The main proof method suggested is that of temporal reasoning in which the time dependence of events is the basic concept. Two formal systems are presented for providing a basis for temporal reasoning. One forms a formalization of the method of intermittent assertions, while the other is an adaptation of the tense logic system Kb, and is particularly suitable for reasoning about concurrent programs. Article :. However, since introducing a framing operator destroys monotonicity, a canonical model may no longer capture the intended meaning of a program.

Hence, a minimal model theory is developed. Within this model, negation by default is used to manipulate frame operator.

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Further, the temporal semantics of framed programs is captured by means of the minimal models. The existence of a minimal model for a given framed program is also proved. An example is given to illustrate how the semantics of framed programs can be captured. Tense Logic is obtained by adding the tense operators to an existing logic; above this was tacitly assumed to be the classical Propositional Calculus. Other tense-logical systems are obtained by taking different logical bases.

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Of obvious interest is tensed predicate logic, where the tense operators are added to classical First-order Predicate Calculus. This enables us to express important distinctions concerning the logic of time and existence. For example, the statement A philosopher will be a king can be interpreted in several different ways, such as.

The interpretation of such formulae is not unproblematic, however. The problem concerns the domain of quantification.

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For the second two formulae above to bear the interpretations given to them, it is necessary that the domain of quantification is always relative to a time: thus in the semantics it will be necessary to introduce a domain of quantification D t for each time t. These problems are related to the so-called Barcan formulae of modal logic, a temporal analogue of which is. This formula can only be guaranteed to be true if there is a constant domain that holds for all points in time; under this assumption, bare existence as expressed by the existential quantifier will need to be supplemented by a temporally restricted existence predicate which might be read 'is extant' in order to refer to different objects existing at different times.

For more on this and related matters, see van Benthem, , Section 7. Some important examples are the following:. These were introduced by Kamp The intended meanings are.

It is possible to define the one-place tense operators in terms of S and U as follows:. The importance of the S and U operators is that they are expressively complete with respect to first-order temporal properties on continuous, strictly linear temporal orders which is not true for the one-place operators on their own. Metric tense logic. We can define the general, non-metric operators by. This operator assumes that the time series consists of a discrete sequence of atomic times. The formula O p is then intended to mean that p is true at the immediately succeeding time step.

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## Temporal logic - Wikipedia

This can only mean the time immediately following the present in a discrete temporal order. In discrete time, the future-tense operator F is related to the next-time operator by the equivalence. One could similarly define a past-time version of O ; but since the main usefulness of this particular operator has been in relation to the logic of computer programming, where one is mainly interested in execution sequences of programs extending into the future, this has not so often been done.

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The standard model-theoretic semantics of Tense Logic is closely modelled on that of Modal Logic. An interpretation of the tense-logical language assigns a truth value to each atomic formula at each time in the temporal frame. Given such an interpretation, the meanings of the weak tense operators can be defined using the rules.

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We can now provide a precise characterisation of system K t of Minimal Tense Logic. The theorems of K t are precisely those formulae which are true at all times under all interpretations over all temporal frames. Many tense-logical axioms have been suggested as expressing this or that property of the flow of time, and the semantics gives us a precise way of defining this correspondence between tense-logical formulae and properties of temporal frames.

A formula p is said to characterise a set of frames F if. A tense-logical formula p corresponds to a first-order formula q just so long as p characterises the class of frames for which q is true. Some well-known examples of such pairs of formulae are:. For details, see van Benthem