Abstract
In this paper we develop a novel propositional semantics based on the framework of branching time. The basic idea is to replace the moment-history pairs employed as parameters of truth in the standard Ockhamist semantics by pairs consisting of a moment and a consistent, downward closed set of so-called transitions. Whereas histories represent complete possible courses of events, sets of transitions can represent incomplete parts thereof as well. Each transition captures one of the alternative immediate future possibilities open at a branching point. The transition semantics exploits the structural resources a branching time structure has to offer and provides a fine-grained picture of the interrelation of modality and time. In addition to temporal and modal operators, a so-called stability operator becomes interpretable as a universal quantifier over the possible future extensions of a given transition set. The stability operator allows us to specify how and how far time has to unfold for the truth value of a sentence at a moment to become settled and enables a perspicuous treatment of future contingents. We show that the semantics developed along those lines generalizes and extends extant approaches: both Peirceanism and Ockhamism can be viewed as limiting cases of the transition approach that build on restricted resources only, and on both accounts, stability collapses into truth.
Highlights
The interaction of modality and time became a topic of formal investigation early on in the development of tense logic by Prior (1957)
We show that the semantics developed along those lines generalizes and extends both Peirceanism and Ockhamism: both accounts can be viewed as limiting cases of the transition approach, and Peircean and Ockhamist validity are definable in the transition semantics
Both Peirceanism and Ockhamism are obtained by restricting the range of transition sets that are taken into account in the semantic evaluation, and on both accounts, stability collapses into truth
Summary
The interaction of modality and time became a topic of formal investigation early on in the development of tense logic by Prior (1957). Our future operator has both Peircean and Ockhamist traits: it demands a future witness in every possible future continuation of the moment of evaluation that is an extension of the given transition set. At any moment before the drawing, the sentences “Your ticket will win” and “Your ticket will lose” are neither stably-true nor stably-false but contingent with respect to the past course of events up to that moment Their truth values only stabilize as the future unfolds. We show that the semantics developed along those lines generalizes and extends both Peirceanism and Ockhamism: both accounts can be viewed as limiting cases of the transition approach, and Peircean and Ockhamist validity are definable in the transition semantics Both Peirceanism and Ockhamism are obtained by restricting the range of transition sets that are taken into account in the semantic evaluation, and on both accounts, stability collapses into truth.
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