From Linear to Branching-Time Temporal Logics: Transfer of Semantics and Definability

This 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 tree

Ockhamist Propositional Dynamic Logic: a natural link between PDL and CTL

International audienceWe present a new logic called Ockhamist Propositional Dynamic Logic, OPDL, which provides a natural link between PDL and CTL*. We show that both PDL and CTL* can be polynomially embedded into OPDL in a rather simple and direct way. More generally, the semantics on which OPDL is based provides a unifying framework for making the dynamic logic family and the temporal logic family converge in a single logical framework. Decidability of the satisfiability problem for OPDL is studied in the paper

History relativism as extreme assessment relativism : a note on Prior's Ockhamism

Since the early days of Ockhamist semantics, it has been recognized that the history - relative notion of truth which the theory postulates is problematic : it is unclear what it means that a sentence is true relative to a possible course of events; it is also unclear how such a notion of relative truth relates to the everyday notion of truth simpliciter. To rationalize the Ockhamist notion of truth I compare two relativistic theories : the assessment relativism of John MacFarlane and the history relativism of Belnap et al. In the end, I suggest that we may understand the history-relative notion of truth as the truth assessed relative to an end of time. On the formal level, I introduce a doomsday extension of a branching model and prove that history - relative truth in any given model is equivalent to doomsday - relative truth in the extended model. It turns out that this equivalence holds in general only if the end of time is also, in a sense, beyond time

Agency and fictional truth: a formal study on fiction-making

Fictional truth, or truth in fiction/pretense, has been the object of extended scrutiny among philosophers and logicians in recent decades. Comparatively little attention, however, has been paid to its inferential relationships with time and with certain deliberate and contingent human activities, namely, the creation of fictional works. The aim of the paper is to contribute to filling the gap. Toward this goal, a formal framework is outlined that is consistent with a variety of conceptions of fictional truth and based upon a specific formal treatment of time and agency, that of so-called stit logics. Moreover, a complete axiomatic theory of fiction-making TFM is defined, where fiction-making is understood as the exercise of agency and choice in time over what is fictionally true. The language L of TFM is an extension of the language of propositional logic, with the addition of temporal and modal operators. A distinctive feature of L with respect to other modal languages is a variety of operators having to do with fictional truth, including a \u2018fictionality\u2019 operator M (to be read as \u201cit is a fictional truth that\u201d). Some applications of TFM are outlined, and some interesting linguistic and inferential phenomena, which are not so easily dealt with in other frameworks, are accounted for

The master argument and branching time

It is argued that reconstructions of the so-called ‘Master Argument’ of Dideros Cronos to the effect that possibility should be understood as present or future truth, essentially relies on two axioms: i) that every true proposition concerning the past is necessary, and ii) that it follows necessarily from a proposition being true that it always has been the case that it would be true. It is furthermore argued that these two axioms are inconsistent in the sense that any tense/modal semantics which incorporates both collapses either modally (fails to distinguish between truth simpliciter and modalised truth) or temporally (fails to offer a plausible semantical account for propositions about the future). This finding is, furthermore, taken as indicator for the more generel claim that there are principled difficulties involved in construing semantics for combined tense/modal logical systems

Back to the actual future

The purpose of the paper is to rethink the role of actuality in the branching model of possibilities. We investigate the idea that the model should be enriched with an additional factor - the so-called Thin Red Line - which is supposed to represent the single possible course of events that gets actualized in time. We believe that this idea was often misconceived which prompted some unfortunate reactions. On the one hand, it suggested problematic semantic models of future tense and and on the other, it provoked questionable lines of criticism. We reassess the debate and point to potential pitfalls, focusing on the semantic dimension of the Thin Red Line theory. Our agenda transcends the semantics, however. We conclude that semantic considerations do not threaten the Thin Red Line theory and that the proper debate should be carried in the domain of metaphysics

A topological perspective for branching-time logics

Questa tesi è suddivisa in due parti: - nella prima parte, abbiamo descritto la *Logica Temporale* e, in particolare, alcune logiche del tempo ramificato; - nella seconda parte abbiamo analizzato da un punto di vista topologico alcune proprietà algebriche degli alberi, su cui sono basate molte semantiche per la logica temporale. L'obiettivo principale della Logica temporale è la definizione di un linguaggio formale (e di una semantica) che possa esprimere proposizioni con tempi verbali, come "Ha piovuto", "Ho dormito, in passato", "In futuro impareremo a volare". Si osserva immediatamente che il valore di verità di queste proposizioni dipende dal momento in cui sono considerate, a differenza delle proposizioni della Logica Classica. Perciò, il primo passo per una definizione di verità in questo contesto è quello di costruire una adeguata struttura matematica che rappresenti il tempo. Nei primi due capitoli, analizzeremo varie scelte sintattiche e semantiche per la Logica Temporale, e differenti assunzioni ontologiche (una su tutte, l'*Indeterminismo*) che modificheranno il modello del tempo considerato. Seguendo l'articolo *Topological Aspects of Branching-Time Semantics *(2003) di M. Sabbadin e A. Zanardo, nel terzo capitolo abbiamo preso in considerazione una semantica inusuale per la Logica Temporale, basata su una naturale struttura topologica aggiunta alla semantica Ockhamista. In particolare, lo spazio dei rami massimali (o storie) della rappresentazione ad albero del tempo diventa uno spazio topologico *non-Archimedeo*. In questo capitolo vengono analizzate in dettaglio le proprietà di questo spazio, e viene dimostrato un importante risultato di validità. Infine il quarto capitolo, che è in parte un lavoro di ricerca, contiene la "traduzione" in questo nuovo linguaggio topologico di varie proprietà algebriche degli alberi. Vengono descritte le proprietà topologiche dello spazio delle storie degli alberi lineari, degli alberi finitamente ramificati, di quelli ben fondati, e di altre classi più particolari, come gli alberi ω-cofinali e gli alberi *jointed*. Il capitolo si conclude con un'analisi degli alberi di Souslin e degli alberi speciali, strettamente collegati al noto *Problema di Souslin*