-1,c_i),..., $\neg$E(c₁,c_m), E(x,z)$\land$E(z,y) $\Rightarrow$E(x,y).

From such a theory, by suitable applications of the $\land$-introduction rule, particularization and MP, we can prove E(c₁,c_m) and this contradicts the hypothesis $\neg$E(c₁,c_m). Consider a value λ very close to 1 and such that λ^(m-1) = 0. Then in fuzzy logic we can formalize Poincaré's argument as follows:

Step 1.

Since E(c₁,c₂) [at degree λ]

and E(c₂,c₃) [at degree λ]

we can state E(c₁,c₂)$\land$E(c₂,c₃) [at degree λ⁽²⁾].

Therefore, since

E(c₁,c₂)$\land$E(c₂,c₃)$\Rightarrow$ E(c₁,c₃) [at degree 1]

we can state

E(c₁,c₃) [at degree λ⁽²⁾].

Step 2.

Since E(c₁,c₃) [at degree λ⁽²⁾]

and E(c₃,c₄) [at degree λ]

we can state

E(c₁,c₃)$\land$E(c₃,c₄) [at degree λ⁽³⁾]

Therefore, since

E(c₁,c₃)$\land$E(c₃,c₄)$\Rightarrow$ E(c₁,c₄) [at degree 1]

we can state E(c₁,c₄) [at degree λ⁽³⁾]

...

Step m-2.

Since E(c₁, c_m-1) [at degree λ^(m-2)]

and E(c_m-1,c_m) [at degree λ]

we can state E(c₁, c_m-1)$\land$E(c_m-1, c_m) [at degree λ^(m-1)]

Therefore, since E(c₁, c_m-1)$\land$E(c_m-1, c_m)$\Rightarrow$E(c₁, c_m) [at degree 1]

we can state E(c₁, c_m) [at degree λ^(m-1)].

Thus, such a proof entails that the conclusion E(c₁,c_m) is true at least at degree λ^(m-1) = 0 (no information). This is not paradoxical.

The Liar paradox

In the Liar paradox one considers a sentence S that asserts its own untruth. More precisely, if S* is a name for such a sentence, then such a sentence satisfies

S ↔ ¬True(S*).

If we admit, in accordance with Tarski's point of view, the equivalence

True(S*) ↔ S,

we can infer

(*) True(S*) ↔ ¬True(S*).

This is a contradiction in classical logic. In multi-valued logic (and therefore in fuzzy logic) such a conclusion is not paradoxical at all since it is sufficient to admit that the truth value of True(S*) is 0.5 (see Zadeh 1979). Indeed in fuzzy logic usually one interprets the negation with the function 1-x and therefore in such a case both the sentences True(S*) and ¬True(S*) assume the value 0.5. In accordance the formula True(S*) ↔ ¬True(S*) holds true.

It is well known that the proof of Gödel's incompleteness theorem uses a self-referential statement which is similar to the one in the Liar paradox. Namely, instead of True(S*) one considers Theorem(S*). A depth analysis of such a such a sentence in the framework of fuzzy logic and related with Peano arithmetic can be find in Hájek et al. 2000.

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

This very simple fuzzy logic is obtained by an obvious fuzzyfication of first order classical logic. Indeed, assume, for example, 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 and denote by $\vdash$ the related consequence relation. Then a fuzzy deduction system is obtained by considering as fuzzy subset of logical axioms the characteristic function of la and as fuzzy inference rules the extension of MP obtained by assuming that ʘ is the minimum operator $\wedge$. Moreover, an extension of the Generalization Rule is obtained by assuming that if we prove α at degree λ then we obtain $\forall$xα(x) at the same degree λ. Assume that D is the deduction operator of such a fuzzy logic and that s is a fuzzy theory. Then D(s)(α) = 1 for every logically true formula α and, otherwise,

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

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 T be a system of axioms for set theory and assume 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.

See also

• Fuzzy subalgebra
• Fuzzy associative matrix
• Fuzzy logic programming
• Fuzzy set
• Paradoxes
• Rough set
• Similarity logic
• Necessity logic
• MV-algebras
• Basic logic

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