The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK-logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK-logic with negation by a family of connectives implicitly defined by equations and compatible with BCK-congruences. Many of the logics in the current literature are natural expansions of BCK-logic with negation. The validity (...) of the analogous of Glivenko theorem in these logics is equivalent to the validity of a simple one-variable formula in the language of BCK-logic with negation. (shrink)

Along the same line as that in Ono (Ann Pure Appl Logic 161:246–250, 2009), a proof-theoretic approach to Glivenko theorems is developed here for substructural predicate logics relative not only to classical predicate logic but also to arbitrary involutive substructural predicate logics over intuitionistic linear predicate logic without exponentials QFLe. It is shown that there exists the weakest logic over QFLe among substructural predicate logics for which the Glivenko theorem holds. Negative translations of substructural predicate logics are studied (...) by using the same approach. First, a negative translation, called extended Kuroda translation is introduced. Then a translation result of an arbitrary involutive substructural predicate logics over QFLe is shown, and the existence of the weakest logic is proved among such logics for which the extended Kuroda translation works. They are obtained by a slight modification of the proof of the Glivenko theorem. Relations of our extended Kuroda translation with other standard negative translations will be discussed. Lastly, algebraic aspects of these results will be mentioned briefly. In this way, a clear and comprehensive understanding of Glivenko theorems and negative translations will be obtained from a substructural viewpoint. (shrink)

A constructivisation of the cut-elimination proof for sequent calculi for classical, intuitionistic and minimal infinitary logics with geometric rules—given in earlier work by the second author—is presented. This is achieved through a procedure where the non-constructive transfinite induction on the commutative sum of ordinals is replaced by two instances of Brouwer’s Bar Induction. The proof of admissibility of the structural rules is made ordinal-free by introducing a new well-founded relation based on a notion of embeddability of derivations. Additionally, conservativity for (...) classical over intuitionistic/minimal logic for the seven (finitary) Glivenko sequent classes is here shown to hold also for the corresponding infinitary classes. (shrink)

It is well known that classical propositional logic can be interpreted in intuitionistic propositional logic. In particular Glivenko's theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko's theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via a double negation interpretation. Our presentation is algebraic and is formulated in the context of residuated lattices. In the (...) last part of the paper, we also discuss some extended forms of the Kolmogorov translation and we compare it to the Glivenko translation. (shrink)

In this article we deal with Glivenko type theorems for intuitionistic modal logics over Prior's MIPC. We examine the problems which appear in proving Glivenko type theorems when passing from the intuitionistic propositional logic Intto MIPC. As a result we obtain two different versions of Glivenko's theorem for logics over MIPC. Since MIPCcan be thought of as a one-variable fragment of the intuitionistic predicate logic Q-Int, one of the versions of Glivenko's theorem for logics over MIPCis (...) closely related to that for intermediate predicate logics obtained by Umezawa [27] and Gabbay [15]. Another one is rather surprising. (shrink)

Glivenko-type theorems for substructural logics are comprehensively studied in the paper [N. Galatos, H. Ono, Glivenko theorems for substructural logics over FL, Journal of Symbolic Logic 71 1353–1384]. Arguments used there are fully algebraic, and based on the fact that all substructural logics are algebraizable 279–308] and also [N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, in: Studies in Logic and the Foundations of Mathematics, vol. 151, Elsevier, 2007] for the (...) details). As a complementary work to the algebraic approach developed in [N. Galatos, H. Ono, Glivenko theorems for substructural logics over FL, Journal of Symbolic Logic 71 1353–1384], we present here a concise, proof-theoretic approach to Glivenko theorems for substructural logics. This will show different features of these two approaches. (shrink)

We give a simple proof-theoretic argument showing that Glivenko’s theorem for propositional logic and its version for predicate logic follow as an easy consequence of the deduction theorem, which also proves some Glivenko type theorems relating intermediate predicate logics between intuitionistic and classical logic. We consider two schemata, the double negation shift (DNS) and the one consisting of instances of the principle of excluded middle for sentences (REM). We prove that both schemata combined derive classical logic, while each (...) one of them provides a strictly weaker intermediate logic, and neither of them is derivable from the other. We show that over every intermediate logic there exists a maximal intermediate logic for which Glivenko’s theorem holds. We deduce as well a characterization of DNS, as the weakest (with respect to derivability) scheme that added to REM derives classical logic. (shrink)

In 1968, Orevkov presented proofs of conservativity of classical over intuitionistic and minimal predicate logic with equality for seven classes of sequents, what are known as Glivenko classes. The proofs of these results, important in the literature on the constructive content of classical theories, have remained somehow cryptic. In this paper, direct proofs for more general extensions are given for each class by exploiting the structural properties of G3 sequent calculi; for five of the seven classes the results are (...) strengthened to height-preserving statements, and it is further shown that the constructive and minimal proofs are identical in structure to the classical proof from which they are obtained. (shrink)

We give several new equivalences of NIP for formulas and new proofs of known results using Talagrand (Ann Probab 15:837–870, 1987) and Haydon et al. (in: Functional Analysis Proceedings, The University of Texas at Austin 1987–1989, Lecture Notes in Mathematics, Springer, New York, 1991). We emphasize that Keisler measures are more complicated than types (even in the NIP context), in an analytic sense. Among other things, we show that for a first order theory T and a formula $$\phi (x,y)$$, the (...) following are equivalent. (shrink)

The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given (generalizing the results of [9]) in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the title. We (...) include examples to show how we can apply the results to describe free algebras in some well known varieties of involutive MTL-algebras and of pseudocomplemented MTL-algebras. (shrink)

We study how to postpone the application of the reductio ad absurdum rule ) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of \, which induces a negative translation from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

This paper is concerned with the logical structure of intuitionistic equality theories. We prove that De Jongh theorem holds for the theory of decidable equality, but uniform De Jongh theorem fails even for the theory of weakly decidable equality. We also show that the theory of weakly decidable equality is the weakest equality theory which enjoys Glivenko theorem.

In a classical paper [15] V. Glivenko showed that a proposition is classically demonstrable if and only if its double negation is intuitionistically demonstrable. This result has an algebraic formulation: the double negation is a homomorphism from each Heyting algebra onto the Boolean algebra of its regular elements. Versions of both the logical and algebraic formulations of Glivenko’s theorem, adapted to other systems of logics and to algebras not necessarily related to logic can be found in the literature (...) (see [2, 9, 8, 14] and [13, 7, 14]). The aim of this paper is to offer a general frame for studying both logical and algebraic generalizations of Glivenko’s theorem. We give abstract formulations for quasivarieties of algebras and for equivalential and algebraizable deductive systems and both formulations are compared when the quasivariety and the deductive system are related. We also analyse Glivenko’s theorem for compatible expansions of both cases. (shrink)

Bosbach and Riečan states on residuated lattices both are generalizations of probability measures on Boolean algebras. Just from the observation that both of them can be defined by using the canonical structure of the standard MV-algebra on the unit interval [0, 1], generalized Riečan states and two types of generalized Bosbach states on residuated lattices were recently introduced by Georgescu and Mureşan through replacing the standard MV-algebra with arbitrary residuated lattices as codomains. In the present paper, the Glivenko theorem (...) is first extended to residuated lattices with a nucleus, which gives several necessary and sufficient conditions for the underlying nucleus to be a residuated lattice homomorphism. Then it is proved that every generalized Bosbach state (of type I, or of type II) compatible with the nucleus on a nucleus-based-Glivenko residuated lattice is uniquely determined by its restriction on the nucleus image of the underlying residuated lattice, and every relatively generalized Riečan state compatible with the double relative negation on an arbitrary residuated lattice is uniquely determined by its restriction on the double relative negation image of the residuated lattice. Our results indicate that many-valued probability theory compatible with nuclei on residuated lattices reduces in essence to probability theory on algebras of fixpoints of the underlying nuclei. (shrink)

In this paper, we mainly investigate the existence of states based on the Glivenko theorem in bounded semihoops, which are building blocks for the algebraic semantics for relevant fuzzy logics. First, we extend algebraic formulations of the Glivenko theorem to bounded semihoops and give some characterizations of Glivenko semihoops and regular semihoops. The category of regular semihoops is a reflective subcategory of the category of Glivenko semihoops. Moreover, by means of the negative translation term, we characterize (...) the Glivenko variety. Then we show that the regular semihoop of regular elements of a free algebra in the variety of Glivenko semihoops is free in the corresponding variety of regular semihoops. Similar results are derived for the semihoop of dense elements of free Glivenko semihoops. Finally, we give a purely algebraic method to check the existence of states on Glivenko semihoops. In particular, we prove that a bounded semihoop has Bosbach states if and only if it has a divisible filter, and a bounded semihoop has Riečan states if and only if it has a semi-divisible filter. (shrink)

Glivenko’s theorem says that classical provability of a propositional formula entails intuitionistic provability of the double negation of that formula. This stood right at the beginning of the success story of negative translations, indeed mainly designed for converting classically derivable formulae into intuitionistically derivable ones. We now generalise this approach: simultaneously from double negation to an arbitrary nucleus; from provability in a calculus to an inductively generated abstract consequence relation; and from propositional logic to any set of objects whatsoever. (...) In particular, we give sharp criteria for the generalisation of classical logic to be a conservative extension of the one of intuitionistic logic with double negation. (shrink)

Alongside the analogy between maximal ideals and complete theories, the Jacobson radical carries over from ideals of commutative rings to theories of propositional calculi. This prompts a variant of Lindenbaum’s Lemma that relates classical validity and intuitionistic provability, and the syntactical counterpart of which is Glivenko’s Theorem. The Jacobson radical in fact turns out to coincide with the classical deductive closure. As a by-product we obtain a possible interpretation in logic of the axioms-as-rules conservation criterion for a multi-conclusion Scott-style (...) entailment relation over a single-conclusion one. (shrink)

We generalize the double negation construction of Boolean algebras in Heyting algebras to a double negation construction of the same in Visser algebras. This result allows us to generalize Glivenko’s theorem from intuitionistic propositional logic and Heyting algebras to Visser’s basic propositional logic and Visser algebras.

We examine second order intuitionistic propositional logic, IPC². Let $F_\exists $ be the set of formulas with no universal quantification. We prove Glivenko's theorem for formulas in $F_\exists $ that is, for φ € $F_\exists $ φ is a classical tautology if and only if ¬¬φ is a tautology of IPC². We show that for each sentence φ € $F_\exists $ (without free variables), φ is a classical tautology if and only if φ is an intuitionistic tautology. As a (...) corollary we obtain a semantic argument that the quantifier V is not definable in IPC² from ⊥, V, ^, →. (shrink)

A natural deduction formulation is given for the intermediate logic called MH by Gabbay in [4]. Proof-theoretic methods are used to show that every deduction can be normalized, that MH is the weakest intermediate logic for which the Glivenko theorem holds, and that the Craig-Lyndon interpolation theorem holds for it.

This paper is devoted to the investigation of term-definable connexive implications in substructural logics with exchange and, on the semantical perspective, in sub-varieties of commutative residuated lattices (FL ${}_{\scriptsize\mbox{e}}$ -algebras). In particular, we inquire into sufficient and necessary conditions under which generalizations of the connexive implication-like operation defined in [6] for Heyting algebras still satisfy connexive theses. It will turn out that, in most cases, connexive principles are equivalent to the equational Glivenko property with respect to Boolean algebras. Furthermore, (...) we provide some philosophical upshots like, e.g., a discussion on the relevance of the above operation in relationship with G. Polya’s logic of plausible inference, and some characterization results on weak and strong connexivity. (shrink)

A sequent root-first proof-search procedure for intuitionistic propositional logic is presented. The procedure is obtained from modified intuitionistic multi-succedent and classical sequent calculi, making use of Glivenko’s Theorem. We prove that a sequent is derivable in a standard intuitionistic multi-succedent calculus if and only if the corresponding prefixed-sequent is derivable in the procedure.

Conditional logics have traditionally been intended to formalize various intuitively correct modes of reasoning involving conditional expressions in natural language. Although conditional logics have by now been thoroughly studied in a classical context, they have yet to be systematically examined in an intuitionistic context, despite compelling philosophical and technical reasons to do so. This paper addresses this gap by thoroughly examining the basic intuitionistic conditional logic ICK, the intuitionistic counterpart of Chellas’ important classical system CK. I give ICK both worlds (...) semantics and algebraic semantics, and prove that these are equivalent. I give a Gödel-type embedding of ICK into CK and a Glivenko-type embedding of CK into ICK. I axiomatize ICK and prove soundness, completeness, and decidability results. Finally, I discuss extending ICK. (shrink)

A deterministic weakening \ of the Belnap–Dunn four-valued logic \ is introduced to formalize the acceptance and rejection of a proposition at a state in a linearly ordered informational frame with persistent valuations. The logic \ is formalized as a sequent calculus. The completeness and decidability of \ with respect to relational semantics are shown in terms of normal forms. From an algebraic perspective, the class of all algebras for \ is described, and found to be a subvariety of Berman’s (...) variety \. Every linearly ordered frame is logically equivalent to its dual algebra. It is proved that \ is the logic of a nine-element distributive lattice with a negation. Moreover, \ is embedded into \ by Glivenko’s double-negation translation. (shrink)

A classical paraconsistent logic, which is regarded as a modified extension of first-degree entailment logic, is introduced as a Gentzen-type sequent calculus. This logic can simulate the classical negation in classical logic by paraconsistent double negation in CP. Theorems for syntactically and semantically embedding CP into a Gentzen-type sequent calculus LK for classical logic and vice versa are proved. The cut-elimination and completeness theorems for CP are also shown using these embedding theorems. Similar results are also obtained for an intuitionistic (...) paraconsistent logic, and several versions of Glivenko and Gödel-Gentzen translation theorems are proved for CP and IP. (shrink)

In this paper we prove that the free algebras in a subvariety equation image of the variety equation image of semi-Heyting algebras are directly decomposable if and only if equation image satisfies the Stone identity.

In this study, we introduce Gentzen-type sequent calculi BDm and BDi for a modal extension and an intuitionistic modification, respectively, of De and Omori’s extended Belnap–Dunn logic BD+ with classical negation. We prove theorems for syntactically and semantically embedding BDm and BDi into Gentzen-type sequent calculi S4 and LJ for normal modal logic and intuitionistic logic, respectively. The cut-elimination, decidability, and completeness theorems for BDm and BDi are obtained using these embedding theorems. Moreover, we prove the Glivenko theorem for (...) embedding BD+ into BDi and the McKinsey–Tarski theorem for embedding BDi into BDm. (shrink)

Consider the problem which set V of propositional variables suffices for whenever, where, and ⊢c and ⊢i denote derivability in classical and intuitionistic implicational logic, respectively. We give a direct proof that stability for the final propositional variable of the (implicational) formula A is sufficient; as a corollary one obtains Glivenko's theorem. Conversely, using Glivenko's theorem one can give an alternative proof of our result. As an alternative to stability we then consider the Peirce formula. It is an (...) easy consequence of the result above that adding a single instance of the Peirce formula suffices to move from classical to intuitionistic derivability. Finally we consider the question whether one could do the same for minimal logic. Given a classical derivation of a propositional formula not involving ⊥, which instances of the Peirce formula suffice as additional premises to ensure derivability in minimal logic? We define a set of such Peirce formulas, and show that in general an unbounded number of them is necessary. (shrink)

A cut-free Gentzen sequent calculus for Ewald’s intuitionistic tense logic \ is established. By the proof-theoretic method, we prove that, for every set of strictly positive implications S, the classical tense logic \ is embedded into its intuitionistic analogue \ via Kolmogorov, Gödel–Genzten and Kuroda translations respectively. A sufficient and necessary condition for Glivenko type theorem in tense logics is established.

G3-style sequent calculi are introduced for a family of logics with strong negation: Gurevich logic, Nelson logic, intuitionistic propositional logic, Avron logic, De-Omori logic, and classical propositional logic. Structural properties including cut elimination are established for these calculi. In addition, a Glivenko theorem for embedding classical propositional logic into Gurevich logic is shown.

We identify the \-ideals of a distributive demi-pseudocomplemented algebra L as the kernels of the boolean congruences on L, and show that they form a complete Heyting algebra which is isomorphic to the interval \ of the congruence lattice of L where G is the Glivenko congruence. We also show that the notions of maximal \-ideal, prime \-ideal, and falsity ideal coincide.

Intuitionistic logic formalises the foundational ideas of L.E.J. Brouwer’s mathematical programme of intuitionism. It is one of the earliest non-classical logics, and the difference between classical and intuitionistic logic may be interpreted to lie in the law of the excluded middle, which asserts that either a proposition is true or its negation is true. This principle is deemed unacceptable from the constructive point of view, in whose understanding the law means that there is an effective procedure to determine the truth (...) of all propositions. This understanding of the distinction between the two logics supports the view that negation plays a vital role in the formulation of intuitionistic logic.Nonetheless, the formalisation of negation in intuitionistic logic has not been universally accepted, and many alternative accounts of negation have been proposed. Some seek to weaken or strengthen the negation, and others actively supporting negative inferences that are impossible with it.This thesis follows this tradition and investigates various aspects of negation in intuitionistic logic. Firstly, we look at a problem proposed by H. Ishihara, which asks how effectively one can conserve the deducibility of classical theorems into intuitionistic logic, by assuming atomic classes of non-constructive principles. The classes given in this section improve a previous class given by K. Ishii in two respects: (a) instead of a single class for the law of the excluded middle, two classes are given in terms of weaker principles, allowing a finer analysis and (b) the conservation now extends to a subsystem of intuitionistic logic called Glivenko’s logic. This section also discusses the extension of Ishihara’s problem to minimal logic.Secondly, we study the relationship between two frameworks for weak constructive negation, the approach of D. Vakarelov on one hand and the framework of subminimal negation by A. Colacito, D. de Jongh, and A. L. Vargas on the other hand. We capture a version of Vakarelov’s logic with the semantics of the latter framework, and clarify the relationship between the two semantics. This also provides proof-theoretic insights, which results in the formulation of a cut-free sequent calculus for the aforementioned system.Thirdly, we investigate the ways to unify the formalisations of some logics with contra-intuitionistic inferences. The enquiry concerns paraconsistent logics by R. Sylvan and A. B. Gordienko, as well as the logic of co-negation by G. Priest and of empirical negation by M. De and H. Omori. We take Sylvan’s system as basic, and formulate the frame conditions of the defining axioms of the other systems. The conditions are then used to obtain cut-free labelled sequent calculi for the systems.Finally, we consider L. Humberstone’s actuality operator for intuitionistic logic, which can be seen as the dualisation of a contra-intuitionistic negation. A compete axiomatisation of intuitionistic logic with actuality operator is given, and comparisons are made for some related operators.Abstract prepared by Satoru Niki.E-mail:[email protected]. (shrink)

A residuated lattice is said to be integrally closed if it satisfies the quasiequations \ and \, or equivalently, the equations \ and \. Every integral, cancellative, or divisible residuated lattice is integrally closed, and, conversely, every bounded integrally closed residuated lattice is integral. It is proved that the mapping \\backslash {\mathrm {e}}\) on any integrally closed residuated lattice is a homomorphism onto a lattice-ordered group. A Glivenko-style property is then established for varieties of integrally closed residuated lattices with (...) respect to varieties of lattice-ordered groups, showing in particular that integrally closed residuated lattices form the largest variety of residuated lattices admitting this property with respect to lattice-ordered groups. The Glivenko property is used to obtain a sequent calculus admitting cut-elimination for the variety of integrally closed residuated lattices and to establish the decidability, indeed PSPACE-completenes, of its equational theory. Finally, these results are related to previous work on BCI-algebras, semi-integral residuated pomonoids, and Casari’s comparative logic. (shrink)

A unified and modular falsification-aware single-succedent Gentzen-style framework is introduced for classical, paradefinite, paraconsistent, and paracomplete logics. This framework is composed of two special inference rules, referred to as the rules of explosion and excluded middle, which correspond to the principle of explosion and the law of excluded middle, respectively. Similar to the cut rule in Gentzen’s LK for classical logic, these rules are admissible in cut-free LK. A falsification-aware single-succedent Gentzen-style sequent calculus fsCL for classical logic is formalized based (...) on the proposed framework. The calculus fsCL is obtained from the existing falsification-aware single-succedent Gentzen-style sequent calculus GN4 for Nelson’s paradefinite (or paraconsistent) four-valued logic N4 by adding the rules of explosion and excluded middle. A falsification-aware single-succedent Gentzen-style sequent calculus GN3 for Nelson’s paracomplete three-valued logic N3 is also obtained from GN4 by adding the rule of explosion. The cut-elimination theorems for fsCL, GN3, and some of their neighbors as well as the Glivenko theorem for fsCL are proved. (shrink)

This paper shows that the inhabitation problem in the lambda calculus with negation, product, polymorphic, and existential types is decidable, where the inhabitation problem asks whether there exists some term that belongs to a given type. In order to do that, this paper proves the decidability of the provability in the logical system defined from the second-order natural deduction by removing implication and disjunction. This is proved by showing the quantifier elimination theorem and reducing the problem to the provability in (...) propositional logic. The magic formulas are used for quantifier elimination such that they replace quantifiers. As a byproduct, this paper also shows the second-order witness theorem which states that a quantifier followed by negation can be replaced by a witness obtained only from the formula. As a corollary of the main results, this paper also shows Glivenko’s theorem, Double Negation Shift, and conservativity for antecedent-empty sequents between the logical system and its classical version. (shrink)

In this work, an intuitionistic propositional logic with a Galois connection is introduced. In addition to the intuitionistic logic axioms and inference rule of modus ponens, the logic contains only two rules of inference mimicking the performance of Galois connections. Both Kripke-style and algebraic semantics are presented for IntGC, and IntGC is proved to be complete with respect to both of these semantics. We show that IntGC has the finite model property and is decidable, but Glivenko's Theorem does not (...) hold. Duality between algebraic and Kripke semantics is presented, and a representation theorem for Heyting algebras with Galois connections is proved. In addition, an application to rough L-valued sets is presented. (shrink)

This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these asvariant semanticsand present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we demonstrate (...) that failure via specific finite examples. Using these, we prove that the syntactic version of these translations will fail to satisfy Troelstra’s criteria in the corresponding substructural logical setting. (shrink)

The introduction and elimination rules for material implication in natural deduction are not complete with respect to the implicational fragment of classical logic. A natural way to complete the system is through the addition of a new natural deduction rule corresponding to Peirce's formula → A) → A). E. Zimmermann [6] has shown how to extend Prawitz' normalization strategy to Peirce's rule: applications of Peirce's rule can be restricted to atomic conclusions. The aim of the present paper is to extend (...) Seldin's normalization strategy to Peirce's rule by showing that every derivation Π in the implicational fragment can be transformed into a derivation Π' such that no application of Peirce's rule in Π' occurs above applications of →-introduction and →-elimination. As a corollary of Seldin's normalization strategy we obtain a form of Glivenko's theorem for the classical {→}-fragment. (shrink)

In this paper we continue the study of the variety \ of monadic Gödel algebras. These algebras are the equivalent algebraic semantics of the S5-modal expansion of Gödel logic, which is equivalent to the one-variable monadic fragment of first-order Gödel logic. We show three families of locally finite subvarieties of \ and give their equational bases. We also introduce a topological duality for monadic Gödel algebras and, as an application of this representation theorem, we characterize congruences and give characterizations of (...) the locally finite subvarieties mentioned above by means of their dual spaces. Finally, we study some further properties of the subvariety generated by monadic Gödel chains: we present a characteristic chain for this variety, we prove that a Glivenko-type theorem holds for these algebras and we characterize free algebras over n generators. (shrink)

Ishihara's problem of decidable variables asks which class of decidable propositional variables is sufficient to warrant classical theorems in intuitionistic logic. We present several refinements to the class proposed by Ishii for this problem, which also allows the class to cover Glivenko's logic. We also treat the extension of the problem to minimal logic, suggesting a couple of new classes.

In this paper, we give an algebraic completeness theorem for constructive logic with strong negation in terms of finite rough set-based Nelson algebras determined by quasiorders. We show how for a quasiorder R, its rough set-based Nelson algebra can be obtained by applying Sendlewski’s well-known construction. We prove that if the set of all R-closed elements, which may be viewed as the set of completely defined objects, is cofinal, then the rough set-based Nelson algebra determined by the quasiorder R forms (...) an effective lattice, that is, an algebraic model of the logic E 0, which is characterised by a modal operator grasping the notion of “to be classically valid”. We present a necessary and sufficient condition under which a Nelson algebra is isomorphic to a rough set-based effective lattice determined by a quasiorder. (shrink)

States have been introduced on commutative and non-commutative algebras of fuzzy logics as functions defined on these algebras with values in [0,1]. Starting from the observation that in the definition of Bosbach states there intervenes the standard MV-algebra structure of [0,1], in this paper we introduce Bosbach states defined on residuated lattices with values in residuated lattices. We are led to two types of generalized Bosbach states, with distinct behaviours. Properties of generalized states are useful for the development of an (...) algebraic theory of probabilistic models for non-commutative fuzzy logics. (shrink)