We consider the two‐variable fragment of first‐order logic with counting, subject to the stipulation that a single distinguished binary predicate be interpreted as an equivalence. We show that the satisfiability and finite satisfiability problems for this logic are both NExpTime‐complete. We further show that the corresponding problems for two‐variable first‐order logic with counting and two equivalences are both undecidable.

We explore the finite model theory of the characterisation theorems for modal and guarded fragments of first-order logic over transition systems and relational structures of width two. A new construction of locally acyclic bisimilar covers provides a useful analogue of the well known tree-like unravellings that can be used for the purposes of finite model theory. Together with various other finitary bisimulation respecting model transformations, and Ehrenfeucht–Fraïssé game arguments, these covers allow us to upgrade finite approximations for full bisimulation equivalence (...) towards approximations for elementary equivalence. These techniques are used to prove several ramifications of the van Benthem–Rosen characterisation theorem of basic modal logic for refinements of ordinary bisimulation equivalence, both in the sense of classical and of finite model theory. (shrink)

Consider any logical system, what is its natural repertoire of logical operations? This question has been raised in particular for first-order logic and its extensions with generalized quantifiers, and various characterizations in terms of semantic invariance have been proposed. In this paper, our main concern is with modal and dynamic logics. Drawing on previous work on invariance for first-order operations, we find an abstract connection between the kind of logical operations a system uses and the kind of invariance conditions the (...) system respects. This analysis yields (a) a characterization of invariance and safety under bisimulation as natural conditions for logical operations in modal and dynamic logics, and (b) some new transfer results between first-order logic and modal logic. (shrink)

In van Benthem (2008), van Benthem proposes a dynamic consequence relation defined as${\psi _1}, \ldots,{\psi _n}{ \models ^d}\phi \,{\rm{iff}}{ \models ^{pa}}[{\psi _1}] \ldots [{\psi _n}]\phi,$where the latter denotes consequence in public announcement logic, a dynamic epistemic logic. In this paper we investigate the structural properties of a conditional dynamic consequence relation$\models _{\rm{\Gamma }}^d$extending van Benthem’s proposal. It takes into account a set of background conditions Γ, inspired by Makinson (2003) wherein Makinson calls this reasoning ‘modulo’ a set Γ. In the (...) presence of common knowledge, conditional dynamic consequence is definable from (unconditional) dynamic consequence. An open question is whether dynamic consequence is compact. We further investigate a dynamic consequence relation for soft instead of hard announcements. Surprisingly, it shares many properties with (hard) dynamic consequence. Dynamic consequence relations provide a novel perspective on reasoning about protocols in multi-agent systems. (shrink)

A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬BOXp, read as “p is true but you do not know p.” Not only is the (...) Moore sentence unsuccessful, it is self-refuting, for it never remains true as described. We show that in logics of knowledge and belief for a single agent (extended by S5), Moorean phenomena are the source of all self-refutation; moreover, in logics for an introspective agent (extending KD45), Moorean phenomena are the source of all unsuccessfulness as well. This is a distinctive feature of such logics, for with a non-introspective agent or multiple agents, non-Moorean unsuccessful formulas appear. We also consider how successful and self-refuting formulas relate to the Cartesian and learnable formulas, which have been discussed in connection with Fitch’s “paradox of knowability.” We show that the Cartesian formulas are exactly the formulas that are not eventually self-refuting and that not all learnable formulas are successful. In an appendix, we give syntactic characterizations of the successful and the self-refuting formulas. (shrink)

There are several known Lindström-style characterization results for basic modal logic. This paper proves a generic Lindström theorem that covers any normal modal logic corresponding to a class of Kripke frames definable by a set of formulas called strict universal Horn formulas. The result is a generalization of a recent characterization of modal logic with the global modality. A negative result is also proved in an appendix showing that the result cannot be strengthened to cover every first-order elementary class of (...) frames. This is shown by constructing an explicit counterexample. (shrink)

We study first-order logic with two variables FO² and establish a small substructure property. Similar to the small model property for FO² we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO² under constraints that require several binary relations to be interpreted as equivalence relations. With a single equivalence relation, FO² has the finite model property and is complete for non-deterministic (...) exponential time, just as for plain FO² With two equivalence relations, FO² does not have the finite model property, but is shown to be decidable via a construction of regular models that admit finite descriptions even though they may necessarily be infinite. For three or more equivalence relations, FO² is undecidable. (shrink)

Additionally, the text shows how to develop computationally feasible methods to mesh with this framework.

Logic is not just about single-agent notions like reasoning, or zero-agent notions like truth, but also about communication between two or more people. What we tell and ask each other can be just as 'logical' as what we infer in Olympic solitude. We show how such interactive phenomena can be studied systematically by merging epistemic and dynamic logic.

This is a survey article on algebraic logic. It gives a historical background leading up to a modern perspective. Central problems in algebraic logic (like the representation problem) are discussed in connection to other branches of logic, like modal logic, proof theory, model-theoretic forcing, finite combinatorics, and Gödel’s incompleteness results. We focus on cylindric algebras. Relation algebras and polyadic algebras are mostly covered only insofar as they relate to cylindric algebras, and even there we have not told the whole story. (...) We relate the algebraic notion of neat embeddings (a notion special to cylindric algebras) to the metalogical ones of provability, interpolation and omitting types in variants of first logic. Another novelty that occurs here is relating the algebraic notion of atom-canonicity for a class of boolean algebras with operators to the metalogical one of omitting types for the corresponding logic. A hitherto unpublished application of algebraic logic to omitting types of first order logic is given. Proofs are included when they serve to illustrate certain concepts. Several open problems are posed. We have tried as much as possible to avoid exploring territory already explored in the survey articles of Monk [93] and Németi [97] in the subject. (shrink)

The paper considers the set ML 1 of first-order polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability criterion for formulas in ML 1 , which reduces the modal satisfiability to the classical one. The criterion is then used to single out a number of new, in a sense optimal, decidable fragments of various modal predicate logics.

Controlled languages are regimented fragments of natural languagedesigned to make the processing of natural language more efficient andreliable. This paper defines a controlled language, E2V, whose principalgrammatical resources include determiners, relative clauses, reflexivesand pronouns. We provide a formal syntax and semantics for E2V, in whichanaphoric ambiguities are resolved in a linguistically natural way. Weshow that the expressive power of E2V is equal to that of thetwo-variable fragment of first-order logic. It follows that the problemof determining the satisfiability of a set (...) of E2V sentences is NEXPTIMEcomplete. We show that E2V can be extended in various ways withoutcompromising these complexity results; however, relaxing our policy onanaphora resolution renders the satisfiability problem for E2Vundecidable. Finally, we argue that our results have a bearing on thebroader philosophical issue of the relationship between natural andformal languages. (shrink)

By a fragment of a natural language we mean a subset of thatlanguage equipped with semantics which translate its sentences intosome formal system such as first-order logic. The familiar conceptsof satisfiability and entailment can be defined for anysuch fragment in a natural way. The question therefore arises, for anygiven fragment of a natural language, as to the computational complexityof determining satisfiability and entailment within that fragment. Wepresent a series of fragments of English for which the satisfiabilityproblem is polynomial, NP-complete, EXPTIME-complete,NEXPTIME-complete (...) and undecidable. Thus, this paper represents a casestudy in how to approach the problem of determining the logicalcomplexity of various natural language constructions. In addition, wedraw some general conclusions about the relationship between naturallanguage and formal logic. (shrink)

Abstract State Machines (ASMs) provide a formal method for transparent design and specification of complex dynamic systems. They combine advantages of informal and formal methods. Applications of this method motivate a number of computability and decidability problems connected to ASMs. Such problems result for example from the area of verifying properties of ASMs. Their high expressive power leads rather directly to undecidability respectively uncomputability results for most interesting problems in the case of unrestricted ASMs. Consequently, it is rather natural to (...) ask whether there exist expressive classes of ASMs for which we can prove positive decidability and computability results. In this work, we introduce such a class of ASMs. The concept is similar to the one of the guarded fragment of first-order logic. We analyze the expressive power of this class and prove that it is stronger than Datalog LITE and the guarded fragment of first-order fixed point logic. Some decidability and computability results have been proven in earlier works. (shrink)

We expand first order models with a tolerance relation on thedomain. Intuitively, two elements stand in this relation if they arecognitively close for the agent who holds the model. This simplenotion turns out to be very powerful. It leads to a semanticcharacterization of the guarded fragment of Andréka, van Benthemand Németi, and highlights the strong analogies between modallogic and this fragment. Viewing the resulting logic – tolerance logic– dynamically it is a resource-conscious information processingalternative to classical first order logic. The (...) differences areindicated by several examples. (shrink)

We provide a canonical construction of conformal covers for finite hypergraphs and present two immediate applications to the finite model theory of relational structures. In the setting of relational structures, conformal covers serve to construct guarded bisimilar companion structures that avoid all incidental Gaifman cliques-thus serving as a partial analogue in finite model theory for the usually infinite guarded unravellings. In hypergraph theoretic terms, we show that every finite hypergraph admits a bisimilar cover by a finite conformal hypergraph. In terms (...) of relational structures, we show that every finite relational structure admits a guarded bisimilar cover by a finite structure whose Gaifman cliques are guarded. One of our applications answers an open question about a clique constrained strengthening of the extension property for partial automorphisms (EPPA) of Hrushovski, Herwig and Lascar. A second application provides an alternative proof of the finite model property (FMP) for the clique guarded fragment of first-order logic CGF, by reducing (finite) satisfiability in CGF to (finite) satisfiability in the guarded fragment, GF. (shrink)

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

Preservation and interpolation results are obtained for L ∞ω and sublogics $\mathscr{L} \subseteq L_{\infty\omega}$ such that equivalence in L can be characterized by suitable back-and-forth conditions on sets of partial isomorphisms.

Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the `finite base property' and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.

Sentences \(\varphi\) and \(\psi\) are _cognitive synonyms_ for one when they play the same role in one’s cognitive life. The notion is pervasive (Sect. 1 ), but elusive: it is bound to be hyperintensional (Sect. 2 ), but excessive fine-graining would trivialize it and there are reasons for some coarse-graining (Sect. 2.1 ). Conceptual limitations stand in the way of a natural algebra (Sect. 2.2 ), and it should be sensitive to subject matters (Sect. 2.3 ). A cognitively adequate individuation (...) of content may be intransitive (Sect. 3 ) due to ‘dead parrot’ series: sequences of sentences \(\varphi _1, \ldots, \varphi _n\) where adjacent \(\varphi _i\) and \(\varphi _{i+1}\) are cognitive synonyms while \(\varphi _1\) and \(\varphi _n\) are not (Sect. 3.1 ). Finding an intransitive account is hard: Fregean equipollence won’t do (Sect. 3.2 ) and a result by Leitgeb shows that it wouldn’t satisfy a minimal compositionality principle (Sect. 3.3 )._Sed contra_, there are reasons for transitivity, too (Sect. 3.4 ). In Sect. 4, we come up with a formal semantics capturing this jumble of desiderata, thereby showing that the notion is coherent. In Sect. 5, we re-assess the desiderata in its light. (shrink)

In this paper we introduce the modelwise interpolation property of a logic that states that whenever \(\models\phi\to\psi\) holds for two formulas \(\phi\) and \(\psi\), then for every model \(\mathfrak{M}\) there is an interpolant formula \(\chi\) formulated in the intersection of the vocabularies of \(\phi\) and \(\psi\), such that \(\mathfrak{M}\models\phi\to\chi\) and \(\mathfrak{M}\models\chi\to\psi\), that is, the interpolant formula in Craig interpolation may vary from model to model. We compare the modelwise interpolation property with the standard Craig interpolation and with the local interpolation (...) property by discussing examples, most notably the finite variable fragments of first order logic, and difference logic. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic. (shrink)

In this paper, we introduce a variant of second-order propositional modal logic interpreted on general (or Henkin) frames, \(SOPML^{\mathcal {H}}\), and present a decidable fragment of this logic, \(SOPML^{\mathcal {H}}_{dec}\), that preserves important expressive capabilities of \(SOPML^{\mathcal {H}}\). \(SOPML^{\mathcal {H}}_{dec}\) is defined as a _modal loosely guarded fragment_ of \(SOPML^{\mathcal {H}}\). We demonstrate the expressive power of \(SOPML^{\mathcal {H}}_{dec}\) using examples in which modal operators obtain (a) the epistemic interpretation, (b) the dynamic interpretation. \(SOPML^{\mathcal {H}}_{dec}\) partially satisfies the principle of (...) non-Fregean logic: two different _atomic_ propositions with the same truth value can have different contents. In \(SOPML^{\mathcal {H}}_{dec}\), we also define _relating connectives_ and show that the _weak Boethius’ Thesis_ built using these connectives is a valid formula of \(SOPML^{\mathcal {H}}_{dec}\). (shrink)

I examine notions of equivalence between logics (understood as languages interpreted model-theoretically) and develop two new ones that invoke not only the algebraic but also the string-theoretic structure of the underlying language. As an application, I show how to construe modal operator languages as what might be called typographical notational variants of _bona fide_ first-order languages.

Fix 2 < n < ω and let C A n denote the class of cylindric algebras of dimension n. Roughly, C A n is the algebraic counterpart of the proof theory of first-order logic restricted to the first n var...

In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these (...) accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic. (shrink)

This paper develops and refines the suggestion that logical systems are conceptual artefacts that are the outcome of a design-process by exploring how a constructionist epistemology and meta-philosophy can be integrated within the philosophy of logic.

Letn > 2. A weakly representable relation algebra that is not strongly representable is constructed. It is proved that the set of all n by n basic matrices forms a cylindric basis that is also a weakly but not a strongly representable atom structure. This gives an example of a binary generated atomic representable cylindric algebra with no complete representation. An application to first order logic is given.

This book illustrates the program of Logical-Informational Dynamics. Rational agents exploit the information available in the world in delicate ways, adopt a wide range of epistemic attitudes, and in that process, constantly change the world itself. Logical-Informational Dynamics is about logical systems putting such activities at center stage, focusing on the events by which we acquire information and change attitudes. Its contributions show many current logics of information and change at work, often in multi-agent settings where social behavior is essential, (...) and often stressing Johan van Benthem's pioneering work in establishing this program. However, this is not a Festschrift, but a rich tapestry for a field with a wealth of strands of its own. The reader will see the state of the art in such topics as information update, belief change, preference, learning over time, and strategic interaction in games. Moreover, no tight boundary has been enforced, and some chapters add more general mathematical or philosophical foundations or links to current trends in computer science. The theme of this book lies at the interface of many disciplines. Logic is the main methodology, but the various chapters cross easily between mathematics, computer science, philosophy, linguistics, cognitive and social sciences, while also ranging from pure theory to empirical work. Accordingly, the authors of this book represent a wide variety of original thinkers from different research communities. And their interconnected themes challenge at the same time how we think of logic, philosophy and computation. Thus, very much in line with van Benthem's work over many decades, the volume shows how all these disciplines form a natural unity in the perspective of dynamic logicians (broadly conceived) exploring their new themes today. And at the same time, in doing so, it offers a broader conception of logic with a certain grandeur, moving its horizons beyond the traditional study of consequence relations. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

The guarded fragment with transitive guards, [GF+TG], is an extension of the guarded fragment of first-order logic, GF, in which certain predicates are required to be transitive, transitive predicate letters appear only in guards of the quantifiers and the equality symbol may appear everywhere. We prove that the decision problem for [GF+TG] is decidable. Moreover, we show that the problem is in 2E. This result is optimal since the satisfiability problem for GF is 2E-complete 1719–1742). We also show that the (...) satisfiability problem for two-variable [GF+TG] is N-hard in contrast to GF with bounded number of variables for which the satisfiability problem is E-complete. (shrink)

In this paper, we introduce a new fragment of the first-order temporal language, called the monodic fragment, in which all formulas beginning with a temporal operator have at most one free variable. We show that the satisfiability problem for monodic formulas in various linear time structures can be reduced to the satisfiability problem for a certain fragment of classical first-order logic. This reduction is then used to single out a number of decidable fragments of first-order temporal logics and of two-sorted (...) first-order logics in which one sort is intended for temporal reasoning. Besides standard first-order time structures, we consider also those that have only finite first-order domains, and extend the results mentioned above to temporal logics of finite domains. We prove decidability in three different ways: using decidability of monadic second-order logic over the intended flows of time, by an explicit analysis of structures with natural numbers time, and by a composition method that builds a model from pieces in finitely many steps. (shrink)

We show that the satisfiability problem for the monodic guarded, loosely guarded, and packed fragments of first-order temporal logic with equality is 2Exptime-complete for structures with arbitrary first-order domains, over linear time, dense linear time, rational number time, and some other classes of linear flows of time. We then show that for structures with finite first-order domains, these fragments are also 2Exptime-complete over real number time and hence over most of the commonly used linear flows of time, including the natural (...) numbers, integers, rationals, and any first-order definable class of linear flows of time. (shrink)

We prove that the existence of atomic models for countable atomic theories does not hold for Ln the first order logic restricted to n variables for finite n > 2. Our proof is algebraic, via polyadic algebras. We note that Lnhas been studied in recent times as a multi-modal logic with applications in computer science. 2000 MATHEMATICS SUBJECT CLASSIFICATION. 03C07, 03G15.

While second-order quantifiers have long been known to admit nonstandard, or interpretations, first-order quantifiers (when properly viewed as predicates of predicates) also allow a kind of interpretation that does not presuppose the full power-set of that interpretationgeneral” interpretations for (unary) first-order quantifiers in a general setting, emphasizing the effects of imposing various further constraints that the interpretation is to satisfy.

For any finite n 3 there are two atomic n-dimensional cylindric algebras with the same atom structure, with one representable, the other, not.Hence, the complex algebra of the atom structure of a representable atomic cylindric algebra is not always representable, so that the class RCAn of representable n-dimensional cylindric algebras is not closed under completions. Further, it follows by an argument of Venema that RCAn is not axiomatisable by Sahlqvist equations, and hence nor by equations where negation can only occur (...) in constant terms.Similar results hold for relation algebras. (shrink)

We give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order (...) logic as long as the number of variables available is > 2 and we have a binary relation symbol in our language. We also prove a stronger result to the effect that there is no finite upper bound for the extra variables needed in the witness formulas. This result further emphasizes the ongoing interplay between algebraic logic and first order logic. (shrink)

This note exploits back-and-forth characteristics to construct, using a single method, short proofs for ten classics of first-order and modal logic: interpolation theorems, preservation theorems, and Lindström's theorem.

Three notions of definability in multimodal logic are considered. Two are analogous to the notions of explicit definability and implicit definability introduced by Beth in the context of first-order logic. However, while by Beth’s theorem the two types of definability are equivalent for first-order logic, such an equivalence does not hold for multimodal logics. A third notion of definability, reducibility, is introduced; it is shown that in multimodal logics, explicit definability is equivalent to the combination of implicit definability and reducibility. (...) The three notions of definability are characterized semantically using (modal) algebras. The use of algebras, rather than frames, is shown to be necessary for these characterizations. (shrink)

In the 1970s Codd introduced the relational algebra, with operators selection, projection, union, difference and product, and showed that it is equivalent to first-order logic. In this paper, we show that if we replace in Codd’s relational algebra the product operator by the “semijoin” operator, then the resulting “semijoin algebra” is equivalent to the guarded fragment of first-order logic. We also define a fixed point extension of the semijoin algebra that corresponds to μGF.

It is shown that the many-dimensional modal logic K n , determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any $n > 2$ . On the other hand, K n is determined by a class of frames satisfying a single first-order sentence.

We prove ExpTime-membership of the satisfiability problem for loosely ∀-guarded first-order formulas with a bounded number of variables and an unbounded number of constants. Guarded fragments with constants are interesting by themselves and because of their connection to hybrid logic.

This chapter contains sections titled: Enriching Extensional Logic with Intensional Notions Changing Views of Modal Logic A Précis of Basic Modal Logic The Major Applications Fine‐Structure of Expressive Power System Combination: Action and Information Back to the Heartland Conclusion.

This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are discussed: continuous (...) dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games. (shrink)

In the literature, there have been several methods and definitions for working out whether two theories are “equivalent” or not. In this article, we do something subtler. We provide a means to measure distances between formal theories. We introduce two natural notions for such distances. The first one is that of axiomatic distance, but we argue that it might be of limited interest. The more interesting and widely applicable notion is that of conceptual distance which measures the minimum number of (...) concepts that distinguish two theories. For instance, we use conceptual distance to show that relativistic and classical kinematics are distinguished by one concept only. (shrink)

The paper presents a family of propositional epistemic logics such that languages of these logics are extended by quantification over modal operators or over agents of knowledge and extended by predicate symbols that take modal operators as arguments. Denote this family by \}\). There exist epistemic logics whose languages have the above mentioned properties :311–350, 1995; Lomuscio and Colombetti in Proceedings of ATAL 1996. Lecture Notes in Computer Science, vol 1193, pp 71–85, 1996). But these logics are obtained from first-order (...) modal logics, while a logic of \}\) can be regarded as a propositional multi-modal logic whose language includes quantifiers over modal operators and predicate symbols that take modal operators as arguments. Among the logics of \}\) there are logics with a syntactical distinction between two readings of epistemic sentences: de dicto and de re. We show the decidability of logics of \}\) with the help of the loosely guarded fragment of first-order logic. Namely, we generalize LGF to a higher-order decidable loosely guarded fragment. The latter fragment allows us to construct various decidable propositional epistemic logics with quantification over modal operators. The family of this logics coincides with \}\). There are decidable propositional logics such that these logics implicitly contain quantification over agents of knowledge, but languages of these logics are usual propositional epistemic languages without quantifiers and predicate symbols :345–378, 1993). Some logics of \}\) can be regarded as counterparts of logics defined in Grove and Halpern :345–378, 1993). We prove that the satisfiability problem for these logics of \}\) is Pspace-complete using their counterparts in Grove and Halpern :345–378, 1993). (shrink)

This paper investigates the “general” semantics for first-order logic introduced to Antonelli, 637–58, 2013): a sound and complete axiom system is given, and the satisfiability problem for the general semantics is reduced to the satisfiability of formulas in the Guarded Fragment of Andréka et al. :217–274, 1998), thereby showing the former decidable. A truth-tree method is presented in the Appendix.