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Modalities in Temporal Logic
Full text linkIn logics of branching-time, ‘possibility’ can be conceived as ‘existence of a suitable set of histories’ passing through the moment under consideration. A particular limit case of this is the Ockhamist notion of possibility, which is explained as truth at some history. The tree-like representation of time offers other ways of defining possibility as, for instance, truth at any history in some equivalence class modulo undividedness. In general, we can consider representations of time in which, at any moment t, the set of histories passing through t can be decomposed into indistinguishability classes. This yields to a new general notion of possibility including, as particular cases, other notions previously considered
Branching-Time as a relative closeness relation among histories
No full textIn the traditional approaches to branching-time, histories are dened as linearly
ordered and maximal sets of moments. The `geometrical' approach considers
both moments and histories as primitive entities with no set-theoretical and ontological
dependency of the latter on the former. In the a topological approach
the original perspective is inverted: only histories are primitive entities and moments
are dened as sets of histories. Moreover, these particular sets of histories
can be dened also by means of a relative closeness relation among histories
Quantification over Sets of Possible Worlds in Branching-Time Semantics
No full textTemporal logic is one of the many areas in which a possible world semantics
is adopted. Prior's Ockhamist and Peircean semantics for branching-time, though, depart
from the genuine Kripke semantics in that they involve a quanti ̄cation over histories,
which is a second-order quanti ̄cation over sets of possible worlds. In the paper, variants
of the original Prior's semantics will be considered and it will be shown that all of them
can be viewed as ̄rst-order counterparts of the original semantics
Indistinguishability, choices, and logics of agency
No full textThis paper deals with structures in which T is a tree and I is a function assigning each moment a partition of the set of histories passing through it. The function I is called indistinguishability and generalizes the notion of undividedness. Belnap’s choices are particular indistinguishability functions. Structures provide a semantics for a language L with tense and modal operators. The first part of the paper investigates the set-theoretical properties of the set of indistinguishability classes, which has a tree structure. The significant relations between this tree and T are established within a general theory of trees. The aim of second part is testing the expressive power of the language L . The natural environment for this kind of investigations is Belnap’s seeing to it that (stit). It will be proved that the hybrid extension of L (with a simultaneity operator) is suitable for expressing stit concepts in a purely temporal language
Moment/History Duality in Prior's Logics of Branching-Time
No full textThe basic notions in Prior's Ockhamist and Peircean logics of branching-time are the notion of moment and that of history (or Course of events). In the tree semantics, histories are defined as maximal linearly ordered sets of moments. In the geometrical approach, both moments and histories are primitive entities and there is no set theoretical (and ontological) dependency of the latter on the former. In the topological approach, moments can be defined as the elements of a rank 1 base of a non-Archimedean topology on the set of histories. In this paper, it will be shown that the topological approach, and hence the other approaches, can be reconstructed in a framework in which the basic notions are those of history and of relative closeness relation among histories
Logiche Temporali
No full textViene presentata una panoramica, anche in prospettiva storica, della logica temporale
Topological Aspects of Branching-Time Semantics
No full textThe aim of this paper is to present a new perspective under which branching-time semantics can be viewed. The set of histories (maximal linearly ordered sets) in a tree structure can be endowed in a natural way with a topological structure. Properties of trees and of bundled trees can be expressed in topological terms. In particular, we can consider the new notion of topological validity for Ockhamist temporal formulae. It will be proved that this notion of validity is equivalent to validity with respect to bundled trees
Combining Linear Orders with Modalities for Possible Histories
No full textTwo main areas of temporal logics are those of linear time and of branching time. Linear orders, though, play a crucial role also in logics for branching time. Prior’s Ockhamist and Peircean semantical rules for branching time, in fact, involve quantification over histories in tree-like structures, where histories are maximal linearly ordered sets of moments. Moreover, this quantification can be viewed as the result of the application of a modal operator. This means that the language and semantics for branching time can be obtained as the combination of languages and semantics for linear time with a modality for possible histories. In this paper, we study various degrees of combining linear time and modal operators and semantics, and we discuss the problem of transferring logical properties from linear to branching time
From linear to branching-time temporal logics: transfer of semantics and definability
Full text linkThis paper investigates logical aspects of combining linear orders as semantics for modal and temporal logics, with modalities for possible paths, resulting in a variety of branching time logics over classes of trees. Here we adopt a unified approach to the Priorean, Peircean and Ockhamist semantics for branching time logics, by considering them all as fragments of the latter, obtained as combinations, in various degrees, of languages and semantics for linear time with a modality for possible paths. We then consider a hierarchy of natural classes of trees and bundled trees arising from a given class of linear orders and show that in general they provide different semantics. We also discuss transfer of definability from linear orders to trees and introduce a uniform translation from Priorean to Peircean formulae which transfers definability of properties of linear orders to definability of properties of all paths in trees
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