Tense and temporal logic

A special kind of logic is needed to represent the valid kinds of arguments involving tensed sentences. The first significant presentation of a tense logic appeared in Prior (1957). Sentential tense logic, in its simplest form, adds to classical sentential logic two tense operators, P and F. The basic idea is to analyse past and future tenses in terms of prefixes ‘It was true that’ and ‘It will be true that’, attached to present-tensed sentences. (Present-tensed sentences do not need present tense operators, since ‘It is true that Jane is walking’ is equivalent to ‘Jane is walking’.) Translating the symbols into English is merely a preliminary to a semantics for tense logic; we may translate ‘P’ as ‘it was true that’ but we still have the question of the meaning of ‘it was true that’. There are at least two versions of the tensed theory of time – the minimalist version and the maximalist version – that can be used for the interpretation of the tense logic symbols.

The minimalist version implies that there are no past or future particulars, and thus no things or events that have properties of pastness or futurity. What exists are the things, with their properties and relations, that can be mentioned in certain present-tensed sentences. If ‘Jane is walking’ is true, then there is a thing, Jane, which possesses the property of walking. ‘Socrates was discoursing’, even if true, does not contain a name that refers to a past thing, Socrates, since there are no past things. The ontological commitments of past and future tensed sentences are merely to propositions, which are sentence-like abstract objects that are the meanings or senses of sentences. ‘Socrates was discoursing’ merely commits us to the proposition expressed by the sentence ‘It was true that Socrates is discoursing’.

The maximalist tensed theory of time, by contrast, implies that there are past, present and future things and events; that past items possess the property of pastness, present items possess the property of presentness, and future items possess the property of being future. ‘Socrates was discoursing’ involves a reference to a past thing, Socrates, and implies that the event of Socrates discoursing has the property of being past.