@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.}
}