In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godo''s logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in.

The category \(\mathbb {DRDL}{'}\), whose objects are c-differential residuated distributive lattices satisfying the condition \(\textbf{CK}\), is the image of the category \(\mathbb {RDL}\), whose objects are residuated distributive lattices, under the categorical equivalence \(\textbf{K}\) that is constructed in Castiglioni et al. (Stud Log 90:93–124, 2008). In this paper, we introduce weak monadic residuated lattices and study some of their subvarieties. In particular, we use the functor \(\textbf{K}\) to relate the category \(\mathbb {WMRDL}\), whose objects are weak monadic residuated distributive lattices, (...) and the category \(\mathbb {WMDRDL}{'}\), whose objects are pairs formed by an object of \(\mathbb {DRDL}{'}\) and a center weak universal quantifier. (shrink)

In this paper, we investigate universal and existential quantifiers on NM-algebras. The resulting class of algebras will be called monadic NM-algebras. First, we show that the variety of monadic NM-algebras is algebraic semantics of the monadic NM-predicate logic. Moreover, we discuss the relationship among monadic NM-algebras, modal NM-algebras and rough approximation spaces. Second, we introduce and investigate monadic filters in monadic NM-algebras. Using them, we prove the subdirect representation theorem of monadic NM-algebras, and characterize simple and subdirectly irreducible monadic NM-algebras. (...) Finally, we present the monadic NM-logic and prove its completeness with respect to monadic NM-algebras. These results constitute a crucial first step for providing an algebraic foundation for the monadic NM-predicate logic. (shrink)

Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP -based. This presentation is then used to obtain, in a uniform way applicable to most substructural logics, a form (...) of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics. Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved. (shrink)

Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0.1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) ∨ ((B → A) ∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MTL and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are (...) provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1]. (shrink)

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...) (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

A symmetric residuated lattice is an algebra such that is a commutative integral bounded residuated lattice and the equations x=x and =xy are satisfied. The aim of the paper is to investigate the properties of the unary operation ε defined by the prescription εx=x→0. We give necessary and sufficient conditions for ε being an interior operator. Since these conditions are rather restrictive →0)=1 is satisfied) we consider when an iteration of ε is an interior operator. In particular we consider the (...) chain of varieties of symmetric residuated lattices such that the n iteration of ε is a boolean interior operator. For instance, we show that these varieties are semisimple. When n=1, we obtain the variety of symmetric stonean residuated lattices. We also characterize the subvarieties admitting representations as subdirect products of chains. These results generalize and in many cases also simplify, results existing in the literature. (shrink)

Results on arithmetical complexity of important sets of formulas of several fuzzy predicate logics are surveyed and some new results are proven.

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms . We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and (...) rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind , where φ has no additional truth-constants. (shrink)

Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. GMV-algebras are a non-commutative generalization of MV-algebras and are an algebraic counterpart of the non-commutative Łukasiewicz infinite valued logic. We introduce monadic GMV-algebras and describe their connections to certain couples of GMV-algebras and to left adjoint mappings of canonical embeddings of GMV-algebras. Furthermore, functional MGMV-algebras are studied and polyadic GMV-algebras are introduced and discussed.

The first section of this paper is concerned with the intrinsic properties of elementary monadic logic (EM), and characterizations in the spirit of Lindström [2] are given. His proofs do not apply to monadic logic since relations are used, and intrinsic properties of EM turn out to differ in certain ways from those of the elementary logic of relations (i.e., the predicate calculus), which we shall call EL. In the second section we investigate connections between higher-order monadic and polyadic logics.EM (...) is the subsystem of EL which results by the restriction to one-place predicate letters. We omit constants (for simplicity) but take EM to contain identity. Let atypebe any finite sequence (possibly empty) of one-place predicate letters. A modelMof typehas a nonempty universe ∣M∣ and assigns to each predicate letterPofa subsetPMof ∣M∣.Let us take a monadic logicLto be any collection of classes of models, calledL-classes, satisfying the following:1. All models in a givenL-class are of the same type (called the type of the class).2. Isomorphic models lie in the sameL-classes.3. IfandareL-classes of the same type, thenandareL-classes. (shrink)

The present paper deals with the predicate version MTL of the logic MTL by Esteva and Godo. We introduce a Kripke semantics for it, along the lines of Ono''s Kripke semantics for the predicate version of FLew (cf. [O85]), and we prove a completeness theorem. Then we prove that every predicate logic between MTL and classical predicate logic is undecidable. Finally, we prove that MTL is complete with respect to the standard semantics, i.e., with respect to Kripke frames on the (...) real interval [0,1], or equivalently, with respect to MTL-algebras whose lattice reduct is [0,1] with the usual order. (shrink)

Two variants of monadic fuzzy predicate logic are analyzed and compared with the full fuzzy predicate logic with respect to finite model property (properties) and arithmetical complexity of sets of tautologies, satisfiable formulas and of analogous notion restricted to finite models.

The monoidal t-norm based logic MTL is obtained from Hájek''s Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...) (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication $\phi \rightarrow \psi $ with no interpolant. Importantly, this result does not contradict the unfortunately named ‘Craig interpolation’ theorem established by Rauszer in [24] since that article is about the property more correctly named ‘deductive interpolation’ (see (...) Galatos, Jipsen, Kowalski and Ono’s use of this term in [5]) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent. (shrink)

Solovay proved the arithmetical completeness theorem for the system GL of propositional modal logic of provability. Montagna proved that this completeness does not hold for a natural extension QGL of GL to the predicate modal logic. Let Th(QGL) be the set of all theorems of QGL, Fr(QGL) be the set of all formulas valid in all transitive and conversely well-founded Kripke frames, and let PL(T) be the set of all predicate modal formulas provable in Tfor any arithmetical interpretation. Montagna’s results (...) are described as Th(QGL) ⊊ (Fr(QGL), PL(PA) ⊈ Fr(QGL), and Th(QGL) ⊊ PL(PA). In this paper, we prove the following three theorems: (1) Fr(QGL) ⊈ PL(T) for any Σ1-sound recursively enumerable extension T of I Σ1, (2) PL(T) ⊈ Fr(QGL) for any recursively enumerable S1755020312000275_inline1 A -theory T extending I Σ1, and (3) Th(QGL) ⊊ Fr(QGL) ∩ PL(T) for any recursively enumerable S1755020312000275_inline2 A -theory T extending I Σ2. To prove these theorems, we use iterated consistency assertions and nonstandard models of arithmetic, and we improve Artemov’s lemma which is used to prove Vardanyan’s theorem on the Π0 2-completeness of PL(T). (shrink)

We define and study monadic MV-algebras as pairs of MV-algebras one of which is a special case of relatively complete subalgebra named m-relatively complete. An m-relatively complete subalgebra determines a unique monadic operator. A necessary and sufficient condition is given for a subalgebra to be m-relatively complete. A description of the free cyclic monadic MV-algebra is also given.

Rational Pavelka logic extends Lukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0, 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.