On fuzzy truth-values and quasi-standard completeness

Abstract

Propositional many-valued logics constitute formalisation of fuzzy logics, as the intended set of truthvalues is the real unit interval [0, 1], or meaningful subsets of it. In this paper we propose to frame some intuitive notion about fuzzy truth-values in formal logic and algebraic de!nitions, inducing some re"ections about the usual notion of standard completeness. The main feature of fuzzy logic is to have fuzzy truth-values, that is, the classical notion of false/true (or 0/1) membership of an element to a set is generalised to a wide spectrum of values in [0, 1]. From a mathematical point of view, in the last decades several many-valued logics have been introduced, whose natural semantics is evaluated in the real unit interval equipped with reasonable generalisations of classical two-valued connectives. In particular, the hierarchy of schematic extensions of Esteva and Godo's Monoidal t-norm based logic M T L is widely considered as a mature framework for studying truth-functional, [0, 1]-valued fuzzy logic from a purely formal, algebraic-logical approach. As a matter of fact M T L is sound and complete with respect to standard structures, that is, algebraic systems whose universe is the real unit interval [0, 1], equipped with a left-continuous t-norm as conjunction, and its residuum as implication (and the constants 0 and 1 with the obvious meaning of, resp., crisp falsity and crisp truth). This property, motivating the introduction of M T L, is known as standard completeness of the logic M T L, and it can be applied to its schematic extensions, too. De!nition 1.1. A schematic extension L of M T L is standard complete i# there is a class Std(L) of standard structures such that for any formula ϕ, it holds that ϕ is a theorem of L i# ϕ = 1 is a valid identity in every algebra in Std(L). We refer to Std(L) as the standard models of L. Theorem 1.2. M T L is standard complete. Std(M T L) can be chosen as the class of all standard structures.

Keywords

• eol>Fuzzy logic
• truth values
• standard completeness
• MTL

1. Introduction

De!nition 1.1 seems justly to capture adequately the notion of a logic whose truth-values are fuzzy, in the sense that the set of truth-values coincides with the real unit interval [ 0, 1 ], totally ordered in the natural way. But this is not the end of the story for what concerns logics and their truth-value sets. Some observations are in order.

First, not all schematic extensions of M T L are standard complete. Foremost examples are !nite-valued logics, among which, classical two-valued logic, which is then nicely considered as a particular case of fuzzy logic. Actually, !nitely valued logics are generally considered authentic fuzzy logics, both for their extensive use in applications, and for the general tameness of their treatment, both in applicative contexts and in more theoretical ones. But there are other examples, where the schematic extension of M T L considered cannot be given a full [ 0, 1 ] semantics, while being actually in!nitely-valued. These logics are usually considered only by those theoreticians which explore the structure of the whole lattices of subvarieties of M T L, but they are seldom considered as actual fuzzy-valued logics.

In this work we propose some observations on the notion of standard completeness showing that it can be reasonably strengthened and also weakened, providing us with a sort of classi!cation of schematic extension of M T L for what regards their !tness with respect to [ 0, 1 ]-valued semantics. In particular we shall argument that there is a very strong notion of being [ 0, 1 ]-valued, which is satis!ed exactly by one schematic extension of M T L, namely Łukasiewicz logic. On the other hand we shall propose sound and complete semantics for some standard complete extensions which are very far from having the whole interval [ 0, 1 ] as intended truth-value set. Further, we shall consider a weakening of the notion of standard completeness to show that some non-standard complete extensions of M T L, which are for this reason usually not considered as actual fuzzy logics, are indeed very close, in a precise technical sense, to have full [ 0, 1 ]-valued semantics.

2. Preliminaries

A t-norm is a binary operation from [ 0, 1 ]2 into [ 0, 1 ] that is associative, commutative, nondecreasing in both arguments, and has 0 as absorbing element and 1 as unit. Given a leftcontinuous t-norm ⊙, its associated residuum is the binary operation x → y = max{z | z ⊙ x ≤ y}. The algebra [ 0, 1 ]⊙ = ([ 0, 1 ], ⊙, →, ∧, 0), where x ∧ y = min(x, y) , is called a standard algebra and it is completely determined by the left-continuous t-norm ⊙.

A t-norm ⊙ is Archimedean if it has the Archimedean property, that is, if for each x, y ∈ (0, 1) there is a natural number n such that xn ≤ y, where by xn we mean x ⊙ · · · ⊙ x, n times. A t-norm ⊙ is nilpotent if for each x ∈ [0, 1) there is a natural number n such that xn = 0. Clearly, each nilpotent t-norm is Archimedean1.

Two t-norms ⊙1 and ⊙2 are isomorphic if there is a strictly increasing bijective map f : [ 0, 1 ] → [ 0, 1 ] such that f (x ⊙1 y) = f (x) ⊙2 f (y) for every x, y ∈ [ 0, 1 ]. Two standard algebras are isomorphic if their t-norms are isomorphic.

Among the examples of t-norms and corresponding residua (hence of standard algebras), we mention the following:

1In the t-norm literature (see [ 1 ]), the de!nitions of Archimedean and nilpotent t-norms are applied only to continuous ones. Here, we generalise these de!nitions to all t-norms.

• Gödel t-norm a ⊙G b = min{a, b} with residuum a →G b = 1 if a ≤ b and a →G b = b otherwise. The algebra [ 0, 1 ]G = ([ 0, 1 ], ⊙G, →G, ∧, 0) is the standard Gödel algebra. • Product t-norm a ⊙P b = a · b (that is the usual product), that is a strictly monotone continuous t-norm having residuum a →P b = 1 if a ≤ b and a →P b = b/a otherwise.

The algebra [ 0, 1 ]P = ([ 0, 1 ], ⊙P , →P , ∧, 0) is the standard Product algebra. • Łukasiewicz t-norm a ⊙L b = max{0, a + b − 1}, that is a nilpotent continuous t-norm having residuum a →L b = min{0, 1 − a + b}. The algebra [ 0, 1 ] = ([ 0, 1 ], ⊙, →, ∧, 0) is the standard Łukasiewicz algebra, also called the standard M V -algebra. • Nilpotent minimum, that is a non-continuous but left-continuous t-norm that, despite its name, is not a nilpotent t-norm: a ⊙NM b = min(a, b) if a + b > 1 and a ⊙NM b = 0 otherwise, with residuum a →NM b = 1 if a ≤ b and a →NM b = max{1 − a, b}, otherwise. The algebra [ 0, 1 ]NM = ([ 0, 1 ], ⊙NM , →NM , ∧, 0) is the standard NMalgebra. • Drastic product t-norm, that is a non-continuous but right-continuous and as such it does not have a residuum: a ⊙DP b = b if a = 1, a ⊙DP b = a if b = 1 and a ⊙DP b = 0 otherwise.

Proposition 2.1. Any left-continuous nilpotent t-norm is isomorphic with Łukasiewicz t-norm. Proof. In [2] it is proved that any left-continuous Archimedean t-norm is continuous. Since nilpotent t-norms are archimedean, any left-continuous nilpotent t-norm is continuous. In [ 1 ], Prop. 5.10 it is proved that any continuous nilpotent t-norm is isomorphic with Łukasiewicz t-norm.

Monoidal t-norm based logic (M T L, for short), axiomatized in [3], was proved in [4] to be complete with respect to the set of all standard algebras (this is stated as Theorem 1.2 in the introduction). The algebraic counterpart of M T L, via the usual Lindenbaum construction, is the variety V(M T L) of M T L-algebras. An MTL-algebra (A, ∗, →, ∧, 0) is a prelinear commutative bounded integral residuated lattice. Any standard algebra ([ 0, 1 ], ⊙, →, ∧, 0) is an M T L-algebra and by Theorem 1.2 V(M T L) is generated by the set of standard algebras. In any M T L-algebra we set 1 := 0 → 0.

A #lter F of an M T L-algebra A = (A, ⊙, →, ∧, 0) is a subset of A containing 1 and such that if a ≤ b and a ∈ F then also b ∈ F and if a, b ∈ F also a ⊙ b ∈ F . A proper !lter p of A is prime i# for each pair of elements x, y ∈ A either x → y ∈ p or y → x ∈ p. The set of prime !lters of A is called its prime spectrum Spec(A) and can be topologised by setting as a base of closed sets all subsets of the form {p ∈ Spec(A) | a ∈ p}, for a ∈ A. We denote by M ax(A) the set of !lters of A that are maximal with respect to set inclusion, endowed with the topology inherited by restriction from Spec(A). An M T L-algebra is simple if its only proper !lter is {1}. Each axiomatic extension L of M T L determines a subvariety V(L) of V(M T L). We shall denote the free n-generated algebra in a variety V(L) by Fn(L).

Hájek’s Basic logic (BL for short, [5]) is the axiomatic extension of M T L by means of the divisibility axiom (ϕ ∧ ψ) → (ϕ ⊙ (ϕ → ψ)). The algebraic counterpart of BL is the variety V(BL) of BL-algebras. BL is the logic of all continuous t-norms and their residua, in the sense that V(BL) is generated by all standard algebras [ 0, 1 ]⊙ for ⊙ any continuous t-norm [6].

Gödel logic (G for short) is the axiomatic extension of BL given by adding the idempotency axiom ϕ → (ϕ ⊙ ϕ). The variety V(G) of Gödel algebras is formed by the BL-algebras satisfying the equation x ⊙ x = x. Gödel logic is standard complete and further, the standard Gödel algebra generates V(G).

Nilpotent Minimum logic (N M for short) is the axiomatic extension of M T L obtained by adding the involutiveness axiom ¬¬ϕ → ϕ and the so-called weak nilpotent minimum axiom ¬(ϕ ⊙ ψ) ∨ ((ϕ ∧ ψ) → (ϕ ⊙ ψ)). In [3] it is proved that N M is standard complete since the standard algebra [ 0, 1 ]NM generates V(N M ). N M − is the extension of N M by the axiom (¬(¬ϕ ⊙ ¬ϕ)) ⊙ (¬(¬ϕ ⊙ ¬ϕ)) → ¬(¬(ϕ ⊙ ϕ) ⊙ ¬(ϕ ⊙ ϕ)).

While Drastic product t-norm is not residuated, there are M T L-chains obtained by restricting this t-norm to suitable subsets of [ 0, 1 ]. These chains generate the variety V(DP ), associated with the logic DP , axiomatised by ϕ ∨ ¬(ϕ ⊙ ϕ).

Łukasiewicz logic (Ł for short) is the axiomatic extension of BL given by adding the axiom ¬¬ϕ → ϕ. The variety V(Ł) of M V -algebras is formed by the BL-algebras satisfying the equation ¬¬x = x. We refer the reader to [7, 8] for all background on M V -algebras. Łukasiewicz logic is standard complete and further, by Chang’s algebraic completeness, the standard M V algebra generates V(Ł). Every M V -algebra is the interval of some lattice-ordered group. Indeed, the functor Γ implements the equivalence between the category of M V -algebras and the category of lattice-ordered abelian groups (abelian ℓ-groups) with strong unit. For every abelian ℓ-group (G, +, 0, ≤) with strong unit u the functor Γ equips the unit interval [0, u] = {0 ≤ x ≤ u | x ∈ G} with the operations x ⊙ y = max(0, x + y − u) and x → y = min(u − x + y, u). It is easy to see that the resulting structure Γ(G, u) = ([0, u], ⊙, →, ∧, u) is an M V -algebra.

We are particularly interested in the simple M V -algebra Sn = Γ(Z, n − 1) and in the nonsimple M V -algebras Snω = Γ(Z ×~ Z, (n − 1, 0)) and Snc = Γ(Z ×~ R, (n − 1, 0)) for n ≥ 2, where ×~ stands for the lexicographic product (i.e., the direct product with the order relation de!ned lexicographically: (n, m) ≤ (n′, m′) if and only if n < n′ or n = n′ and m ≤ m′). We denote the operations of Snω and Snc respectively by ⊙nω, →nω and ⊙cn, →cn.

Komori fully classi!ed all subvarieties of M V -algebras. In particular, a proper variety of M V -algebras is generated by a set of chains I ∪ J where I is a !nite set of chains of the form Sk and J a !nite set of chains of the form Skω. Notice that Skc generates the same variety as Skω. 3. Single standard completeness, and truly [ 0, 1 ]-valued logics

We start strengthening the notion of standard completeness, as follows.

De!nition 3.1. A schematic extension L of M T L is singly standard complete i# there exists a single standard structure S(L) such that the set of standard models of L can be chosen as Std(L) = {S(L)}.

Proposition 3.2. A schematic extension L of M T L is singly standard complete i$, for all integers n ≥ 0, the free n-generated algebra in V(L), Fn(V(L)), is isomorphic with the subalgebra of the algebra of all functions [ 0, 1 ]n → [ 0, 1 ], generated by the projections xi : (t1, . . . , tn) 7→ ti, using the operations of a standard algebra.

When concrete representation matters, we shall identify Fn(V(L)) with the algebra of functions given in Proposition 3.2.

M T L itself is standard complete while is not known if it is singly standard complete, and most likely it is not. On the other hand BL is singly standard complete, and one can choose S(BL) in several distinct, not mutually isomorphic ways. A rather canonical choice is the ordinal sum of ω copies of the standard M V -algebra, which is used in [9], by applying Proposition 3.2, to characterise the free BL-algebras. Many other extensions of M T L are standard complete but not singly so. Some examples could be found in the paper [10], as subvarieties of DN M G. On the other hand there are singly standard complete extensions of M T L which are formally [ 0, 1 ]-valued, but we shall argument here that these values are not fully satisfactory truth values.

We begin this discussion recalling what happens in classical propositional logic, where truth-values are in bijection with maximal theories, and in turn with maximal !lters of the Lindenbaum algebra, that is, the free Boolean algebra over a denumerable in!nite set of free generators.

Theorem 3.3. Let v : V ar → {0, 1} be a truth-value assignment in classical propositional logic and let Θv be {xi | v(xi) = 1} ∪ {¬xi | v(xi) = 0}. Then there is a unique maximal theory extending Θv and moreover, the correspondence v 7→ Θv is a bijection between the set of all truth-value assignments and maximal theories.

Clearly, truth-value assignments can be identi!ed with points in {0, 1}ω, and maximal theories are in bijection with quotients of the Lindenbaum algebra over maximal !lters. Further, these correspondences still hold on all the fragments with a !nite number of variables: let V arn, F ormn and Fn(B) denote respectively the set of the !rst n variables, the set of all formulas over these variables, and the free Boolean algebra over n free generators.

Theorem 3.4. For each n ∈ ω ∪ {ω}, each pair of the following sets are in bijective correspondence.

• Points p ∈ {0, 1}n. • Truth-value assignments v : V arn → {0, 1}. • Maximal theories Θ ⊂ F ormn.

• Maximal #lters p ∈ M ax(Fn(B)).

Notice in particular that for each truth-value assignment v : V arn → {0, 1}, the quotient of Fn(B) over the !lter pv = {f ∈ Fn(B) | f (v) = 1} is isomorphic with the set of restrictions of elements in Fn(B), thought as functions f : {0, 1}n → {0, 1}, to the singleton {(v(x1), . . . , v(xn))}. So, truth-values are the same as points in the domain of the functions forming the free Boolean algebra, and the evaluation of a formula under a given truth-value assignment is the same as restricting the function corresponding to that formula to the singleton formed by the point corresponding to that truth-value assignment. Given distinct assignments, there are formulas distinguishing them. These observations can be applied to the [ 0, 1 ]-valued setting, by requiring that each point in [ 0, 1 ]n behaves as a unique truth-value assignment, and versa vice, each assignment corresponds uniquely to a point. Further, the evaluation of formulas should undergo the same above-mentioned process, and, in particular, distinct points should be discerned by formulas.

De!nition 3.5. A standard complete schematic extension L of M T L is truly [ 0, 1 ]-valued i#, for every integer n > 0 and for every point v ∈ [ 0, 1 ]n the correspondence v 7→ {f ∈ Fn(V(L)) | f (v) = 1}, is a homeomorphism between [ 0, 1 ]n and M ax(Fn(V(L))). Theorem 3.6. The only logic truly [ 0, 1 ]-valued is Łukasiewicz in#nite-valued logic. Proof. It is well known that the map v 7→ {f ∈ Fn(M V ) | f (v) = 1} is a homeomorphism between [ 0, 1 ]n and M ax(Fn(M V )), whence Łukasiewicz logic is truly [ 0, 1 ]-valued.

Now, assume L is truly [ 0, 1 ]-valued. Since L is singly standard complete, V(L) is generated by a single standard structure S(L), and by Proposition 3.2, Fn(V(L)) is identi!able with the subalgebra of the algebra of all the functions [ 0, 1 ]n → [ 0, 1 ] generated by the projections. Then in particular, maximal !lters of Fn(V(L)) are in bijection with points of [ 0, 1 ]n via the map v 7→ {f ∈ Fn(V(L)) | f (v) = 1}. This implies that, for each v ∈ [ 0, 1 ]n, each algebra A(v) of the form {f ∈ Fn(V(L)) | f ↾ {v}}, being isomorphic to Fn(V(L)) modulo the maximal !lter determined by the point v, is a simple algebra. But A(v) is obtained by substituting in Fn(V(L)) each generator xi with the element vi ∈ [ 0, 1 ]. Whence A(v) is the chain generated by {v1, . . . , vn} ⊆ [ 0, 1 ]. Since A(v) is simple, for each vi 6= 1 there is an integer ki such that viki = 0, otherwise vi would generate a proper !lter. But this property must hold for any n and any set {v1, . . . , vn} ⊆ [ 0, 1 ], that is, for each v ∈ [0, 1) there is k such that vk = 0. Then the conjunction of S(L) is a nilpotent t-norm, which in turn implies, by Proposition 2.1 that L is Łukasiewicz logic.

4. Quasi-standard complete logics

While in the previous section we have dealt with a reasonable strenghtening of the notion of standard completeness, in this section we propose and apply a weakening of the same notion in order to stress that some schematic extensions of M T L, which are usually considered only from the purely technical algebraic point of view, are almost as fuzzy-valued logic as major mathematical fuzzy logics such as Product and Gödel logics. We start by emphasising that singly standard complete logics may be given a sound and complete semantics which is very far from being [ 0, 1 ]-valued. We recall that, when concrete representation matters, we identify free algebras Fn(L), for L a singly standard complete logic, with the algebra of functions f : [ 0, 1 ]n → [ 0, 1 ] generated by the projections. Free Gödel algebras Fn(G) have been described in [11], and free product algebras Fn(P ) in [12].

Fix any real number ǫ ∈ (0, 1). Let ιǫ(0) = {0} and ιǫ(1) = [1 − ǫ, 1] be subsets of [ 0, 1 ]. For each integer n ≥ 0, with each point b ∈ {0, 1}n we associate the subset ιǫ(b) = {p ∈ [ 0, 1 ]n | pi ∈ ιǫ(i)}.

Example 4.1. Let Gn be the Gödel algebra of restrictions of functions in Fn(G) to the set Sb∈{0,1}n ιǫ(b). Then Gn =∼ Fn(G).

Let Pn be the Product algebra of restrictions of functions in Fn(P ) to the set Sb∈{0,1}n ιǫ(b). Then Pn =∼ Fn(P ).

Examples in 4.1 show that a formula is a theorem of Gödel or of Product logic i# it evaluates identically to 1 for all assignments v : V arn → Sb∈{0,1}n ιǫ(b), for arbitrary small values for ǫ. Actually, if we consider in!nitesimal elements, living in the non-standard real interval [ 0, 1 ], in the sense of non-standard analysis, we obtain sound and complete semantics for both Gödel and Product logics, where the interval ιǫ(1) is replaced by an in!nitesimal left neighbourhood of 1. This kind of semantics is actually not a novelty, since it is considered the usual semantics of, for instance, the logic associated with Chang’s M V -algebra S2ω (with the only di#erence that ιǫ(0) is replaced by an in!nitesimal right neighbourhood of 0). But this semantics is seldom, if ever, considered for Gödel and Product logics. The close relationship of V(S2ω) with product algebras amount to a categorial equivalence between the two varieties, as shown in [13], [14]. Notice that the representation of Gödel and Product logics with in!nitesimal truth-values around 1 suggests to consider these logics as variants of Boolean logic, as the non-in!nitesimal values are exactly the Boolean truth-values. Approaching this observation from a topos-theoretic approach, in [15] is recalled that the subobject classi!er in a category dually equivalent to !nite Gödel algebras is very close in structure to the subobject classi!er in the category of sets, that is the familiar notion of characteristic function of a set. Reversing the traditional interpretation, we shall now propose a semantics for S2ω which is very close to be a full standard semantics, and we shall generalise this to a family of other extensions of Łukasiewicz logics. De!nition 4.2. A schematic extension L of M T L is quasi-standard complete i# V(L) is generated by a class of algebras Q(L) such that each A ∈ Q(L) has a universe which is a dense subset of [ 0, 1 ]. L is singly quasi-standard complete i# Q(L) can be chosen as a singleton. Theorem 4.3. For each integer k > 1, the logic associated with Skω is singly quasi-standard complete.

Proof. We start recalling that Skc = Γ(Z ×~ R, (k − 1, 0)) generates the same variety as Sω. Now k we !x an arbitrarily chosen monotonically non-decreasing bijection fk : R → ( 2k−−12 , 2k1−2 ) such that fk(0) = 0: for sake of concreteness let fk(x) = aπrc(tka−n1(x)) . Let now h : Skc → [ 0, 1 ] be the function de!ned by (m, x) 7→

+ fk(x) .

m k − 1 It is easy to check that h is non-decreasing and injective. The range h[Skc] of h is [ 0, 1 ] \ { 22ki+−12 | i = 0, . . . , k − 2}, which is a dense subset of [ 0, 1 ]. It remains to equip the range of h with the structure of an M T L-chain isomorphic with Skc. To achieve this it su&ces to de!ne the conjunction ∗k by going back and forth, as x ∗k y = h(h−1(x) ⊙ck h−1(y)), and, analogously, its residuum ⇒k as x ⇒k y = h(h−1(x) →ck h−1(y)). We conclude that V(Skω) is generated by (h[Skc], ∗k, ⇒k, 0) which is an M T L-chain whose universe is dense in [ 0, 1 ].

No logic associated with a variety generated by Skω, for all integers k > 1, is standard complete, as each subset of the form {(m, x) | x ∈ R}, for a !xed m, is unbounded. The paper [16] introduces a family of t-norms de!ned by combining together chains of the form h[Skc] and Sk+1.

Corollary 4.4. The logic of each variety generated by a set of M V -chains of the form Skω, for any integer k > 1, is quasi-standard complete. The only standard complete among them is Łukasiewicz in#nite-valued logic.

Theorem 4.5. Let V be any variety of M V -algebras. Then exactly one of the following holds. 1. V is #nitely valued, that is V is generated by a #nite set of #nite M V -chains. 2. V is quasi-standard complete.

3. V is the join of one #nitely valued variety with one quasi-standard complete variety.

Proof. By Komori’s complete classi!cation of varieties of M V -algebras.

Proposition 4.6. DP is not quasi-standard complete. N M is singly standard complete and N M − is not standard complete but it is singly quasi-standard complete.

Proof. Any DP chain has a coatom. Each such a chain singly generates DP . Whence, any non-trivial DP chain whose universe is a subset of [ 0, 1 ] must omit an interval of the form (c, 1), where c ∈ [0, 1) is the coatom. Therefore, every set of chains generating DP omits such an interval, that is, their universe is not dense in [ 0, 1 ], and then DP is not quasi-standard complete. Recall that N M is generated by the standard nilpotent minimum algebra, while N M − is generated by the subalgebra of the standard nilpotent minimum algebra obtained by removing from the universe [ 0, 1 ] the point 12 (see [14, 18, 19, 20]). Then the logic N M is singly standard complete, and the logic N M − is singly quasi-standard complete.

Canonical DP chains are de!ned in [17] as having universe of the form [0, c] ∪ {1}.

5. Conclusion and future work

The full [ 0, 1 ] semantics provided by standard completeness lends itself to several applications. Just to mention one such application on the theoretical side, a logic L that is singly standard complete could be endowed with a notion of !nitely additive measure over the space of truthvalue assignments, allowing the development of a probability theory of non-classical events (or, states), where the events are modeled as formulas living in the non-classical, fuzzy, logic L (see [21, 22, 23, 24, 25]). Actually, full [ 0, 1 ] semantics can be too strong a requirement for such a development: in future works we shall show how to develop states over logics which are only quasi-standard complete.

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