A non-associative generalization of continuous t-norm-based logics

Abstract

This paper investigates a non-associative generalization, more exactly, a weak t-associative (wta) generalization, of continuous t-norm-based logics. First, the wta-uninorm logic WA_tBL and its axiomatic extensions WA_tŁ, WA_tΠ, and WA_tG are introduced as [0, e]-continuous wta-uninorm-based analogues of the most famous continuous t-norm-based logics, BL (Basic fuzzy logic), Ł (Łukasiewicz logic), Π (Product logic), and G (Gödel logic), respectively. Their corresponding algebraic structures and algebraic completeness results are considered. Next, completeness with respect to the algebras whose lattice reduct is [0, 1], known as standard completeness, is established for these systems by a construction in the style of Jenei–Montagna.

Keywords

Fuzzy logic micanorm wta-uninorm uninorm t-norm

1 Introduction

The concept of a triangular norm (t-norm for short) was first introduced by Menger [22] to generalize the triangular inequality of a metric. However, the current notions of a t-norm and its dual operator t-conorm are due to Schweizer and Sklar [27, 28]. T-norms and t-conorms, which are commutative, associative, monotonic, binary functions on [0, 1] with identity elements 1 and 0, respectively, generalize the conjunctive and and the disjunctive or aggregation operators (see [7, 16]). As is well known, these operators have been used as connectives for many-valued or fuzzy logics. In particular, logics with (multiplicative) conjunctions and implications interpreted by continuous t-norms and their residua have been extensively investigated in recent years. The infinite-valued systems Ł (Łukasiewicz logic), G (Gödel–Dummett logic), and Π (Product logic) are the most famous examples of logics based on continuous t-norms: Hájek’s BL (Basic fuzzy logic) [14] is the most basic continuous t-norm (based) logic.

Although t-norms and t-conorms play an important role in fuzzy logic (theory), these operators do not admit a compensating behavior. Specifically, t-norms do not allow low values to be compensated by high values, and t-conorms do not allow high values to be compensated by low values (see [33]). For this reason, Yager and Rybalov [34] introduced uninorms as a generalization of t-norms and t-conorms. Uninorm operators have their identity lying somewhere in [0, 1]. After introducing uninorms, many interesting properties of uninorms and their applications, such as full reinforcement, compensation behavior, and bipolar problems, have been studied (see [2, 9 , 33]). Furthermore, several uninorm-based logics have been recently introduced. In particular, uninorm-based analogues of the logics of continuous t-norms and their residua were introduced by Gabbay and Metcalfe in [11]. They introduced BUL (Basic uninorm logic) and its axiomatic extensions as fuzzy logics based on [0, 1)-continuous uninorms. 1

T-norms and uninorms require associativity. In fact, associative operations are abundant in mathematics. For instance, the addition and multiplication of real numbers are such operations. However, many important and interesting operations are non-associative. Subtraction and exponentiation are such examples. Among the non-associative operations, operations with weak forms of associativity, called here weak associative operations, have been extensively investigated. For instance, in abstract algebra, power associativity is a property of a binary operation which is a weak form of associativity (see [1 , 30] for many examples). Exponentiation to the power of any positive integer can be defined consistently whenever multiplication is power-associative. 2 Moreover, quasigroup [18, 23], weakly associative relation algebras [17 , 29], weakly associative lattices [10 , 25], hypergroups [3], and hyperstructures [21, 31, 32] are famous examples.

The systems BL and BUL are axiomatic extensions of MTL (Monoidal t-norm logic) and UL (Uninorm logic), each of which is the logic of left-continuous t-norms and the logic of conjunctive left-continuous uninorms, respectively. Thus, all of these logics are associative logics. Recently, non-associative logics have been introduced. Cintula, Horčík, and Noguera [4 –6] have introduced non-associative generalizations of uninorm (based) logics, i.e., non-associative substructural fuzzy logics along with their corresponding completeness properties. Yang [35] introduced micanorms and micanorm-based logics. A micanorm is a uninorm omitting associativity from its definition. However, logics based on weakly associative algebraic structures have not yet been fully investigated. As far as the author knows, only Yang [36] has introduced three forms of weakly associative uninorms and their corresponding logics, WA_tMUL, A_tMUL, and SA_tMUL, and proved that WA_tMUL is standard complete.

We have at least two interesting facts in Yang’s work. One is that the weak associativity introduced in weakly associative relation algebras can be regarded as an example of Yang’s one weak form of associativity, called strong t-associativity in [36]. The other is that the system WA_tMUL is the logic based on conjunctive and left-continuous weak t-associative uninorms (briefly, wta-uninorms). Note that while the system BUL and its axiomatic extensions introduced in [12] are logics satisfying some continuity criterion, the logics introduced in [36] are not. These are the logics for left-continuous wta-uninorms, but not for continuous ones. Then, a natural concern arises for fuzzy logics based on wta-uninorms meeting some criterion of continuity. This paper investigates such fuzzy logics as weakly associative analogues of the most famous continuous t-norm-based logics BL, Ł, G, and Π . This will give us a chance to study many weakly associative algebraic properties corresponding to the associative ones that the algebraic structures of BL, Ł, G, and Π have.

The paper is organized as follows. In Section 2, we introduce the basic wta-uninorm-based logic WA_tBL, which is intended to cope with the tautologies of [0, e]-continuous conjunctive wta-uninorms and their residua, its axiomatic extensions, and their corresponding algebraic semantics. These logics are introduced as [0, e]-continuous wta-uninorm-based logics. In Section 3, we recall micanorms and wta-uninorms, and introduce wta-uninorm analogues of the well-known Łukasiewicz, Product, and Gödel t-norms. In Section 4, we finally establish the standard completeness for these logics using the Jenei–Montagna–style construction introduced in [8, 15 , 36].

2 The logic WA_tBL and its axiomatic extensions

The term [0, e]-continuous wta-uninorm-based logics refers to substructural fuzzy logic systems with wta-uninorm-based semantics, where the (strong) conjunction and implication connectives ‘&’ and ‘→’ are interpreted by a [0, e]-continuous wta-uninorm and its residuum. In this framework, the weakest logic is WA_tBL. This logic and its axiomatic extensions (henceforth referred to as extensions) can be based on a countable propositional language with formulas Fm built inductively (as usual) from a set of propositional variables VAR, binary connectives →, & , ∧ , ∨, and constants ⊤, ⊥ , t, f 3 , with defined connectives:

df1. ¬A:= A → f,

df2. A ↔ B:= (A → B) ∧ (B → A), and

df3. -φ : = φ→ ⊥.

We start with the following axiomatizations of WA_tBL, the most basic [0, e]-continuous wta-uninorm-based logic, and its extensions.

Definition 1.

Let φ_t be φ ∧ t. WA_tBL consists of the following axiom schemes and rules:

$\begin{array}{l} φ \to φ (s e l f - i m p l i c a t i o n, S I) \\ (φ \land ψ) \to φ, (φ \land ψ) \to ψ (\land - e l i m i n a t i o n, \land - E) \\ ((φ \to ψ) \land (φ \to χ)) \to (φ \to (ψ \land χ)) (\land - i n t r o d u c t i o n, \land - I) \\ φ \to (φ \lor ψ), ψ \to (φ \lor ψ) (\lor - i n t r o d u c t i o n, \lor - I) \\ ((φ \to χ) \land (ψ \to χ)) \to ((φ \lor ψ) \to χ) (\lor - e l i m i n a t i o n, \lor - E) \\ ⊥ \to φ (e x f a l s u m q u o d l i b e t, E F) \\ (φ & ψ) \to (ψ & φ) (& - c o m m u t a t i v i t y, & - C) \\ (t \to φ) \leftrightarrow φ (p u s h a n d p o p, P P) \\ φ \to (ψ \to (ψ & φ)) (& - a d j u n c t i o n, & - A d j) \\ (φ_{t} & ψ_{t}) \to (φ \land ψ) (& \land) \\ (ψ & (φ & (φ \to (ψ \to χ)))) \to χ (r e s i d u a t i o n, R e s^{'}) \\ (φ \to ((φ & (φ \to ψ)) & (ψ \to χ))) \to (φ \to χ) (T^{'}) \\ ((δ & ε) \to (δ & (ε & {(φ \to ψ)}_{t}))) \lor (δ^{'} \to (ε^{'} \to ((ε^{'} & δ^{'}) & {(ψ \to φ)}_{t}))) (P L) \\ (φ_{t} & (ψ_{t} & χ_{t})) \leftrightarrow ((φ_{t} & ψ_{t}) & χ_{t}) (w e a k t - a s s o c i a t i v i t y, w A S_{t}) \\ (t \to φ) \lor (φ \to ψ) \lor (ψ \to (φ & (φ \to ψ))) (R D I V_{t}^{w}) \\ φ \to ψ, φ ⊢ ψ (m o d u s p o n e n s, m p) \\ φ ⊢ φ_{t} (a d j_{u}) \\ φ ⊢ (δ & ε) \to (δ & (ε & φ)) (α) \\ φ ⊢ δ \to (ε \to ((ε & δ) & φ)) (β) \end{array}$

$\begin{array}{l} [(ii)] The following are three extensions of W A_{t} B L : \\ • Involutive {WA}_{t} BL, W A_{t} Ł, is W A_{t} B L plus(DNE) \neg \neg φ \to φ; and (D N E_{t}) {(- - φ)}_{t} \to φ . \\ • Cancellative {WA}_{t} BL, W A_{t} Π, is W A_{t} B L plus (GN) (φ \land - φ) \to ⊥; (CP) (φ \to ψ) \to (- ψ \to - φ); and {(RCAN}_{t}^{w}) {(- - φ)}_{t} \to ({(φ_{t} \to (φ_{t} & ψ))}_{t} \to ψ) . \\ • Idempotent {WA}_{t} BL, W A_{t} G, is W A_{t} B L plus ({ID}_{t}) {(φ & φ)}_{t} \leftrightarrow φ_{t} . \end{array}$

For easy reference, we let Ls be the set of the wta-uninorm-based logics defined previously, i.e, Ls = {WA_tBL, WA_tŁ, WA_tΠ, WA_tG}.

A theory is a set of formulas. A proof in a theory T over L (∈ Ls) is a sequence s of formulas such that each element of s is either an axiom of L, a member of T, or is derivable from previous elements of s by means of a rule of L. T ⊢ φ, more exactly T ⊢ _Lφ, means that φ is provable in T with respect to (w.r.t.) L, i.e., there is an L-proof of φ in T. A theory T is inconsistent if T⊢ ⊥; otherwise, it is consistent.

The deduction theorem for L (∈ Ls) is as follows:

Theorem 1. [4, 5] For each set Γ ∪ {φ, ψ} of formulas, the following holds: Γ, φ ⊢ _Lψ iff Γ ⊢ _Lγ (φ) → ψ for some γ ∈ Π (bDT^∗). 4

For convenience, ‘¬,’ ‘-,’ ‘∧,’ ‘∨,’ ‘→,’ ‘⊤,’ and ‘⊥’ are used ambiguously as propositional connectives and constants, as well as algebraic operators and special elements, but context should clarify their meanings.

Suitable algebraic structures for Ls are obtained as varieties of residuated lattice-ordered groupoids with unit (rlu-groupoids) in the sense of [12].

Definition 2. [36]

A pointed, bounded, commutative rlu-groupoid is a structure (A, ⊤ , ⊥ , t, f, ∧ , ∨ , ∗ , →) such that: (A, ⊤ , ⊥ , ∧ , ∨) is a bounded lattice with a top element ⊤ and a bottom element ⊥; (A, ∗ , t) is a commutative groupoid with unit; f is an arbitrary element, i.e., a point, of A; and y ≤ x → z iff x ∗ y ≤ z, for all x, y, z ∈ A (residuation).

A wta-monoidal residuated lattice is a pointed, bounded, commutative rlu-groupoid satisfying: x, y, z ≤ t implies x ∗ (y ∗ z) = (x ∗ y) ∗ z (weak t-associativity).

A WA_tMUL-algebra is a wta-monoidal residuated lattice satisfying: t ≤ ((z ∗ w) → (z ∗ (w ∗ (x → y) _t))) ∨ (z′ → (w′ → ((w′ ∗ z′) ∗ (y → x) _t))), for all x, y, z, w, z′, w′ ∈ A (PL^𝒜).

Definition 3. (L-algebras) Let x_t, ¬x, and -x be x ∧ t, x → f, and x→ ⊥, respectively.

A WA_tBL-algebra is a WA_tMUL-algebra satisfying: t ≤ (t → x) ∨ (x → y) ∨ (y → (x ∗ (x → y))), for all x, y ∈ A ( ${RDIV}_{t}^{w 𝒜}$ ).

A WA_tŁ-algebra is a WA_tBL-algebra satisfying: for all x ∈ A, t ≤ ¬¬ x → x (DNE^𝒜) and t ≤ (-- x) _t → x (DNE_t^𝒜).

A WA_tΠ-algebra is a WA_tBL-algebra satisfying: for all x, y ∈ A, t≤ (x ∧ - x) → ⊥ (GN^𝒜), t ≤ (x → y) → (- y → - x) (CP^𝒜), and t ≤ (-- x) _t → ((x_t → (x_t ∗ y)) _t → y) ( ${RCAN}_{t}^{w 𝒜}$ ).

A WA_tG-algebra is a WA_tBL-algebra satisfying: (x ∗ x) _t = x_t (ID_t^𝒜).

We call all of these algebras L-algebras.

Let $A$ be an L-algebra, T a theory, φ a formula, and $K$ a class of L-algebras. An $A$ -evaluation is a function $v : Fm \to A$ satisfying: $v (♯ (φ_{1}, \dots, φ_{m})) = ♯^{A} (v (φ_{1}), \dots, v (φ_{m}))$ , where ♯ ∈ {→ , ∧ , ∨ , & , ⊤ , ⊥ , t, f} and $♯^{A}$ ∈ {→, ∧, ∨, ∗, ⊤, ⊥ , t, f}. A formula φ is valid in $A$ if v (φ) ≥ t for each $A$ -evaluation v. An $A$ -evaluation v is an $A$ -model of T if v (φ) ≥ t for each φ ∈ T. We denote the class of $A$ -models of T by Mod(T, $A$ ). φ is a semantic consequence of T w.r.t. $K$ , denoted by $T ⊨_{K} φ$ , if $Mod (T, A) = Mod (T \cup {φ}, A)$ for each $A \in K$ . $A$ is an L-algebra if and only if (iff) whenever φ is L-provable in any T (i.e., T ⊢ _Lφ, L an L logic), it is a semantic consequence of T w.r.t. { $A$ } (i.e., $T ⊨_{{A}} φ$ , $A$ a corresponding L-algebra). By MOD(L), we denote the class of L-algebras, and by MOD^ℓ(L), the class of linearly ordered L-algebras. Finally, we write T ⊨ _Lφ and $T ⊨_{L}^{ℓ} φ$ in place of T ⊨ _MOD(L)φ and T ⊨ _MOD^ℓ(L)φ, respectively.

Theorem 2.

(Finite strong completeness) Let T be a finite theory over L (∈ Ls) and φ a formula. Then T ⊢ _Lφ iff T ⊨ _Lφ iff $T ⊨_{L}^{ℓ} φ$ .

(Strong completeness) Let L be WA_tG, T a theory over L, and φ a formula. Then T ⊢ _Lφ iff T ⊨ _Lφ iff $T ⊨_{L}^{ℓ} φ$ .

Proof. We obtain this theorem as a corollary of Theorem 3.1.8 in [6]. □

3 [0, e]-continuous residuated wta-uninorms

In this section, by ‘1,’ ‘0,’ ‘e,’ and ‘∂,’ we denote ⊤, ⊥, t, and f, respectively, on the real unit interval [0, 1]. First, we introduce standard L-algebras, micanorms, and wta-uninorms. An L-algebra is standard iff its lattice reduct is [0, 1]. In standard L-algebras, the wta-monoid operator ∗ is a micanorm. A micanorm ([35]) is a function ∘ : [0, 1] ² → [0, 1] such that for some e ∈ [0, 1] and for all x, y, z ∈ [0, 1]: (a) x ∘ y = y ∘ x (commutativity), (b) e ∘ x = x = x ∘ e (identity), and (c) x ≤ y implies x ∘ z ≤ y ∘ z and z ∘ x ≤ z ∘ y (monotonicity). An associative micanorm is a uninorm; a uninorm satisfying e = 1 is a t-norm. A micanorm ∘ is called conjunctive if 0 ∘ 1 =1 ∘ 0 =0, and disjunctive if 0 ∘ 1 =1 ∘ 0 =1.

A wta-uninorm is a uninorm with a weak form of associativity instead of associativity itself.

Definition 4.

[36] A wta-uninorm is a micanorm satisfying that (weAS) if x, y, z ≤ e, then x ∗ (y ∗ z) = (x ∗ y) ∗ z.

The following are some particular wta-uninorms.

An Ł-wta-uninorm is a wta-uninorm satisfying: x ∗ y = max (x + y - e, 0) if x, y ≤ e.

A Π-wta-uninorm is a wta-uninorm satisfying: x ∗ y = xy if x, y ≤ e.

A G-wta-uninorm is a wta-uninorm satisfying: x ∗ y = min (x, y) if x, y ≤ e.

The famous Łukasiewicz, Product, and Gödel t-norms T^Ł (x, y) = max {0, x + y - 1}, T^Π (x, y) = xy, and T^G (x, y) = min {x, y} are examples of Ł-wta-, Π-wta-, and G-wta-uninorms, respectively. Moreover, the following proposition ensures that these wta-uninorms are weakly associative generalizations of the Łukasiewicz, Product, and Gödel t-norms.

Proposition 1. Ł-wta-, Π-wta-, and G-wta-uninorms are [0, e]-continuous.

Proof. Let [0, 1] ⌈ _[0,e] : = {x ∈ [0, 1] :0 ≤ x ≤ e}. As is well-known, uninorms have subalgebras isomorphic to t-norms: for an arbitrary uninorm ∗ with identity e ∈ (0, 1), ∗ : [0, 1] ⌈ _[0,e] × [0, 1] ⌈ _[0,e] → [0, 1] ⌈ _[0,e] is isomorphic to a t-norm (see [9, 20]). Since wta-uninorms are associative on [0, e], we can ensure that wta-uninorms also have subalgebras isomorphic to t-norms. In particular, the definitions of Ł-wta-, Π-wta-, and G-wta-uninorms ensure that each [0, 1] ⌈ _[0,e] part is isomorphic to the Łukasiewicz, Product, and Gödel t-norms, respectively. Thus, since Łukasiewicz, Product, and Gödel t-norms are continuous, it is clear that Ł-wta-, Π-wta-, and G-wta-uninorms are [0, e]-continuous. □

As in t-norms and uninorms, suitable functions for implications are obtained using residual operators of wta-uninorms. A wta-uninorm ∘ is said to be residuated if there is a binary function → : [0, 1] ² → [0, 1], called the residuum of ∘, such that x ∘ y ≤ z iff y ≤ x → z for all x, y, z ∈ [0, 1]. Then, given a wta-uninorm ∘, residuated implication → determined by ∘ is defined as x → y : = sup {z ∈ [0, 1] : x ∘ z ≤ y} for all x, y ∈ [0, 1].

Let L-wta-uninorms be the wta-uninorms corresponding to L-algebras. The operation ∗ of any L-algebra on [0, 1] is a conjunctive L-wta-uninorm with identity t and residuum →; conversely any residuated L-wta-uninorm ∘ gives rise to an L-algebra on [0, 1]. We finally show this, i.e., that any [0, e]-continuous residuated L-wta-uninorm ∘ gives rise to an L-algebra on [0, 1].

Lemma 1. Let ∘ be a [0, e]-continuous residuated wta-uninorm.

If y ≤ x < 1, then x ∘ (x → y) = y for all x, y ∈ [0, 1].

If x ≤ e, then x ≤ y or x ∘ (x → y) = y for all x, y ∈ [0, 1].

If e ≤ x < 1, then y ∧ e ≤ x ∘ (x → y) ≤ y for all x, y ∈ [0, 1].

Proof. For (a), let y ≤ x < 1. Note first that 0 = x ∘ 0 ≤ y ≤ x = x ∘ e. Then, by [0, e]-continuity, there is z ∈ [0, 1] such that x ∘ z = y and so z = x → y. Hence, we have x ∘ (x → y) = y. (b) is immediate from (a). For (c), we consider the case e ≤ x < y (noting that the case y ≤ x < 1 holds by (a)). Then, since e ≤ x → y, we obtain e ≤ x = x ∘ e ≤ x ∘ (x → y); therefore, y ∧ e ≤ x ∘ (x → y) ≤ y. □

In general, a function n : [0, 1] → [0, 1] is said to be a negation function iff it is non-increasing and satisfies n (0) =1 and n (1) =0. This works on t-norms very well, but not as well on uninorms because, e.g., in (conjunctive) uninorms, a non-increasing function n may not satisfy n (1) =0. We henceforth call a non-increasing function a negation function. A negation function (briefly stated as negation) n satisfying n (n (x)) ≥ x for all x ∈ [0, 1] is said to be a super-involutive negation; a super-involutive negation n is said to be a weak negation if n further satisfies n (0) =1 and n (1) =0; a weak negation n is said to be a strong negation (or involutive negation) if n further satisfies n (n (x)) = x for all x ∈ [0, 1]. We call an involutive negation n on [0, e] e-involutive.

Proposition 2.

If ∘ is a [0, e]-continuous residuated wta-uninorm with identity e, then for any ∂ ∈ [0, 1], ([0, 1] , 1, 0, e, ∂, min, max, ∘ , →) is a standard WA_tBL-algebra.

If ∘ is a [0, e]-continuous residuated wta-uninorm with an involutive negation n₁ and an e-involutive negation n₂, then ([0, 1] , 1, 0, e, ∂, min, max, ∘, →) is a standard WA_tŁ-algebra.

Let n be the negation satisfying (GN^𝒜), i.e., Gödel negation. If ∘ is a [0, e]-continuous residuated wta-uninorm with n and identity e and strictly monotone on [0, e], then ([0, 1] , 1, 0, e, ∂, min, max, ∘, →) is a standard WA_tΠ-algebra.

If ∘ is a [0, e]-continuous, [0, e]-idempotent, residuated wta-uninorm, then ([0, 1], 1, 0, e, ∂, min, max, ∘, →) is a standard WA_tG-algebra.

Proof. Note that any residuated wta-uninorm gives rise to a WA_tMUL-algebra (see [36]). Thus, for (i), it suffices to prove ( ${RDIV}_{t}^{w 𝒜}$ ), i.e., the condition: e ≤ max {e → x, x → y, y → (x ∘ (x → y))}. By residuation, it suffices instead to prove either e ≤ x, x ≤ y, or y ≤ x ∘ (x → y). Here, we consider the case y < x < e. By Lemma 1 (b), we obtain y = x ∘ (x → y); therefore, y ≤ x ∘ (x → y). The other cases are obvious.

It directly follows from the involutiveness of n₁ and the e-involutiveness of n₂ that (ii) holds.

For (iii), it suffices to prove ( ${RCAN}_{t}^{w 𝒜}$ ), i.e., the condition: e ≤ min {-- x, e} → (min {(min {x, e} → (min {x, e} ∘ y)) , e} → y). If x = 0, then -- x = 0 and so the condition holds. We consider the case x ≠ 0. Then, since -- x = 1 and so min {-- x, e} = e, by residuation (twice), it suffices instead to prove min {min {x, e} → (min {x, e} ∘ y) , e} ≤ y. Here, we consider the case x ≤ e. If e < y, then the condition holds because min {x → (x ∘ y) , e} ≤ y. Let y < e. Then, since x ∘ y ≤ min {x, y} ≤ y and ∘ is strictly monotone on [0, e], the condition also holds.

It also follows from the [0, e]-idempotence of ∘ that (iv) holds. □

4 Standard completeness

In this section, we provide standard completeness results for Ls using the Jenei–Montagna–style construction in [8 , 36]. We first show that finite or countable, linearly ordered WA_tBL-algebras are embeddable into a standard algebra (For convenience, we add the ‘less than or equal to’ relation symbol “≤” to such algebras.)

Proposition 3. For every finite or countable, linearly ordered WA_tBL-algebra A = (A, ≤ _A, ⊤ , ⊥ , t, f, ∧ , ∨ , ∗ , →), there is a countable ordered set X, a binary operation ∘ on X, and a map h from A into X such that the following conditions hold:

X is densely ordered and has a maximum Max, a minimum Min, and special elements e and ∂.

(X, ∘ , ⪯ , e) is a linearly ordered, monotonic, commutative weak e-associative groupoid with unit.

∘ is conjunctive and left-continuous w.r.t. the order topology on (X, ⪯), and continuous on {x ∈ X : Min ⪯ x ⪯ e}.

h is an embedding of the structure (A, ≤ _A, ⊤ , ⊥ , t, f, ∧ , ∨ , ∗) into (X, ⪯, Max, Min, e, ∂, min, max, ∘), and for all m, n ∈ A, h (m → n) is the residuum of h (m) and h (n) in (X, ⪯ , Max, Min, e, ∂, max, min, ∘).

Proof. The proof is analogous to that of Proposition 6 in [36]. For convenience, we assume A to be a subset of Q ∩ [0, 1] with finite or countable elements, where 0 and 1 are the least and greatest elements and some e and any ∂ are special elements, each of which corresponds to ⊤, ⊥ , t, and f, respectively. Let X = {(m, x) : m ∈ A \ {0 (= ⊥)} and x ∈ Q ∩ (0, m]} ∪ {(0, 0)}. For (m, x) , (n, y) ∈ X, we define: (m, x) ⪯ (n, y) iff either m < _An, or m = _An and x ≤ y.

Here, we prove that (X, ∘ , ⪯ , e) is a weak e-associative groupoid in (II) and ∘ is continuous on ({x ∈ X : Min ⪯ x ⪯ e} , ⪯) in (III). For convenience, we henceforth drop the index A in ≤_A and = _A, if we need not distinguish them. Context should clarify the intention.

Define, for (m, x) , (n, y) ∈ X:

$(m, x) \circ (n, y) = {\begin{matrix} \max {(m, x), (n, y)} & if m * n = m \lor n, m \neq n, and \\ (m, x) ⪯ e or (n, y) ⪯ e; \\ \min {(m, x), (n, y)} & if m * n = m \land n, and \\ (m, x) ⪯ e or (n, y) ⪯ e; \\ (m * n, m * n) & otherwise. \end{matrix}$

First, we verify weak e-associativity, i.e., we prove that, for all (l, x), (m, y), (n, z) ⪯ (e, e), we have: (WeAS) (l, x) ∘ ((m, y) ∘ (n, z)) = ((l, x) ∘ (m, y)) ∘ (n, z). We distinguish several cases:

Case (i). l ∗ (m ∗ n) = ∧ (l, m, n). Both sides of (WeAS) are equal to min{(l, x), (m, y), (n, z)}.

Case (ii). l ∗ (m ∗ n) ≠ ∧ (l, m, n), and l ∗ (m ∗ n) ∈ {l, m, n}. This is not the case because ∗ ensures l ∗ (m ∗ n) ≤ ∧ (l, m, n).

Case (iii). l ∗ (m ∗ n) ∉ {l, m, n} and l ∗ (m ∗ n) = l ∧ (m ∗ n) = m ∗ n. Since ∗ ensures m ∗ n < m ∧ n, l ≤ e, both sides of (WeAS) are equal to (m ∗ n, m ∗ n).

Case (iv). l ∗ (m ∗ n) ∉ {l, m, n} and l ∗ (m ∗ n) ≠ l ∧ (m ∗ n). We need to consider the case l ∗ (m ∗ n) ≤ e. Then, since l ∗ (m ∗ n) < ∧ (l, m, n), both sides of (WeAS) are equal to (l ∗ (m ∗ n) , l ∗ (m ∗ n)).

This completes the proof of weak e-associativity.

Let X|_e:= {(m, x) ∈ X : (0, 0) ⪯ (m, x) ⪯ (e, e)}. We then prove ∘ is continuous on X|_e. For this, we prove that, if 〈 (m_i, x_i) : i ∈ N〉 and 〈 (m_j, x_j) : j ∈ N〉 is any increasing and decreasing sequences (w.r.t. ⪯) of elements of X such that sup {(m_i, x_i) : i ∈ N} = (m, x) = inf {(m_j, x_j) : j ∈ N}, then, for all (n, y) ∈ X|_e, we have sup {(m_i, x_i) ∘ (n, y) : i ∈ N} = (m, x) ∘ (n, y) = inf {(m_j, x_j) ∘ (n, y) : j ∈ N}. Note that, for almost all i, j, m_i = m = m_j (otherwise, $(m, \frac{x}{2}) ≺ (m, x)$ would be an upper bound of the sequence 〈 (m_i, x_i) : i ∈ N〉, and similarly for the sequence 〈 (m_j, x_j) : j ∈ N〉). By deleting a finite number of elements of the sequences 〈 (m_i, x_i) : i ∈ N〉 and 〈 (m_j, x_j) : j ∈ N〉, we can suppose that, for all i, j, m_i = m = m_j and that sup {x_i : i ∈ N} = x = inf {x_j : j ∈ N}. Then, we consider the following cases:

Case (i). m ∗ n = m ∧ n. We have (m, x) ∘ (n, y) = min {(m, x) , (n, y)}, (m_i, x_i) ∘ (n, y) = min {(m_i, x_i) , (n, y)}, and (m_j, x_j) ∘ (n, y) = min {(m_j, x_j) , (n, y)}. Then, continuity follows from continuity of the min operation.

Case (ii). m ∗ n ≠ m ∧ n. Since (m, x) ∘ (n, y) = (m ∗ n, m ∗ n) and, for all i, j, (m_i, x_i) ∘ (n, y) = (m_i ∗ n, m_i ∗ n) = (m ∗ n, m ∗ n) and (m_j, x_j) ∘ (n, y) = (m_j ∗ n, m_j ∗ n) = (m ∗ n, m ∗ n), we have (m_i, x_i) ∘ (n, y) = (m, x) ∘ (n, y) = (m_j, x_j) ∘ (n, y).

This completes the proof of continuity on X|_e.

For the proof of the other conditions, see the proof of Proposition 2 in [35]. □

Proposition 4. Every countable linearly and densely ordered WA_tBL-algebra can be embedded into a standard algebra.

Proof. The proof is analogous to that of Proposition 3 in [35]. Let X, A, etc. be as in Proposition 3. We prove continuity on [0, e]. Suppose that increasing sequences of reals in [0, e] 〈α_i : i ∈ N〉, 〈β_i : i ∈ N〉 and decreasing sequences of such reals 〈α_j : j ∈ N〉, 〈β_j : j ∈ N〉 are such that sup {α_i : i ∈ N} = α = inf {α_j : j ∈ N} and sup {β_i : i ∈ N} = β = inf {β_j : j ∈ N}. By the monotonicity of $\hat{\circ}$ , $\sup {α_{i} \hat{\circ} β_{i}} = α \hat{\circ} β = \inf {α_{j} \hat{\circ} β_{j}}$ . Since the restriction of $\hat{\circ}$ to Q ∩ [0, e] is continuous, we obtain $α \hat{\circ} β = \sup {q \hat{\circ} r : q, r \in Q \cap [0, e], q \leq α, r \leq β} = \sup {q \hat{\circ} r : q, r \in Q \cap [0, e], q < α, r < β} = \inf {s \hat{\circ} t : s, t \in Q \cap [0, e], α \leq s, β \leq t} = \inf {s \hat{\circ} t : s, t \in Q \cap [0, e], α < s, β < t}$ . For each q < α < s, r < β < t, there is an n such that q < α_n < s and r < β_n < t. Thus, $\sup {α_{n} \hat{\circ} β_{n} : n \in N} \geq \sup {q \hat{\circ} r : q, r \in Q \cap [0, e], q < α, r < β} = α \hat{\circ} β = \inf {s \hat{\circ} t : s, t \in Q \cap [0, e], α < s, β < t} \geq \inf {α_{n} \hat{\circ} β_{n} : n \in N}$ . Hence, $\hat{\circ}$ is continuous on [0, e].

The proof of the remainder of this proposition is almost the same as Proposition 3 in [35]. □

Theorem 3. (Finite strong standard completeness) Let T be a finite theory over WA_tBL. For WA_tBL, the following are equivalent:

For every standard WA_tBL-algebra and evaluation v, if v (ψ) ≥ e for all ψ ∈ T, then v (φ) ≥ e.

Proof. (1) → (2): This follows from definition.

(2) → (1): Let φ be a formula such that $T ⊢_{W A $_{t} $ B L} φ$ , A a linearly ordered WA_tBL-algebra, and v an evaluation in A such that v (ψ) ≥ t for all ψ ∈ T and v (φ) < t. Let h′ be the embedding of A into the standard WA_tBL-algebra as in Proposition 4. Then, h′ ∘ v is an evaluation into the standard WA_tBL-algebra such that h′ ∘ v (ψ) ≥ e and yet h′ ∘ v (φ) < e. □

In an analogy to the proof of standard completeness for WA_tBL, we can prove the following:

Theorem 4. Let T be a theory over WA_tG and a finite theory over WA_tŁ and WA_tΠ. For L $\in {W A_{t} Ł, W A_{t} Π, W A_{t} G}$ , the following are equivalent: for a (finite) theory over L,

T ⊢ _Lφ.

For every standard L-algebra and evaluation v, if v (ψ) ≥ e for all ψ ∈ T, then v (φ) ≥ e.

Proof. First, w.r.t. WA_tΠ, the definition of ∘ is the same as in the proof of Proposition 3. Thus, for WA_tΠ, we simply prove the conditions (GN^𝒜) and ( ${RCAN}_{t}^{w 𝒜}$ ) as examples. Given a set X, we define Gödel negation - as follows: - (m, x) = (0, 0) if (m, x) ≠ (0, 0) and otherwise - (m, x) = (1, 1). For (GN^𝒜), we need to show min {(m, x) , - (m, x)} = (0, 0). We need to consider the case (m, x) ≠ (0, 0). From the definition, it is immediate that - (m, x) = (0, 0). We then prove ( ${RCAN}_{t}^{w 𝒜}$ ), i.e., the condition: (e, e) ⪯ min {-- (m, x) , (e, e)} → (min {(min {(m, x) , (e, e)} → (min {(m, x) , (e, e)} ∘ (n, y))) , (e, e)} → (n, y)). First, note that if (m, x) = (0, 0), we have -- (m, x) = (0, 0), and otherwise -- (m, x) = (1, 1). Then, the proof of the condition is almost the same as ( ${RCAN}_{t}^{w 𝒜}$ ) in Proposition 2 (iii).

We next give the definition of ∘ w.r.t WA_tG and prove (ID_t^𝒜). Define, for (m, x) , (n, y) ∈ X: $(m, x) \circ (n, y) = {\begin{matrix} \max {(m, x), (n, y)} & if m * n = m \lor n, and \\ (m, x) ≻ e or (n, y) ≻ e; \\ \min {(m, x), (n, y)} & if m * n = m \land n, and \\ (m, x) ⪯ e or (n, y) ⪯ e; \\ (m * n, m * n) & otherwise. \end{matrix}$

We need to prove that, for (m, x) ⪯ (e, e), we have (m, x) = (m, x) ∘ (m, x): Since m = m ∗ m, we have (m, x) = (m, x) ∘ (m, x).

For WA_tŁ, first take A as in the proof of Proposition 3. For each m ∈ A, let m⁺ denote the successor of m if it exists, otherwise m⁺ = m. Note first that since the negation in A, defined as ∼m:= m → ∂, is involutive, we have that m = (∼ n) ⁺ iff n = (∼ m) ⁺; moreover, if m < m⁺, then (∼ (m⁺)) ⁺ = ∼m. Here, we take Y below in place of the X defined in the proof of Proposition 3. Let (Y, ⪯) be the linearly ordered set defined by Y = {(m, m) : m ∈ A} ∪ {(m, x) : ∃ m′ ∈ A such that m = _Am^′+ > _Am′ and x ∈ Q ∩ (0, m)}, with ⪯ being the corresponding lexicographic ordering as above. It is clear that (Y, ⪯) is a subset of the ordered set (X, ⪯) defined in Proposition 3 with the same bounds and special elements e and ∂. Note that Y is closed under ∘. First, it is obvious that, as in the proof of Proposition 3, we can prove condition (I) (but with Y in place of X) in Proposition 3.

Now we define a new operation ⊙ on Y, based on ∘, as follows: $(m, x) ⊙ (n, y) = {\begin{matrix} (m + n - e, x + y - e) & if m, n \leq e, \\ \min {\partial, (m, x) \circ (n, y)} & if m > e or n > e, and either \\ m = (\sim n)^{+} and \frac{p}{q} + \frac{p^{'}}{q^{'}} \leq 1, \\ where x = m \frac{p}{q} and y = n \frac{p^{'}}{q^{'}}, or \\ m < (\sim n)^{+}; \\ (m, x) \circ (n, y) & otherwise . \end{matrix}$

We verify that ⊙ satisfies the condition (DNE_t^𝒜), i.e., we prove that for (m, x) ∈ X, min {-- (m, x) , (e, e)} ⪯ (m, x). Let (l, x) → (m, y) be sup {(n, z) ∈ X : (l, x) ∘ (n, z) ≤ (m, y)}. If (e, e) ≺ (m, x), we have (m, x) ⪯ -- (m, x) since (m, x) → (0, 0) = (0, 0) and so (m, x) ⊙ ((m, x) → (0, 0)) = (0, 0). Thus, we have min {-- (m, x) , (e, e)} = (e, e) ≺ (m, x). Otherwise, we can first obtain (m, x) → (n, y) = (e - m + n, e - x + y), for m, n ≤ e. Then, we have min {-- (m, x) , (e, e)} = (m, x) since - (m, x) = (e - m + 0, e - x + 0) = (e - m, e - x) and so -- (m, x) = (e - (e - m) +0, e - (e - x) =0) = (m, x). The proofs of the other conditions are similar to those in Theorem 5 in [35].

The remainder of the proof for standard completeness of L is almost the same as in WA_tBL. □

We show that every WA_tBL-chain contains a BL-chain, and similarly for WA_tŁ-chains and Ł-chains; WA_tΠ-chains and P-chains; and WA_tG-chains and G-chains. Let A be the universe and A_t : = {x : ⊥ ≤ x ≤ t}. Define x → _ty : = (x → y) ∧ t.

Proposition 5.

For every WA_tBL-chain, (A_t, ∧ , ∨ , ∗ , → _t, ⊥ , t) is a BL-chain.

For every WA_tŁ-chain, (A_t, ∧ , ∨ , ∗ , → _t, ⊥ , t) is an Ł-chain.

For every WA_tΠ-chain, (A_t, ∧ , ∨ , ∗ , → _t, ⊥ , t) is a P-chain.

For every WA_tG-chain, (A_t, ∧ , ∨ , ∗ , → _t, ⊥ , t) is a G-chain.

Proof.

If x, y ∈ A_t, then it is clear that x ∧ y, x ∨ y, x ∗ y, x → _ty ∈ A_t. Moreover, (A_t, ∧ , ∨ , ⊥ , t) is a bounded lattice and (A_t, ∗ , t) is a commutative wta-monoid. It is also easy to prove the residuation and prelinearity properties. We check the divisibility property. We need to show that t ≤ (x → _ty) ∨ (y → _t (x ∗ (x → _ty))) for all x, y ∈ A_t. Let x ≤ y. Then since t ≤ x → _ty by residuation, we are done. Let x > y. Then, since y < x ≤ t, we have t ≤ y → (x ∗ (x → y)) by ( ${RDIV}_{t}^{w 𝒜}$ ). Since y < x = x ∗ t, we have x → y = x → _ty; therefore, t ≤ y → _t (x ∗ (x → _ty)).

By (i), it suffices to show that the negation is involutive. Define ¬_tx = x→ _t ⊥. Then, for all x, y ∈ A_t, we can define x ∗ y : = ¬ _t (x → _t ¬ _ty) and x → _ty : = ¬ _t (x ∗ ¬ _ty). Since ¬_t (x → _t ⊥) = x ∗ t = x, we have ¬_t ¬ _tx = x. Note that ¬_t =- since f =⊥.

Gödel negation - is defined as ¬_t satisfying (GN^𝒜). Thus, it suffices to check the cancellation property. We need to show that t ≤ -- x → _t ((x → _t (x ∗ y)) → _ty) for all x, y ∈ A_t. If x =⊥, then -- x =⊥ and so we are done. Let x≠ ⊥. We have -- x = t. We need to show that t ≤ (x → _t (x ∗ y)) → _ty. If x ≤ x ∗ y, then we have t ≤ ((x → (x ∗ y)) ∧ t) → y by ( ${RCAN}_{t}^{w 𝒜}$ ); therefore, t ≤ (x → _t (x ∗ y)) → _ty. If x > x ∗ y, then we have t ≤ (x → (x ∗ y)) → y by ( ${RCAN}_{t}^{w 𝒜}$ ). Since x ∗ y < x = x ∗ t, we also have x → (x ∗ y) = x → _t (x ∗ y); therefore, t ≤ (x → _t (x ∗ y)) → _ty.

By (i), it suffices to show that ∗ is idempotent. Since x ∗ x = (x ∗ x) ∧ t = x ∧ t = x for all x, y ∈ A_t, it is immediate that ∗ is idempotent. □

We finally check that every standard L-algebra is indeed based on a [0, e]-continuous wta-uninorm.

Proposition 6. If $A = ([0, 1], 1, 0, e, \partial, \min, \max, \circ, \to)$ is an L-algebra, then ∘ is [0, e]-continuous.

Proof. Note that a wta-uninorm is residuated iff it is left-continuous and conjunctive (see [36]). Thus, it suffices to show that ∘ is right-continuous on [0, e]. Let x, y ∈ [0, e] and let (x_i) _i≥0 be a non-increasing sequence in [0, e] such that $x = \inf_{i} x_{i}$ . Let $w = \inf_{i} (x_{i} \circ y)$ . Clearly, w ≥ x ∘ y. Thus, we just need to show w ≤ x ∘ y. Since the monoid operation of a standard BL-algebra, (Ł-algebra, P-algebra, and G-algebra, respectively) is a continuous t-norm, ∘ is continuous on [0, e] by Proposition 5. □

Theorem 5.

(Finite strong standard completeness) Let T be a finite theory over L (∈ Ls) and φ a formula. T ⊢ _Lφ iff for every standard [0, e]-continuous L-algebra and evaluation v, if v (ψ) ≥ e for each ψ∈ T, then v (φ) ≥ e.

(Strong standard completeness) Let L be WA_tG, T a theory over L, and φ a formula. T ⊢ _Lφ iff for every standard [0, e]-continuous L-algebra and evaluation v, if v (ψ) ≥ e for each ψ∈ T, then v (φ) ≥ e.

Proof. (i) and (ii) follow from Propositions 3 and 6, and Theorem 4. □

Gabbay and Metcalfe [11] introduced [0, 1)-continuous uninorm-based analogues of the logics of continuous t-norms and their residua. We may also consider a wta generalization of the [0, 1)-continuous uninorm-based logics. To introduce such logics remains an open problem.

Footnotes

Note that the reason why they do not introduce [0, 1]-continuous uninorm-based logics is that the only examples of continuous residuated uninorms are t-norms.

For example, there is no need to distinguish whether x³ should be defined as (xx) x or as x (xx), since these are equal (see e.g. Wikipedia).

Here, the constant f corresponds to the negation of t. Using the constant f and the connective →, we can obtain a new connective negation ¬ (see df1) and define t as f → f.

For γ and Π (bDT^∗), see [4 , ].

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A non-associative generalization of continuous t-norm-based logics

Abstract

Keywords

1 Introduction

2 The logic WA t BL and its axiomatic extensions

3 [0, e]-continuous residuated wta-uninorms

4 Standard completeness

Footnotes

References

2 The logic WA_tBL and its axiomatic extensions