Modal logics are studied in their algebraic disguise of varieties of so-called modal algebras. This enables us to apply strong results of a universal algebraic nature, notably those obtained by B. Jonsson. It is shown that the degree of incompleteness with respect to Kripke semantics of any modal logic containing the axiom □ p → p or containing an axiom of the form $\square^mp \leftrightarrow\square^{m + 1}p$ for some natural number m is 2 ℵ 0 . (...) Furthermore, we show that there exists an immediate predecessor of classical logic (axiomatized by $p \leftrightarrow \square p$ ) which is not characterized by any finite algebra. The existence of modal logics having 2 ℵ 0 immediate predecessors is established. In contrast with these results we prove that the lattice of extensions of S4 behaves much better: a logic extending S4 is characterized by a finite algebra iff it has finitely many extensions and any such logic has only finitely many immediate predecessors, all of which are characterized by a finite algebra. (shrink)

The present paper investigates the groups of automorphisms for some lattices of modal logics. The main results are the following. The lattice of normal extensions of S4.3, NExtS4.3, has exactly two automorphisms, NExtK.alt1 has continuously many automorphisms. Moreover, any automorphism of NExtS4 fixes all logics of finite codimension. We also obtain the following characterization of pretabular logics containing S4: a logic properly extends a pretabular logic of NExtS4 iff its lattice of extensions is finite (...) and linear. (shrink)

Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on: • introducing the classes of so-called explosive, complete and classical Belnapian modal logics; • assigning to every normal modal logic three special conservative extensions in these classes; • associating with every Belnapian modal logic its explosive, complete and classical counterparts. (...) We investigate the relationships between special extensions and counterparts, provide certain handy characterisations and suggest a useful decomposition of the lattice of logics containing BK. (shrink)

Most material below is ranked around the splittings of lattices of normal modal logics. These splittings are generated by nite subdirect irreducible modal algebras. The actual computation of the splittings is often a rather delicate task. Rened model structures are very useful to this purpose, as well as they are in many other respects. E.g. the analysis of various lattices of extensions, like ES5, ES4:3 etc becomes rather simple, if rened structures are used. But this point will (...) not be touched here. The variety T BA , which corresponds to S4, is congruence- distributive. Hence any every tabular extension L S4 has nite many extensions only. Around 1975 it has been proved by several authors, that also the converse is true: If L S4 has nitely many extensions only, then L is necessarely tabular. These and other results will be extended to much richer lattices. For simplicity we state the results explicitely only for the lattice N of modal logics with one modal operator. But most of them carry over literally to the lattice Nk of k-ramied normal modal logics . It is easy to construct examples of 2-ramied modal logics which are P OST-complete but not tabular. Whether this will be possible for normal modal logic seems to be an open problem. In view of results below it seems very unlikely that such examples exist. (shrink)

This paper investigates the structure of lattices of normal mono- and polymodal subframelogics, i.e., those modal logics whose frames are closed under a certain type of substructures. Nearly all basic modal logics belong to this class. The main lattice theoretic tool applied is the notion of a splitting of a complete lattice which turns out to be connected with the “geometry” and “topology” of frames, with Kripke completeness and with axiomatization problems. We investigate in (...) detail subframe logics containing K4, those containing polymodal Altn and subframe logics which are tense logics. (shrink)

In this paper we analyze the propositional extensions of the minimal classical modal logic system E, which form a lattice denoted as CExtE. Our method of analysis uses algebraic calculations with canonical forms, which are a generalization of the normal forms applicable to normal modal logics. As an application, we identify a group of automorphisms of CExtE that is isomorphic to the symmetric group S4.

ABSTRACTCorrespondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components,, and, then frames are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated (...) class='Hi'>modal logic. This suggests that a reduction of the first to the latter may be possible, encoding Positive Lattice Logic as a fragment of Two-Sorted, Residuated Modal Logic. The reduction is analogous to the well-known Gödel-McKinsey-Tarski translation of Intuitionistic Logic into the S4 system of normal modal logic. In this article, we carry out this reduction in detail and we derive some properties of PLL from corresponding properties of First-Order Logic. The reduction we present is extendible to the case of lattices with operators, making use of recent results by this author on the relational representation of normal lattice expansions. (shrink)

In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploščica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. We then (...) extend each of the three representations to lattices with a multiplicative unary modality; the representing structures, like so-called graph-based frames, add a second relation of accessibility interacting with compatibility. The three representations generalize possibility semantics for classical modal logics to non-classical modal logics, motivated by a recent application of modal orthologic to natural language semantics. (shrink)

We are concerned with modal logics in the class EM0 of extensions of M0 . G denotes re exive frames. MG the modal logic on G in the sense of Kripke. M is nite if M = MG for some nite G. Finite G's will be drawn as framed diagrams, e.g. G = ! ; G = ! ; the latter shorter denoted by . EM0 is a complete lattice with zero M0 and one M . (...) If M M0 M0 is a succ of M. An ip of M is an immediate predecessor of M. E.g. M ! and M are the only ip's of M in ES4. M[P] denotes the extension of M adding P as a \new axiom", e.g. S4 = M0 [p ! p]. M is nitely axiomatizable if M = M0 [P] for some formula P. One easily shows if M is f.a. then each predecessor is separated from M by an ip of M. It is known that each nite M is f.a. and has only nitely many succ's all of which are nite. (shrink)

Earlier algebraic semantics for Belnapian modal logics were defined in terms of twist-structures over modal algebras. In this paper we introduce the class of BK -lattices, show that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety. We prove that the lattice of subvarieties of the variety of BK -lattices is dually isomorphic to the lattice of extensions of Belnapian modal logic BK . Finally, we describe (...) invariants determining a twist-structure over a modal algebra. (shrink)

We show that the lattices of the normal extensions of four well-known logics—propositional intuitionistic logic \, Grzegorczyk logic \, modalized Heyting calculus \ and \—can be joined in a commutative diagram. One connection of this diagram is an isomorphism between the lattices of the normal extensions of \ and \; we show some preservation properties of this isomorphism. Two other connections are join semilattice epimorphims of the lattice of the normal extensions of \ onto that of \ and (...) of the lattice of the normal extensions of \ onto that of \. The link between \ and \ is a well-known isomorphism established by the Blok–Esakia theorem. (shrink)

This paper investigates partitions of lattices of modal logics based on superintuitionistic logics which are defined by forming, for each superintuitionistic logic L and classical modal logic , the set L[] of L-companions of . Here L[] consists of those modal logics whose non-modal fragments coincide with L and which axiomatize if the law of excluded middle p V p is added. Questions addressed are, for instance, whether there exist logics with the (...) disjunction property in L[], whether L[] contains a smallest element, and whether L[] contains lower covers of . Positive solutions as concerns the last question show that there are (uncountably many) superclean modal logics based on intuitionistic logic in the sense of Vakarelov [28]. Thus a number of problems stated in [28] are solved. As a technical tool the paper develops the splitting technique for lattices of modal logics based on superintuitionistic logics and ap plies duality theory from [34]. (shrink)

Some results on the upper end of the lattice of all modal propositional logics.

In this work we study the decidability of a class of global modal logics arising from Kripke frames evaluated over certain residuated lattices, known in the literature as modal many-valued logics. We exhibit a large family of these modal logics which are undecidable, in contrast with classical modal logic and propositional logics defined over the same classes of algebras. This family includes the global modal logics arising from Kripke frames evaluated (...) over the standard Łukasiewicz and Product algebras. We later refine the previous result, and prove that global modal Łukasiewicz and Product logics are not even recursively axiomatizable. We conclude by closing negatively the open question of whether each global modal logic coincides with its local modal logic closed under the unrestricted necessitation rule. (shrink)

A polymodal lattice is a distributive lattice carrying an n-place operator preserving top elements and certain finite meets. After exploring some of the basic properties of such structures, we investigate their freely generated instances and apply the results to the corresponding logical systems — polymodal logics — which constitute natural generalizations of the usual systems of modal logic familiar from the literature. We conclude by formulating an extension of Kripke semantics to classical polymodal logic and proving (...) soundness and completeness theorems. (shrink)

Much work has been done on specific instances of residuated lattices with modal operators . In this paper, we develop a general framework that subsumes three important classes of modal residuated lattices: interior algebras, Abelian ℓ-groups with conuclei, and negative cones of ℓ-groups with nuclei. We then use this framework to obtain results about these three cases simultaneously. In particular, we show that a categorical equivalence exists in each of these cases. The approach used here emphasizes the role (...) played by reducts in the proofs of these categorical equivalences. Lastly, we develop a connection between translations of logics and images of modal operators. (shrink)

We generalize the notion of a monadic algebra to that of a pseudomonadic algebra. In the same way as monadic algebras serve as algebraic models of epistemic modal system S5, pseudomonadic algebras serve as algebraic models of doxastic modal system KD45. The main results of the paper are: Characterization of subdirectly irreducible and simple pseudomonadic algebras, as well as Tokarz's proper filter algebras; Ordertopological representation of pseudomonadic algebras; Complete description of the lattice of subvarieties of the variety (...) of pseudomonadic algebras. (shrink)

We define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For (...) all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K and the propositional fragment of Wijesekera’s Constructive Concurrent Dynamic Logic. (shrink)

Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply (...) taking each individual finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula phi a formula f(phi) such that for every formula psi, phi -> BOX psi is provable in L if and only if f(phi) -> psi is provable in L? We answer this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic. (shrink)

We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To (...) be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics. (shrink)

A many-valued modal logic, called linear abelian modal logic \(\rm {\mathbf{LK(A)}}\) is introduced as an extension of the abelian modal logic \(\rm \mathbf{K(A)}\). Abelian modal logic \(\rm \mathbf{K(A)}\) is the minimal modal extension of the logic of lattice-ordered abelian groups. The logic \(\rm \mathbf{LK(A)}\) is axiomatized by extending \(\rm \mathbf{K(A)}\) with the modal axiom schemas \(\Box(\varphi\vee\psi)\rightarrow(\Box\varphi\vee\Box\psi)\) and \((\Box\varphi\wedge\Box\psi)\rightarrow\Box(\varphi\wedge\psi)\). Completeness theorem with respect to algebraic semantics and a hypersequent calculus admitting cut-elimination are established. Finally, (...) the correspondence between hypersequent calculi and axiomatization is investigated. (shrink)

Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply (...) taking each individual finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula φ a formula f(φ) such that for every formula ψ, φ → []ψ is provable in L if and only if f(φ) → ψ is provable in L? We answer this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic. (shrink)

We study modal logics in the setting of varieties of modal algebras. Any variety of modal algebras generated by a finite algebra — such, a variety is called tabular — has only finitely many subvarieties, i.e. is of finite height. The converse does not hold in general. It is shown that the converse does hold in the lattice of varieties of K4-algebras. Hence the lower part of this lattice consists of tabular varieties only. We (...) proceed to show that there is a continuum of pretabular varieties of K4-algebras — those are the non-tabular varieties all of whose proper subvarieties are tabular — in contrast with Maksimova's result that there are only five pretabular varieties of S4-algebras. (shrink)

We consider a modal language over crisp frames and formulas evaluated on a finite MTL-chain (a linearly ordered commutative integral residuated lattice). We first show that the basic modal abstract logic with constants for the values of the MTL-chain is the maximal abstract logic satisfying Compactness, the Tarski Union Property and strong invariance for bisimulations. Finally, we improve this result by replacing the Tarski Union Property by a relativization property.

We investigate the expressivity of many-valued modal logics based on an algebraic structure with a complete linearly ordered lattice reduct. Necessary and sufficient algebraic conditions for admitting a suitable Hennessy–Milner property are established for classes of image-finite and modally saturated models. Full characterizations are obtained for many-valued modal logics based on complete BL-chains that are finite or have the real unit interval [0, 1] as a lattice reduct, including Łukasiewicz, Gödel, and product modal (...) logics. (shrink)

This paper is devoted to showing certain connections between normal modal logics and those strictly regular modal logics which have as a theorem. We extend some results of E. J. Lemmon (cf. [66]). In particular we prove that the lattice of the strictly regular modal logics with the axiom is isomorphic to the lattice of the normal modal logics.

This paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their "A General Algebraic Semantics for Sentential (...) class='Hi'>Logics". The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo, Landini and Ziliani, at the suggestion of the late Antonio Monteiro. (shrink)

The Handbook of Modal Logic contains 20 articles, which collectively introduce contemporary modal logic, survey current research, and indicate the way in which the field is developing. The articles survey the field from a wide variety of perspectives: the underling theory is explored in depth, modern computational approaches are treated, and six major applications areas of modal logic (in Mathematics, Computer Science, Artificial Intelligence, Linguistics, Game Theory, and Philosophy) are surveyed. The book contains both well-written expository articles, (...) suitable for beginners approaching the subject for the first time, and advanced articles, which will help those already familiar with the field to deepen their expertise. Please visit: http://people.uleth.ca/~woods/RedSeriesPromo_WP/PubSLPR.html - Compact modal logic reference - Computational approaches fully discussed - Contemporary applications of modal logic covered in depth. (shrink)

A well-known result, going back to the twenties, states that, under some reasonable assumptions, any logic can be characterized as the set of formulas satisfied by a matrix 〈,F〉, whereis an algebra of the appropriate type, andFa subset of the domain of, called the set of designated elements. In particular, every quasi-classical modal logic—a set of modal formulas, containing the smallest classical modal logicE, which is closed under the inference rules of substitution and modus ponens—is characterized by (...) such a matrix, wherenow is a modal algebra, andFis a filter of. If the modal logic is in fact normal, then we can do away with the filter; we can study normal modal logics in the setting of varieties of modal algebras. This point of view was adopted already quite explicitly in McKinsey and Tarski [8]. The observation that the lattice of normal modal logics is dually isomorphic to the lattice of subvarieties of a variety of modal algebras paved the road for an algebraic study of normal modal logics. The algebraic approach made available some general results from Universal Algebra, notably those obtained by Jónsson [6], and thereby was able to contribute new insights in the realm of normal modal logics [2], [3], [4], [10].The requirement that a modal logic be normal is rather a severe one, however, and many of the systems which have been considered in the literature do not meet it. For instance, of the five celebrated modal systems, S1–S5, introduced by Lewis, S4 and S5 are the only normal ones, while only SI fails to be quasi-classical. The purpose of this paper is to generalize the algebraic approach so as to be applicable not just to normal modal logics, but to quasi-classical modal logics in general. (shrink)

Given a normal (multi-)modal logic a characterization is given of the finitely presentable algebras A whose logics L A split the lattice of normal extensions of . This is a substantial generalization of Rautenberg [10] and [11] in which is assumed to be weakly transitive and A to be finite. We also obtain as a direct consequence a result by Blok [2] that for all cycle-free and finite A L A splits the lattice of normal extensions (...) of K. Although we firmly believe it to be true, we have not been able to prove that if a logic splits the lattice of extensions of then is the logic of an algebra finitely presentable over ; in this respect our result remains partial. (shrink)

In "An Alternative Semantics for Quantified Relevant Logic" (JSL 71 (2006)) we developed a semantics for quantified relevant logic that uses general frames. In this paper, we adapt that model theory to treat quantified modal logics, giving a complete semantics to the quantified extensions, both with and without the Barcan formula, of every proposi- tional modal logic S. If S is canonical our models are based on propositional frames that validate S. We employ frames in which not (...) every set of worlds is an admissible proposition, and an alternative interpretation of the uni- versal quantifier using greatest lower bounds in the lattice of admissible propositions. Our models have a fixed domain of individuals, even in the absence of the Barcan formula. For systems with the Barcan formula it is possible to preserve the usual Tarskian reading of the quantifier, at the expensive of sometimes losing va- lidity of S in the underlying propositional frames. We apply our results to a number of logics, including S4.2, S4M and KW, whose quantified extensions are incomplete for the standard semantics. (shrink)

In this paper we consider the class of truth-functional modal many-valued logics with the complete lattice of truth-values. The conjunction and disjunction logic operators correspond to the meet and join operators of the lattices, while the negation is independently introduced as a hierarchy of antitonic operators which invert bottom and top elements. The non-constructive logic implication will be defined for a subclass of modular lattices, while the constructive implication for distributive lattices (Heyting algebras) is based on relative (...) pseudo-complements as in intuitionistic logic. We show that the complete lattices are intrinsically modal, with banal identity modal operator. We define the autoreferential set-based representation for the class of modal algebras, and show that the autoreferential Kripke-style semantics for this class of modal algebras is based on the set of possible worlds equal to the complete lattice of algebraic truth-values. The philosophical assumption is based on the consideration that each possible world represents a level of credibility, so that only propositions with the right logic value (i.e., level of credibility) can be accepted by this world, then we connect it with paraconsistent properties and LFI logics. The bottom truth value in this complete lattice corresponds to the trivial world in which each formula is satisfied, that is, to the world with explosive inconsistency. The top truth value corresponds to the world with classical logics, while all intermediate possible worlds represent the different levels of paraconsistent logics. (shrink)

In this paper we define an augmentation mHC of the Heyting propositional calculus HC by a modal operator ?. This modalized Heyting calculus mHC is a weakening of the Proof-Intuitionistic Logic KM of Kuznetsov and Muravitsky. In Section 2 we present a short selection of attractive (algebraic, relational, topological and categorical) features of mHC. In Section 3 we establish some close connections between mHC and certain normal extension K4.Grz of the modal system K4. We define a translation of (...) mHC into K4.Grz and prove that this translation is exact, i. e. theorem-preserving and deducibility-invariant. We have established (however, in this note we do not present a proof of this) that the lattice of all extensions of mHC is isomorphic to the lattice of normal extensions of K4.Grz (a generalization of the Kuznetsov and Muravitsky theorem). (shrink)

We will outline the contributions of A.V. Kuznetsov to modal logic. In his research he focused mainly on semantic, i.e. algebraic, issues and lattices of extensions of particular modal logics, though his proof of the Full Conservativeness Theorem for the proof-intuitionistic logic KM (Theorem 17 below) is a gem of proof-theoretic art.

We define the concepts of minimal p-morphic image and basic p-morphism for transitive Kripke frames. These concepts are used to determine effectively the least number of variables necessary to axiomatize a tabular extension of K4, and to describe the covers and co-covers of such a logic in the lattice of the extensions of K4.

We try to translate the intuitionistic propositional logic INT into Brouwer's modal logic KTB. Our translation is motivated by intuitions behind Brouwer's axiom p →☐◊p The main idea is to interpret intuitionistic implication as modal strict implication, whereas variables and other positive sentences remain as they are. The proposed translation preserves fragments of the Rieger-Nishimura lattice which is the Lindenbaum algebra of monadic formulas in INT. Unfortunately, INT is not embedded by this mapping into KTB.

In many applications we have a set of objects together with their properties. Since the available information is usually incomplete and/or imprecise, the true knowledge about subsets of objects can be determined approximately only. In this paper, we discuss a fuzzy generalisation of two pairs of relation-based operators suitable for fuzzy set approximations, which have been recently investigated by Düntsch and Gediga. Double residuated lattices, introduced by Orlowska and Radzikowska, are taken as basic algebraic structures. Main properties of these operators (...) are presented. (shrink)

Logical implications are closely related to modal operators. Lattice-valued logic LL and quantum logic QL were formulated in Titani S (1999) Lattice Valued Set Theory. Arch Math Logic 38:395–421, Titani S (2009) A Completeness Theorem of Quantum Set Theory. In: Engesser K, Gabbay DM, Lehmann D (eds) Handbook of Quantum Logic and Quantum Structures: Quantum Logic. Elsevier Science Ltd., pp. 661–702, by introducing the basic implication → which represents the lattice order. In this paper, we fomulate (...) a predicate orthologic provided with the basic implication, which corresponds to complete ortholattices, and then formulate a quantum logic which is equivalent to QL, by using a modal operator instead of the basic implication. (shrink)

This article shows that although Fitch’s paradox has been extremely widely studied, up to now no correct formalization of the problem has been proposed. The purpose of this article is to present the paradox front the viewpoint of combining logics. It is argued that the correct minimal logic to state the paradox is composed by a fusion of modal frames, and a fusion of modal languages and logics.

The paper studies many-dimensional modal logics corresponding to products of Kripke frames. It proves results on axiomatisability, the finite model property and decidability for product logics, by applying a rather elaborated modal logic technique: p-morphisms, the finite depth method, normal forms, filtrations. Applications to first order predicate logics are considered too. The introduction and the conclusion contain a discussion of many related results and open problems in the area.