## Temporal Logic

N. Rescher and A. Urquhart. *Temporal Logic*, Springer, 1971.

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#### Abstract:

This book is a useful contribution to that part of philosophical logic that deals with the formal analysis of temporal concepts. The authors would seem to be successful in their stated aim of producing a book on the subject that does not merely duplicate the writings of its founder, Arthur Prior. They provide a survey of the state of the field at the time of writing, together with new approaches to established areas, and some provocative suggestions for further developments. The first part of the book is largely devoted to the development of various calculi that serve to express temporally-relativised truth of statements and to analyse the structure of time in relation to the validity of particular tense logical principles. A system of topological logic is set out that has quantification over individuals that are to be understood as positions, and a parametrised operator $P\sb t$ of positional realisation ($P\sb t(p)$ is to mean "proposition $p$ is realised (true) at position $t$"). By construing positions as moments of time and introducing a special indexical symbol $n$ ("now") the authors construct a specifically temporal system $R$ as a special type of positional logic. They establish the completeness and decidability of $R$ by a translation into monadic first-order logic. Subsequent chapters deal with extensions of $R\sb 1$ including a temporal-group logic in which the temporal variables range over the elements of a commutative group, the group operation being used to constraint the iteration of realisation operators. Further refinements involve ordered groups, and the introduction of a metric space structure on the set of times (temporal positions). The other major type of calculus considered is of the propositional variety, with propositional variables, the classical connectives, and the "always will be" and "always has been" operators on propositions. Lemmon's minimal logic $K\sb t$ is shown to be complete for the standard Kripke semantics by a semantic tableaux method, and the analysis is extended to the logics corresponding to various structural features of the temporal ordering (branching, linear, infinite, beginning, ending, dense, etc.). A chapter is then devoted to temporal modality in which the "omnitemporal truth" and "truth now and always in the future" interpretations of necessity are studied. The modal logics corresponding to particular tense systems are studied, and some new results and open problems set out (the conjecture on p. 129 about the omnitemporal interpretation in branching time has been refuted by K. Segerberg [cf. $P$ and $Q$. Mini-essays in honor of Risto Hilpinen on his thirtieth birthday, March 9, 1973, pp. 18--23, Group in Logic and Methodology of Real Finland, Abo, 1973]. In the latter parts of the book the machinery thus far developed is applied to a wide range of philosophically motivated issues. These include: the theory of descriptions in the temporal context; the analysis of processes (stochastic and deterministic) and events; "world-state" and "world history" propositions that completely describe the world at, or up to, a particular moment; the dimension of time; the Master Argument of Diodorus; temporal determinism versus future contingency; many-valued logics; propositional quantification; and quantification over, and existence and identity of, individuals. Probably the major omissions from the book are the post-Henkin method of model construction by means of maximal-consistent sets of sentences, and the techniques of Bull and Segerberg for analysing the various kinds of linear temporal ordering. However, in the reviewer's opinion the work is best regarded not as a text of complex proofs, but rather as a sourcebook of facts, concepts, ideas and suggestions. The mathematician is shown how his subject finds fruitful application in this branch of logical analysis. The philosopher is shown how his concerns may be clarified by, and benefit from, the use of formal methods. And the student who simply wishes to know the "what" and "why" of tense logic is given an extremely readable response to that inquiry.

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@BOOK{rescher71:_tempor_logic, title = {Temporal Logic}, publisher = {Springer}, year = {1971}, author = {N. Rescher and A. Urquhart}, abstract = { This book is a useful contribution to that part of philosophical logic that deals with the formal analysis of temporal concepts. The authors would seem to be successful in their stated aim of producing a book on the subject that does not merely duplicate the writings of its founder, Arthur Prior. They provide a survey of the state of the field at the time of writing, together with new approaches to established areas, and some provocative suggestions for further developments. The first part of the book is largely devoted to the development of various calculi that serve to express temporally-relativised truth of statements and to analyse the structure of time in relation to the validity of particular tense logical principles. A system of topological logic is set out that has quantification over individuals that are to be understood as positions, and a parametrised operator $P\sb t$ of positional realisation ($P\sb t(p)$ is to mean "proposition $p$ is realised (true) at position $t$"). By construing positions as moments of time and introducing a special indexical symbol $n$ ("now") the authors construct a specifically temporal system $R$ as a special type of positional logic. They establish the completeness and decidability of $R$ by a translation into monadic first-order logic. Subsequent chapters deal with extensions of $R\sb 1$ including a temporal-group logic in which the temporal variables range over the elements of a commutative group, the group operation being used to constraint the iteration of realisation operators. Further refinements involve ordered groups, and the introduction of a metric space structure on the set of times (temporal positions). The other major type of calculus considered is of the propositional variety, with propositional variables, the classical connectives, and the "always will be" and "always has been" operators on propositions. Lemmon's minimal logic $K\sb t$ is shown to be complete for the standard Kripke semantics by a semantic tableaux method, and the analysis is extended to the logics corresponding to various structural features of the temporal ordering (branching, linear, infinite, beginning, ending, dense, etc.). A chapter is then devoted to temporal modality in which the "omnitemporal truth" and "truth now and always in the future" interpretations of necessity are studied. The modal logics corresponding to particular tense systems are studied, and some new results and open problems set out (the conjecture on p. 129 about the omnitemporal interpretation in branching time has been refuted by K. Segerberg [cf. $P$ and $Q$. Mini-essays in honor of Risto Hilpinen on his thirtieth birthday, March 9, 1973, pp. 18--23, Group in Logic and Methodology of Real Finland, Abo, 1973]. In the latter parts of the book the machinery thus far developed is applied to a wide range of philosophically motivated issues. These include: the theory of descriptions in the temporal context; the analysis of processes (stochastic and deterministic) and events; "world-state" and "world history" propositions that completely describe the world at, or up to, a particular moment; the dimension of time; the Master Argument of Diodorus; temporal determinism versus future contingency; many-valued logics; propositional quantification; and quantification over, and existence and identity of, individuals. Probably the major omissions from the book are the post-Henkin method of model construction by means of maximal-consistent sets of sentences, and the techniques of Bull and Segerberg for analysing the various kinds of linear temporal ordering. However, in the reviewer's opinion the work is best regarded not as a text of complex proofs, but rather as a sourcebook of facts, concepts, ideas and suggestions. The mathematician is shown how his subject finds fruitful application in this branch of logical analysis. The philosopher is shown how his concerns may be clarified by, and benefit from, the use of formal methods. And the student who simply wishes to know the "what" and "why" of tense logic is given an extremely readable response to that inquiry.} }

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