Tense Logic

95 Pages·1976·2.769 MB·English

by  Robert P. McArthur (auth.)

TENSE LOGIC
SYNTHESE LIBRARY
MONOGRAPHS ON EPISTEMOLOGY,
LOGIC, METHODOLOGY, PHILOSOPHY OF SCIENCE,
SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE,
AND ON THE MA THEMA TICAL METHODS OF
SOCIAL AND BEHAVIORAL SCIENCES
Managing Editor:
J AAKKO HINTIKKA, Academy of Finland and Stanford University
Editors:
ROBER T S. COHEN, Boston University
DONALD DAVIDSON, Rockefeller University and Princeton University
GABRIEL NUCHELMANS, University of Leyden
WESLEY C. SALMON, University of Arizona
VOLUME 111
ROBERT P. McARTHUR
Colby College, Waterville, Maine, U.S.A.
TENSE LOGIC
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
ISBN 978-90-481-8345-6 ISBN 978-94-017-3219-2 (eBook)
DOI 10.1007/978-94-017-3219-2
AlI Rights Reserved
Copyright © 1976 by Springer Science+Business Media Dordrecht
Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1976
No part of the material protected by this copyright notice may be reproduced or
utilized in any form by or any means, electronic or mechanical,
including photocopying, recording or by any informational storage and
retrieval system, without written permission from the copyright owner
PREFACE
This monograph is designed to provide an introduction to the principal areas
of tense logic. Many of the developments in this ever-growing field have been
intentionally excluded to fulfill this aim. Length also dictated a choice
between the alternative notations of A. N. Prior and Nicholas Rescher - two
pioneers of the subject. I choose Prior's because of the syntactical parallels
with the language it symbolizes and its close ties with other branches of logi
cal theory, especially modal logic.
The first chapter presents a wider view of the material than later chapters.
Several lines of development are consequently not followed through the
remainder of the book, most notably metric systems. Although it is import
ant to recognize that the unadorned Prior-symbolism can be enriched in vari
ous ways it is an advanced subject as to how to actually carry off these
enrichments. Readers desiring more information are referred to the appropri
ate literature. Specialists will notice that only the first of several quantifi
cational versions of tense logic is proven complete in the final chapter. Again
constraints of space are partly to blame. The proof for the 'star' systems is
wildly complex and at the time of this writing is not yet ready for publi
cation.
It is a pleasure to thank Professors Hugues Leblanc, Henry Kyburg, Frank
Parker, Mr. Daniel Cohen, and Mr. Russell Wahl for their generous assistance
and helpful comments on earlier versions of this work. I should also like to
thank Colby College's Humanities Grant Committee for a research grant
during the summer of 1974.
R.P.M.
TABLE OF CONTENTS
PREFACE v
CHAPTER l/AN OVERVIEW OF TENSE LOGIC 1
1.1. Introduction 1
1.2. The Formalities of Tensed Statements 2
1.3. Interpreting Tensed Statements. I 7
1.4. Interpreting Tensed Statements. II 12
Notes 15
CHAPTER 2/LINEAR TENSE LOGICS 17
2.1. The Minimal System K 17
t
2.2. Linear Enlargements of K 24
t
Notes 36
CHAPTER 3/BRANCHING TENSE LOGIC AND
TEMPORAL MODALITY 37
3.1. Branching Tense Logics 37
3.2. Ten1poral Modality 42
Notes 51
CHAPTER 4/QUANTIFICATIONAL TENSE LOGICS 53
4.1. The Q Systems 53
4.2. The Q* and Q** Systems 60
Notes 66
CHAPTER S/SOUNDNESS AND COMPLETENESS
THEOREMS FOR TENSE LOGIC 67
Notes 78
APPENDIX 1/ SC TAUTOLOGIES USED IN PROOFS 79
APPENDIX II/SUMMARY OF THE SYSTEMS 80
BIBLIOGRAPHY 83
CHAPTER 1
AN OVERVIEW OF TENSE LOGIC
1.1. INTRODUCTION
As P. F. Strawson noted in his Introduction to Logical Theory, 1 standard
logic seems ill-adapted to deal with statements containing tensed verbs or
explicit temporal references. For example, we say things like 'There were
women among the survivors' but the existential quantifier 3 ~ is read 'There is
(timelessly) an x such that', and the relation of predicate expressions to
their subjects is also presumed to be timeless.2 Hence, trying to symbolize
this statement results in the awkward
(1) (3 x )(x is a woman & x is a person who was among the survivors)
where both occurrences of 'is' are taken to be timeless. One of the conse
quences of this deficiency in standard logic is that arguments whose premisses
and conclusion contain tensed verbs may be pronounced invalid, even though
they are intuitively valid. So, in the following case
(2) John is running
(therefore) John will have run
most would agree that the conclusion follows from the premiss, but this can
not be represented by standard logical means.
Responses to complaints such as Strawson's have fallen, until recently,
into two classes. Some, such as W. V. Quine, argue that all statements con
taining tensed verbs can be paraphrased into an atemporal form and repre
sented in an extended version of standard logic.3 Others take the situation
outlined above as evidence of the inherent limitation of formal logic, showing
that it is incapable of representing the statements of ordinary language. This
is Strawson's position.4
In the last few years a third response has appeared which takes the form of
developing a (formal) logic of tense and temporality. Slipping between the
alternatives posed by Quine and Strawson, such logics offer a neat compro
mise. On one hand, temporality is preserved against Quine's atemporal
2 CHAPTER 1
paraphrases. And on the other hand, the scope of formal logic is extended so
as to meet, at least in this area, Strawson's misgivings about the limits of logic.
In addition to the formalization of tensed statements, the systematization
of inferences involving such statements has been a primary aim of tense
logicians. But here, as in all non-standard areas of logical theory, a difficult
question arises as to which assumptions about tense - and, as we shall soon
point out, about time - should be taken axiomatically in the development of
deductive systems of tense logic. Without an exploration of various possi
bilities any proposed answer to this question will inevitably seem arbitrary.
Thus many systems of tense logic have been created which incorporate differ
ing tense logical principles. Such systems allow the investigation of these prin
ciples taken singly and in "groups. And, in this manner, the covert assumptions
behind intuitive appraisals of tensed arguments are brought to the surface.
1.2. THE FORMALITIES OF TENSED STATEMENTS
As regards temporal reference, two distinct categories of statements can be
distinguished. These are:
(i) Temporally definite
s
(ii) Temporally .indefinite.
The first category includes all those statements having no temporal reference
whatsoever - so-called atemporal statements - such as '5 is a prime number'
as well as those with specific temporal references, e.g.,
(3) It always rains in Boston.
The essential feature of temporally definite statements is that their truth
values are independent of the time of their utterance. Or, to put the matter in
a somewhat different manner, such statements do not convey any positional
information concerning the temporal relation of the speaker and the event or
state of affairs depicted. Of course, hybrids of the sort
(4) It rained in Boston on May 12, 1973
are not unusual. Although some positional information is imparted by (4),
i.e., that the time of utterance is later than May 12, 1973, to convey this fact
is generally not the primary intent of the speaker. In such cases, statements
like (4) can be thought of as if they were expressed with atemporal verbs
(e.g., It rains in Boston on May 12, 1973). When the positional information is
AN OVERVIEW OF TENSE LOGIC 3
the point of the statement, it can be parsed as a conjunction of a temporally
definite statement and a temporally indefinite statement, e.g.,
(4') It rains in Boston on May 12, 1973 and May 12, 1973 is now past.
Since the tense of the verb, as already noted, does not play an important
role in temporally definite statements, it is not surprising that tense logic is
more concerned with the second category: temporally indefinite statements.
Several examples of this sort of statement are:
(5) Richmond was the capital of the Confederacy.
(6) It is sunny and warm today.
(7) Montreal will be the site of the next Olympic Games.
Note that the truth-value of all these is very much dependent upon the time
of utterance, a property we can take as essential to this category of state
ment.
English has three basic tenses which may enter into the composition of a
temporally indefinite statement. These are the (simple) past, present, and
future tenses. Following the analogy of the formal analysis of negation in
ordin~ry logic, iOt is convenient to think of statements in these three tenses as
compounds consisting of a one-place statement operator - which picks up
the tense - and an interior statement. Utilizing the symbol F for the future
tense operator, P for the past tense operator, and T for the present tense
operator, statements (5)-(7) above can be partially symbolized as follows.
(5 ') ~Richmond is the capital of the Confederacy)
(6') T(It is sunny and warm today)
(7') F(Montreal is the site of the next Olympic Games)
Note however that construing the tense operators as statement operators
requires the verb in the (temporally indefinite) interior statement to have
some tense or other. Hence a certain redundancy is present in (6'); not only
does the statement operator T signify present tense but the interior statement
in is the present tense as well. Thus the T operator is superfluous and can be
dispensed with. This leaves us with only two tense operators - F and P.
The function of F and P is to strip all but present tenseness from the main
verb in the original statement and pack the tense into the prefix. Conse
quently, reading F as 'It will be the case that' and P as 'It has been the case
4 CHAPTER 1
that' seems appropriate. When symbols are translated back into English this
results in some artificiality, e.g., 'It will be the case that it rains in Boston',
but no more than arises from the formal reading of the negation operator as
'It is not the case that'.
Utilizing the two tense operators in combination allows the symbolization
of other tenses as well. Below are statements in two common perfect tenses
(past and future) with symbolizations.
(8) It had rained in Boston = PPp
FPP
(9) It will have rained in Boston =
(Note that p stands for the atomic statement 'It rains in Boston'). With the
addition of the truth-functional connectives & (conjunction) and ~ (negation)
other constructions are possible, e.g., 'It did rain in Boston but isn't now'
beingpp & ~p.
Negations in statements containing tense operators must be handled with
care, for there is a considerable difference between
and
(11)
~Fp.
The best rendition of (10) (with p still doing duty for 'It rains in Boston') is
(10') It won't always rain in Boston
whereas (11) would be
(11 ') It will never rain in Boston.
The reasons for these paraphrases are straightforward. Fp translates 'p at
sometime in the future' and, hence F~p translates '~p sometime in the
future'. But ~ Fp, being the denial of Fp, translates 'p at no time in the
future'. The same considerations apply, mutatis mutandis, to P.
The Law of Double Negation operatires as usual, as shown by the equiv~
alences (= is the biconditional symbol)
(12) =Fp
~~Fp
=pp
(13) ~~pp
(14) =Fp
F~~p