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We introduce dual counterpart intuitionistic logic (or DCInt): a constructive logic that is a conservative extension of intuitionistic logic, a sublogic of bi-intuitionistic logic, has the logical duality property of classical logic, and also retains the modal character of its interpretation of the connective dual to intuitionistic implication. We define its Kripke semantics along with the corresponding notion of a bisimulation, and then prove that it has both the disjunction property and (its dual) the constructible falsity property. Also, for any class |$ {\mathcal{C}}$| of Kripke frames from our semantics, we identify a condition such that |$ {\mathcal{C}}$| will have the disjunction property if it satisfies the condition. This provides a method for generating extensions of DCInt that retain the disjunction property. [ABSTRACT FROM AUTHOR]

The theory of stratified institutions is a general axiomatic approach to model theories where the satisfaction is parameterized by states of the models. In this paper we further develop this theory by introducing a new technique for representing stratified institutions, which is based on projecting to such simpler structures. On the one hand this can be used for developing general results applicable to a wide variety of already existing model theories with states, such as those based on some form of Kripke semantics. On the other hand this may serve as a template for defining new such model theories. In this paper we emphasize the former application of this technique by developing general results on model amalgamation and on the existence diagrams for stratified institutions. These are two most useful properties to have in institution theoretic model theory. [ABSTRACT FROM AUTHOR]

This paper proposes an intuitionistic generalization of van Benthem and Liu's dynamic logic of relation changers, where relation changers are dynamic operators that rewrite each agent's accessibility relation. We employ Nishimura's Kripke semantics for a constructive propositional dynamic logic to define the semantics of relation changers. Strong completeness results of Hilbert-style axiomatizations for constructive dynamic logic of relation changers are established by two types of rewriting strategies via recursion axioms: inside-out and outside-in strategies. To establish the semantic completeness by outside-in strategy, we propose the notion of composition of relation changers and an appropriately defined notion of length of a formula. Moreover, we reveal that recursion axioms have additional two faces: semantic and proof-theoretic ones. As for the semantic face, we introduce an alternative semantics for dynamic logic of relation changers to specify the semantic meaning of recursion axioms used in the inside-out strategy. This leads us to a semantic completeness proof of the axiomatization for the original semantics, which does not require a rewriting strategy based on recursion axioms. As for the proof-theoretic face, we transform each of recursion axioms required in the outside-in strategy into two (left and right) inference rules to provide a sequent calculus for dynamic logic of relation changers. Since the resulting sequent calculus is cut-free but semi-analytic, it still enjoys the Craig interpolation theorem by Maehara's method. [ABSTRACT FROM AUTHOR]

The idealizations resulting from the use of Kripke semantics in Epistemic Logic are inherited by formalizations of group epistemic notions. For example, distributed knowledge is often taken to reflect the potential knowledge of a group: what agents would know if they had unbounded means of communication and deductive ability. However, this does not specify whether/how this potential can be actualized, especially since real people are not unbounded reasoners. Inspired by experiments on group reasoning, we identify two dimensions of actualizing distributed knowledge: communication and inference. We build a dynamic framework with effortful actions accounting for both, combining impossible-worlds semantics and action models , and we provide a method for extracting a sound and complete axiomatization. [ABSTRACT FROM AUTHOR]

Two algebraic structures, the contrapositionally complemented Heyting algebra (ccHa) and the contrapositionally |$\vee $| complemented Heyting algebra (c |$\vee $| cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHa s and its extension |${\textrm {ILM}}$| - |${\vee }$| for c |$\vee $| cHa s are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirce's logic and Vakarelov's logic MIN. With a focus on properties of the two negations, relational semantics for ILM and |${\textrm {ILM}}$| - |${\vee }$| are obtained with respect to four classes of frames, and inter-translations between the classes preserving truth and validity are provided. ILM and |${\textrm {ILM}}$| - |${\vee }$| are shown to have the finite model property with respect to these classes of frames and proved to be decidable. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunn's logical framework for the purpose, two logics |$K_{im}$| and |$K_{im-{\vee }}$| are discussed, where the language does not include implication. The |$K_{im}$| -algebras are reducts of ccHa s and are different from relevant algebraic structures having two negations. The negations in the |$K_{im}$| -algebras and |$K_{im-{\vee }}$| -algebras are shown to occupy distinct positions in an enhanced form of Dunn's kite of negations. Relational semantics for |$K_{im}$| and |$K_{im-{\vee }}$| is provided by a class of frames that are based on Dunn's compatibility frames. It is observed that this class coincides with one of the four classes giving the relational semantics for ILM and |${\textrm {ILM}}$| - |${\vee }$|⁠. [ABSTRACT FROM AUTHOR]

We study a version of public announcement logic with common knowledge based on the relevant logic |$\textsf {R}$|⁠. Public announcements, as represented in our framework, are not necessarily truthful and accepted by all agents, nor is it assumed that beliefs are preserved under announcements. We establish a completeness result with respect to a relational semantics, and we show that an alternative semantics based on information states is dual to the relational one. We add a question-forming inquisitive disjunction operator to the language and prove a completeness result with respect to the information semantics. [ABSTRACT FROM AUTHOR]

In this article, I consider the positive logic of Boolean groups (i.e. Abelian groups where every non-identity element has order 2), where these are taken as frames for an operational semantics à la Urquhart. I call this logic BG. It is shown that the logic over the smallest nontrivial Boolean group, taken as a frame, is identical to the positive fragment of a quasi-relevance logic that was developed by Robles and Méndez (an extension of this result where negation is included is also discussed). It is proved that BG satisfies the variable sharing property and is Halldén complete, while failing to satisfy the disjunction property. Sound and complete subscripted tableaux are presented for BG. An axiomatization (in the positive language extended with fusion) is presented, which is sound with respect to BG and, with respect to a related ternary relational semantics, complete; the problem of identifying a complete axiomatization for BG itself is left open. Connections between BG and the quasi-relevance logic KR are also discussed. [ABSTRACT FROM AUTHOR]

This special issue contains extended versions of selected papers presented at the 3rd International Workshop on Dynamic Logic: New Trends and Applications (DaLí 2020). The submissions to DaLí 2020 and to the special issue span across various areas including dynamic logic and dynamic algebra, dynamic logic and public announcement logic and epistemic and modal logic in general. Strong completeness results concerning Hilbert-style axiomatizations of constructive dynamic logic of relation changers are established using two types of rewriting strategies via recursion axioms: inside-out and outside-in strategies. [Extracted from the article]

Let a priority micro pluralist be someone who holds that particles or other microscopic objects are fundamental. Rivals to priority micro pluralism include priority monism (the view that the only fundamental concrete object is the entire cosmos) as well as the Aristotelian view that some ordinary macroscopic objects are fundamental. Although priority micro pluralism is popular, I show that it encounters great difficulty in even the most straightforward cases. For example, this tennis ball is spherical; how is this fact to be grounded in facts about the microscopic realm? I consider a number of possible answers to this question. The most promising proposals attempt to exploit the close connection emphasized by Kripke between objects and their original material constituents. I argue that these proposals fail. I conclude that it is worth seriously considering alternatives to this sort of pluralism. [ABSTRACT FROM AUTHOR]

We explore two ways of formalizing Kreisel's addendum to the Brouwer–Heyting–Kolmogorov interpretation. To do this, we compare Artemov's justification logic with simply typed |$\lambda $| calculus. First, we provide a completeness result for Kripke style semantics of the implicational fragment of the intuitionistic logic of proofs. Then we introduce a map from justification terms into |$\lambda $| terms, which can be viewed as a method of extracting the computational content of the justification terms. Then we examine the interpretation of Kreisel's addendum in justification logic along with the image of the resulting justification terms under our map. [ABSTRACT FROM AUTHOR]

Information transmission in social networks is riddled with issues of reliability and trustworthiness. One of the main sources of disinformation can be traced back to agents—human or artificial—whose political or cultural agenda is guided by conspiracy theories. Modelling and understanding the behaviour of such agents within social networks is therefore crucial to approach the disinformation problem. In the present paper, we formulate the logic (un)SecureNDsim* , equipped with a proof-theory and a relational semantics in which negative trust relations are defined formalizing the attitude of paranoid agents, i.e. agents distrusting any information originating from the authority and thereby spreading what can be characterized as the content of conspiracy theories. The logic is implemented in a multi-agent simulation aimed at analysing the effects of conspiracy theorists in networks of agents. In particular, we analyse consensus reaching scenarios and the ability of paranoid agents to induce the spread of potentially false information. [ABSTRACT FROM AUTHOR]

We offer tableaux systems for logics of essence and accident and logics of non-contingency, showing their soundness and completeness for Kripke semantics. We also show an interesting parallel between these logics based on the semantic insensitivity of the two non-normal operators by which these logics are expressed. [ABSTRACT FROM AUTHOR]

In this work we examine some of the possibilities of combining a simple probability operator with other modal operators, in particular with a belief operator. We will examine the semantics of two possible situations for expressing probabilistic belief or the lack of it, a simple subjective probability operator (SPO) versus the composition of a belief operator, plus an objective modal operator (BOP). We will study their interpretations in two probabilistic semantics: a relational Kripkean one and a variation of neighbourhood semantics, showing that the latter is able to represent the lack of probabilistic belief more directly, just with the SPO, whereas relational semantics needs the combination of BOP probability to represent lack of belief. [ABSTRACT FROM AUTHOR]

Labelled proof theory has been famously successful for modal logics by mimicking their relational semantics within deductive systems. Simpson in particular designed a framework to study a variety of intuitionistic modal logics integrating a binary relation symbol in the syntax. In this paper, we present a labelled sequent system for intuitionistic modal logics such that there is not only one but two relation symbols appearing in sequents: one for the accessibility relation associated with the Kripke semantics for normal modal logics and one for the pre-order relation associated with the Kripke semantics for intuitionistic logic. This puts our system in close correspondence with the standard birelational Kripke semantics for intuitionistic modal logics. As a consequence, it can be extended with arbitrary intuitionistic Scott–Lemmon axioms. We show soundness and completeness, together with an internal cut elimination proof, encompassing a wider array of intuitionistic modal logics than any existing labelled system. [ABSTRACT FROM AUTHOR]

We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms |${T}$|⁠ , |${P}$| and |${D}$| and for every |$n \geq 1$|⁠ , rule |${RD}_n^+$|⁠. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatization by a syntactic proof of cut elimination. Then, we define a terminating proof search strategy in the hypersequent calculi and show that it is optimal for coNP -complete logics. Moreover, we show that from every failed proof of a formula or hypersequent it is possible to directly extract a countermodel of it in the bi-neighbourhood semantics of polynomial size for coNP logics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube. [ABSTRACT FROM AUTHOR]

Full simply typed lambda calculus is the simply typed lambda calculus extended with product types and sum types. We propose a Kripke-style semantics for full simply typed lambda calculus. We then prove soundness and completeness of type assignment in full simply typed lambda calculus with respect to the proposed semantics. The key point in the proof of completeness is the notion of a canonical model. [ABSTRACT FROM AUTHOR]

We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On one hand, it is well known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e. a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on 'natural' propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted. [ABSTRACT FROM AUTHOR]

Dynamic epistemic logic is a logic that is aimed at formally expressing how a person's knowledge changes. We provide a cut-free labelled sequent calculus (⁠|$\textbf{GDEL}$|⁠) on the background of existing studies of Hilbert-style axiomatization |$\textbf{HDEL}$| of dynamic epistemic logic and labelled calculi for public announcement logic. We first show that the |$cut$| rule is admissible in |$\textbf{GDEL}$| and show that |$\textbf{GDEL}$| is sound and complete for Kripke semantics. Moreover, we show that the basis of |$\textbf{GDEL}$| is extended from modal logic K to other familiar modal logics including S5 with keeping the admissibility of cut, soundness and completeness. [ABSTRACT FROM AUTHOR]

Lie between classical logic and Brouwer-Heyting intuitionistic logic proposed in the first third of Rewriting logic is both a logical framework where many logics can be naturally represented and a Represented in the rewriting logic framework symbolic methods are of great importance. Is however limited to finitary logics, i.e. logics with Hilbert-style axiomatization using finitary rules Besides the better-known Nelson's logic and paraconsistent Nelson's logic, in 'Negation and. [Extracted from the article]

This paper explores the relationship between dynamic and truth conditional semantics for epistemic modals. It provides a generalization of a standard dynamic update semantics for modals. This new semantics derives a Kripke semantics for modals and a standard dynamic semantics for modals as special cases. The semantics allows for new characterizations of a variety of principles in modal logic, including the inconsistency of 'p and might not p'. Finally, the semantics provides a construction procedure for transforming any truth conditional semantics for modals into a dynamic semantics for modals with similar properties. [ABSTRACT FROM AUTHOR]

We study basic propositional logic |$\operatorname{\textbf{(BPC)}}$| augmented with the law of the weak excluded middle |$\operatorname{\textbf{(WEM)}}$|⁠, i.e. |$\operatorname{\textbf{BPW}} = \operatorname{\textbf{BPC}} + \operatorname{\textbf{WEM}}$|⁠. We show that the variety of the algebraic models of |$\operatorname{\textbf{BPW}}$| is canonical, and its Kripke completeness is proved via cononicity. Moreover, it is also proved that |$\operatorname{\textbf{BPW}}$| has the finite model property and is decidable. It is shown that |$\operatorname{\textbf{BPC}}$| and |$\operatorname{\textbf{BPW}}$| have the same behaviour on the |$\bot $| -free formulas and that |$\operatorname{\textbf{CPC}}$| and |$\operatorname{\textbf{BPW}}$| have the same behaviour on the negated formulas. [ABSTRACT FROM AUTHOR]

We can look at a first-order (or propositional) intuitionistic Kripke model as an ordered set of classical models. In this paper, we show that for a finite-depth Kripke model in an arbitrary first-order language or propositional language, local (classical) truth of a formula is equivalent to non-classical truth (truth in the Kripke semantics) of a Friedman's translation of that formula, i.e. |$ \alpha \Vdash A^\rho \Leftrightarrow \mathfrak{M}_\alpha \models A$|⁠. We introduce some applications of this fact. We extend the result of Ardeshir and Hesaam (2002, Math. Logic Quart. 48, 391–395) and show that semi-narrow Kripke models of Heyting Arithmetic |$ \textsf{HA} $| are locally |$ \textsf{PA} $|⁠. [ABSTRACT FROM AUTHOR]

The classification-cum-explanation Kripke assigns to rigidity requires the notion to apply to singular and general terms alike. But Kripke's own notion of rigidity is tailor-made for singular terms, and an extensive debate has not secured a general notion of rigidity apt to provide the classification-cum-explanation Kripke aims for. I propose that we look for a Kripkean alternative to generalized rigidity. I argue that on Kripkean premises, natural kind terms and proper names belong to the meta-semantic category of paradigm terms. I also argue that the property of being a paradigm term effects precisely the classification-cum-explanation Kripke attributes to rigidity. I conclude that Kripkeans should acknowledge that being a paradigm term, rather than designating rigidly, plays the role Kripke delineates, and that we can stop worrying about generalizing rigidity. Kripke's notion works fine for singular terms. And there is no need for rigidity to apply beyond these. [ABSTRACT FROM AUTHOR]

This article studies two ways of updating neighbourhood models: updating by taking the intersection of the neighbourhoods with the announced proposition, and updating by selecting all the neighbourhoods that can entail the announced proposition. For each of these two ways, we establish reduction axioms, some model-theoretic results and algebraic semantics for public announcement logic, and dynamic epistemic logic of product update. We also study various concepts of group knowledge, including general knowledge, common knowledge, and distributed knowledge over neighbourhood models. [ABSTRACT FROM AUTHOR]

In this paper, we answer a natural question concerning the relation between forcing and satisfaction in Kripke models of intuitionistic predicate logic. We define a class of formulas denoted by $$\mathsf{R}^{*}$$, with the property that forcing of any $$\mathsf{R}^{*}$$ -formula in a node of a Kripke model of intuitionistic predicate logic implies its satisfaction in the classical structure attached to that node. We also prove that any formula with this property is an $$\mathsf{R}^{*}$$ -formula. [ABSTRACT FROM AUTHOR]

The Stone duality for bounded lattices by this author, with J.M. Dunn, is lifted in this article to a duality for lattices with operators. The dual frames of lattice expansions are two-sorted frames (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y), further equipped with an (n + 1)-ary relation $$\textit{R}_{\delta }$$ and a dual relation $$\textit{R}^{\partial }_{\delta }$$ for each n -ary lattice operator of some distribution type $$\delta $$. The closures of the generalized image operators generated by the relations are shown to be precisely the $$\sigma $$ -extensions of the corresponding lattice operators and thus the full complex algebra of Galois-stable sets of the frame constitutes a concrete canonical extension of the lattice expansion. Thereby, the results presented in this article extend to the non-distributive case the classical Jónsson–Tarski results for Boolean algebras with operators and their extension to mere distributive lattices with operators. Consequently, the duality based approach to relational logic semantics is extended here to the case of logics dropping distribution. As an application example, we model the full BCK calculus. Both plain Kripke-type (two-sorted) frame semantics, as well as general (two-sorted) frame semantics are presented, the distinctive feature of the latter choice being that the interpretation of additional lattice operators (such as modal operators) is typically verbatim the same as in the distributive case, which is desirable in intended applications (such as temporal, or dynamic extensions of non-distributive lattice logic). [ABSTRACT FROM AUTHOR]

The normal modal system S 4. 2 is a useful tool in Epistemic Logic: it has been advocated as the ‘ correct ’ logic of knowledge by Lenzen (1979, Erkenntnis , 14, 33–56) and Stalnaker (2006, Philos. Stud., 128, 169–199) and it has been declared as the logic of justified true belief (JTB) by Voorbraak (1993, PhD Thesis). Goldblatt (1980, Stud. Logica , 39, 219–236) has proved that S 4. 2 is the temporal logic of relativistic spacetime , in particular it is the logic of the Diodorean modality when time is modelled by $$\mathbb{T}^{4}$$ , the four-dimensional Minkowskian geometry forming the basis of Einstein’s theory of special relativity. More recently, J. D. Hamkins and B. Löwe (2008, Trans. Am. Math. Soc., 360, 1793–1817) have employed S 4. 2 in the metamathematics of Set Theory and in particular in the study of Paul Cohen’s method of forcing : S 4. 2 is exactly the modal logic of forcing , the collection of valid principles emerging when $$\Box \varphi $$ is interpreted as ‘ $$\varphi $$ is true in every forcing extension’. In this survey paper we provide a guided tour to S 4. 2 focusing on its epistemic interpretation and its model-theoretic characterizations. We briefly review the results of Lenzen (1979, Erkenntnis , 14, 33–56) and Stalnaker (2006, Philos. Stud., 128, 169–199) and provide an epistemic interpretation of S 4. 2 models. We survey the frame completeness results for S 4. 2 , contributing also a new simple proof that it is determined by the class of partial pre-orders with a final cluster. En route , we provide a quick glimpse to other semantics for this modal system. Our goal is to provide a useful guide to researchers interested in the basic facts about the modal logic S 4. 2. [ABSTRACT FROM AUTHOR]

Dynamic Epistemic Logic (DEL) can model complex information scenarios in a way that appeals to logicians. However, existing DEL implementations are ad-hoc, so we do not know how the framework really performs. For this purpose, we want to hook up with the best available model checking and SAT techniques in computational logic. We do this by first providing a bridge: a new faithful representation of DEL models as so-called knowledge structures that allow for symbolic model checking. For more complex epistemic change we introduce knowledge transformers analogous to action models. Next, we show that we can now solve well-known benchmark problems in epistemic scenarios much faster than with existing methods for DEL. We also compare our approach to model checking for temporal logics. Finally, we show that our method is not just a matter of implementation, but that it raises significant issues about logical representation and update. [ABSTRACT FROM AUTHOR]

Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication usually called 'exclusion'. A standard-style sequent calculus for this logic is easily obtained by extending multiple-conclusion sequent calculus for intuitionistic logic with exclusion rules dual to the implication rules (in particular, the exclusion-left rule restricts the premise to be single-assumption). However, similarly to standard-style sequent calculi for non-classical logics like S5 , this calculus is incomplete without the cut rule. Motivated by the problem of proof search for propositional bi-intuitionistic logic (BiInt), various cut-free calculi with extended sequents have been proposed, including (i) a calculus of nested sequents by Goré et al. which includes rules for creation and removal of nests (called 'nest rules', resp. 'unnest rules') and (ii) a calculus of labelled sequents by the authors, derived from the Kripke semantics of BiInt , which includes 'monotonicity rules' to propagate truth/falsehood between accessible worlds. In this paper, we develop a proof-theoretic study of these three sequent calculi for BiInt grounded on translations between them. We start by establishing the basic meta-theory of the labelled calculus (including cut-admissibility), and use then the translations to obtain results for the other two calculi. The translation of the nested calculus into the standard-style calculus explains how the unnest rules encapsulate cuts. The translations between the labelled and the nested calculi reveal the two formats to be very close, despite the former incorporating semantic elements, and the latter being syntax-driven. Indeed, we single out (i) a labelled calculus whose sequents have a 'label in focus' and which includes 'refocusing rules' and (ii) a nested calculus with monotonicity and refocusing rules, and prove these two calculi to be isomorphic (in a bijection both at the level of sequents and at the level of derivations). [ABSTRACT FROM AUTHOR]

This article presents results in modal correspondence theory for possibility semantics, a generalization of Kripke semantics introduced by Humberstone and later developed by Holliday and others. First, it is shown by an algebraic argument that all Sahlqvist formulae have first-order correspondents over possibility frames. Second, model-theoretic aspects of correspondence over possibility frames are examined in the style of van Benthem. [ABSTRACT FROM AUTHOR]

In this article, we give a cut-free sequent calculus LLn for the logic SIL(Ln) characterized by the family of finite linear Kripke frames whose heights are at most n. We also give a proof-theoretic decision procedure for SIL(Ln) by the use of LLn. [ABSTRACT FROM AUTHOR]

Positive modal logic, as introduced by Dunn in 1995, is the negation-free fragment of the standard modal logic of all Kripke frames. Positive coalgebraic logic, introduced by the authors in a previous work, expands the above result from Kripke frames to more general transition systems, namely to coalgebras of weak-pullback preserving functors. We show that this construction is both modular and uniform in the functor giving the type of coalgebra. More precisely, we formalize both Set and Pos-based coalgebraic modal logic as institutions, and we exhibit a morphism of institutions between them giving the positive fragment of coalgebraic modal logic. [ABSTRACT FROM AUTHOR]

In this article, a non-standard informational semantics for superintuitionistic modal logics is introduced and studied. It is based on algebraic structures that are interpreted as algebras of information states. The proposed semantics combines into one framework various features of standard relational and algebraic semantics. Especially, the connection to the algebraic semantics is explored in detail. The framework can be viewed as a generalization of inquisitive semantics that enables us to add inquisitive disjunction to any superintuitionistic modal propositional logic. [ABSTRACT FROM AUTHOR]

We propose stratified institutions (a decade old generalized version of the theory of institutions of Goguen and Burstall) as a fully abstract model theoretic approach to modal logic. This allows for a uniform treatment of model theoretic aspects across the great multiplicity of contemporary modal logic systems. Moreover, Kripke semantics (in all its manifold variations) is captured in an implicit manner free from the sometimes bulky aspects of explicit Kripke structures, also accommodating other forms of concrete semantics for modal logic systems. The conceptual power of stratified institutions is illustrated with the development of a modal ultraproducts method that is independent of the concrete details of the actual modal logical systems. Consequently, a wide array of compactness results in concrete modal logics may be derived easily. [ABSTRACT FROM AUTHOR]

We give a separable axiomatization of the Gabbay-de Jongh logic which is different from the Gabbay and de Jongh's one by using an algebraic method with Jankov's characteristic formula (Jankov, 1969, USSR-Izvestiya, 3 17-35). We also show that a variant of the Gabbay and de Jongh's well-known axiomatization is not separable. [ABSTRACT FROM AUTHOR]

A system WF of subintuitionistic logic is introduced, weaker than Corsi's basic subintuitionistic system F. A derivation system with and without hypotheses is given in line with the authors' derivation system for F. A neighbourhood semantics is introduced with a somewhat more complex definition than the neighbourhood semantics for non-normal modal logics. Completeness is proved for WF with respect to this neighbourhood semantics, and similarly for some logics between WF and F which characterize nice frame classes. The study by the authors of the conservativity of IPC over F with respect to some classes of implications is extended to WF, and shows clearly the difference in strength between the two logics. Study of translations of these weak subintuitionistic logics into non-normal modal logics turned out to be hard because of the difference between their respective neighbourhood structures and leaves us with some open problems. [ABSTRACT FROM AUTHOR]

We extend unified correspondence theory to Kripke frames with impossible worlds and their associated regular modal logics. These are logics the modal connectives of which are not required to be normal: only the weaker properties of additivity ◇x∨◇y=◇(x∨y) and multiplicativity □x∧□y=□(x∧y) are required. Conceptually, it has been argued that their lacking necessitation makes regular modal logics better suited than normal modal logics at the formalization of epistemic and deontic settings. From a technical viewpoint, regularity proves to be very natural and adequate for the treatment of algebraic canonicity Jónsson-style. Indeed, additivity and multiplicativity turn out to be key to extend Jónsson's original proof of canonicity to the full Sahlqvist class of certain regular distributive modal logics naturally generalizing distributive modal logic. Most interestingly, additivity and multiplicativity are key to Jónsson-style canonicity also in the original (i.e. normal) DML. Our contributions include: the definition of Sahlqvist inequalities for regular modal logics on a distributive lattice propositional base; the proof of their canonicity following Jónsson's strategy; the adaptation of the algorithm ALBA to the setting of regular modal logics on two non-classical (distributive lattice and intuitionistic) bases; the proof that the adapted ALBA is guaranteed to succeed on a syntactically defined class which properly includes the Sahlqvist one; finally, the application of the previous results so as to obtain proofs, alternative to Kripke's, of the strong completeness of Lemmon's epistemic logics E2-E5 with respect to elementary classes of Kripke frames with impossible worlds. [ABSTRACT FROM AUTHOR]

In this article, we study modal extensions of Product fuzzy logic with both algebraic semantics and relational semantics based on Kripke structures with crisp accessibility relations, when the underlying product fuzzy logic is expanded with truthconstants, theΔoperator and with two infinitary inference rules. We provide completeness results for both kinds of semantics. Finally, we also consider a generalization of possibilistic logic evaluated over product algebras. [ABSTRACT FROM AUTHOR]

We consider the language with temporal operators G, H and coderivative operator [d]. G means 'always in the future', H means 'always in the past' and [d] means 'around now but possibly not now'. We finitely axiomatize the logic of this language interpreted over the real line, we prove that this logic is PSPACE-complete, and that it is neither strongly complete nor Kripke-complete (and consequently that it fails the finite model property). [ABSTRACT FROM AUTHOR]

Primal infon logic (PIL) was introduced in 2009 in the framework of policy and trust management. In the meantime, some generalizations appeared, and there have been some changes in the syntax of the basic PIL. This article is on the basic PIL, and one of our purposes is to 'institutionalize' the changes.We prove a small-model theorem for the propositional fragment of basic primal infon logic (PPIL), give a simple proof of the PPIL locality theorem and present a linear-time decision algorithm (announced earlier) for PPIL in a form convenient for generalizations. For the sake of completeness, we cover the universal fragment of basic PIL. We wish that this article becomes a standard reference on basic PIL. [ABSTRACT FROM AUTHOR]

The standard first-order reading of modality does not bind individual variables, i.e. if x is free in F(x), then x remains free in F(x). Accordingly, if stands for 'provable in arithmetic,' ∀x F(x) states that F(n) is provable for any given value of n=0,1,2,...; this corresponds to a de re reading of modality. The other, de dicto meaning of F(x), suggesting that F(x) is derivable as a formula with a free variable x, is not directly represented by a modality, though, semantically, it could be approximated by compound constructions, e.g. ∀xF(x). We introduce the first-order logic FOS4* in which modalities can bind individual variables and, in particular, can directly represent both de re and de dicto modalities. FOS4* extends first-order S4 and is the natural forgetful projection of the first-order logic of proofs FOLP. The same method of introducing binding modalities obviously works for other modal logics as well. [ABSTRACT FROM AUTHOR]

We consider intermediate predicate logics defined by fixed well-ordered (or dually well-ordered) linear Kripke frames with constant domains where the order-type of the well-order is strictly smaller than ωω. We show that two such logics of different order-type are separated by a first-order sentence using only one monadic predicate symbol. Previous results by Minari, Takano and Ono, as well as the second author, obtained the same separation but relied on the use of predicate symbols of unbounded arity. [ABSTRACT FROM AUTHOR]

We study the expressive power of fragments of inclusion and independence logic defined by restricting the number k of universal quantifiers in formulas. Assuming the so-called strict semantics for these logics, we relate these fragments of inclusion and independence logic to sublogics ESOf (k∀) of existential second-order logic, which in turn are known to capture the complexity classes NTIMERAM(nk). [ABSTRACT FROM AUTHOR]

Motivated from reasoning in rough set theory, we have introduced and studied proof-theoretic properties of a class of logics L(T,I) in the first part of this article. These are logics with models based on 'dynamic I spaces', which are finite sequences of Kripke I frames with a common domain, I being any of the normal modal systems K, K4, T, B, S4, KTB, KB4, S5. This article presents the second part of the study on L(T,I), where we focus on decidability issues. [ABSTRACT FROM AUTHOR]

'Binary Routley-semantics' (bR-semantics) differs from Routley-Meyer semantics (RM-semantics) mainly in that the accessibility relation is binary instead of ternary. Intuitionistic bR-semantics is essentially defined when introducing the Routley operator (used for modelling negation) in Kripke models for positive intuitionistic logic. The aim of this article is to define the minimal logic in intuitionistic bR-semantics, HM, as well as a number of its extension. HM is minimal in intuitionistic bR-semantics in the same sense in which Sylvan and Plumwood's BM is minimal in RM-semantics with a set of designated points. [ABSTRACT FROM AUTHOR]

We consider the Gödel bi-modal logic determined by fuzzy Kripke models where both the propositions and the accessibility relation are infinitely valued over the standard Gödel algebra [0,1], and prove strong completeness of the Fischer Servi intuitionistic modal logic IK plus the prelinearity axiom with respect to this semantics. We axiomatize also the bi-modal analogues of classical T, S4 and S5, obtained by restricting to models over frames satisfying the [0,1]-valued versions of the structural properties which characterize these logics. As an application of the completeness theorems we obtain a representation theorem for bi-modal Gödel algebras. [ABSTRACT FROM AUTHOR]

We explore when finite epistemic models are definable up to simulation or bisimulation, either over the basic multi-agent epistemic language L or over its extension LC with common knowledge operators. Our main results are the following:(1) Bisimulation to finite epistemic states (i.e. pointed models) is not definable in L.(2) Global bisimulation to finite epistemic models is definable using model validity in L.(3) Simulation by finite epistemic states is not always definable, even over LC.(4) We identify a class of epistemic models called well-multifounded for which simulation to finite epistemic states is definable over L.The first two results contrasts with that of van Benthem that finite epistemic states are definable up to bisimulation in LC by so-called characteristic formulae.Our results have applications in the logical specification of multi-agent systems. Characteristic formulae, which define states up to bisimulation, fully capture all modally definable properties of models, and can for example be used to give complete calculi for validity over fixed structures. Meanwhile, simulation is a ‘substructure’ relation that is appropriate for epistemic logic, as it preserves positive existential formulae. Simulation formulae are useful in giving a dynamic description of models subject to epistemic actions. [ABSTRACT FROM PUBLISHER]