tense logic an extension of classical logic introduced by Arthur Prior (Past, Present, and Future, 1967), involving operators P and F for the past and future tenses, or ‘it was the case tha. . .’ and ‘it will be the case tha. . .’. Classical or mathematical logic was developed as a logic of unchanging mathematical truth, and can be applied to tensed discourse only by artificial regimentation inspired by mathematical physics, introducing quantification over ‘times’ or ‘instants.’ Thus ‘It will have been the case that p,’ which Prior represents simply as FPp, classical logic represents as ‘There [exists] an instant t and there [exists] an instant tH such that t [is] later than the present and tH [is] earlier than t, and at tH it [is] the case that pH, or DtDtH (to‹t8tH ‹t8p(tH)), where the brackets indicate that the verbs are to be understood as tenseless. Prior’s motives were in part linguistic (to produce a formalization less removed from natural language than the classical) and in part metaphysical (to avoid ontological commitment to such entities as instants). Much effort was devoted to finding tense-logical principles equivalent to various classical assertions about the structure of the earlier–later order among instants; e.g., ‘Between any two instants there is another instant’ corresponds to the validity of the axioms Pp P PPp and Fp P FFp. Less is expressible using P and F than is expressible with explicit quantification over instants, and further operators for ‘since’ and ‘until’ or ‘now’ and ‘then’ have been introduced by Hans Kamp and others. These are especially important in combination with quantification, as in ‘When he was in power, all who now condemn him then praised him.’
As tense is closely related to mood, so tense logic is closely related to modal logic. (As Kripke models for modal logic consist each of a set X of ‘worlds’ and a relation R of ‘x is an alternative to y’, so for tense logic they consist each of a set X of ‘instants’ and a relation R of ‘x is earlier than y’: Thus instants, banished from the syntax or proof theory, reappear in the semantics or model theory.) Modality and tense are both involved in the issue of future contingents, and one of Prior’s motives was a desire to produce a formalism in which the views on this topic of ancient, medieval, and early modern logicians (from Aristotle with his ‘sea fight tomorrow’ and Diodorus Cronos with his ‘Master Argument’ through Ockham to Peirce) could be represented. The most important precursor to Prior’s work on tense logic was that on many-valued logics by Lukasiewicz, which was motivated largely by the problem of future contingents. Also related to tense and mood is aspect, and modifications to represent this grammatical category (evaluating formulas at periods rather than instants of time) have also been introduced. Like modal logic, tense logic has been the object of intensive study in theoretical computer science, especially in connection with attempts to develop languages in which properties of programs can be expressed and proved; variants of tense logic (under such labels as ‘dynamic logic’ or ‘process logic’) have thus been extensively developed for technological rather than philosophical motives. See also FUTURE CONTINGENTS , MANY- VALUED LOGI. J.Bur.