We prove some embedding theorems for classical conditional logic, covering 'finitely cumulative' logics, 'preferential' logics and what we call 'semi-monotonic' logics. Technical tools called 'partial frames' and 'frame morphisms' in the context of neighborhood semantics are used in the proof.

This paper investigates (modal) extensions of Heyting-Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We first develop matrix as well as Kripke style semantics for those logics. Then, by extending the Gö;del-embedding of intuitionistic logic into S4, it is shown that all (modal) extensions of Heyting-Brouwer logic can be embedded into tense logics (with additional modal operators). An extension of the Blok-Esakia-Theorem is proved for this embedding.

This paper consists of some lecture notes in which conditional logic is treated as an extension of modal logic. Completeness and filtration theorems are provided for some basis systems.

The formal language studied in this paper contains two categories of expressions, terms and formulas. Terms express events, formulas propositions. There are infinitely many atomic terms and complex terms are made up by Boolean operations. Where and are terms the atomic formulas have the form = ( is the same as ), Forb ( is forbidden) and Perm ( is permitted). The formulae are truth functional combinations of these. An algebraic and a model theoretic account of validity are given and (...) an axiomatic system is provided for which they are characteristic.The closure principle, that what is not forbidden is permitted is shown to hold at the level of outcomes but not at the level of events. In the two final sections some other operators are considered and a semantics in terms of action games. (shrink)

This chapter contains sections titled: Introduction Modal propositional Logics (MPLs) Modal Quantificational Logics(QMLs).

Questions connected with the admissibility of rules of inference and the solvability of the substitution problem for modal and intuitionistic logic are considered in an algebraic framework. The main result is the decidability of the universal theory of the free modal algebra imageω extended in signature by adding constants for free generators. As corollaries we obtain: there exists an algorithm for the recognition of admissibility of rules with parameters in the modal system Grz, the substitution problem for Grz and for (...) the intuitionistic calculus H is decidable, intuitionistic propositional calculus H is decidable with respect to admissibility . A semantical criterion for the admissibility of rules of inference in Grz is given. (shrink)

One way to obtain a comprehensive semantics for various systems of modal logic is to use a general notion of non-normal world. In the present article, a general notion of modal system is considered together with a semantic framework provided by such a general notion of non-normal world. Methodologically, the main purpose of this paper is to provide a logical framework for the study of various modalities, notably prepositional attitudes. Some specific systems are studied together with semantics using non-normal worlds (...) of different kinds. (shrink)

This article concerns the metatheory of a class of modal logics whose language includes propositional constants of various kinds. The main novelties are the use of general frames with specific restrictions and the definition of the strict range of a formula. Many examples from the literature are treated within the framework provided and some traditional model-theoretic issues such as preservation results concerning the validity of formulas and definability results concerning frame properties are addressed.

In their presentation of canonical models for normal systems of modal logic, Hughes and Cresswell observe that some of these models are based on a frame which can be also thought of as a collection of two or more isolated frames; they call such frames ‘non-cohesive’. The problem of checking whether the canonical model of a given system is cohesive is still rather unexplored and no general decision procedure is available. The main contribution of this article consists in introducing a (...) method which is sufficient to show that canonical models of some relevant classes of normal monomodal and bimodal systems are always non-cohesive. (shrink)

The main purpose of this paper is to define and study a particular variety of Montague-Scott neighborhood semantics for modal propositional logic. We call this variety the first-order neighborhood semantics because it consists of the neighborhood frames whose neighborhood operations are, in a certain sense, first-order definable. The paper consists of two parts. In Part I we begin by presenting a family of modal systems. We recall the Montague-Scott semantics and apply it to some of our systems that have hitherto (...) be uncharacterized. Then, we define the notion of a first-order indefinite semantics, along with the more specific notion of a first-order uniform semantics, the latter containing as special cases the possible world semantics of Kripke. In Part II we prove consistency and completeness for a broad range of the systems considered, with respect to the first-order indefinite semantics, and for a selected list of systems, with respect to the first-order uniform semantics. The completeness proofs are algebraic in character and make essential use of the finite model property. A by-product of our investigations is a result relating provability in S-systems and provability in T-systems, which generalizes a known theorem relating provability in the systems S 2° and C 2. (shrink)

A finitely alternative normal tense logic \ is a normal tense logic characterized by frames in which every point has at most n future alternatives and m past alternatives. The structure of the lattice \\) is described. There are \ logics in \\) without the finite model property, and only one pretabular logic in \\). There are \ logics in \\) which are not finitely axiomatizable. For \, there are \ logics in \\) without the FMP, and infinitely many pretabular (...) extensions of \. (shrink)

In this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some L and some cannot be separated from by a suitably wide class of complete algebras — usually Kripke algebras. We are going to show that classical examples of incomplete logics, e.g., Fine logic, are not complete with respect to any class of complete BAOs. (...) Even above Grz it is possible to find a continuum of such logics, which immediately implies the existence of a continuum of neighbourhood-incomplete Grz logics. Similar results can be proved for Löb logics. In addition, completely incomplete logics above Grz may be found uniformly as a result of failures of some admissible rule of a special kind. (shrink)

The present paper is based on [11], where a number of conjectures are made concerning the structure of the lattice of normal extensions of the tense logicKt. That paper was mainly dealing with splittings of and some sublattices, and this is what I will concentrate on here as well. The main tool in analysing the splittings of will be the splitting theorem of [8]. In [11] it was conjectured that each finite subdirectly irreducible algebra splits the lattice of normal extensions (...) ofK4t andS4t. We will show that this is not the case and that on the contrary only very few and trivial splittings of the mentioned lattices exist. (shrink)

The paper is a brief survey of the most important semantic constructions founded on the concept of possible world. It is impossible to capture in one short paper the whole variety of the problems connected with manifold applications of possible worlds. Hence, after a brief explanation of some philosophical matters I take a look at possible worlds from rather technical standpoint of logic and focus on the applications in formal semantics. In particular, I would like to focus on the fruitful (...) marriage of possible world semantics and algebra and its evolution leading to very general construction of Wójcicki called referential semantics and some of its refinements. The presentation is informal and sketchy; the main purpose is to put in one place a short, and readable I hope, description of the most important constructions and to point out the main sources of these solutions. (shrink)

We illustrate, with three examples, the interaction between boolean and modal connectives by looking at the role of truth-functional reasoning in the provision of completeness proofs for normal modal logics. The first example (§ 1) is of a logic (more accurately: range of logics) which is incomplete in the sense of being determined by no class of Kripke frames, where the incompleteness is entirely due to the lack of boolean negation amongst the underlying non-modal connectives. The second example (§ 2) (...) focusses on the breakdown, in the absence of boolean disjunction, of the usual canonical model argument for the logic of dense Kripke frames, though a proof of incompleteness with respect to the Kripke semantics is not offered. An alternative semantic account is developed, in terms of which a completeness proof can be given, and this is used (§ 3) in the discussion of the third example, a bimodal logic which is, as with the first example, provably incomplete in terms of the Kripke semantics, the incompleteness being due to the lack of disjunction (as a primitive or defined boolean connective). (shrink)

In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem, “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, ${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question (...) in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to ${\cal V}$-baos, namely the provability logic $GLB$. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is ${\cal V}$-complete. After these results, we generalize the Blok Dichotomy to degrees of ${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness. (shrink)

The contribution of this paper lies with providing a systematically specified and intuitive interpretation pattern and delineating a class of relational structures and models providing a natural interpretation of logical operators on an underlying propositional calculus of Positive Lattice Logic and subsequently proving a generic completeness theorem for the related class of logics, sometimes collectively referred to as Generalized Galois Logics.

This article addresses and resolves some issues of relational, Kripke-style, semantics for the logics of bounded lattice expansions with operators of well-defined distribution types, focusing on the case where the underlying lattice is not assumed to be distributive. It therefore falls within the scope of the theory of Generalized Galois Logics, introduced by Dunn, and it contributes to its extension. We introduce order-dual relational semantics and present a semantic analysis and completeness theorems for non-distributive lattice logic with n -ary additive (...) or multiplicative operators, with negation operators modally interpreted as impossibility and unnecessity, as well as with implication connectives. Order-dual relational semantics shares with the generalized Kripke frames, or the bi-approximation semantics approach, the use of both a satisfaction and a co-satisfaction relation, but it also responds to the recently voiced concerns of Craig, Haviar and Conradie about the relative non-intuitiveness of the 2-sorted semantics of the aforementioned approaches. In this article, we provide a standard interpretation of modalities and natural interpretations of both negation and implication, despite the absence of distribution. Thereby, our results contribute in creating the necessary background for research in non-distributive logics with modalities variously interpreted as dynamic, temporal etc, by analogy to the classical case. (shrink)

We introduce game-theoretic semantics for systems without the conveniences of either a De Morgan negation, or of distribution of conjunction over disjunction and conversely. Much of game playing rests on challenges issued by one player to the other to satisfy, or refute, a sentence, while forcing him/her to move to some other place in the game’s chessboard-like configuration. Correctness of the game-theoretic semantics is proven for both a training game, corresponding to Positive Lattice Logic and for more advanced games for (...) the logics of lattices with weak negation and modal operators. (shrink)

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson and Tarski (...) representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality. (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)

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)

We develop a method for showing that various modal logics that are valid in their countably generated canonical Kripke frames must also be valid in their uncountably generated ones. This is applied to many systems, including the logics of finite width, and a broader class of multimodal logics of ‘finite achronal width’ that are introduced here.

Robert Stalnaker has recently advocated propositional contingentism, the claim that it is contingent what propositions there are. He has proposed a philosophical theory of contingency in what propositions there are and sketched a possible worlds model theory for it. In this paper, such models are used to interpret two propositional modal languages: one containing an existential propositional quantifier, and one containing an existential propositional operator. It is shown that the resulting logic containing an existential quantifier is not recursively axiomatizable, as (...) it is recursively isomorphic to second-order logic, and a natural candidate axiomatization for the resulting logic containing an existential operator is shown to be incomplete. (shrink)

Here, I combine the semantics of Mares and Goldblatt [20] and Seki [29, 30] to develop a semantics for quantified modal relevant logics extending ${\bf B}$. The combination requires demonstrating that the Mares–Goldblatt approach is apt for quantified extensions of ${\bf B}$ and other relevant logics, but no significant bridging principles are needed. The result is a single semantic approach for quantified modal relevant logics. Within this framework, I discuss the requirements a quantified modal relevant logic must satisfy to be (...) “sufficiently classical” in its modal fragment, where frame conditions are given that work for positive fragments of logics. The roles of the Barcan formula and its converse are also investigated. (shrink)

The semantics of e-models for tense logics with binary operators for `until' and `since' (US-logics) was introduced by Bellissima and Bucalo in 1995. In this paper we show the adequacy of these semantics by proving a general Henkin-style completeness theorem. Moreover, we show that for these semantics there holds a Stone-like duality theorem with the algebraic structures that naturally arise from US-logics.

A new characterization of tabularity in tense logic is established, namely, a tense logicLis tabular if and only if$\mathsf {tab}_n^T\in L$for some$n\geq 1$. Two characterization theorems for the Post-completeness in tabular tense logics are given. Furthermore, a characterization of the Post-completeness in the lattice of all tense logics is established. Post numbers of some tense logics are shown.

In this paper we introduce Hilbert algebras with Hilbert–Galois connections (HilGC-algebras) and we study the Hilbert–Galois connections defined in Heyting algebras, called HGC-algebras. We assign a categorical duality between the category HilGC-algebras with Hilbert homomorphisms that commutes with Hilbert–Galois connections and Hilbert spaces with certain binary relations and whose morphisms are special functional relations. We also prove a categorical duality between the category of Heyting Galois algebras with Heyting homomorphisms that commutes with Hilbert–Galois connections and the category of spectral Heyting (...) spaces endowed with a binary relation with certain special continuous maps. (shrink)

The title reflects my conviction that, viewed semantically,modal logic is fundamentally dialogical; this conviction is based on the key role played by the notion of bisimulation in modal model theory. But this dialogical conception of modal logic does not seem to apply to modal proof theory, which is notoriously messy. Nonetheless, by making use of ideas which trace back to Arthur Prior (notably the use of nominals, special proposition symbols which ‘name’ worlds) I will show how to lift the dialogical (...) conception to modal proof theory. I argue that this shift to hybrid logic has consequences for both modal and dialogical logic, and I discuss these in detail. (shrink)

We show that monadic intuitionistic quantifiers admit the following temporal interpretation: “always in the future” (for$\forall $) and “sometime in the past” (for$\exists $). It is well known that Prior’s intuitionistic modal logic${\sf MIPC}$axiomatizes the monadic fragment of the intuitionistic predicate logic, and that${\sf MIPC}$is translated fully and faithfully into the monadic fragment${\sf MS4}$of the predicate${\sf S4}$via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension${\sf TS4}$of${\sf S4}$and provide a full and faithful translation (...) of${\sf MIPC}$into${\sf TS4}$. We compare this new translation of${\sf MIPC}$with the Gödel translation by showing that both${\sf TS4}$and${\sf MS4}$can be translated fully and faithfully into a tense extension of${\sf MS4}$, which we denote by${\sf MS4.t}$. This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for${\sf MS4.t}$using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered. (shrink)

The problem of future contingents is regarded as an important philosophical problem in connection with determinism and it should be treated by tense logic. Prior’s early work focused on the problem, and later Prior studied branching-time tense logic which was invented by Kripke. However, Prior’s idea to use three-valued logic for the problem seems to be still alive. In this paper, we consider partial and paraconsistent approaches to the problem of future contingents. These approaches theoretically meet Aristotle’s interpretation of future (...) contingents. (shrink)