], we constructed three semilinear substructural logics
by adding one simple axiom:
to Metcalfe and Montagna’s uninorm logic
, involutive uninorm logic
], and a suitable extension
] of Metcalfe, Olivetti, and Gabbay’s pseudo-uninorm logic
], respectively. Especially, we show that
are complete with respect to finite
-algebras, respectively. That is, they are logics for finite UL
In this paper, we prove that
are standard complete by Wang’s constructions in [5
] and [6
], which are some generalizations of the Jenei and Montagna-style approach for proving standard completeness for monoidal t
] and the proof of the standard completeness for
given by Esteva, Gispert, Godo, and Montagna in [8
]. These constructions have been extended by Yang in [9
Substructural logics are logics that lack some of the three basic structural rules of contraction, weakening, and exchange. For a survey, see [13
]. Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval
, i.e., logics that are standard complete [2
]. Our result in this paper thus shows that
are substructural fuzzy logics.
As pointed out in [6
], our construction in Lemma 5 also presents a method to construct uninorms and involutive uninorms. Then, the standard completeness for
gives a characterization of uninorms and involutive uninorms and their residua constructed by finite
-algebras from our constructions. That is, the identity
holds in all these standard
-algebras; see Definition 5. These new classes of uninorms and involutive uninorms may be used in the theory of evaluation as the aggregation operators or combining functions [14
We have proven that
is standard complete in [16
]. However, we are unable to prove whether
is standard complete or complete with respect to finite
-algebras and left them as open problems. In addition, we have proven that
is standard complete in [17
], which is a longstanding open problem in the circle of fuzzy logic. Unfortunately, such a great work has not been accepted by our community since 2015, although one referee thought that the central ideas in our proof are reasonable and could find no significant flaws in the reasoning. The referee also said that he would also not be confident that the proof is correct if the proof were his own and he had spent many months laboring over it.
2. and Algebras Involved
The Hilbert system
is the logic of bounded representable residuated lattices, which is based on a countable propositional language with formulas built inductively as usual from a set of propositional variables, binary connectives
, and constants
, with definable connectives:
Definition 1. consists of the following axioms and rules : Definition 2
]). A logic is a schematic extension (extension for short) of if it results from by adding axioms in the same language. In particular,
is plus (WCM) ;
is plus ;
is plus .
New extensions of are defined as follows.
and are and plus , respectively.
Let in the remainder of this section. A proof in of a formula from a set of formulas is defined as usual. We write if such a proof exists.
]). An -algebra is a bounded residuated lattice with universe A, binary operations , and constants such that:
is a bounded lattice with top element ⊤ and bottom element ⊥;
is a monoid;
iff iff ;
, where, for any
We use the convention that · binds stronger than other binary operations, and we shall often omit·; we will thus write instead of , for example. Suitable classes of algebras of extensions of are defined as follows.
]). Let be an -algebra. For , an extension of , is an -algebra if all axioms of are valid in . An -chain is an -algebra that is linearly ordered. In particular:
is an -algebra if the weak commutativity (Wcm) holds: iff for all ;
is a -algebra if for all ;
is an -algebra if it is a -algebra such that for all ;
is an -algebra ( or -algebra) if it is an -algebra ( or -algebra) such that the following identity holds: .
]). Let be an -algebra. (i) An -valuation v is a homomorphism from the term algebra determined by formulas in to ; (ii) A formula φ is valid in if holds for any -valuation v; (iii) The relation of semantic consequence holds if each -evaluation that validates all formulae in a theory Γ validates φ as well.
]). iff for every -chain , i.e., is a semilinear substructural logic.
Let be an -algebra. Then, is an -algebra if and only if iff for all .
For the proof of the necessity part, see Lemma 2.4(i) of [1
]. For the sufficiency part, assume that
. Suppose that
by Definition 4 (iii), and hence,
by (Wcm). Therefore,
by the assumption. Then,
, a contradiction, and thus,
is proven by a similar way. Hence,
Let be an -chain and . Then:
implies and ;
implies and ;
Only (1) is proven as follows; for the others, see [1
by Proposition 1 and (Wcm). Thus,
. The case of
is proven in the same way. □
Clearly, Lemma 1 holds for all and -chains.
3. Wang’s Construction and Standard Completeness
In this section, let , be a finite or countable -chain and be arbitrary elements of A.
]). Let be an -chain. For each , t is the immediate predecessor of s in A if: (i) , ; (ii) implies . For each , let denote the immediate predecessor of s in A if it exists, otherwise take .
Let we define:
iff either , or and and,
Now define, for :
where by and are meant and with respect to , respectively. We will omit the index if it does not cause confusion.
Let be an -chain. Then, iff for all in X.
Let . Since for some by Definition 7, then . Thus, by (Fin). Hence, . The sufficiency part of the lemma is proven in the same way. □
]). Let be an -algebra. Let:
Let be an -chain and . (i) If , , , then ; (ii) if and , then ; (iii) .
(i) If , then by Proposition 1, and thus, . If , then by . Thus, let and in the following.
by . by . Then, . Thus, . Hence, by . by Lemma 1(3) and . Therefore, by . Then, . Thus, . Hence, .
Suppose that . Then, . Thus, by Proposition 1, a contradiction, and, hence . Therefore, . by . Then, .
Suppose that , then . Thus, by Proposition 1, a contradiction, and hence, .
Therefore, . by . Then, . Then, by . Thus, by Lemma 1(1).
Suppose that , then , a contradiction with , and hence, by . Then, .
Thus, . Hence, . Then, and . Thus, by , a contradiction with . Thus, the case of and does not exist. This completes the proof of (i).
(ii) It follows from that by Proposition 1. Then, by , and thus, .
(iii) See Proposition 3.7 (2) of [6
Let be a finite -chain. Then, if and only if for all in X.
Let . There are three cases to be considered.
Case 1. and . Then, . Thus, , . Then, by Proposition 1. If , then by Lemma 3(iii). Let in the following. If , then . Otherwise, or . Then, by Lemmas 3(i) and 3(ii). Thus, .
Case 2. , then and . Thus, . Hence, by Proposition 1 and (Wcm). Therefore,
Case 3. , then . Thus, by Proposition 1. Hence, by Lemma 3(iii), .
By a similar procedure, we prove that if . □
Let be an -chain, X, and the binary operation ∘ on X be as in Definition 7. The following conditions hold:
X is densely ordered and has a maximum and a minimum .
is a linearly-ordered monoid, where .
∘ is left-continuous with respect to the order topology on .
There is a map Φ from A into X such that Φ is an embedding of the structure into , and for all is the residuum of and in , respectively.
Claim (e) has been proven by Lemma 2. As pointed out in [3
], the associativity of ∘ is mainly dependent on Lemma 1(1)∼(2). Other claims are proven in the same way as that of Theorem 4.5 in [3
Every countable -chain can be embedded into a standard -algebra.
, etc., be as in Definition 7. We can assume, without loss of generality, that
. Now, define for
. The proof of the weak commutativity, the monotonicity, associativity, left-continuity, etc., of * is the same as that of Theorem 4.6 in [3
]. The neutral element of * is
. By the left-continuity of *, the following property holds.
(P) , .
We prove that iff for any in . Given , then for all . Let . Then, . Thus, by Lemma 5(e). Hence, . Therefore, by (P). The sufficient part of the claim is proven in a similar way. □
By Lemma 1, Definition 8, Lemma 4, we can prove the claims similar to Lemma 5 and 6 for
-algebras. As a consequence of these lemmas, and extending Theorem 3.3 of [7
] in the obvious way, we obtain the following standard completeness.
and are complete with respect to the class of standard algebras involved.