ABSTRACT We introduce a two-dimensional temporal logic as a formalism which enables the description of both the history of a world and the evolution of an observer's views about the history. We apply such formalism to the description of certain problems that occur in historical database systems due to updates. The historical dimension describes the history of a world according to an observer's view at a certain moment in time. The transaction dimension describes the evolution of an observer's view; changes (...) can occur in the observer's view of the present and future as well as in his views of the past. The database is seens as an observer of the history of the world. Besides describing the history of the world, the database system can also execute temporal specifications in the form of temporal actions. The two-dimensional formalism is used to describe the effects of updates in the execution of temporal actions. The detection of some problematic effects and its implementation are discussed. (shrink)

We introduce a methodology whereby an arbitrary logic system L can be enriched with temporal features to create a new system T(L). The new system is constructed by combining L with a pure propositional temporal logic T (such as linear temporal logic with Since and Until) in a special way. We refer to this method as adding a temporal dimension to L or just temporalising L. We show that the logic system T(L) preserves several properties of the original temporal logic (...) like soundness, completeness, decidability, conservativeness and separation over linear flows of time. We then focus on the temporalisation of first-order logic, and a comparison is make with other first-order approaches to the handling of time. (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.

We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness (...) for the corresponding Hilbert-style axiomatization, but also in the development of a mosaic-based tableau system. We further consider extensions for dealing with the case when interactions between the two dimensions exist, thus covering a wide class of bundled Ockhamist branching-time logics, and present for them some partial results, such as a non-analytic version of the tableau system. (shrink)

The present paper studies formal properties of so-called modal information logics (MILs)—modal logics first proposed in (van Benthem 1996 ) as a way of using possible-worlds semantics to model a theory of information. They do so by extending the language of propositional logic with a binary modality defined in terms of being the supremum of two states. First proposed in 1996, MILs have been around for some time, yet not much is known: (van Benthem 2017, 2019 ) pose two central (...) open problems, namely (1) axiomatizing the two basic MILs of suprema on preorders and posets, respectively, and (2) proving (un)decidability. The main results of the first part of this paper are solving these two problems: (1) by providing an axiomatization [with a completeness proof entailing the two logics to be the same], and (2) by proving decidability. In the proof of the latter, an emphasis is put on the method applied as a heuristic for proving decidability ‘via completeness’ for semantically introduced logics; the logics lack the FMP w.r.t. their classes of definition, but not w.r.t. a generalized class. These results are build upon to axiomatize and prove decidable the MILs attained by endowing the language with an ‘informational implication’—in doing so a link is also made to the work of (Buszkowski 2021 ) on the Lambek Calculus. (shrink)

We introduce and study hierarchies of extensions of the propositional modal and temporal languages with pairs of new syntactic devices: point of reference-reference pointer which enable semantic references to be made within a formula. We propose three different but equivalent semantics for the extended languages, discuss and compare their expressiveness. The languages with reference pointers are shown to have great expressive power (especially when their frugal syntax is taken into account), perspicuous semantics, and simple deductive systems. For instance, Kamp's and (...) Stavi's temporal operators, as well as nominals (names, clock variables), are definable in them. Universal validity in these languages is proved undecidable. The basic modal and temporal logics with reference pointers are uniformly axiomatized and a strong completeness theorem is proved for them and extended to some classes of their extensions. (shrink)

This paper investigates modular combinations of temporal logic systems. Four combination methods are described and studied with respect to the transfer of logical properties from the component one-dimensional temporal logics to the resulting combined two-dimensional temporal logic. Three basic logical properties are analyzed, namely soundness, completeness, and decidability. Each combination method comprises three submethods that combine the languages, the inference systems, and the semantics of two one-dimensional temporal logic systems, generating families of two-dimensional temporal languages with varying expressivity and varying (...) degrees of transfer of logical properties. The temporalization method and the independent combination method are shown to transfer all three basic logical properties. The method of full join of logic systems generates a considerably more expressive language but fails to transfer completeness and decidability in several cases. So a weaker method of restricted join is proposed and shown to transfer all three basic logical properties. (shrink)

In Ockhamist branching-time logic [Prior 67], formulas are meant to be evaluated on a specified branch, or history, passing through the moment at hand. The linguistic counterpart of the manifoldness of future is a possibility operator which is read as `at some branch, or history (passing through the moment at hand)'. Both the bundled-trees semantics [Burgess 79] and the $\langle moment, history\rangle$ semantics [Thomason 84] for the possibility operator involve a quantification over sets of moments. The Ockhamist frames are (3-modal) (...) Kripke structures in which this second-order quantification is represented by a first-order quantification. The aim of the present paper is to investigate the notions of modal definability, validity, and axiomatizability concerning 3-modal frames which can be viewed as generalizations of Ockhamist frames. (shrink)

We give a Hilbert style axiomatization for the set of formulas in the temporal language with Until and Since which are valid over the real number flow of time. The axiomatization, which is orthodox in the sense of only having the usual temporal rules of inference, is complete with respect to single formulas.

We present an axiomatisation for the first-order temporal logic with connectives Until and Since over the class of all linear flows of time. Completeness of the axiom system is proved.We also add a few axioms to find a sound and complete axiomatisation for the first order temporal logic of Until and Since over rational numbers time.

In this paper, we explore the relation between an agent’s doxastic attitude and her arguments in support of a given claim. Our main contribution is the design of a logical setting that allows us reason about binary arguments which are either in favour or against a certain claim. This is a setting in which arguments and propositions are the basic building blocks so that the concept of argument-based belief emerges in a straightforward way. We work against the background of Dung’s (...) abstract argumentation framework, extending it to a new setting in which we can study the formal properties of binary arguments as well as the larger structures they establish. This paper introduces a formal ‘two-dimensional’ language to talk about propositions and arguments, for which a sound and complete axiom system is provided. (shrink)

This paper exhibits a general and uniform method to prove axiomatic completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ, where x occurs only positively in γ, we obtain the flat modal fixpoint language by adding to the language of polymodal logic a connective γ for each γΓ. The term γ is meant to be interpreted as the least fixed point of the functional interpretation of the term γ. We consider the following (...) problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language on Kripke structures. We prove two results that solve this problem.First, let be the logic obtained from the basic polymodal by adding a Kozen–Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective γ. Provided that each indexing formula γ satisfies a certain syntactic criterion, we prove this axiom system to be complete.Second, addressing the general case, we prove the soundness and completeness of an extension of . This extension is obtained via an effective procedure that, given an indexing formula γ as input, returns a finite set of axioms and derivation rules for γ, of size bounded by the length of γ. Thus the axiom system is finite whenever Γ is finite. (shrink)

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finite relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an (...) algebra is ω-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this approach are looked at, and include the step by step method. (shrink)

Information about the powers and abilities of acting entities is used to coordinate their actions in societies, either physical or digital. Yet, the commonsensical meaning of an acting entity being deemed able to do something is still missing from the existing specification languages for the web or for multi-agent systems. We advance a general purpose abstract logical account of evidence-based ability. A basic model can be thought of as the ongoing trace of a multi-agent system. Every state records systemic confirmations (...) and disconfirmations of whether an acting entity is able to bring about something. Qualitative inductive reasoning is then used in order to infer what acting entities are deemed able to bring about in the multi-agent system. A temporalised modal language is used to talk about deemed ability, actual agency, and confirmation and disconfirmation of deemed ability. What constitutes a confirmation and a disconfirmation is left to the modeller as in general it depends on the application at hand. So to illustrate the methodology we propose two extended examples, one in practical philosophy, the other in system engineering. We first use a logic of agency and ability to obtain a version of Mele’s general practical abilities. Then, we look at the management of abilities in a supervised system. (shrink)