In the paper we consider complexity of intuitionistic propositional logic and its natural fragments such as implicative fragment, finite-variable fragments, and some others. Most facts we mention here are known and obtained by logicians from different countries and in different time since 1920s; we present these results together to see the whole picture.

This paper develops an order-theoretic generalization of Blok and Pigozziʼs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation (...) called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic . Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: it is injective, it is isotonic, it commutes with the inverse image operator of any algebraic homomorphism, and it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties –. (shrink)

We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.

If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of (...) sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate □, we tackle both problems. Given a frame (W, R) consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret □ at every world in such a way that □ $\ulcorner A \ulcorner$ holds at a world ᵆ ∊ W if and only if A holds at every world $\upsilon$ ∊ W such that ᵆR $\upsilon$ . The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of □ at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic. (shrink)

This paper presents a generalization of Fine’s completeness theorem for transitive logics of finite width, and proves the Kripke completeness of transitive logics of finite “suc-eq-width”. The frame condition for each finite suc-eq-width axiom requires, in rooted transitive frames, a finite upper bound of cardinality for antichains of points with different proper successors. The paper also presents a generalization of Rybakov’s completeness theorem for transitive logics of prefinite width, and proves the Kripke completeness of transitive logics of prefinite “suc-eq-width”. The (...) frame condition for each prefinite suc-eq-width axiom requires, in rooted transitive frames, a finite upper bound of cardinality for antichains of points that have a finite lower bound of depth and have different proper successors. We will construct continuums of transitive logics of finite suc-eq-width but not of finite width, and continuums of those of prefinite suc-eq-width but not of prefinite width. This shows that our new completeness results cover uncountably many more logics than Fine’s theorem and Rybakov’s theorem respectively. (shrink)

We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.

All extensions of the modal Grzegorczyk logic Grz possessing projective Beth's property PB2 are described. It is proved that there are exactly 13 logics over Grz with PB2. All of them are finitely axiomatizable and have the finite model property. It is shown that PB2 is strongly decidable over Grz, i.e. there is an algorithm which, for any finite system Rul of additional axiom schemes and rules of inference, decides if the calculus Grz+Rul has the projective Beth property.

In this paper we provide frame definability results for weak versions of classical modal axioms that can be expressed in Fitting's many-valued modal languages. These languages were introduced by M. Fitting in the early '90s and are built on Heyting algebras which serve as the space of truth values. The possible-worlds frames interpreting these languages are directed graphs whose edges are labelled with an element of the underlying Heyting algebra, providing us a form of many-valued accessibility relation. Weak axioms of (...) the form we treat here have been examined from the completeness perspective and further explored for applications in non-monotonic reasoning. Here, we introduce more weak many-valued modal axioms and prove a frame correspondence result for all of them. The classes of corresponding labelled frames possess algebraic properties which are strongly reminiscent of many classical ones, such as the Church-Rosser property, reflexivity, transitivity, partial functionality, etc. (shrink)

In a previous work we introduced the algorithm \SQEMA\ for computing first-order equivalents and proving canonicity of modal formulae, and thus established a very general correspondence and canonical completeness result. \SQEMA\ is based on transformation rules, the most important of which employs a modal version of a result by Ackermann that enables elimination of an existentially quantified predicate variable in a formula, provided a certain negative polarity condition on that variable is satisfied. In this paper we develop several extensions of (...) \SQEMA\ where that syntactic condition is replaced by a semantic one, viz. downward monotonicity. For the first, and most general, extension \SSQEMA\ we prove correctness for a large class of modal formulae containing an extension of the Sahlqvist formulae, defined by replacing polarity with monotonicity. By employing a special modal version of Lyndon's monotonicity theorem and imposing additional requirements on the Ackermann rule we obtain restricted versions of \SSQEMA\ which guarantee canonicity, too. (shrink)

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that (...) restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction-and cut-free sequent calculus equivalent to the Hubert system for basic modal logic shows the standard failure argument untenable by proving the underivability of DA from A. (shrink)

Este livro marca o início da Série Investigação Filosófica. Uma série de livros de traduções de textos de plataformas internacionalmente reconhecidas, que possa servir tanto como material didático para os professores das diferentes subáreas e níveis da Filosofia quanto como material de estudo para o desenvolvimento pesquisas relevantes na área. Nós, professores, sabemos o quão difícil é encontrar bons materiais em português para indicarmos. E há uma certa deficiência na graduação brasileira de filosofia, principalmente em localizações menos favorecidas, com relação (...) ao conhecimento de outras línguas, como o inglês e o francês. Tentamos, então, suprir essa deficiência, ao introduzirmos traduções de textos importantes ao público de língua portuguesa, sem nenhuma finalidade comercial e meramente pela glória da filosofia. O presente volume é constituído de três traduções de verbetes importantes sobre lógica, da Enciclopédia de Filosofia da Stanford: (1) A Lógica de Aristóteles, (2) Lógica Clássica, (3) Lógica Modal. (shrink)

This paper is concerned with counterfactual logic and its implications for the modal status of mathematical claims. It is most directly a response to an ambitious program by Yli-Vakkuri and Hawthorne (2018), who seek to establish that mathematics is committed to its own necessity. I claim that their argument fails to establish this result for two reasons. First, their assumptions force our hand on a controversial debate within counterfactual logic. In particular, they license counterfactual strengthening— the inference from ‘If A (...) were true then C would be true’ to ‘If A and B were true then C would be true’—which many reject. Second, the system they develop is provably equivalent to appending Deduction Theorem to a T modal logic. It is unsurprising that the combination of Deduction Theorem with T results in necessitation; indeed, it is precisely for this reason that many logicians reject Deduction Theorem in modal contexts. If Deduction Theorem is unacceptable for modal logic, it cannot be assumed to derive the necessity of mathematics. (shrink)

The literature on quantum logic emphasizes that the algebraic structures involved with orthodox quantum mechanics are non distributive. In this paper we develop a particular algebraic structure, the quasi-lattice ( $${\mathfrak{I}}$$ -lattice), which can be modeled by an algebraic structure built in quasi-set theory $${\mathfrak{Q}}$$. This structure is non distributive and involve indiscernible elements. Thus we show that in taking into account indiscernibility as a primitive concept, the quasi-lattice that ‘naturally’ arises is non distributive.

This paper investigates a generalized version of inquisitive semantics. A complete axiomatization of the associated logic is established, the connection with intuitionistic logic and several intermediate logics is explored, and the generalized version of inquisitive semantics is argued to have certain advantages over the system that was originally proposed by Groenendijk (2009) and Mascarenhas (2009).

It is natural to ask under what conditions negating a conditional is equivalent to negating its consequent. Given a bivalent background logic, this is equivalent to asking about the conjunction of Conditional Excluded Middle (CEM, opposite conditionals are not both false) and Weak Boethius' Thesis (WBT, opposite conditionals are not both true). In the system CI.0 of consequential implication, which is intertranslatable with the modal logic KT, WBT is a theorem, so it is natural to ask which instances of CEM (...) are derivable. We also investigate the systems CIw and CI of consequential implication, corresponding to the modal logics K and KD respectively, with occasional remarks about stronger systems. While unrestricted CEM produces modal collapse in all these systems, CEM restricted to contingent formulas yields the Alt2 axiom (semantically, each world can see at most two worlds), which corresponds to the symmetry of consequential implication. It is proved that in all the main systems considered, a given instance of CEM is derivable if and only if the result of replacing consequential implication by the material biconditional in one or other of its disjuncts is provable. Several related results are also proved. The methods of the paper are those of propositional modal logic as applied to a special sort of conditional. (shrink)

In his 1964 note, ‘Two Additions to Positive Implication’, A. N. Prior showed that standard axioms governing conjunction yield a nonconservative extension of the pure implicational intermediate logic OIC of R. A. Bull. Here, after reviewing the situation with the aid of an adapted form of the Kripke semantics for intuitionistic and intermediate logics, we proceed to illuminate this example by transposing it to the setting of modal logic, and then relate it to the propositional logic of what have been (...) called ‘Hilbert algebras with infimum’. (shrink)

ABSTRACT The paper studies propositional logics in a bimodal language, in which the first modality is interpreted as the local truth, and the second as the universal truth. The logic S4UC is introduced, which is finitely axiomatizable, has the f.m.p. and is determined by every connected separable metric space.

ABSTRACT George Gargov was an active pioneer in the ‘Sofia School’ of modal logicians. Starting in the 1970s, he and his colleagues expanded the scope of the subject by introducing new modal expressive power, of various innovative kinds. The aim of this paper is to show some general patterns behind such extensions, and review some very general results that we know by now, 20 years later. We concentrate on simulation invariance, decidability, and correspondence. What seems clear is that ‘modal logic’ (...) as a genre of logical systems has a much wider scope than originally conceived, and that we have not reached its limits yet. (shrink)

A logic is called metacomplete if formulas that are true in a certain preferred interpretation of that logic are theorems in its metalogic. In the area of relevant logics, metacompleteness is used to prove primeness, consistency, the admissibility of γ and so on. This paper discusses metacompleteness and its applications to a wider class of modal logics based on contractionless relevant logics and their neighbours using Slaney’s metavaluational technique.

The aim of the present book is to give a comprehensive account of the ‘state of the art’ of substructural logics, focusing both on their proof theory and on their semantics (both algebraic and relational. It is for graduate students in either philosophy, mathematics, theoretical computer science or theoretical linguistics as well as specialists and researchers.

We establish the dichotomy property for stable canonical multi-conclusion rules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics.

MIPC is a well-known intuitionistic modal logic of Prior and Bull . It is shown that every normal intuitionistic modal logic L over MIPC has the finite model property whenever L is Kripke-complete and universal.

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. The translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. It is practically relevant because it makes it possible to use a decision procedure for the guarded fragment in order to decide regular grammar logics with (...) converse. The class of regular grammar logics includes numerous logics from various application domains. A consequence of the translation is that the general satisfiability problem for every regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Other logics that can be translated into GF2 include nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed-point operators. (shrink)

In this paper we give an example of intertranslatability between an ontology of individuals (nominalism), an ontology of properties (realism), and an ontology of facts (factualism). We demonstrate that these three ontologies are dual to each other, meaning that each ontology can be translated into, and recaptured from, each of the others. The aiin of the enterprise is to raise the possibility that, at least in some settings, there may be no need for considerations of ontological primacy. Whether the world (...) is made up of things, or properties, or facts, may be no more than a matter of how we look at it. (shrink)

S4.2 is the modal logic of directed partial pre-orders and/or the modal logic of reflexive and transitive relational frames with a final cluster. It holds a distinguished position in philosophical...

We prove strong completeness of the □-version and the ◊-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this (...) logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants. (shrink)

Here, we provide a detailed description of the mutual relation of formulas with finite propositional variables p1, …, pm in modal logic S4. Our description contains more information on S4 than those given in Shehtman (1978) and Moss (2007); however, Shehtman (1978) also treated Grzegorczyk logic and Moss (2007) treated many other normal modal logics. Specifically, we construct normal forms, which behave like the principal conjunctive normal forms in the classical propositional logic. The results include finite and effective methods to (...) find a normal form equivalent to a given formula A by clarifying the behavior of connectives and giving a finite method to list all exact models. (shrink)

The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several (...) weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency. (shrink)

We introduce and investigate the notion of uniform Lyndon interpolation property which is a strengthening of both uniform interpolation property and Lyndon interpolation property. We prove several propositional modal logics including \, \, \ and \ enjoy ULIP. Our proofs are modifications of Visser’s proofs of uniform interpolation property using layered bisimulations Gödel’96, logical foundations of mathematics, computer science and physics—Kurt Gödel’s legacy, Springer, Berlin, 1996). Also we give a new upper bound on the complexity of uniform interpolants for \ (...) and \. (shrink)

Trying to overcome Dugundji’s result on uncharacterisability of modal logics by finite logical matrices, Kearns and Ivlev proposed, independently, a characterisation of some modal systems by means of four-valued multivalued truth-functions , as an alternative to Kripke semantics. This constitutes an antecedent of the non-deterministic matrices introduced by Avron and Lev . In this paper we propose a reconstruction of Kearns’s and Ivlev’s results in a uniform way, obtaining an extension to another modal systems. The first part of the paper (...) is devoted to four-valued Nmatrices, including Kearns’s and Ivlev’s. Besides proving with full details Kearns’s results for T, S4 and S5, we also obtain a characterisation of the system B by four-valued Nmatrices with level valuations. Concerning Ivlev’s results, two new modal systems are introduced and char.. (shrink)

The Local Computation Framework has been used to improve the efficiency of computation in various uncertainty formalisms. This paper shows how the framework can be used for the computation of logical deduction in two different ways; the first way involves embedding model structures in the framework; the second, and more direct, way involves embedding sets of formulae. This work can be applied to many of the logics developed for different kinds of reasoning, including predicate calculus, modal logics, possibilistic logics, probabilistic (...) logics and non-monotonic logics. (shrink)

The aim of this paper is to give new kinds of modal logics suitable for reasoning about regions in discrete spaces. We call them dynamic logics of the region-based theory of discrete spaces. These modal logics are linguistic restrictions of propositional dynamic logic with the global diamond E. Their formulas are equivalent to Boolean combinations of modal formulas like E(A ∧ ⟨α⟩ B) where A and B are Boolean terms and α is a relational term. Examining what we can say (...) about dynamic models when we use formulas to describe them, we successively address the axiomatization/completeness issue and the decidability/complexity issue of our dynamic logics of the region-based theory of discrete spaces. (shrink)

Jean-Yves Béziau (‘Classical Negation can be Expressed by One of its Halves’, Logic Journal of the IGPL 7 (1999), 145–151) has given an especially clear example of a phenomenon he considers a sufficiently puzzling to call the ‘paradox of translation’: the existence of pairs of logics, one logic being strictly weaker than another and yet such that the stronger logic can be embedded within it under a faithful translation. We elaborate on Béziau’s example, which concerns classical negation, as well as (...) giving some additional background (especially from intuitionistic logic) to the example. Our interest is more on the logical exploration of the phenomenon Béziau’s case exemplifies than on the question of whether that phenomenon is (even prima facie ) paradoxical, though in Section 5 we do approach the latter question – somewhat obliquely – by considering an analogous phenomenon which it is hard to find puzzling. (shrink)

Standard models for model predicate logic consist of a Kripke frame whose worlds come equipped with relational structures. Both modal and two-sorted predicate logic are natural languages for speaking about such models. In this paper we compare their expressivity. We determine a fragment of the two-sorted language for which the modal language is expressively complete on S5-models. Decidable criteria for modal definability are presented.

This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...) to weak Heyting algebras and implicative semilattices. The book also dives into the Blok-Esakia theorem and provides an outline of the intuitionistic modal logic KM which is closely related to the Gödel-Löb provability logic GL. One chapter scrutinizes Esakia’s work interpreting modal diamond as the derivative of a topological space within the setting of point-free topology. The final chapter in the volume is dedicated to the derivational semantics of modal logic and other related issues. (shrink)

Stalnaker, 169–199 2006) introduced a combined epistemic-doxastic logic that can formally express a strong concept of belief, a concept of belief as ‘subjective certainty’. In this paper, we provide a topological semantics for belief, in particular, for Stalnaker’s notion of belief defined as ‘epistemic possibility of knowledge’, in terms of the closure of the interior operator on extremally disconnected spaces. This semantics extends the standard topological interpretation of knowledge with a new topological semantics for belief. We prove that the belief (...) logic KD45 is sound and complete with respect to the class of extremally disconnected spaces and we compare our approach to a different topological setting in which belief is interpreted in terms of the derived set operator. We also study belief revision as well as belief dynamics by providing a topological semantics for conditional belief and belief update modalities, respectively. Our setting based on extremally disconnected spaces, however, encounters problems when extended with dynamic updates. We then propose a solution consisting in interpreting belief in a similar way based on hereditarily extremally disconnected spaces, and axiomatize the belief logic of hereditarily extremally disconnected spaces. Finally, we provide a complete axiomatization of the logic of conditional belief and knowledge, as well as a complete axiomatization of the corresponding dynamic logic. (shrink)

Gödel logic is the extension of intuitionistic logic by the linearity axiom. Symmetric Gödel logic is a logical system, the language of which is an enrichment of the language of Gödel logic with their dual logical connectives. Symmetric Gödel logic is the extension of symmetric intuitionistic logic. The proof-intuitionistic calculus, the language of which is an enrichment of the language of intuitionistic logic by modal operator was investigated by Kuznetsov and Muravitsky. Bimodal symmetric Gödel logic is a logical system, the (...) language of which is an enrichment of the language of Gödel logic with their dual logical connectives and two modal operators. Bimodal symmetric Gödel logic is embedded into an extension of Gödel–Löb logic, the language of which contains disjunction, conjunction, negation and two modal operators. The variety of bimodal symmetric Gödel algebras, that represent the algebraic counterparts of bimodal symmetric Gödel logic, is investigated. Description of free algebras and characterization of projective algebras in the variety of bimodal symmetric Gödel algebras is given. All finitely generated projective bimodal symmetric Gödel algebras are infinite, while finitely generated projective symmetric Gödel algebras are finite. (shrink)

This paper partly answers the question “what a frame may be exactly like when it characterizes a pretabular logic in NExtK4”. We prove the pretabularity crieria for the logics of finite depth in NExtK4. In order to find out the criteria, we create two useful concepts—“pointwise reduction” and “invariance under pointwise reductions”, which will remain important in dealing with the case of infinite depth.

We give an example of a variety of Heyting algebras and of a splitting algebra in this variety that is not finitely presentable. Moreover, we show that the corresponding splitting pair cannot be defined by any finitely presentable algebra. Also, using the Gödel-McKinsey-Tarski translation and the Blok-Esakia theorem, we construct a variety of Grzegorczyk algebras with similar properties.

The trilattice SIXTEEN₃ is a natural generalization of the wellknown bilattice FOUR₂. Cut-free, sound and complete sequent calculi for truth entailment and falsity entailment in SIXTEEN₃, are presented.

The modal logic of Gödel sentences, termed as GS, is introduced to analyze the logical properties of 'true but unprovable' sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk's Logic, where modality can be interpreted as 'true and provable'. As we show, GS and Grzegorczyk's Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS. GS is also an extended system of the logic of 'Essence and Accident' proposed by Marcos (...) (Bull Sect Log 34(1):43-56, 2005). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic. (shrink)

Only propositional logics are at issue here. Such a logic is contra-classical in a superficial sense if it is not a sublogic of classical logic, and in a deeper sense, if there is no way of translating its connectives, the result of which translation gives a sublogic of classical logic. After some motivating examples, we investigate the incidence of contra-classicality (in the deeper sense) in various logical frameworks. In Sections 3 and 4 we will encounter, originally as an example of (...) what (in Section 2) we call a contra-classical modal logic, an unusual logic boasting a connective (" demi-negation" ) whose double application is equivalent to a single application of the negation connective. Pondering the example points the way to a general characterization of contra-classicality (Theorems 3.3 and 4.6). In an Appendix (Section 5), we look at one alternative to classical logic as the target for such translational assimilation, intuitionistic logic, calling logics which resist the assimilation, in this case, contra- intuitionistic. We will show that one such logic is classical logic itself, thereby strengthening a result of Wojcicki's to the effect that the consequence relation of classical logic cannot be faithfully embedded by any connective-by-connective translation into that of intuitionistic logic. (What the "faithfully" means here is that not only is the translation of anything provable in the 'source' logic.. (shrink)

By introducing the intensional mappings and their properties, we establish a new semantical approach of characterizing intermediate logics. First prove that this new approach provides a general method of characterizing and comparing logics without changing the semantical interpretation of implication connective. Then show that it is adequate to characterize all Kripke_complete intermediate logics by showing that each of these logics is sound and complete with respect to its (unique) ‘weakest characterization property’ of intensional mappings. In particular, we show that classical (...) logic has the weakest characterization property , which is the strongest among all possible weakest characterization properties of intermediate logics. Finally, it follows from this result that a translation is an embedding of classical logic into intuitionistic logic, iff. its semantical counterpart has the property. (shrink)

The trilattice SIXTEEN3 introduced in Shramko & Wansing (2005) is a natural generalization of the famous bilattice FOUR2. Some Hilbert-style proof systems for trilattice logics related to SIXTEEN3 have recently been studied (Odintsov, 2009; Shramko & Wansing, 2005). In this paper, three sequent calculi GB, FB, and QB are presented for Odintsovs coordinate valuations associated with valuations in SIXTEEN3. The equivalence between GB, FB, and QB, the cut-elimination theorems for these calculi, and the decidability of B are proved. In addition, (...) it is shown how the sequent systems for B can be extended to cut-free sequent calculi for Odintsov’s LB, which is an extension of B by adding classical implication and negation connectives. (shrink)

We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (∧, →) homomorphisms, (∧, →, 0) homomorphisms, and (∧, →, ∨) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we obtain a new simplified proof (which is algebraic in nature) of Zakharyaschev’s (...) theorem that each intermediate logic can be axiomatized by canonical formulas. (shrink)

In this paper we study the expressive power and definability for modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt–Thomason definability theorem in terms of the well-established first-order topological language.

In this paper, we extend the canonicity methodology in Ghilardi & Meloni (1997) to arbitrary lattice expansions, and syntactically describe canonical inequalities for lattice expansions consisting of -meet preserving operations, -multiplicative operations, adjoint pairs, and constants. This approach gives us a uniform account of canonicity for substructural and lattice-based logics. Our method not only covers existing results, but also systematically accounts for many canonical inequalities containing nonsmooth additive and multiplicative uniform operations. Furthermore, we compare our technique with the approach in (...) Dunn et al. (2005) and Gehrke et al. (2005). (shrink)

Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are not normal. The set of extensions (...) of S4.3 is radically altered by the addition of a constant: we use it to construct continuum many such normal extensions of S4.3, and continuum many non-normal ones, none of which have the finite model property. But for logics with weakly transitive frames there are only eight maximally normal ones, of which five extend K4 and three extend S4. (shrink)