Tense and Tense Logic 9783110871036, 9789027934536

Table of contents :
ACKNOWLEDGMENTS
TABLE OF CONTENTS
GLOSSARY OF SYMBOLS AND ABBREVIATIONS
1. INTRODUCTION
2. TENSE
3. TENSE LOGIC AND TIME STRUCTURES
4. THE TENSES OF LOGIC AND THE LOGIC OF TENSES
5. BEYOND TENSE LOGIC
APPENDIX: A MORE RIGOROUS SYNTAX AND SEMANTICS FOR LEP
BIBLIOGRAPHY
INDEX

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JANUA LINGUARUM STUDIA MEMORIAE NICOLAI VAN WIJK DEDICATA edenda curat C. H. VAN S C H O O N E V E L D Indiana University

Seties Minor,

215

TENSE AND TENSE LOGIC by J O H N E. CLIFFORD ASSISTANT PROFESSOR OF PHILOSOPHY University of Missouri - St. Louis

MOUTON PUBLISHERS • THE HAGUE • PARIS • NEW YORK

First edition: 1975 Second printing: 1980 LIBRARY OF CONGRESS CATALOG CARD NUMBER: 74-81949 ISBN 90 279 3453 3 © Copyright 1975 by Mouton Publishers, The Hague, The Netherlands No part of this book may be translated or reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publishers

Printed in Great Britain

In loving memory of Mama B.

ACKNOWLEDGMENTS

There are many people without whose aid and encouragement this work would not have been completed or would not have possessed what virtues it does. I am pleased to acknowledge my debts : In the area of content, To the late Professor Arthur N. Prior for the inspiration of his teaching and his advice and encouragement during the inchoative aspect of the writing of this work. To Professor William E. Bull, Spanish, and Professor David B. Kaplan, Philosophy, UCLA, for their patience in explaining to me what was obvious to them and for forcing me to explain what was to them obscure. What this work has of accuracy and intelligibility is due primarily to their efforts. To Professors Barbara H. Partee, Linguistics, and Ernest A. Moody, Philosophy, UCLA, for reading drafts of this work and making many valuable suggestions, the incorporation of more of which would doubtless have improved the work. To Doctor Tora Kay Lanto Bikson for her critical reading of many drafts, for sound advice and needed encouragement throughout. In the area of material support, To Linda Janguard Deutsch and Tanya Aron of the Philosophy Department, UCLA, Carol Hatch, Faculty Secretary at Webster College, and Janiece Fister of the Philosophy Department at the University of Missouri-St. Louis for supplies, equipment, and cheerful assistance in the preparation of manuscripts. To Grace Valenti and Kathy Austin, Philosophy Department,

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ACKNOWLEDGMENTS

UMSL, and Tony Sarver, Mel Cobb's Paperworkshop, Los Angeles, for fast and accurate typing of "filial" drafts and endless corrections. To Betty Ann Johns of the Thomas Jefferson Library, UMSL, for her careful rendition of the diagrams for this work. On the personal level, To my colleagues, especially Henry Shapiro an$ James. Doyle at UNfSL and Thomas Bikson at Webster College for enduring my moods and lightening my" load while I worked on this. To Naomi and f t u g o Kunkel for putting -up with me and putting .me up during my frequent forays to Los Angeles. And for financial aid, To the Committee on Faculty Research and Publication, to Robert S. Bader, Dean of the College of Arts and Sciences, and to the Philpsophy Departmerit, University of Missouri-St. Louis, for grants which enabled this work to come to publication.

TABLE OF CONTENTS

Acknowledgments

7

Glossary of Symbols and Abbreviations

11

1. Introduction.

17

2. Tense 2.1. The Meaning of Tense 2.2. The Structure of Tense 2.3. Tense in Use 2.4. English Tense Markers

22 Ï2 33 36 45

3. Tense Logic and Time Structures 3.1. Preliminary : English and the Language of Logic . . 3.2. The Grammar of the Language of Logic 3.3. The Semantics of Logic

57 58 62 65

4. The 4.1. 4.2. 4.3.

91 91 94

Tenses of Logic and the Logic of Tenses Tense in Natural and Logical Languages Logical Languages with Natural Tenses Tenses in Logical and Natural Languages Reconsidered

121

5. Beyond Tense Logic 5.1. Metrical Tense Logic 5.2. Explicit Tense Logic 5.3. Explicit Tense Logic with a Functional Base . . . . Appendix: A more Rigorous Syntax and Semantics for LEP Bibliography Index

123 123 136 151 163 168 170

GLOSSARY O F SYMBOLS AND ABBREVIATIONS Object language items are marked (o), those in both languages (om).

A A a

AP B B b

C C C c

D D d

Page where introduced the past segment of time 36 reference before speech 40 event to begin - inchoative aspect (o) 107 subsegment anterior to event in segment 37 event anterior to reference 40 variable for streams (om) 78,137 axis posterior to speech 42 moment of speech 36 reference simultaneous with speech 40 event is beginning - initiative aspect (o) 103 event in segment 37 event simultaneous with reference 40 variable for streams (om) 78,138 future segment of time 36 reference posterior to speech 40 event going on - imperfective aspect (o) 102 chronicle, assigns truth values to atomic formulae . 76 chronicle, assigns classes to predicates 160 subsegment posterior to event in segment 37 event posterior to reference 40 variable for streams (om) 78,138 event is finished - perfective aspect (o) 107 function, distance between points along a stream. . 129 nonce variable for numbers - day of the month . . 135

12

GLOSSARY OF SYMBOLS

Page where introduced E event is ending - terminative aspect (o) 106 Ë •> elector - selects representatives of sets 147 E! "exists" (o) 154 F "it will be the case that" (o) 62,128,137 F "it will be the case that" (vector) (o) 95 F function giving sets of future moments 76 as above, ordered by distance 128 F! "it will definitely be the case that" (o) 150 F" "it will be the case at a certain time that" (o) . . . 116 / numerical index on formulae i . . . . 141 / function, assigns numbers to metrical variables . . 130 /[ ] like / except for the replacement indicated 131 G "it will always be the case that" (o) 64 G "it will always be the case that" (vector) (o) . . . . 101 g function, assigns streams to stream variables. . . . 145 g(ijjpy like g except /? assigned the same stream at i and j . 145 g( ) like g except for replacement indicated 145 H "it always was/has been the case that" (o) 64 H "it has always been the case that" (vector) (o) . . . 101 E function, assigns individuals to individual variables. 160 E[] like E except for replacement indicated 161 / cover for interpretations 86, 131 / variable for moments 71 index for individual terms 158 iff if and only if 63 j variable for moments 71 k variable for moments 80 variable for numbers (om) . 125 LEF language for explicit tense logic on a functional basé 151 LEP language for explicit tense logic on a propositional base 136 LMP language for metrical tense logic on a propositional base 123 LTP language for topological tense logic on a propositional

GLOSSARY OF SYMBOLS

S3 m

N N n 0 P P P P' P" P„ PP PPF p q R R R RAP RP r S S S s T T

13

Page where introduced base . 63 the set of moments 70 variable for numbers (om) 126 nonce variable for nuiiibers - month 135 index for metrical terms 141 "it is now (date)" (o) 135 the set of numbers (measures) 129 variable for numbers (om) 126 marker which is the absence of overt markers . . . 45 "it was/has been the case that" (o) 62,124,137 "it has been the case that" (vector) (o) 94 function giving past moments > 76 as above, ordered by distance 128 "it was the case as near as you like to now that" (o) 103 "it was the case at a certain time that" (o) 116 "it was the case the nth time ago that... that..." (o) 111 42 axis - moment of speech . Past, Present and Future by A. N. Prior 19 atomic formula (o) 61 code for purpose of an expression 141 atomic formula (o) 61 predicate for individuals (o) 158 "it is the case at (date) that" (o) 134 function, assigns sets to predicates 147 axis posterior to RP 42 axis anterior to speech 42 atomic formula (o) 61 predicate for individuals (o) 158 "since" (o) 102 the set of streams 70 atomic formula (o) 61 index for stream terms 141 predicate for individuals (o) 158 function, assigns operations to operators 147

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GLOSSARY OF SYMBOLS

Page where introduced T" cover for and rF"n 120 TM Time and Modality by A. N. Prior 19 U "until" (o) 102 U universe - the set of individuals 160 x,y,z variables for individuals (o) 159 y nonce variable for numbers - year 135 Z cover for set of linguistic items 63 a P y covers for stream terms 138 a subsubsegment anterior to event in subsegment . . 37 P event in subsegment 37 r cover for predicates of individuals 154 y subsubsegment posterior to event in subsegment . . 37 A repeating event 110 d subordinate repeating event 112 £ "the selected object such that" (o) 142 C cover for individual terms 159 cover for terms 166 tj cover for individual terms 159 0 cover for operators 126 1 definite descriptor - "the unique object such that" (o) 142 ¡xv covers for superscripts 116 cover for metrical terms 126 £ set of nonampliating places 158 n "it is always the case that" (o) 108 cover for predicates 127 n cover for atomic formulae 59 the real number pi (o) 126 E "it is sometimes the case that" 108 T cover for tautologies (om) 103 < c o v e r s for formulae 59 Q unique event (calendar base) (o) 134 { } enclosed is a morpheme or morpheme cluster . . . 26 *-X previous event relevant at X 38 ->X previous event continues to X 38

GLOSSARY OF SYMBOLS

X-> X*— + 0 / / r n

~ & v () t= ¡ h ¥tj = if < = + / V V A = < x

This would yield, as a first step toward giving the corresponding formula, It will be the case that: if it is not the case that p, then, if it is not the case that q, then r. The stock connectives of the tense logical language are given in the following list (the Greek letters indicate places where the formulae are to be inserted):

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"if 0, then 5P" "either 0 or V" "both 0 and v Wy r ( r ) y . This formula displays the form of many other sentences, e.g. of If Nashua does not run, then Swap will win, unless Canonero's leg heals. The differences between this sentence and the original sample are not logically relevant under this analysis and so disappear into the rp~t, rq~>, and rr~1.

3.2 THE GRAMMAR OF THE LANGUAGE OF LOGIC

Implicit in the above paragraph is the claim that rF( ~ p -* q - > r ) y is a formula of the language of tense logic. This will be true if it can be shown to be generated by the grammar of that language. This grammar has a particularly simple structure, since the internal structure of the simple sentences can be ignored, leaving only rules for building up compound formulae to be dealt with. It is simpler to specify the language by defining the class of its formulae than by giving the rules for constructing them explicitly. However, the class will be defined in such a way that the rules for formula construction can be read off the definition. Informally, the class of formulae is that class which contains all of the atomic formulae (the residua markers, rp~l, etc.) and all of the strings built up from them by joining them by the connectives, joining the results of such compoundings, and so on. The definition envisions the possibility that one might start with some base that contained more than just the atomic formulae, and that there might be ways of

TENSE LOGIC AND TIME STRUCTURES

63

compounding other than those involving the connectives listed. Both of the possibilities will be realized in later chapters where, for example, formulae of the form rRxy~> may occur ("jc is related to / ' ) , and compounds may be formed with the connective rP~\ Yet these later languages still contain the language to be described here, for its formulae are all of the members of the smallest class which satisfies the definition to be given. The classes of formulae of these later languages also satisfy this definition but include other formulae as well and so are not all in this language because not all are in the smallest class. This smallest class is included in every class that satisfies this definition because any larger class can be gotten only by adding more things to those specified exactly by the definition. The language to be described in this way is called LTP. This is a shorthand way of saying that it is a language (L) based upon that for propositional logic (P) to be used to represent the topological (T) features of time. The topological features are those which do not require any measure: order, direction, divergence, etc. Thus, this language makes no reference to dates or duration. Formally, that is an LTP formula 1 is defined in the following: 0 is an LTP formula if and only if (abbreviated i f f ) 0 is a member of every set Z, such that (1) all of the atomic formulae are in Z (2) if W is in Z , then r ~ if"1 is in Z (3) if W and X are in Z , then (a) r (W &.XY is in Z (b) r ( i P v x y is in Z (c) r ( W - * x y is in Z (d) r ( y * - * X y is in Z (4) if V is in Z, then (a) rPW is in Z (b) rFW> is in Z 1 Because of a difference in symbols, these formulae will not look exactly like those in Prior's works. To change from one symbolization to the other is, however, purely mechanical.

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TENSE LOGIC AND TIME STRUCTURES

The formula derived from the sample sentence above can now be shown to be an LTP formula. This is done step by step. First, it is clear that rp~1, rq~[, and V are formulae by clause (1), since they are atomic formulae by inspection. Since these are formulae, so are r ~ / ? n and r ~ is "it will always be the case that", r ~ f is "it will never be the case that", and r F ~ n is "it will not be the case that". In what follows, another form of abbreviation will also be used. Some of the parentheses required by clause (3) of the definition will be omitted. Since this abbreviation must not introduce any ambiguity, the omissions will be made only in the following situations, which permit restoring parentheses in a (substantially) unique way. (1) If the first symbol of the formula is a left parenthesis, that parenthesis and its matching right parenthesis (the last symbol in the formula) may be dropped. For example, rp-*q~> will be written for r { p - + q y . (2) If a formula is certified by clause (3a) and a component is also certified by (3a), the parentheses around the component may be dropped. This also applies to formulae certified by (3b) with (3b) components. Thus, rp & q & r"1 may be written for rp &(q & r)~\ This abbreviation may also stand for r(p & q) & r"1. The two un-

TENSE LOGIC AND TIME STRUCTURES

65

abbreviated formulae are equivalent and, thus, a unique restoration of parentheses is not required. This rule may also be applied to constituents as well as whole formulae. Thus, rp->(q &(r & i))"1 may be written as rp-+(q & r & j)"1. (3) If a formula certified by clause (3a) or (3b) is a component of a formula certified by clause (3c) or (3d), the parentheses around that component may be dropped. Thus, rp Siq-*^ may be written for r(p & g)->r~". Combining this abbreviation with the preceding permits rp++q v r v i 1 for rp(q v (r v i))"1.

3.3 THE SEMANTICS OF LOGIC

As just indicated, the formulae of LTP are meant to be either true or false, like the statements made by declarative sentences of natural languages. Like statements, the formulae are, by and large, such that they are sometimes true and sometimes false. That is, just as the statement that it is not raining is true now but was not a few months ago, is true at some points of time but not at others. Among the formulae of LTP, there may be some which are true at all points of time, formulae which are like the statements of laws of nature, for example, in a natural language. There are also some formulae which must be true, whose truth follows from their structure, which are such that there is no way to make them false. It is less easy to find such sentences in natural languages but Either it is raining or it is not may do. To be sure, we may be unable to decide which it is — does a heavy mist count as rain or not? — but whichever way we decide, the sentence stands as true. Logic is particularly interested in formulae of this last sort (unfalsifiable), but tense logic is also interested in the second sort — those which happen to be true at all times. By means of these sempiternal (true at all times) formulae, tense logicians try to express the features of time structures. Of course, no formula of LTP says what the structure of time is. If there were a formula which said that the structure of time is so-and-so, asserting this formula would not guarantee that time is

66

TENSE LOGIC AND TIME STRUCTURES

so-and-so. The formula might be false, even though asserted. Only a true formula about time could say what the structure of time is, but no formula of LTP can say of itself (or any other formula of LTP) that it is true; at least, not in any way that might not also be false. That is, the formula which says, in effect, "this formula is true" might be false, just as the formula referred to might be. Further, there are real problems of allowing in the object language formulae which say of formulae in the object language that they are true. To have such formulae leaves the way open for the classical paradoxes of truth, e.g. The Liar (what is the truth value of This very sentence is false?). However, no formula says what the structure of time is for another reason. No formula is about time. Indeed, the formulae of LTP are not about anything. The sentences from which the formulae are derived are about things and say that these things are so-and-so, but the parts of the sentences which do this are lost in the transition to LTP, replaced by atomic formulae. Atomic formulae, since they may represent any simple sentence, represent no particular simple sentences, and, thus, are not about anything, time included. The formula derived from the sample sentence is not about either the weather or race horses, since it might be derived from a sentence about either, and many other things as well. What does express the structure of time is the fact that all formulae of a certain form are always true. This can not be said in LTP since, as noted, LTP has no way of talking about its formulae or their form, or of saying that they are true. It can, however, be said in English, which does have means of talking about the formulae of LTP (witness the discussion so far) and of saying of these formulae whether they are true or not at a given time or at all times. English also has the means of saying that time has a certain structure. Thus, in English, it can be said that formulae of a certain form are true at all times if time has a certain structure. How can the structure of time be expressed in LTP, then? It is not possible in the language taken in abstraction from its use. When LTP is being used, however, it is possible. LTP in use formulates a

TENSE LOGIC AND TIME STRUCTURES

67

theory. A theory is formulated by laying down a set of rules for going from one formula to another and a set of initial formulae (axioms and postulates). This specification is done in English (the metalanguage). The formulae taken as initial are some of those which are always true according to the theory being formulated. The rules are such that, if the formulae with which one begins are true, the formula with which one ends is also true. It is in a certain theory, then, that LTP can express that time has a certain structure. Using that theory is claiming that time has that structure. The postulates of that theory are some of the formulae which are always true if time has that structure (the axioms of the theory are some of the formulae which are true regardless of the structure of time — they are common to all theories formulated in LTP)} The structure of time is expressed in LTP in use by the fact that certain formulae have a privileged status: they may be asserted without fear of contradiction and certain rules apply to them. The formulae which have this status are, first, the axioms and postulates of the theory being used, and, second, all of those formulae derived from these by the rules, the theorems of the theory. The axioms and postulates were so picked that from them all of the formulae which are true at all times if time has the structure which the theory expresses may be derived by the rules. 3.3.1.

The Structures in the Metatheory of Tense Logic

In the metalanguage, English, it is possible to say what structure of time a given theory in LTP expresses. This involves two factors: '

Prior gives the following as axioms for tense logic: (0) the axioms for prepositional logic

(1) rG(j> -+q) -> (Fp -+Fqf (2) 'H(p -+q) (Pp -+Pqy

(3) -*HFp^ 1 (4) 'p GPp or, rather, formulae equivalent to these. In the exposition given here, (3) and (4) will turn out not to be axioms, but will be postulates for most theories. Prior's rules, aside from those for propositional logic, are:

RG\ if i> is a theorem, so is rG is a theorem, so is r H 0 1 tPPFi 32-58, 176).

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TENSE LOGIC AND TIME STRUCTURES

a description of the structure of time and a means of determining what formulae are always true if time has that structure and are, thus, fit to be theorems (here using that term to include the axioms and postulates) in that theory. This last, in turn, requires a means of determining whether a given formula is true at a point of time (a moment). It is no part of the task of tense logic to discover what the structure of time really is (or whether there is such a thing as time to have a structure).3 Tense logic merely presents a number of theories and determines what formulae will be theorems of each of these theories. Thus, should someone discover that there is such a thing as time and that it has such-and-such a structure, the logic for dealing with it will be ready for use. In keeping with this, the process of determining whether a given formula is true at a moment is different from that of determining the truth of a sentence. For a sentence, the process is to find out what the sentence says about the world and then find whether the world is the way the sentence says it is. Since the formulae of LTP say nothing about the way the world is, there is no way to find out what they say about the world or whether the world is that way. Rather, the atomic formulae, the parts which derive from those parts of sentences which say something about the world, are arbitrarily said to be true or false at each moment. There are, of ' Prior, PPF, 14-16, denies that the use of time structures in the theory in the metalanguage needs to be more than a device. It involves no metaphysical commitment to the existence of some entity, Time, nor to moments. The use of such a theory is merely a convenient way of dealing with tense relations, an aid to calculation. Prior suggests that what are taken as moments fixed independently of the events which occur at them might also be considered as classes of formulae referring to events. This would not be quite the same as taking a moment as a class of simultaneous events, since the class of formulae would include tensed formulae as well as untensed (present tense). However, a moment could be taken as class of simultaneous intentional events: the untensed formulae representing attentive events, those with rP ' representing retrotentive events, and those with r F ' protentive. This would give a Sautrantika model, in which the whole of history is cognized at each moment by the enlightened mind. Of course, as will be seen, if debilities are in force, not all of history may be present at each moment.

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course, a number of ways in which this assignment of truth or falsity may be carried out. Since rp~1 can derive from any sentence, there is a different assignment for each sentence: rpy can be taken as mirroring the truth behavior of any sentence from which it could derive. The same is true for any other atomic formula. But the structure of time does not depend upon the way in which this assignment is made. Thus, to say that a formula is always true says more than that it is true at every moment. It says that the formula is true at each moment, regardless of what atomic formulae are taken to be true at that moment. Thus, no atomic formula (or negation of an atomic formula) can be always true, as its truth would depend upon the choice of assignment of truth values (truth and falsity). However, only the truth values of the atomic formulae are assigned, the values for compound formulae can be derived from these. The fundamental notion to be defined, then, appears to be "this formula is true at this moment with respect to a certain assignment of truth values to the atomic formulae". Even this is not quite sufficient, however. Since L TP is a tensed language, the truth of some formulae at a given moment depends upon the truth of other formulae at other times: rP) but its consequent were not (not N, rPPp~>). Suppose, then, that the antecedent is true, that rPp~[. Then there is a moment, j, in P(i), such that I r p ~ [ , rp~[ is true at j. But the time structure is dense, so, between i and j, there must be another moment k. Now k is in P{i) since it is closer to i than j is and j is within the scope of the debility (if any). Similarly, j is in P(k), since it is closer to k than it is to i yet is in P{i). Thus, there is a moment, k, in P(i) such that there is a moment (J) in its past (P(k)) at which rp~l is true. That is, l=, rPPp~[. Thus the formula cannot be falsified at i and so is true at /. The argument makes no use of any feature of the time structure other than its density and no use of any special feature of history or debility, so it can be repeated at every point in every dense time structure. The English reading for this density postulate is somewhat awkward. A more idiomatic reading would be "if it was the case that p, then it had been the case that p". However, this reading is not appropriate for the formula as a postulate for density. The point about density is that the two rP~"s of the consequent take one back to the same point as the single r P n of the antecedent. If the English reading just given were taken uncritically, however, then the past marker on had would take one to the same moment as that on was, meaning that the anteriority marker in been indicated a still earlier time. In short, the density postulate would also hold in all repetitive histories (including cyclic ones), provided that no debilities interfered (repetitive 1= rPp-* PPp~*). In general, it is difficult to find a formula which is uniquely identified with a given feature; most hold in other structures than the ones which tense logicians have taken as their primary source. As a rule, however, they would not be taken as postulates for other features, as they could be derived from other formulae which were more closely bound to the feature to be expressed. In repetitive histories, rPp-*PPp~' can be derived from rp -» Ppn, a simpler and more direct statement of repetitivity. An extreme case of a formula holding in a structure different

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from the one it characterizes is the linearity postulate: rPp &Pq-+ P(p 8tq) v P{p & Pq) v P(q & Pp)~". This might be read, colloquially, as "If it was once the case that p and it was once the case that q, then either it was once the case that p and q together, or else, when p occurred, q had already occurred, or, when q occurred, p had already occurred." This sentence is linear-valid because, in a linear structure, the two points in P(i) at which p and q are true must be ordered with respect to one another. Either they are, in fact, the same point, in which case the first disjunct of the consequent holds, or one is earlier than the other, in which case one of the two other disjuncts holds, as the case may be. The only question is whether the earlier moment is in the past of the later, but this is guaranteed, as before, by the fact that it is in P(i) and i is further away still. But this same formula also holds in total histories. Indeed, in a total history, all of rPp"\ rPq~', rP(p & q f , rP(p & Pqf, and rP(q & PpY will be true at i, since none of them is contradictory. Indeed, there will be streams into the past on which not one but any combination of the last three items (and, hence, both of the first two) will hold. But a total history requires the extreme opposite of a linear time structure, all possible streams must be used (or at least as many as can be distinguished by sequences of events). The theory of total history would probably be given by a new rule: if 0 is not contradictory, then. rP (P"1 is true at each moment. Not all proposed postulates deviate so much from their intended features as linearity does, or even as much as density. Some do not deviate at all. The infinity postulate is valid only in infinite structures : (for an infinite past, which is enough to guarantee that the structure is infinite) r ~ Pp-*P ~p~*. If time has a beginning, then this sentence is false at the beginning of time. At the beginning of time r ~ Pp"1 will be true because there is no moment in the past (so rPp~' must be false). If there were a moment in the past then this would not be the beginning of time. For this same reason, rP ~ p'1 must be false then. Indeed, at the beginning of time every formula of the form rPn is false and every one of the form r ~ P 0 n is, consequently, true. But if time is finite in the past, there must be a

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TENSE LOGIC AND TIME STRUCTURES

beginning moment, so the infinity postulate is not true at every moment in that structure. The formula rH(p & ~ pY which is derivable from the negation of the infinity postulate is true only at the beginning moment of time and always true then. So the infinity postulate (for past time) is true just in case time has no beginning. To say that time has a beginning is just to say that either this is the beginning moment or that that moment has passed: r H(p & ~p) v PH(p & ~ py. It remains to show that this language, despite the metalanguage theory which it expresses and which is so different from any theory which might have arisen as a conceptualization of tense, does represent tense. This is done in the next chapter.

4 THE TENSES O F LOGIC AND THE LOGIC O F TENSES

In this, the central chapter of this study, it is to be shown that the tense markers of LTP do represent tense, in the sense defined in chapter 2. That this is so may already be clear but the sense in which it is so is less obvious. That is, given that the tense markers do express tense, the attempt will be made to show that they do so in exactly the way in which the overt markers of English do. This leads to the construction of two languages like LTP which contain markers that behave more like English markers than do the markers of LTP itself. Both of these languages will have to be rejected as purely tense languages in the end. But their construction will be useful for several other purposes. In one of these languages, the relation between various conceptualizations of tense — aspect and the temporal relations, before and after, since and until — can be explored, as well as some of the notions related to measuring time. The other language shows, in its failure, that LTP is not to be understood as representing the surface tense structure of a natural language like English, but as representing its deep structure.

4.1 TENSE IN NATURAL AND LOGICAL LANGUAGES

As noted in chapter 2, the practical function of tense markers is to establish the order relationship of events to the event of uttering the tense marker in question. Events are thereby classified into those the relevant aspect of which is anterior to the utterance (although, if the aspect is not terminative, the event may still be

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going on), those which are going on at the moment of utterance, and those the relevant aspect of which is potential, anticipated (although, if this aspect is not initiative, they may be going on already). Formally, this is done by indicating the relation of the relevant portion of the event to the moment of utterance as being either anterior, simultaneous, or posterior. This pattern may then be extended to relate events to events other than the utterance, events already placed with respect to utterance. The semantics of LTP presents a different picture. There is in this no reference to events as defining relationships, rather, all relations are between points of time, moments. What happens at these moments plays no part in defining the relationships. The formal semantics is done sub specie aeternitatis — from the point of view of eternity, that is, external to time. All of time is present at one moment and all may be taken in at a glance. The absolute relations between moments may thus be described, and, with them, the relationships between the events which happen at those moments. Insofar as LTP reflects this semantics, then, it does not express tense. However, in use, LTP does not reflect this semantics in any obvious way. In LTP there is no mention of moments of time or of their relations to one another. Rather, the reference is to events (formulae being true) and their order around, as it were, the utterance of a formula. What corresponds to the utterance of a sentence in LTP is the point of evaluation. To be sure, in the semantics this is identified with a moment of time at which the formula is evaluated, but the evaluation of the formula at that time treats the formula as being within time, not external to it. The utterance (evaluation) of a formula divides time into past and future relative to the point of evaluation, just as the utterance of a sentence divides time relative to the moment of utterance. The division is described from outside of time but functions within time. Once the force of these remarks is clear, it is also apparent that the tense markers of LTP mark tense under a time conceptualization. The formulae of LTP do not say that events happen at earlier or later moments, although that is how the function of

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the tense markers is described: they relate events to the moment of evaluation. That is, they classify events in the same way that tense does. Thus they express tense, since, as noted in chapter 2, there may be many ways of expressing tense, any one of which may be used so long as it provides the practical classification and shows the order relationships. LTP merely provides one such way, perhaps one not found in any natural language. The method of expressing tense in LTP does provide certain advantages over that of most natural languages, however. Most natural languages deal with tense by a variety of means. Even when a language has explicit tense markers, there will usually be a number of other devices which also express tense, the English intentional verbs and words for temporal relations, for example. These various systems interact with one another in a number of ways, creating redundancies (explicit past markers with recollect, for example) or producing clusters which contain tense information not in any of their components (as when the French possession of an activity in the abstract came to mean that the activity was future: infinitive + avoir). In LTP, the redundancies are removed and all tense indication is done in a single system, in which all of the tense features are explicit. Further, the system in LTP is used only for the expression of tense, whereas the tense markers of natural languages, even the systematic tense markers, may have a number of uses not involving tense (as the English future marker will is used to characterize someone in the present: Hindus will not eat beef—historically, the source of the use of will as a tense marker). Finally the tense systems of natural languages are limited in several ways. For purely morphotactic reasons, the systematic tense markers cannot be freely combined and other systems, also of limited usefulness, must be introduced to reach an event separated by three events of interest (He had left before she had called). The markers of natural languages are limited in their syntactic scope so that it is difficult to say explicitly that a complex of events, requiring several simple sentences to express, took place at one moment (the various will's in If it will not rain, then, if it will not be foggy, then it will be sunny are all to refer to same time, but it is difficult to explain what it is

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that entitles one to say they do). In LTP, the tense markers are not restricted to single simple sentences or to preset combinations of these but may be used freely to coordinate events. In short, LTP provides a simpler yet more precise means of expressing certain features of tense than is to be found in any natural language. This claim is true, however, only for those features which LTP does express. If LTP is to express tense like a natural language, it is found to be deficient. LTP has only one marker for each direction: r P~' for anteriority, rF~1 for posteriority, and the absence of either for simultaneity. The distinction between axis and vector can, thus, not be expressed systematically in LTP. To be sure, the system of tense markers in LTP is, in this respect, no poorer than many natural languages, many of which lack the axis-vector distinction in at least one direction, as does English for the future, and some of which do not even have systematic markers for all three relations. However, these languages can make up the deficiencies of their systematic markers by using other systems. LTP has no other system to use.

4.2 LOGICAL LANGUAGES WITH NATURAL TENSES

4.2.1. Vectors Added The readings given for the tense markers of LTP suggest that rP~> is to be taken as an axis marker, since that reading uses was, which carries the systematic RP marker. If that is the case, the logic of tenses requires a form to represent the anterior vector, the correlate of {have + -N} as a tense marker. A symbol for this purpose might be r F n or some such modification of rP~1, but the real problem is to describe how it is to function. Given the standard interpretation, that the form to be described represents pure anteriority, it follows that no further tense marker may occur within the scope of the suggested rP~\ To permit this would allow a vector from a vector without an intervening axis for the second vector to attach to, or, equally strange, an axis located with respect to a vector. The one would be like a direction from a

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direction and also violates what has been said of tense — that a vector is always a vector from some axis. The second would be locating a point as in a direction from a direction, which is to fail to locate the point at all. The formation rules for formulae containing rP~l, then, should distinguish between tense-free formulae, to which rP'1 may be prefixed, and tensed formulae, to which it may not be attached. This corresponds to the distinction in English (and, necessarily, in all languages in which the axis-vector distinction is systematically marked): no tense marker may occur in the scope of a vector marker, but a vector may occur in the scope of an axis marker. In specifying the language with rP~l one adds, for symmetry and because some language might distinguish it, the plus vector marker r F~l, subject to the same syntactic restrictions as rF~1. 4.2.1.1. Syntax The set of tense-free sentences is the least set which contains all of the atomic formulae and is closed under compounding with the nontense connectives (i.e. the set of formulae of propositional logic), while the set of sentences of LTP-2 (as the new language might be called) is the least set which contains all tense-free formulae, all of the results of prefixing rP~l and rF"1 to tense-free formulae, and then is closed under compounding using the connectives of LTP (-1), tense and nontense. The pattern of the definition is the same as that for LTP:

is a member of every set Z such that (1) all atomic formulae are in Z (2) if 5Fis in Z, r ~ V is in Z (3) if W and X are in Z, then (a) r (W & X y is in Z (b) r ( ¥ v X y is in Z (c) r ( V - * X y is in Z (d) r (V & V y iff Nu 0 and W I i f f h j t f o r ¥u r(

The vector markers also set up a new point at which the remainder of the formula is to be evaluated. Since, as LTP-2 is now formulated, there will be no tense markers in this remainder, the previous axis need not be referred to. The reference is kept because required by the symbolism and because it will be useful for further developments. t=u rP . The effect of r P"\ say, can often be represented in indirect ways. It has been claimed that the rS — V system can be used to define all tense connectives and that its basic tense connectives cannot be defined in terms of any one-place tense connectives. This only shows that rP~' and T" 1 are not tense connectives in the sense intended, a sense which requires metric notions in the metalanguage. While the development of the rS—V system is of interest in its own right, it is introduced here only because it may be useful in dealing with the aspect system. While whatever is definable in the rS — V system is definable in LTP-2, the definitions in terms of rS~l are sometimes more readable. For example, the notion expressed by "p has held as near as you like to now", i.e. that there is no past time since which r ~p~ l has held continuously, is expressed simply as r~S (r, ~ p Y - This is abbreviated r P'/> n . All of this has been leading up to an attempt to define the other aspects of an event, its beginning and its end. These definitions seem fraught with difficulties. What is wanted for the beginning of an event r 2T is to pick out a moment before which there is a period when that event is not occurring and after which there is a period during which the event is occurring. If rp~l is the sentence which refers to the event, rBp~l should be rPG ~ p & FH p'1. This definition comes up against a number of problems, descending either from the structure of time or from the nature of events. The most drastic problem comes from the ordering which defines because of the restriction on r J 51 in the scope of r F 1 : rp1 might be true before now but not after. The rule RG can be applied to formulae involving r P 1 only in special cases, approximately when either all occurrences of are in the scope of other tense markers or when, in the equivalent formula using r 1 r 1 r only & and ~ as propositional connectives, either every occurrence of P~[ is in the scope of an odd number of negations or every evenly negated occurrence is matched by an oddly negated one. This same restriction also applies to RH with formulae containing rF1.

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the streams of a time structure. If that ordering is merely dense, there does not need to be a moment which satisfies the above condition, for there are segments which have no greatest lower bound. On the other hand, if the ordering is discrete, every moment satisfies it vacuously. In the logic for discrete time rPG$~' and r FH 0"1 are theorems for every formula 1 at a point. (The natural language restriction seems to involve the fact that at the moment of beginning one cannot notice that the event will go on. Perhaps it also involves the fact that by the time one can say that something is beginning, it is already under way. It is the moment of utterance, not of cognition, which is the axis.) Finally, the suggestion that not every interval of time is to be considered an event, even when every moment in that interval is characterized by the same formula, only moves the problem over to another area — the concept of an event. The attempt to remove this problem by using rF' ~ in place r of PG ~ p~* will not work either. The new expression, like the old, picks out the first member of the p interval or the last of the r~p~' interval. It is the existence of such a member that is in question here.

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On the other hand, the problem with discrete time is easily solved. Here rB, ~ r) & is true now but was not at the last moment". The formula r S( t F, ~ T)"1 means " W was true at the last moment" since, with respect to that moment, any formula of the form r G W is true, while, from any other moment r G ~ r"1 will be false because there will be intervening moments and, at them, i is true. The formula, then, is true at the first moment at which 0 is true after a period in which ))"' "

1 iff, for every k in M, 1=«

IIp1 and rIIFp ->•lip\ In any case, there would be vnTIp*-y IJp' and the rule that if

If, then, P~ is taken as the anterior vector marker and is read as "it has been the case that", the problem which remains is to make LTP into a language system by introducing markers for axes. At first glance, it would seem that this could be done by putting a different semantics on the language with the same syntax as LTP-2. However, this would fail to catch one important feature of axis reference in natural languages. In natural languages, there are ways of indicating whether two different uses of the same axis marker ({-D} for example) refer to the same moment or not. There are even means to indicate the temporal order of the several RP's referred t o : I walked in. Mike turned his chair, rose, walked to the bar and poured a double. Then he offered me a chair. When I walked in, Mike turned in his chair and rose. He offered me a chair while he moved to the bar and poured a double. Mike turned in his chair and rose. As I walked in, he walked to the bar. He offered me a chair. LTP-2 has no such means since it would have only the one RP marker (now), rP~1. So long as LTP languages contain only markers for tense, further information about order, other than by putting one marker within the scope of another, is not expressible. Since the order is not always indicated yet one also wants to

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be able to say that two events occur at the same time, the LTP language being constructed must have more than one RP marker. For this purpose, primed rP~"s will be used. Two occurrences of r > 1 7 ~ with the same number of primes would refer to the same moment. Two occurrences with different numbers of primes might refer to different moments. For convenience, the number of primes will be superscripted, rather than the primes themselves: rP3~' rather than rP'"~[. There is no upper limit to the number of different MP's which might be referred to in a given context, and this device prevents the superscripts from growing too long too fast. These numbers are to be understood in their cardinal sense — how many primes. They do not indicate the order of the RP's referred to: rP2~l may refer to a moment before or after that to which rP1 refers, or to the same moment. For general discussion, the Greek letters fx and v will stand for any superscripted number: may be any of r/>1"1, r P 2 n , ..., and ri>v~1 functions in the same way, under the condition that, in a given formula, two occurrences of the same superscript must represent the same number of primes. Natural languages do provide examples of axes determined with respect to axes other than PP, namely, RAP. From this it would appear that there need not be any restrictions on the sentences which appear within the scope of axis markers. The possibility of iteration other than posterior to RP can be seen by the possibility of specifying that point intended by what is marked RP- : He had come the day before. Certainly vector markers can occur within the scope of axis markers; such occurrences were what led to the axisvector distinction in the first place. The occurrence of axis markers within the scope of vector markers is more suspect, but is here included for simplicity. There may be some natural language occurrences: He has come but she had left the day before (although this may be "Always when he has come she has left the day before"). 4.2.2.1. Syntax The syntax of LTP-3, as this new extension of LTP is called, incorporates that of LTP and adds that for the primed rP'1 and

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0 is a formula of LTP-3 iff

, one merely claims that there is a moment at which such-and-such happens: the moment is selected by what happens at it. It is the fixity of the preselected moment throughout a context which allows r P ' " to serve as an axis marker, just as a particular moment picked out by a date would be referred to by an axis marker. To say that the moment referred to by a primed tense marker is selected at the beginning of a context is not quite accurate, however. The moment referred to by rP"~' is fixed as being in the past of the moment of evaluation and, thus, cannot be picked until that moment is fixed. In this way, selection of the reference for an RP marker differs somewhat from the similar selection in other parts of a language. The selection of the referent for nominal expressions which, in a given context, can refer to only one thing can be made at the very beginning of the whole course of utterance. One can begin a long conversation by fixing the reference for John or the red book. For RP markers, the selection must be made anew at various points in the conversation, depending upon the tense context within which the marker is to occur. Thus, all of the RP markers which do not occur in the scope of any other tense marker may be fixed initially; those which occur within the scope of other tense markers cannot be fixed until the referents of these superordinate markers have been fixed.

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The definition for the truth of a formula beginning with a primed tense marker can now be given: )=, rPn iff there is a certain jn in P(i) ¥]n

PPp'")? The LMP density postulate displays the features of LMP which are most different from LTP. In addition to the atomic formulae and connectives from propositional logic and tense markers, LMP has also quantifiers CA-1 "for all" and rV "for some"), predicates on numbers (e.g. r =~', identity), and operators on numbers (e.g. r + addition). All of these may be used to form formulae which have no analogs in LTP. Indeed, there is not one language LMP but many, depending upon what number system is used and which mathematical predicates and operators are used. All of these languages share a common core, however. It is this core which will be described here, although indications of how it might be generalized will be given. The first item which this core has, over and above those items found already in LTP, are terms, the metrical markers. These terms are to refer to the numbers of the measuring system. Since they refer to numbers, the notation of mathematics may be taken over to serve the purpose here. Thus, there will be three types of "

PPF, 56.

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terms: numerals, variables, and complex terms built up with the aid of operators. In the core, the only numerals are strings of digits, excluding only those with r (T at the beginning (the numbers to be used are all positive). If the number system used in the measuring system were the reals (if the ordering in the time structure were continuous), there might also be a number of special symbols for real numbers not expressible in terms of ordinary numerals, e.g. r iC. The variables in the core are rrT, rm~>, and rk~l with or without subscripts (e.g. r m 35 ~'). The set of variables is the same for all LMP languages. In the core, the only operator is r + ~ \ This joins two terms to form a new term, the names of two numbers to form the name of a new number, the sum of the two. If the number system is rational, one would expect also r p which joins two terms to form the name of the ratio between the numbers named by the two. For the reals, one would also expect ry/~l for specifying the positive root. For any number system other than the integers, one would want r."1 for forming names of numbers by giving their decimal expansion. 5.1.1.

Syntax

What constitutes a term can be summed up in the definition (the Greek letters n and v being used to speak about terms in general as 0 and W are used for formulae in general). H is an M-term iff /z is a member of every set Z such that (1) all numerals (strings of digits) are in Z (2) all variables are in Z (3) if vx and v2 are in Z, r v t + v 2 n is in Z. If there are operators other than r +~ 1 in an LMP language, the last clause would have to be modified to cover all of them. Using 0 as a cover symbol for operators, the general clause would be (3') if 0 is an «-place operator and v l5 ..., v„ are in Z, r 0(vu ..., vj" 1 is in Z. Following this clause strictly would give r + (v1( V2)"1 as a term; the

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more familiar form could then be introduced by definition. It is this last clause which permits the creation of complex terms of any degree of complexity. Thus, even in the core, one can have r (n + m) + A:"1 as a term: rn>, rm~1, and rk~* are by clause 2, rn + m"1 and the whole are by clause 3 (the parentheses are, as in the definition of LTP formula, punctuation marks to show grouping). The core also has only one predicate on numbers, r = ~\ In terms of this, others can be defined, e.g. r < n "is less than": r n < v~" = i f r Vn n + n = v"1 "fi is less than v just in case v is the sum of ¡x and some number". In some languages there might be a use for predicates other than r = n and those definable in terms of it; for example, for the integers, is odd and is even. Using 77 as a cover symbol for predicates, there would then need to be a general clause in the definition of M-formula which described how formulae were built up out of terms and predicates: if 77 is an n-place predicate and Vi, ..., v„ are M-terms, then r77(Vi, ..., v„y is an A/-formula. In fact such formulae will be atomic, though treated in a different clause from r/>~1, etc., since they contain no connectives or tense markers. For the core, this general clause is not needed, since there is only the one predicate. The definition of M-formula parallels that for formulae of LTP, except for the added clause for the new kind of atomic formulae and the clauses dealing with quantifiers.

of LMP, take numerical terms, the last is a unit — 0 never varies. The definitions are

Rn ~ l = d f r ( N - v & R - ( n + v)lt ...,4> f are ^-formulae, then

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¥in rn{nu...,

nm, ßu...,

ßs,

..., &fr

iff

S,.... IfiJn, lßJn> •••• lßJh> •••> */> is in R(II) (i) rPßß i»n iff there is a moment j, Ußf'n ' and (9) D(ij lßfT9) = lutjg and ¥ j m m