Action, Comparison and Change: Study in the Semantics of Verbs and Adjectives (Linguistische Arbeiten) [Reprint 2011 ed.] 3484301708, 9783484301702

Table of contents :
Part I. ACTION
I. Introduction
II. Some Theories on the Logic of Action
II.a. von Wright
II.b. Davidson
II.c. Pörn
II.d. Aqvist
III. Some Items from the Theory of Verbs involving Agency
III.a. Evidence for the Presence of Agency in certain Verb Classes
III.b. Agency and the Progressive in English
III.c. The Formal Structure of Verbs
Ill.d. Two Important Classes
IV. Reciprocal Verbs and the Passive
V. DO
V.1. A Formal Semantics
VI. DO and the Semantics of Reciprocal Verbs
VII. Some Left-Over Problems
Part II. COMPARISON AND CHANGE
I. Introduction
II. A Calculus for Adjectives and Nouns
III. Vagueness, Supervaluations and Comparison Classes
IV. Problems with Supervaluations and Comparison Classes
V. An Alternative Proposal
VI. A Language and a Semantics for Adjectives
VII. Subclasses of Adjectives and their Properties
VII.a. Privative Adjectives
VII.b. Affirmative Adjectives
VII.c. Linear Adjectives
VII.d. Strongly Predicative Adjectives
VII.e. Weakly Predicative Adjectives
VII.f. Strongly Extensional Adjectives
VII.g. Weakly Extensional Adjectives
VII.h. One Dimensional Adjectives
VIII. Some Items of the Semantics
IX. An Axiom System
X. On the Order of Adjectives
XI. Some Further Remarks on the Comparative
XII. The Description of Change
XII. 1. The Combination of the Tree-System for Action Sentences with the System for Adjectives
XII.2. The Language L´
XII.3. Semantics for L´
XII.4. Truth Conditions and a Hypothesis
XIII. Coda; A Final Remark on the Logical Form of Comparatives
Footnotes
References

Citation preview

Linguistische Arbeiten

170

Herausgegeben von Hans Altmann, Herbert E. Brekle, Hans Jürgen Heringer, Christian Rohrer, Heinz Vater und Otmar Werner

Jaap Hoepelman

Action, Comparison and Change A Study in the Semantics of Verbs and Adjectives

Max Niemeyer Verlag Tübingen 1986

CIP-Kurztitelaufnahme der Deutschen Bibliothek Hoepelman, Jakob:

Action, comparison and change : a study in the semantics of verbs and adjectives / Jaap Hoepelman. - Tübingen : Niemeyer, 1986. (Linguistische Arbeiten ; 170) NE: GT Hoepelman, Jaap ISBN 3-484-30170-8

ISSN 0344-6727

Max Niemeyer Verlag Tübingen 1986 Alle Rechte vorbehalten. Ohne Genehmigung des Verlages ist es nicht gestattet, dieses Buch oder Teile daraus photomechanisch zu vervielfältigen. Printed in Germany. Druck: Weihert-Druck GmbH, Darmstadt.

CONTENTS

Page Part I . ACTION

1

I.

Introduction

1

II.

Some Theorie« on the Logic of Action

2

H.a.

von Wright

2

II.b.

Davidson

4

II.c.

Porn

10

II.d.

Aqvist

12

III.

Some Items f r o m the Theory of Verbs involving Agency

28

III.a.

Evidence for the Presence of Agency in certain Verb

28

Ill.b.

Classes Agency and the Progressive in English

34

III.c.

The Formal Structure of Verbs

37

I I I . c . l . Dowty

37

I I I . e . 2 . Hoepelman

46

Ill.d.

51

Two Important Classes

I I I . d . i . Accomplishments

51

I I I . d . 2 . Activities

54

IV.

Reciprocal Verbs and the Passive

56

V.

DO

71

V.l.

A Formal Semantics

73

V.l.a.

A Semantics for WANT

77

V.l.b.

A Semantics for

82

V.l.c.

Letting and Forbearing

88

VI.

DO and the Semantics of Reciprocal Verbs

90

VII.

Some Left-Over Problems

94

DO

VI Page Part II.

COMPARISON AND CHANGE

99

I.

Introduction

II.

A Calculus for Adjectives and Nouns

100

III.

Vagueness, Supervaluations

lOb

IV.

Problems w i t h Supervaluations

V.

An A l t e r n a t i v e Proposal

110

VI.

A Language and a Semantics for Adjectives

113

VII.

Subclasses of Adjectives and their Properties

129

VII.a.

Privative Adjectives

129

VII.b.

A f f i r m a t i v e Adjectives

129

VII.c.

Linear A d j e c t i v e s

129

VII.d.

Strongly P r e d i c a t i v e A d j e c t i v e s

131

VII.e.

Weakly Predicative Adjectives

132

VII.f.

Strongly Extensional A d j e c t i v e s

133

VII.g.

Weakly Extensional Adjectives

134

VII.h.

One Dimensional A d j e c t i v e s

134

VIII.

Some Items of the Semantics

135

IX.

An Axiom System

141

X.

On the Order of Adjectives

143

XI.

Some Further Remarks on the Comparative

145

XII.

The Description of Change

158

XII.1.

The Combination of the Tree-System for Action

160

99 and Comparison Classes and Comparison Classes

108

Sentences with the System for Adjectives XII.2.

The Language L ' .

162

XII.3.

Semantics for L ' .

163

XII.4.

T r u t h Conditions and a Hypothesis.

167

XIII.

Coda; A Final Remark on the Logical Form of

176

Comparat ives. Footnotes

179

References

188

PART I. I.

ACTION

Introduction

Action, comparison and change are related notions. An action is generally considered to be a way of bringing about a change, and when a change occurs in or to a certain entity, it often has more or less of a certain property than it had before. In recent linguistics, the notions of agency and action play an increasingly prominent role in studies which try to explain syntactic and semantic phenomena connected with action sentences. Linguistics often borders here on philosophy, borrowing insights obtained in the study of the logic of agency. On the other hand, in philosophy methods and insights used and obtained in linguistics find their way. The present study borrows from both sides. We will give an exposition of some recent theories of action and agency, discuss them and propose some alternatives, based mainly on the requirements of the description of intention which plays a prominent role in the theory of agency, the classification of verbs (which has drawn new interest since Vendler ( 1 9 6 7 ) ) , certain phenomena of the passive and the English progressive tense. The English progressive, moreover, will be the point on which our analysis of action and our analysis of change meet - natural language seems to require a treatment which in some way u n i f i e s these concepts. In order to give a semantics for expressions of natural language describing change, one needs a treatment of the comparative. In this essay a theory of adjectives is proposed, in which the comparative can be defined with relative ease.

2

II. II.a

Some Theories on the Logic of Action von Wright One of the f i r s t to propose a logic of action has been von

Wright in his

(1963) and ( 1 9 6 7 ) . Here he observes that the notion

of action (or at least one important sense of it)

is connected

with the notion of change. To act is to (intentionally) bring about a change, which otherwise would not have happened, or, conversely, to (intentionally) prevent a change, which otherwise would have happened (von Wright, 1967, p. 121 - 123). To give an at least approximate formal account of these i n t u i t i o n s , von Wright develops a propositional logic which, apart from the usual Boolean connectives, has two additional two-place operators: (to be read "and n e x t " ) and

^

T

(to be read "instead o f " ) . Let us

suppose that we have a world, consisting of a set of "states" or "situations" and a linearly ordered set of discrete moments. Let us moreover suppose that each state can be exhaustively described by a f i n i t e set of atomic propositions, symbolized by Ρ _ ι < · · · ' Ρ _ η ' or the negations thereof. At a moment k £iTp_j is the case iff at k pi is the case and at the next moment, k + Ι , ρ ^ is the case. Suppose moreover that we have j u s t one agent. Then, at a moment k Ρ_ί_ϊ£-ϊ can be interpreted as "p^ is the case and p^ would have been the case if the agent had not acted". Now an action (of the agent, because we have only one) can be defined by a formula like the following:

Ρ ή Τ ( - p-; Ip^ )

about - P i " » i.e. "at f i r s t which is the case instead of the agent had not acted". It two elementary states

ρ

"the agent brings

Pi is the case and then - pi p ^ , that would have been the case if is easy to see, that if one has the

and

- ρ , there are 8 d i f f e r e n t ways to

f i l l in the blanks in a formula of the form

... Τ ( . . . I

...).

These eight cases von Wright calls the "elementary modes of act and forbearance". With each mode one can connect a reading in natural language, e.g. -p" and

pT(pIp)

The behaviour of

pT(pI-p)

can be read as "the agent prevents

can be read as "the agent lets Τ

(and of

!_

verned by the following axioms:

which has the

ρ

continue".

same axioms) is

go-

3 AI:

( (p v q) T (r v s) ) ^__> ( p T r ) v (pTs) v ( q T r ) v (qTs)

A2:

(pTq) & (pTs)

A3:

p

A4:

-(pT (q & - q ) )

(pT (q S, s ) )

(pT (q v - q) )

Apart from the operators T and I_ von Wright considers the operators M ( f o r possibility) and £ ( f o r p e r m i s s i b i l i t y ) . It is clear that as a f i r s t attempt von Wright's proposal is open to all kinds of c r i t i c i s m and r e f i n e m e n t . But the basic idea of his theory of actions remains guiding in most subsequent theories: An agent acts if he brings about a change which would not have taken place but for his action, or prevents a change which would have taken place but for his action.^' Two problems, which are partly connected with the fact that von Wright presents his theory in the form of a propositional logic, are noticed by Chisholm and Robison in their comments on von Wright (1967) (Chisholm, 1967) (Robison, 1967). I will vary Chisholms example: Suppose that a person A shoots a person B with a gun which would have gone off anyway at exactly the moment at which A pulls the trigger. Although we cannot say that the change involved in the shooting of B would not have taken place if A had r e f r a i n e d from pulling the trigger, we still want to say that A acted on that occasion and is responsible for shooting B. To overcome this d i f f i c u l t y , Chisholm proposes, loosely speaking, a partition of the action: When A shot B, we imply that there is an event p such that 1) had not been for the agent's interference p would not have occurred and 2) p is a s u f f i c i e n t causal condition for the gun's going o f f . (Chisholm, op.c. p. 137). We will see that the idea of splitting an action returns on quite independent grounds in the linguistic theory of verbs. Robison - among other things - regrets that in von Wright's framework it is not possible to specify the respect in which the agent interfered with the world, where this interference is somehow productive of the change in question and not merely an accompaniment of the change (Robison, op.c. p. 141). This problem of course hangs together with Chisholm's objection and as will be seen a common solution for both problems can be found.

4

lib.

Davidson A quite d i f f e r e n t road to the analysis of action sentences is

chosen by Davidson ( 1 9 6 7 ) , a more linguistically oriented one, one might say. Davidson starts with the observation that in English we seem to be able to r e f e r to events by means of pronouns, as in the famous example: (1)

Strange goings onl Jones did it

slowly, deliberately, in the

bathroom, with a k n i f e , at midnight. What he did was butter a piece of toast. Davidson then proceeds by c r i t i c i z i n g Kenny's (1963) proposal for the logical form of such sentences. According to Kenny we might conceive of the logical structure of (1) as follows: (2)

Jones brought it about that the toast was buttered in the bathroom with a k n i f e at midnight In general, Kenny proposes that action sentences have the

shape of ( 3 ) : (3)

Jones brought it

about that

p

Τη (3) the sentence that replaces

ρ

is to be in the present

tense and it describes the result that the agent has wrought:

it

is a sentence "newly true of the patient", a sentence describing a new state rather than a new event (Davidson op.c. p. 85, r e f e r r i n g to Kenny, op.c. p. 181). Davidson's c r i t i c i s m of Kenny is threefold: In the f i r s t place from ( 4 ) : (4)

Jones buttered the toast in the bathroom with a k n i f e at midnight

we want to i n f e r (5)

(5)

Jones buttered the toast

5

It is by no means clear that (2) accounts for this inference more easily than (4) i t s e l f . In the second place (2) does not solve Kenny's problem of the so-called polyadicity of predication: "buttered" in (4) can be thought of as being a predicate with four argument places - one for "the toast", one for "in the bathroom", one for "with a k n i f e " and one for "at midnight". But of course we can add other adverbial phrases, as "by holding it between the toes of his left foot", which would make a five-place predicate out of "buttered" - and so on, ad libitum. This seems to bar the possibility of accounting for the inference mentioned above by considering "Jones buttered the toast" as short for "Jones buttered something somewhere with something at some place". As Davidson notices this problem obtains also with the predicate "was buttered" in ( 2 ) . The problem of variable polyadicity has lost much of its force in linguistics, as we do no longer think of adverbial expressions like "in the bathroom" as occupying argument places of a predicate corresponding to the verb. In the third place, Davidson objects that in most cases we do not have means to describe in natural language the state denoted by p in (3) other than the ones we have already used: What is newly true of the toast is precisely that it has been buttered by Jones in the bathroom with a k n i f e , at midnight. We seem to be tangled in a vicious circle of explanation here. It is interesting to notice the resemblance of Kenny's proposal to proposals about the structure of verbs which were put forward by generative semanticists at about the same time, as in Lak o f f (1965). The criticism of Davidson applies to their formalism as well. Later on we will see that a solution can be found doing justice to both the ideas of Kenny and those of the generative semanticists . Having dealt with Kenny's theory in this way, and a f t e r brief discussions of ideas of Chisholm and von Wright which are rejected on more or less the same grounds, Davidson turns in quite another direction for the solution of his problems. Deriving inspiration from Reichenbach (1947, § 48) - who's theory he rejects (correctly in my opinion) - he proposes to introduce an extra argument place for every verb that describes an action. Take e.g. ( 6 ) :

6

(6)

Shem kicked Shaun

This sentence is analysed as ( 7 ) : (7)

( E x ) ( K i c k e d (Shem, Shaun, x) )

Likewise (8) could be analysed as ( 9 ) : (8)

I flew my spaceship to the Morning Star

(9)

( E x ) ( F l e w (I,

my spaceship, x) & To (the Morning Star, x ) )

This proposal has several advantages, but its disadvantages are so oppressive, that we shall nevertheless be compelled to reject i t . 2 ) Let us start with the advantages. A formula like ( 7 ) . ( E x ) ( K i c k e d (Shem, Shaun, x ) ) can be read as "There is an event consisting in the kicking of Shaun by Shem". This seems to explain why we can r e f e r to events by means of pronouns. It also seems to explain away the problem of variable polyadicity. Take ( 9 ) , which can be read as: "There is an event x consisting of me flying my spaceship, and x to the Morning Star". It is easy to see that we could have added "and x was at midnight", "and x was from the Earth", etc., etc. However, as the problem of variabale polyadicity has been dealt with in other ways since Davidson's paper, this advantage has lost much of its appeal. Another advantage of the proposal is, for the inference (10)

that it enables us to account

I flew my spaceship to the Morning Star I flew my spaceship

It is easy to see how (9) allows for this inference. Finally, if I flew my spaceship to the Morning Star, then, according to Davidson, one should be able to infer (11)

I flew my spaceship to the Evening Star

The analysis of action sentences based on the operator "bring about" does not allow for this, since, as Davidson argues convin-

7

cingly, "bring about" is not truth functional. There are, however, grounds to seriously doubt the validity of this inference. In any case, if one assumes that "to" is an extensional operator, then the replacement of "the Morning Star" by "the Evening Star" in (9) is of course allowed. Now we come to the disadvantages of Davidson's proposal. In the f i r s t place, his analysis introduces a source of numerous ambiguities in connection with tenses and modalities. These are hidden by the fact, that Davidson does not further analyse the verbs "kicked" and " f l e w " , which are in the past tense. Linguists and philosophers find a certain perverse pleasure in the discovery of ambiguities of sentences of natural language, it is true, but here we seem to be faced with more ambiguities than we need in the study of the phenomena in question. Supposing that we treat the past tense of verbs by means of some sentential past operator Ρ , where shall we put this operator in ( 9 ) 7 The most plausible options are (9' ) and ( 9 " ) : (9')

P { E x ) ( F l y ( I , m y spaceship,x) & To(the Morning S t a r , x »

( 9 ' ' ) ( E x ) P ( F l y ( I , m y spaceship,χ) & To(the Morning S t a r , x ) ) This means that we have to decide on the question whether there was an event in the past such that it consisted in my flying my spaceship to the Morning Star, or whether there is an event now, such that it was the flying by me of my spaceship to the Morning Star. The same problem of course appears when, instead of (9) we try to give a Davidsonian analysis of ( 1 2 ) : (12) If,

I can fly my spaceship to the Morning Star instead of ( 1 2 ) , we take ( 1 2 ' ) :

(12')

I could fly my spaceship to the Morning Star

the set of possible ambiguities is at least doubled. As Lemmon remarks in his comment on Davidson's paper (Lemmon, 1967), the past tense alone makes the inference from "I flew my spaceship to the

8

Morning Star" to "I flew my spaceship to the Evening Star" problematic. What, when at the time of flying the Morning Star was not identical to the Evening Star, but is identical to the Evening Star now? In the second place, from ( 8 ) , we do not only want to infer "I flew my spaceship", but also (13) and (14): 3

(13)

I flew to the Morning Star

)

(14)

My spaceship flew to the Morning Star

(or eventually "I went to the Morning Star" and "My spaceship went to the Morning S t a r " ) . Davidson's analysis gives no clue at all for these inferences. Instead, it is said of a certain event, that it is an event to the Morning Star. How are we to understand this? If I say (15): (15)

I went to the grocery store

I seem to imply that at least during a certain period in the past, I came nearer and nearer to the grocery store. Do we have to interpret (9) as saying that a certain event came nearer and nearer to the Morning Star? In the third place, it is not clear according to which principle To(the Morning S t a r , x ) is separated from the rest of the sentence. If we take a sentence like (16)

Jack saw Jill under the apple tree

and analyse it as (17)

( E x ) s e e ( J a c k , Jill, x) & Under(the apple tree, x)

the result is completely u n i n t u i t i v e , because it need not be the case at all that the action or event of seeing (supposing it is an action or event at a l l ) took place under the apple tree. Jack himself might very well have been under the pear tree, in which case the event of seeing was somewhere between apple and pear tree.

In the fourth place, Davidson's account doesn't seem to be able to account for the so-called "again-ambiguity". This problem is discussed in e.g. Dowty (1976, 1979) and Hoepelman (1978, 1979) and will also be treated in the sequel of this paper. The "againambiguity" amounts to the following: (18)

Yesterday Sean kicked Shaun and today Shem kicked Shaun again

Sentence (18) does not imply that Shem kicked Shaun before, but only that Shaun was kicked before. A good example is given by Dowty (1976, p. 215): (19)

All the k i n g ' s horses and all Humpty Dumpty together again

the k i n g ' s men couldn't put

On its normal reading (19) does not imply that anyone put together Humpty Dumpty, but only that he was "together" before. Kenny's proposal is superior in this respect: Whereas it is unclear how to account for the "again-ambiguity" in Davidson's theory, Kenny's analysis allows us to insert a sentential operator "again" in such a way in two places in the sentence, that the ambiguity is accounted for. In the f i f t h place, Davidson's analysis gives us no clue as to how to handle the passive of action verbs. If we express "Shem kicked Shaun" as ( 7 ) , then, presumably, "Shaun was kicked by Shem" as (20)

(Ex)(Kicked(Shaun,Shem,x))

But it cannot be expressed in this way, that it was Shem who did the kicking, was the agent, and that it was Shaun who remained passive. Finally we remark, that Davidson's analysis is not so much an analysis of action sentences, as of event sentences in general. Every argument in favour of the proposed logical structure of sentences containing action verbs put forward by Davidson, can equally well be applied to sentences containing patently^) non-action verbs, like "find" in (21):

10

(21)

I found my wallet in the hallway

But this of course means giving up the attempt to explain what it is for a verb to express agency at all. There remains, then, the explanation of the possibility of r e f e r r i n g to events by means of pronouns as in ( 1 ) . But brief reflection shows, that this is not even enough to distinguish events from non-events. For we can have sentences like ( 2 2 ) and ( 2 3 ) : (22)

John says, that there is oil in his backyard, but I don't believe it until I've seen it

(23)

Two plus two is f o u r , and don't forget i t l

To explain the logic of action, then, we return to concepts which are much more in the line of Kenny and von Wright and discuss a few more approaches which are written in this tradition.

II.c

Porn

In Porn (1970) we find the sentential operators D^ and C^ , where the indexes i , j , . . . are symbols for agents, combined with an ordinary first-order predicate logic. The resulting language is called LI . If £ is any formula of L^ , then DIP has the following possible readings (among others): (24)

It follows from what i does that i brings it about that p it is a thing done by i that p i

CIP (25)

does

p

p

has the following readings: It is compatible with everything that

i

it is possible for all that i does that what i does allows ( p e r m i t s ) that p

does, that p p

11

L! is interpreted in terms of model-structures in the sense of Hintikka (1962, 1963). An interpretation in terms of possible world semantics would run as follows: Let M = (D, W, Ri , Ι , V) be a model, where D is a domain of individuals, W a set of worlds, I a subset of D to be thought of as the set of agents, V an assignment of values to pairs consisting of an expression of L and an element of W and R^ a set of two-place relations on W indexed by the elements of

I . The truth d e f i n i t i o n for expressions of L^ which are Dand/or C-free is as usual, and will not be reproduced here. Important for us are the following truth conditions for sentences con-

taining (26)

D and/or

C:

V ( D i p , w ) = 1 iff such that wR^w" V ( C i p , w ) = 1 iff such that

V(p,w') = 1

for all w'

V(p,w') = 1

for some w'

wR^w'

It is, moreover, stipulated that for any i , R i is r e f l e x i v e . Thus, for any i the system gives a relativized (to i ) predicate logical version of the modal system T (see e.g. Hughes and Cresswell, 1968). Its axioms are the same as those of T plus a few others having to do with the indices i , j , . . . of D and C . But this is its weakness too. No distinction is made in this way between e.g. an epistemological logic as that of Hintikka (1962), a logic for "wish" or "want" and a logic for agency, like the one presented by Porn. It is easy to see, that D^p can be defined as ^?iZ£ · But it then follows, that i does any logical truth, for it is not compatible with anything that i does, that a logical truth is not the case: therefore any logical truth is done by i * ) . Apart from this, we see that it is not required that i occurs in the formula p of D^£ . So, presumably, we can have a formula like (27)

Dp eter (Henry beats John)

12

which could be read as "Peter brings it about that Henry beats John". But of course we still want to know what Peter did to bring Henry to beat John. That P rn's theory is not a theory of action sentences may also appear from the proposed reading of the operator GI in formulas of the form C^p : "It is compatible with everything i does, that p ". Many things may be compatible with what i does, but we still want to know what i d i d . Explaining what i did in terms of the operator D^ , however, would catch us in a vicious circle. Yet, the attempt to give a semantics for action sentences in terms of possible worlds (or model-structures, as P rn does) has its merits. We will now turn to a treatment in which this method is refined into a system of great subtlety.

II.d

Aqvist

Let us return for a moment to von Wright's formula PiT( - p^I p ^ ) "pi is the case and next -p^ is the case instead of Ρ Ϊ " · i.e. "the agent brings about a change from p^ to -p^". The problem here is to make more precise the notion that -p^ is the case instead of p^ . If a change occurs in such a way that at f i r s t p^ is the case and later on -p^ , where p^ represents a contingent proposition (such as "John is s i t t i n g " ) , then it is always possible to imagine a possible alternative situation in which ρ^ is true instead of -p^ . The point here is, that we are not interested in any such possible alternative, but rather in alternatives which are in some precise sense very similar to the one in which actually -p^ is the case. This notion of similarity has been taken by e.g. Chellas (1969, 1971) and Xqvist (1974, 1978) to be one of historical identity. I.e. if we say of John that he is sitting in a world w at a time t, and that he is not sitting in w at a time t' later than t, we can construct this change in John's position as an act of John, i f , under precisely the same historical preconditions, John could have remained in a sitting position. In the actual world, w, John is not sitting at t ' . But the alternative world w' in which he could still be sitting, should be one, which, up to t ' , has the same history as w, or put d i f f e r e n t l y , in which up to t' precisely the same facts ob-

13

tained, or, d i f f e r e n t l y a g a i n , which is identical to t'.

w up to

This idea is exploited f u l l y in Aqvist (1974, 1978). We will

explain, though not in all

formal d e t a i l , the main ideas of Aqvist

(1978), which constitutes a f u r t h e r development of Aqvist (1974) and is better suited for l i n g u i s t i c purposes. Aqvist defines a world to be a function from a set of time points (which is supposed to be dense, linearly ordered, not beginning and not ending) to a set of situations. The set of worlds forms a tree which is branching towards the f u t u r e but not towards the past. As an illustration we draw the following picture: (28)

w I

:

.

w'

w

w

W:

the set of worlds

In ( 2 8 ) T

represents the time line,

paths in the tree, i.e and w ' ' t'.

t

w'

worlds, and we see that up to

are identical, whereas and t'

w,

w'

and w ' '

and t

w'' w,

are w'

are identical up to

are branching points of the tree. It is possible

to single out subsets of worlds ( i . e . paths of the tree) by means of two kinds of choice functions which we will discuss in the

se-

quel. Trees like ( 2 8 ) serve for the interpretation of a predicate logic

L, which contains predicate and individual constants,

Priorean tense operators, tense constants, modal operators of universal and historical necessity and possibility, interval operators characteristic for the multiple indexing system of Aqvist and Guenthner (1977) and f i n a l l y modal operators expressing optimality according to an agent's intentions or beliefs and maximal probability. Now in this language a two place operator DO is defined, taking as its arguments an individual constant "a" and

14

a sentence consisting of a one place predicate

P" followed by

on its sole argument place. Expression ( 2 9 )

(29)

DO (a, Pa)

is to be read as "a does (or acts) in such a way that Pa" or "a 's action consists in the fact that Pa". E . g . if "John" and "P" formalizes "runs", then

"a" formalizes

DO ( a , Pa)

can be read as

"John acts in such a way that he runs" (Aqvist, 1978, p. 124). A series of d e f i n i t i o n s is proposed, leading to a more and more refined semantics for DO, some of which we will informally present and discuss here. The f i r s t and simplest d e f i n i t i o n centers around the condition which, as we pointed out above, was felt to be characteristic for action sentences already by von Wright, the condition of avoidability. Moreover it contains conditions of beginning, ceasing, and temporal extendedness. First we will draw a picture of a typical situation which makes

DO ( a , Pa )

true:

(30)

Pa - Pa

- Pa

χ

w W

- Pa

The set of worlds of exactly the worlds

w

W of the tree pictured in ( 3 0 ) consists w and

w' . The domain of individuals conis true in w at t 0 ( i . e . in the

tains at least a. D O ( a , P a ) situation w ( t o ) indicated by the

χ

in our picture)

iff

15

I): II):

there is an open interval around

to

in our picture) during which

is true.

Pa

Pa

(between

t

and u

begins to be true at the l e f t coordinate of this inter-

val .

Ill):

Pa

ceases to be true at the right coordinate of this in-

terval . IV):

In an alternative course of events (world)

w'

Pa

is

false during this interval. Condition IV) gives us the notion of avoidability, as it expresses that

Pa

could have been false at the same time as

is actually true in a course of events identical to the property

w up to the moment P .

t

w'

Pa

which is historically

at which

a

began to have

In his second d e f i n i t i o n for the semantics of DO Xqvist introduces an important new consideration about the theory of action: for a property agent

a P ,

(like "runs") to be agentively had by an

it does not s u f f i c e that

avoid having has

P

a

P

a

was in the position to

at the beginning of the period during which he

should also be in the position to stop having

P

at

some moment during that period. This is the so called condition of i n t e r r u p t i b i l i t y V ) , which is pictured more or less accurately in the following f i g u r e :

(30) t0

s

u

Pa . - Pa

Λ

χ

Q

. - Pa . - Pa .

W - Pa

.

w w"

The condition of i n t e r r u p t i b i l i t y can be strengthened to V I ) : the strong condition of i n t e r r u p t i b i l i t y , which requires that not only can

a

stop having

P

at some moment during his period of

a c t i v i t y , but that he can do so at all

moments of this period. See

Figure ( 3 2 ) :

(32) S

S

2

u

K

Pa w T 1 1

I

j - Pa

1 1

- Pa

1

I

1 1 ,

WK

- Pa

- Pa

W

to

W

2

w

l

Jlqvist notices that on the face of it, the strong condition of i n t e r r u p t i b i l i t y may be too strong: Suppose that the action formalized by the predicate

£

is "run". Is it physically pos-

sible to stop running at any moment one pleases during a period of running? However, one notices that a similar problem also obtains with the weak condition of interruptibility: Suppose that the laws of physics are such as they actually are - if someone runs and decides to stop running at a certain moment period a f t e r

to

t o , then for a brief

he will go on moving in a way, that we would

normally describe as " r u n n i n g " . This means that we cannot consider to

to be the point at which an alternative course of events bran-

ches off in which the agent is not running (assuming of course that in this alternative course of events the same laws of physics hold as in the actual one). So it

seems that the alternative

course of events branches off at some point

t±

But then we have not captured the f a c t , that at

later than to

the

to .

agent in-

17

deed stopped to be the agent of running, and that, during a brief period after to he is not running as an agent, but merely as an object of the laws of physics, so to speak. This problem is connected with Chisholm's objection to von Wright's theory of action, mentioned in the beginning of this paragraph, for which Äqvist's theory does not o f f e r a solution either, not even with the weak or strong conditions of interruptibility and not even, as we will shortly see, when his theory is extended to include the notion of intention. Imagine the following situation: Gangster A leans back against a wall and aims and shoots, with the f u l l intention of doing so, at his v i c t i m B. However, at the very moment he does so, his shoulder is hit by a stone which fell down from the wall as an (unintended) result of A's leaning against it. The blow on A's shoulder leads to precisely the same series of events which A is about to produce intentionally: A pulls the trigger and shoots B. In this situation there is no historically identical course of events in which A could not have pulled the trigger, or could have stopped doing so during this action. Still we would want to say that A shot B agentively, and was responsible for doing so (although it would be a hard case in a court of justice perhaps). These problems will be dealt with later on, when we discuss the splitting of actions in connection with the theory of verbs. Another problem which cannot be dealt with by the theory in the state as it is presented up to this point, is connected with the inverses of two-place predicates, and mentioned by Äqvist (1978) on pp. 132 - 133. Consider the sentences (33.a, b). (33.a)

Peter shoots Paul

(33.b)

Paul is shot by Peter

Let us suppose that the transitive verb "shoot" is formalized by the two-place predicate R and the intransitive verb- phrase "shoots Paul" by R^ . Then ( 3 3 . a ) will be formalized by Rba / where a translates "Peter". If we now denote the inverse of R by R^ , then Rba is logically equivalent to R*ab . As a consequence of the semantics given for DO in Aqvist's system, the following is valid as well:

18 (34)

DO(a,R b a)

«-» DO(b,R* a b)

If we consider R^ to be the translation of the phrase "is shot by", this is of course counter-intuitive: (33. a) expresses an action brought about by Peter, but (33. b) does not express an action brought about by Paul. qvist's solution to this problem consists in the introduction of a notion of intention into the d e f i n i n g conditions for the semantics of DO . To this end, to the elements which together make up a model M , qvist adds a f a m i l y of functions indexed by the elements of the set of agents D , such that for each set of worlds (i.e. courses of events) and each agent a subset of worlds is picked out, which we may consider to be the set of best or optimal worlds or courses of events according to the agents intentions or b e l i e f s . More formally, the following family of functions is added: d€D

such that for each d:

optd e

and opt d Y d W:

is a choice function in the sense that for any X,

( 3 5 ) a.

opt d (X) C

X

b.

opt d (X) = JS

only if

c.

opt d (X) Γ\ Υ * X

X = fS

only if

opt d (xO Y) = opt d (X)O Υ-

As a next step an operator (36)

Mus t Ace

is d e f i n e d , such that ( 3 6 )

MustAcc(a,A)

can be read as "it must be the case that A according to the intentions and beliefs of a", or more simply "a wants that A". In ( 3 6 ) a is an individual constant and A a well formed formula of the language L . MustAcc(a,A) is defined in such a way, that it holds in a model M relative to a world w and a time u iff A

19 holds on u in all worlds w' which are historically identical with w up to (but not necessarily including) u and which are optimal according to a's intentions. Put in a more formal way: (36)

MustAcc(a,A)

is true in a model u , iff

M relative to a world

w and a time

opt a (/w/ ~& Q / A / M ( w , u ) where /w/ ^j-' is the equivalence class consisting of the worlds which cire historically identical with w up to (but not necessarily including) u , and /A/i^/ w u > is the set consisting of the worlds w' which are historically identical with w up to (but not necessarily including) u and such that relatively to w' and (Aqvist, 1978, pp. 133 - 136).

u

A is true in

M.

Now the truth conditions for D O ( a , P a ) are strengthened in such a way, that, apart from the conditions mentioned earlier, a condition is included to the e f f e c t that V I I ) : VII):

Any course of events w' which belongs to a's set of optimal worlds and which is historically identical to w up to the l e f t coordinate of the open interval during which Pa is the case, is such that during this interval Pa is the case in w' . Put b r i e f l y : a wanted that Pa .

This condition in its turn can be strengthened to V I I I ) : VIII):

Not only wants a that Pa at the beginning of the interval of his having the property Ρ , he wants this at every moment of the interval as well.

Let us illustrate these ideas by a picture once more. (Aqvist, 1978, p. 137 f . ) .

20

(37) υ

Pa -* - Pa

- Pa

MustAcc(a, EH) Pa) ι

1 Pa - Pa

- Pa

w -

w2 w3 -

In ( 3 7 ) we have chosen the weak condition of intentionality ( V I I ) and weak condition of interruptibility ( V ) . The operator I—> l i n MustAcc(a,i 3 Pa) is one of the interval-operators of the system, indicating that Pa is true in the open interval determined by t and u , so that MustAcc(a,SS3 Pa) being true at t , indicates that in all worlds which are optimal according to a's intentions or b e l i e f s « Pa is the case between t and u . By means of + or - following the w , w ^ , W 2 . W 3 we indicate whether or not these worlds belong to the set of optimal worlds with respect to a . It is easily seen that the situation corresponding to Figure ( 3 7 ) satisfies the condition for MustAce : If w^ is the sole element of a's set of optimal worlds, then indeed Pa is true in w^ between t and u , and even afterwards. And if, apart from w^ , w itself would belong to a's set of optimal worlds as well, the condition still would be satisf i e d . This does not hold, however, if w 2 or W3 would belong to a's set of optimal worlds. Still, it seems to me that as it stands the introduction of the conditions of intentionality does not prevent the implication of DO(b,R a b) by DO(a,R b a) in cases in which we certainly do not want to say that the object of an action sentence acts. Take the sentences (38.a b ) : (38.a)

Mary caresses Peter

(38.b)

Peter is caressed by Mary

21

Let us suppose that "caress Peter" is symbolized by R^ and "is caressed by Mary" by R*a . Let us moreover suppose that Peter likes being caressed by Mary, i.e. that in all the worlds which are optimal according to Peter's intentions Mary goes on caressing him. Then there will be a period in which -R ab is the case, followed by a period during which R*ab is the case, which in turn is followed by a period in which thermore

a

R b

-R ab

is the case. Fur-

could have been avoided and can be interrupted.

Finally MustAcc(b,ESSR a b) is the case. This of course means that DO(b,R* a b) will be true. However, we do not want to say that Peter acted, when he was being caressed, i.e. whatever his actions were when he was being caressed, being caressed was not one of them. This case cannot even be solved by the last r e f i n e ment which Aqvist introduces into the d e f i n i t i o n of DO : As a last condition, a causal connection between the intention of the agent and the produced action is proposed. Moreover, the theory remains unsatisfactory on a few other points, which we will discuss a f t e r an informal exposition of A q v i s t ' s causal condition of intentionality, IX) ( A q v i s t , 1978, pp. 140 - 142). Apart from the functions opt 0 Fs'(t)^0

or

V(S,t) = l

i s the truth value function of

which assigns a value

_/\

V(S,k)

out of [0,l]

S , i.e.

the f u n c t i o n

to any moment

k

of

51

T . The second d i s j u n c t of the second condition in ( 9 9 ) is designed to reflect the i n t u i t i o n that if a sentence tely true, true to the degree 1, it

S

is comple-

cannot become more true, so

that "become S" is false in that case. Actually, matters can be more complicated. The truth value function

Fg

need not always be u n i f o r m l y increasing or decrea-

sing, e.g. jumps may occur. To handle such cases more apparatus must be introduced (see Hoepelman, op. c i t . ) . But as an exposition of the guiding idea the remarks made above may s u f f i c e . However, f u z z y logic is not without philosophical problems (see e.g. Kamp, 1975,

Hoepelman, 1979, p. 38 f f . ) and the machinery I introduce to

d e f i n e the semantics of

Δ

is complicated and not very elegant.

In this paper we will give a much simpler semantics for based on a new approach to the semantics of adjectives.

Δ

,

Let us now

return to the translations of sentences containing expressions of agency.

Ill.d.

Two Important Verb Classes

III.d.1

Accomplishments

One possible translation into intensional logic of "John closed the door", containing the accomplishment verb to close w i l l be (100)+: P((Ex)(Ay)((door'*(vy)

(100) v

x = vy)&(EP)(

H'(DOVP*( j)CA Δ

closeT* ( v x ) )

& closeT*(vx))) In (100) variable of type

IP is

the ordinary past tense operator,

( s, ( (s, e) , t ) ) ,

Ρ

is a

CA is a two-place sentential

operator intended to represent some notion of causation and is the protoverb associated with the English verb to close. It

is a predicate constant of type

(s, ( ( s, e) , t ) ) .

is the interval operator we have discussed above, and

Δ

Potts' operator "become".

Applying well-known reduction rules of Montague (1974).

H' is

52

Formula (100) is intended to express the intuition about the meaning of "John closed the door" put forward above, namely that by doing something during a certain period John caused the door to be more and more closed and to be closed f i n a l l y . In formula (100) we find a subformula (101), which we have intuitively rendered as "by doing something": (101)

(EP)(...(DOVP*(...)

...

The introduction of this phrase is j u s t i f i e d as a generalization of by-phrases which can always occur with accomplishment verbs, like in (102 a, b) : (102.a) (102.b)

John closed the door by pushing a button. John closed the door by blowing on his thumb.

We see that DO is concipiated as a predicate forming operator on one-place predicates. This saves us the trouble of postulating like-subject constraints which is necessary if DO is defined as a sentence-forming operator on sentences. Notice that the proto sentence close^*( v y) in (100) is not under the scope of the DO operator. This means that, if we assume that some part of the meaning of DO Q ( a ) ( f o r some one- place predicate Q and some individual constant Ά) is, that Q ( a ) is under the voluntary control of a , we have not expressed in (100) that the closing of the door was under the voluntary control of John, but only that this closing was some (possibly involuntary) "by-product" of John's performing some voluntary act. It might therefore be asked whether we should not also introduce a translation expressing that the closing of the door was under the voluntary control of John as well. This can easily be done; by means of lamda abstraction we can enlarge the scope of DO in ( 1 0 0 ) , leading to the following formula as a translation of "John closed the door": (103)

P((Ex)(Ay)((door'*(vy) v

x = v y) & ( E P ) ( D O ( ( \ z ) ( H ' ( v P * ( v z ) C A

& closeT*(vx))( j ) ) ) )

Δ

close T *( v x))

53

However, it

seems to me that this would not be correct. If "S

is under the voluntary control of. a" is taken to mean that does not take place if

a

doesn't want it

S

to take place, then

formula (103) is subject to precisely the objection which Chisholm raises against von W r i g h t ' s theory of action, and which, as we have seen, can also be raised against Xqvist's theory. For it may be that John closed the door as an agent, but at the same time, by some device independent of John's w i l l , the door would have been closed anyway, so that we cannot say that the closing of the door would not have taken place if John had not wanted it

to take

place. Still we want to say that John was an agent of the closing. What then, was in fact under the voluntary control of John? Chisholm proposes an analysis to the e f f e c t that there was an event such that had it not been for John's interference, have occurred and such that

p

p

p

would not

is a s u f f i c i e n t causal condition

for the door to close. But now we see that the lexical decomposition of the verb "to open" gives us the possibility to express just this: what (103) expresses, tively had some property

is that the fact that John agen-

P , was a s u f f i c i e n t causal condition

for the closure of the door. One might then say that John was responsible for the closure of the door, if, that having

P

in addition, he knew

is a s u f f i c i e n t causal condition for the closure

of the door. Finally we can express that the closing of the door was not the result of any voluntary act, (104)

as in (104)

John closed the door involuntarily by stumbling on a stone This last, agentless, version of to close can be arrived at

simply by deleting the occurrence of DO in formula (100). Thus we see, that independently of the precise semantics it may be p r o f i t able to d e f i n e for DO, the combination of DO with the device of lexical decomposition of verbs allows us to express Chisholm's solution for the problem of multiple agency. This cannot be done in Aqvists theory, in which for the sake of simplicity it is assumed that the predicates to which DO is applied, are the direct translations of English verbs. It also appears from formulae (100) and (103), that if we have lexical decomposition and operators to express that the event

54 expressed by the verb went on d u r i n g a certain time, then we no longer have to pose the conditions of beginning, ending and temporal

extendedness for the semantics of DO, as Aqvist did: these

conditions can be expressed independently in the Aktionsart of the agentive verb,

III.d.2

if we need them.

Activities

We have seen that in the case of accomplishments, whether transitive or t r a n s i t i v e , the way in which the subject of the

insen-

tence brings about the change in the object of the sentence ( i . e . : himself when the verb is i n t r a n s i t i v e ) can be specified by a by + gerund subordinate clause (Dowty, 1972, p. 104): (105)

John recovered by daily kicking his

(106)

John opened the door by wiggling his

wife ears

It seems that a c t i v i t i e s cannot be specified (107) (108)

in this way:

John walked to London by swimming in the Thames *John laughed at Mary by giggling about Bill On the other hand it

is possible to specify the change taking

place in the object of an a c t i v i t y verb (the subject when the verb is i n t r a n s i t i v e ) , but not the change taking place in the object of an accomplishment verb: (109)

John walked to London

(110)

John laughed himself sick

(111)

John pushed the car into the garage

But:

(112) (113)

John closed the door away *John recovered himself healthy.

(Notice that a f t e r

the addition to an activity verb of an ex-

pression specifying the change that takes place, the newly formed expression is no longer in the category of activity v e r b s ) .

55 This leads us to the hypothesis, that regarding some parts of their structure activity verbs and accomplishment verbs are converses: Whereas accomplishment verbs express that some unspecified property (that may be specified

in a by + gerund clause) causes a

specified change, a c t i v i t i e s express that some specified causes an unspecified change. I.e.

property

where we have analyzed "John

closed the door" in a way which might i n t u i t i v e l y be rendered as "by having some property John caused the door to get the property expressed by the protoverb connected with 'to close'", we w i l l analyze "John pushed the car" in a way which might i n t u i t i v e l y be rendered as "by having the property expressed by the protoverb connected with 'to push' John caused the car to change in some respect". And the sentence "John walked", exhibiting the intransitive a c t i v i t y verb "walk" w i l l be analyzed i n t u i t i v e l y as "by having the property expressed by the protoverb connected with 'to walk' John caused himself to change in some respect" (Hoepelman, 1978, 1979). So the translation of "John pushed the car" into Montague-style intensional logic (adapted to the introduction of the operator Δ ) w i l l be: P((Ex)(Ay)((car'*(vy) 4

(114) v

x =

v

T

>

y) & ( E P ) ( D O p u s h * ( j ) C A Δ

v

P*(vx)))

The translation of "John walked" w i l l be: P ( ( E P ) ( D O walkT*(j)CA ΔνΡ*(:ί)))

(115)

"John pushed the car" does not mean that the car arrived at a specific place ching

Ρ a f t e r a period during which it had been approa-

Ρ more and more. In fact "pushed" can denote the beginning

of an event: (116)

When John arrived, Peter pushed the button. Therefore in the translation of the Aktionsart of unexpanded

activity verbs occurring in non continuous tenses, we have not included the operator verbs.

S , as we did in the case of accomplishment

56

An independent ground for the introduction of an expression indicating that some change is going on into the translation of Aktionsart of activity verbs is the fact that, as we have seen before, there are sentences containing activity verbs which are clearly non-agentive, but occur in the progressive nevertheless, and the hypothesis that one of the sources of the progressive is the expression of change. Regarding formula (115), we remark that it is also possible to extend the scope of the operator DO so as to have the bound variable P in its scope as well (in analogy to what we have done with accomplishments). Formula (115) is interesting in connection with our remarks about interruptibility made earlier. As we have seen, on the one hand it is plausible to suppose that if someone is the agent of some action, he is capable to interrupt this action at some or eventually any moment of its duration. But on the other hand the laws of physics are such, that if someone is engaged in the action of running and decides to stop running at some moment, then he will go on moving in the way he did before for a (perhaps short) period a f t e r the moment of his decision to stop. If we do not lexically decompose activity verbs like "to walk", this of course jeopardises any e f f o r t to d e f i n e the semantics of DO by means of a condition entailing that DO w a l k ( a ) is true if w a l k ( a ) ceases to be true the very moment a wants it to cease to be true. But if we apply lexical decomposition, then, as can be read of from formula (115), we are in a position to say that only the protoverb connected with "to walk" is under the scope of DO , but not the change occurring to the subject of the sentence caused by the fact that the subject agentively has the property expressed by the protoverb. And it is not impossible that this change goes on a f t e r a moment t o at which the subject no longer wants to have the property expressed by the protoverb.

IV.

Reciprocal Verbs and the Passive

In chapter I I I . a we already had occasion to mention the fact that so-called reciprocal verbs, like "to quarrel with", "to play with", "to argue with", do not occur in the passive. Compare in this respect (117) and (118):

57

(117) (118)

John and Mary were kissing John and Mary were quarreling

From (117) (119)

it

follows that

John was kissing Mary and Mary was kissing John

and also

(120)

Mary was being kissed by John and John was being kissed by Mary

From (118) it (121)

follows that

John was quarreling with Mary and Mary was quarreling with John

But we don't have (122)

*Mary was being quarreled with by John and John was being quarreled with by Mary

It is interesting to notice that the verb "to play with" can occur in a reciprocal and in a non-reciprocal sense, and in the latter case it can occur in the passive, compare: (123) (124) (125)

John played with Peter Peter was played with by John John played with the medicine ball

(126)

The medicine ball was played with by John

(124) is acceptable only if it is interpreted as meaning that John played with Peter as an object, as in (127) (128)

The cat played with the mouse The mouse was played with by the

cat

58

Reichenbach already supposed (1966, p. 2 5 3 ) that the passive form of a two-place verb denotes the converse of the relation denoted by the active f o r m . This alone is of course not enough to explain the fact that certain two-place verbs seldom or never occur in the passive. The converse of a two-place relation the relation R

holds of

sis,

R which holds of the pair

(y, x)

R

is

if and only if

(x, y) . In other words - according to this hypothe-

any two-place verb, including the reciprocals, has a con-

verse. Moreover, the fact that in certain languages, e.g. L a t i n , German, and Dutch, there are passive forms of intransitive verbs, cannot be accounted for. Current treatments of the passive can be subdivided in three main groups: 1.

sentential

2.

lexical

3.

phrasal. The sentential treatment d e f i n e s the passive as a rule opera-

ting on f u l l sentence structures. As Bach puts it,

(Bach, 1980, p.

298) in t r a n s f o r m a t i o n a l i s t theory, passive is a transformation defined for structures of the form X - NP - Aux - V ( p r e p ) - NP - Y This hypothesis meets with severe d i f f i c u l t i e s , which are discussed in detail in e.g. Bach (op. c i t . ) and Keenan (1979). We will not repeat their arguments here, but notice that on the force of this hypothesis one would expect reciprocal verbs to occur in the passive. Both Bach and Keenan argue for a phrasal treatment of the passive. As the lexical treatment of the passive can be considered to be a special case of the phrasal treatment, I will ignore it here, and b r i e f l y discuss Bach's and Keenan's proposals. Bach (op. c i t . ) , proposing a treatment in a Montague-framework, supposes that there is a syntactic category of passive verb phrases (PVP) which can be combined with be to form an intransitive verb phrase ( I V P ) , but can occur independently as well. Semantically Bach considers PVP's to be predicatives, denoting sets of individual concepts. The following rules are given (p. 3 1 4 f f . ) :

59

(129)

Agentless Passive Verb Phrases: if y £ TVP, then EN ( γ ) e PVP, where EN ( y ) is the result of making (or choosing) the past participle of the main v e r b ( s ) in ^ . Translation Rule: if ^ translates as # ' , then EN ( ^ ) lates as Xx Ey ( ^ ' ( *\ P P ( x ) ) (y)

(130)

Agentive PVP: if 9 W in£ d : Y)

= supd (X) f^ Υ if

i n f d (Χ) Γ\ Υ

is the case at t

Q —» S

is true in every

world and at every time: in particular at true at

t

(because

given d e f i n i t i o n

S

does not imply

Q —»S

will be

W A N T ( S , a ) , even if

S . To give another example:

Sentence ( 2 2 3 ) (223)

w' :

is true t h e r e ) . So we see that under the

WANT(Q,a)

necessarily implies

is that there is an

Jones gets ten thousand dollar

Q

79 necessarily implies ( 2 2 4 ) (224)

Jones gets ten thousand dollar or dies of pneumonia

We do not want that from ( 2 2 5 ) (225)

Jones wants to get ten thousand dollar

it follows that (226)

Jones wants to get ten thousand dollar or to die of pneumonia I.e.

it

should be possible that ( 2 2 5 ) is t r u e , and ( 2 2 6 )

false. As we saw in the semantics given above for possible: Even if the formulae

a sentence

WANT(Q,a)

and

Q

WANT

this i_s

necessarily implies a sentence S,

-WANT(S,a)

are not contradictory.

Notice, however, that the following formulae are contradictory under these circumstances: This is as it secretary it

WANT(Q, a)

and

WANT(-S,a) .

intuitively should be: If John wants a blonde

cannot be the case that he is opposed to having a

secretary at all,

but it may be the case that his a t t i t u d e towards

having any old secretary is one of i n d i f f e r e n c e . L i k e w i s e , if John wants to get ten thousand dollar, it cannot be the case that he is opposed to getting then thousand dollar as well as to die of pneumonia - by DeMorgans law, from WANT(- Q & - S ,a)

WANT(-(Q v S ) , a )

follows that

.

We also see - and this too is as it that from

it

WANT(-Q,a)

it follows that

intuitively should be -WANT(Q,a) , but not con-

versely (the reader is invited to check this for h i m s e l f ) . What about a repetition of (227)

WANT - operators as in ( 2 2 7 ) 7

WANT(WANT(Q,a),a) It is easy to prove that under certain,

i n t u i t i v e l y accep-

table conditions the following w i l l be true for all agents a, worlds w, times t and assignments g: (228)

WANT(WANT(Q,a),a)

—» WANT(Q,a)

models M,

80

These conditions

(229)

1.

are:

For all if

w' e

then

2.

sup a (* w *t) and w" €

w''e

For all if

a in D a , t in T and w , w ' and w ' ' in W:

w' e

supa(*w'*t),

supa(*w*t)

a in D a , t in T and w , w ' in W: s u p a ( * w * t ) , then

infa(*w*t)

c

in£a(*w'*t) In words: 1.: world

the relation of being superior to is t r a n s i t i v e . If

w'

is superior to world

b e l i e f s ) and world superior to

w''

w

(according to some agent's

is superior to

w' , then

w''

is also

w .

And 2 . :

if we go to a superior world (according

to

etc.)

then the set of i n f e r i o r worlds increases, or remains at least constant. To put it

very i n f o r m a l l y : The more we have got,

the

more we have to loose. One easily sees that formula ( 2 2 8 ) indeed holds under these conditions. To get the converse of ( 2 2 8 ) , i . e . ( 2 3 0 ) : (230)

WANT (S, a) —»· WANT (WANT (S ,a) , a)

we need in addition the f o l l o w i n g conditions: (231)

la. - is like 1. but with i n f a replacing sup a everywhere -

3.

For all a in D a , t in T, w , w ' in W: if

4.

w' c s u p a ( * w * t ) ,

For all if

then w' € sup a ( w ' * t )

a in D a , t in T and w , w ' in W:

w' €. s u p a ( * w * t ) ,

infa(*w*t)

then i n f a ( * w ' * t ) =

31

Instead of 3, we could also have taken 3': 3'. For all etc. if w' e sup a (*w*t), sup a (*w*t)

then sup a (*w'*t) =

Informally 3. says that if one is in a superior world, this world appears a^ a superior world. It is as if someone would express his wish to get 100 dollars and to the suggestion that he would not be happy if he got them, because he then would want 1000 dollars, replies that this would certainly not be the case, that 100 dollars is all he needs, that he would be perfectly happy with them etc. 3' is the condition that from the standpoint of any superior world, the set of superior worlds is identical to the set of superior worlds seen from the standpoint of "this" world, and 4. expresses a special case of ( 2 2 9 ) 2 . , namely, that seen from any superior world the set of i n f e r i o r worlds remains the same as the set of i n f e r i o r worlds as seen from "this" world. It may be debated whether conditions 3. - 4. are i n t u i t i v e . But then, of course, it can be asked, whether we really want the implication (231) to hold. Suppose I want to have 10.000 dollars. We reconstructed this as a statement to the e f f e c t that in some superior world of mine I have 10.000 dollars and in all i n f e r i o r worlds of mine, I don't have 10.000 dollars. Then, if (231) would be true, in none of these i n f e r i o r worlds it would be the case that I wanted 10.000 dollars. This seems highly improbable. We must notice another point here: if we also have as conditions the mirrorimage of conditions ( 2 2 9 ) 1., 2 . , i.e. if in ( 2 2 9 ) 1., 2. we replace all occurrences of supa by inf a and conversely, then any formula of the form (232)

WANT(WANT(S,a),a)

will be false. Finally we might wish that at least the following implications are always true: (233)

WANT(g,a) —*·

-(WANT( - WANT(S,a) , a ) )

(234)

WANT(S,a)—>

-(WANT(WANT(- S , a ) , a ) )

82 It

is easy to see that the f i r s t of these holds if we adopt

conditions (231) 3 ' . , 4 . , and the second if we adopt the condition

(229)

2. or (231) 4..

V.l.b.

A Semantics for DO

Having given a semantics for

WANT , we are in a position to

give d e f i n i t i o n s for the semantics of an operator introduce in our language

L

in the following way: Let

either a one-place predicate of the

form

in which

A x Q ( x ) > where χ

L , or an expression of

Q ( x ) is

L , where

a

P L

be of

a well-formed expression of

is a f r e e variable. Then

expression of

DO , which we

POP ( a )

is a well-formed

is an individual constant or vari-

able. Now we can d e f i n e a stronger and a weaker version of the mantics of (235)

DO .

D e f . strong: Let P and a be as described above. Then

·"«^ = 1 1)

& 2)

M

iff

w

[p(a)] « ·*·« = 1

(Aw ' ) ( w ' C *w*t —> ( Γ- W A N T ( P ( a ) , a ) J M , w ' , t , [-P(a)]M'w'»t'9 = 1))

& 3)

(236)

< E w " ) ( w " e *w*t & [ p ( a ) ] M ' w " ' t ' 9 = 0)

D e f . weak: Let £ and a be as described above. Then [ D O P ( a ) ] M ' w ' t ' 9 = 1 iff 1) [p(a)]M,w,t,g = ι *w*t ·*-*. ( [ w A N T ( - P ( a ) , a ) ] M ' w ' ' t ' 9 = 1

& 2)

(Aw')(w'

& 3)

( E w " ) ( w " e *w*t &

C

L

P i a J M - w " ' ^ = 0)

se-

83

The f i r s t of these d e f i n i t i o n s is strong in the following sense: It al a

a

says of a property

as an agent in a world

has the property

identical with

P

P_

that if

it

is had by an individu-

w at a moment

in a world

w'

w up to the moment

t , then whenever

which is historically

t ,

a

property. By the semantical properties of

wants to have that

WANT , the weak d e f i n i -

tion is a consequence of the strong d e f i n i t i o n : if we adopt the strong d e f i n i t i o n and assume that f w A N T ( - P ( a ) , a ) ] M ' w ' · * ' 9 = 1, f-WANT(P(a) ,a) )]

M w

'

''t'9 = 1

then it

follows that

and thus that

r - p ( a ) l M ' w ' ' t ' 9 = 1. In particular then

a

in

w i t s e l f : if

a

has the property

wants to have that property, and if

the property

Ρ , then

a

does not have

a

wants not to have

it.

The weak d e f i n i t i o n just says that if a property by an individual

a

as an agent , then

Ρ ,

Ρ(a)

£

would (in

is had all

worlds w' historically identical with w, up to t) not have been the case, if

a

had wanted

P ( a ) not to be the case.

Condition 1) of both the strong and the weak d e f i n i t i o n speaks for i t s e l f . It has as a consequence that one cannot ( t r u l y ) do an inconsistent property. Condition 3) expresses that deed is an avoidable property for

a

(in w, at t).

P_

in-

Moreover

it

precludes the possibility that a logical truth can ( t r u l y ) be done. The introduction of a weak and a strong d e f i n i t i o n for agency has been motivated by the following considerations. On the one hand it has been said that if someone acts in a certain way, i.e. has a certain property as an agent, then it must be the case that he wants to perform this act - so e.g. in Aqvist (op. cit., 140,

p.

D e f . 6.) or implicitly in Dowty (1979, p. 1 1 7 f f . ) . This con-

cept has been expressed by the strong d e f i n i t i o n . On the other hand, we have the example discussed above of my walk while being in thoughts. Each step can be considered to be an act,

although not as something I positively want to do. One thing

is sure, however: if I absolutely want to not l i f t my feet any longer, then I will no longer have the property of walking active-

84

This is reflected in the weak d e f i n i t i o n . Concerning condition 3) it might be asked whether it would not be more plausible to introduce a corresponding universal version 3 ' ) : 3')

( A w ' ) ( w " ji w & w"

€

*w*t -* Γ ρ ί β ) * 1 · " " ' * ' ? = θ )

We think that this is not feasible, for the following reason: Suppose that in a world w , at a moment t , an individual a has the property P as an agent, i.e. POP ( a ) is true. In the same situation it is not the case that another individual, b , has the property Q . However, it could have been the case, that under historically exactly the same circumstances, b would have had property Q at t as an agent, i.e. DOQ(b) might have been the case, and, moreover, POP ( a ) might have been the case as well. Condition 3 ' ) does not allow for a situation like this, it only allows for situations in which DOQ(b) would have been true, i.e. we cannot express that two agents do not, but could have acted together. We illustrate our case by means of scheme ( 2 3 7 ) , which would obtain under condition 3 ' ) :

(237)

-P(a)

I 1 0

0 -P(a) I Ο -P(a) DOP( a ) , - Q ( b ) I ο

DOQ(b),

-P(a) -P(a) -P(a)

wi Wo

W3

W

1 1 1

ο

ό 6

W4

*ς Wfi

It may of course be debated whether the weak or the strong d e f i n i t i o n of the semantics of DO is to be preferred. The example of a walk in thoughts, given above, makes me feel that the weak version is more plausible. Therefore I will concentrate on this version.

85

It can easily be proved that under the given semantics an embedding of DO operators connected with j u s t one agent is of no e f f e c t , i.e. that any formula of the form ( 2 2 5 ) : (238)

D0( A x D O P ( x ) ) (a)

is equivalent to just P O P ( a ) . For the proof no special conditions on sup or inf are needed. In one direction the proof is obvious and follows directly from the semantics given for DO . In the other direction the proof is more involved, though not more d i f f i c u l t in principle. We will sketch a part of it, which can safely be skipped by those not interested in the details. Suppose then, that for some

w

in

W and some

t

in

T

it

could be

possible that [uOP(a)J M , w , t , g ( B ) . From (A) it

(A)

= λ

and

f D O ( \ X D O P ( x ) ) (a)1 M ' w ' t »9 = 0

follows that

(1)

(2)

(3)

( A w ' ) ( w ' € *w*t —» ( [ w A N T ( - P ( a ) , a ) 1 M , w ' , t , g f - P ( a ) ] M . w ' . t . g = 1)}

( E w " ) ( w " e *w*t & [-P(a)] M , w " , t , g

= 1

(By the semantics given for the weak version of DO) From (B) it

follows that

(4)

|~DOP(a)l M , w , t , g

(5)

(Ew'")(w'"

C

= 0

*w*t

& &

[WANT(-DOP(a),a)] M , w ' " , t , g fpOP(a)] M , w ' " , t , g = X)

86

(6)

(Aw ---- )(w ---- e *w*t —»[DOP (a)]

M w

'

1)

It can easily be seen that (4) contradicts the assumption ( A ) , and that (6) contradicts (3) ( v i a the semantics of D O ) . Therefore we must assume that (5) holds. If we now work out ( 2 ) and (5) according to the definitions given for the semantics of WANT

and

DO , we get

the r

(C)

following: from

(2)

( E i ) ( i e s u p a *w'*t &

[-P(a)J M , i . t , g ,

( A w ' ) ( w ' e *w*t -*>(· ( A i ' H i ' e i n f a *w'*t -> - P ( a ) '*'·^9 = D)

and from (5) (Dl)

(Ew'") (Dla)

j w ' " e *w*t & ( E i 2 ) [-P(a)] M , i 2 , t , g

=

[i 2

e su

Pa *w'"*

ι ( E w 2 ) ( w 2 « : sup a *i 4 *t

(Dlb)

( E i 4 ) ( i 4 €*i 2 *t & ( A w 3 ) ( w 3 e i n f a *i 4 *t -» [-P(a)] M ' w 3 · 1)

(Die)

(D2)

=

( A i 5 ) ( i 5 c*i 2 *t

(Ai3)

i 3 C i n f a *w'"*t

= 0

87

(D2a)

M, i 3 , t , g

=

( E w 4 ) ( w 4 c s u p a *i 6 *t & (D2b)

[-p(a)]M'w4't'9· = Γ

&

(Ai6)(i6£*i3*t-»( ( A w 5 ) ( w 5 e i n f a *i 6 *t-* [-P(a)] [•_p(a)jM,i6,t,g

(D2c)

= 1)}

M w5

'

' *'9 = 0

&

( E i 7 ) ( i 7 C*i3*t

(D3a)

= l

( E w 6 ) ( w 6 c supa *i 8 *t & (D3b)

(Ai8)(i8eV"*t-»( ( A w 7 ) ( w 7 e i n f a *i8*t

= 0

= 1))

(D3c)

( E i 9 ) ( i 9 e*w'"*t &

= 0)

Now (Dlb) and (C) lead to a contradiction, *w*t

because

* . *. 12 t =

( i 2 € * w * t ) . And ( D i e ) leads to a contradiction with ( 3 ) ,

for the same reason. So we must assume that ( D l a ) holds. From this a n d from ( D 2 ) , ( D 2 a ) i t (E)

follows that ( E ) :

( w " ' e * w * t & ( E i 2 ) ( i 2 € s u p a *w"'*t &

I p ( a ) 1M'i 2 ' t ' 9 = 0)

& ( A i 3 ) ( i 3 C i n f a *w'"*t—*· [ ρ ( Α ) Μ ' ί 3 ' ^ 9 = 1 ) )

According to the semantics for

WANT , (E) is equivalent

(F):

(F)

(w"'e*w*t&

[wANT(-P(a),a)]M,w'",t,g

=

to

88

Then from (2) and (F) it (G)

(w"'c*w*t&

follows that ( G ) :

[wANT(-P(a),a)1 M , w ' " , t , g

However, from (D3a) we also have

=

[p(a)jM'w

» f c » 9 = ι , so

that we f i n a l l y get ( H ) : ( E w ' " ) ( w ' " € * w * t & [-P(af|M'w'"'t'9 - 1

(H)

M,w'",t,g a contradiction. Therefore, if | D O P ( a ) [θΟ( A x D O P ( x ) ) ( a ) ] M . w . t . g = χ .

V . l . c.

M Wft

'

=

&

x)

' 9 = 1 , then

Letting and Forbearing

If we introduce a notion of historically restricted possibility and a corresponding operator

mr

in

language

L (comp.

Aqvist, op. c i t . , p. 123), then we can d e f i n e the operators of letting and forbearing. Say that for two worlds a moment of time t:

t:

v^^ «2

iff for all

w j ^ t ' ) = W 2 ( t ' ) . We can express

that

Wi

W1«"~£-'W2

is historically identical to

sarily including)

t'

w^

and

v/2

such that

and

t'
t ' 9 = ι i f f there is a w' such that [s] M , w ' , t , g = ι where S sentence of

with w' v e K )

CI.

(Av)(veQ+(Z)(K)-»«v^Q-(Z)(K))

CII.

Q+(Z)(K) = jZf->CT(Z)(K) = 0

CIII. Q-(Z)(K) = 0->(Q+(Z)(K) = K

V

Q + ( Z ) ( K ) = 0)

CIV.

( A v ) ( Z ( (v) ) = J * - > ( A K ) ( v X Z ( K ) ) )

CV.

( Z ( K ) = K) «-»· (Av) (Aw) ( v e K & w e K ) — * - Z ( { v , w } ) = {v,w) ) )

CVI.

( Z ( K ) = £ 0 < - ^ . ( A v ) ( A w ) ( ( v e K & w e K ) - » - Z { { v , w } ) = 20)

CO. - CVI. are supposed to hold for all

adjectives and common

nouns. The next group of conditions, CVII. - C I X . , we assume to hold not for all

adjectives, but still for many. In the discussion

of some of the subcategories of the adjectives we w i l l propose some conditions with an even more limited range. CVII.

(Av)(vcQ+(Z)(K)-*v eZ(K))

C V I I I . ( A v ) (Aw) ( (v eK

& weK & veQ+(Z)(K) &

w X Q + ( Z ) ( K ) ) - ^ Q + ( Z ) ( { v , w } ) = {v}). CIX.

(Av)(Aw)

( ( Q + ( Z ) ( { v , w > ) = {v} & v £ K & w f i T K )

+

-^

+

(w€Q (Z)(K)—>veQ (Z)(K))) Finally we have a rule

C+/-

on all

conditions apart from C I I .

and C I I I . : C+/-.

If

in any condition we replace all

occurrences of + by -

and of - by + , then the resulting formula will be a c o n d i t i o n ^ ) . Before discussing these conditions in detail, let

us once more

consider in an i n t u i t i v e way the way in which common nouns function in the theory and the e f f e c t of m o d i f y i n g a common by an adjective. The common noun, e.g.

"tomato", is interpreted as a f u n c t i o n ,

which from any subset of the domain

U

of individuals picks out

the set of things which are tomatoes relative to the subset. In many cases the set singled out by "tomato" relative to some subset w i l l be the empty set.

E . g . we can suppose that "tomato" singles

out the empty set relative to the set of screws. So we see, we can express the non-applicability objects

that

of a predicate to a set of

(whether this predicate is simple or composite), by

stipulating that in case of non-applicability to a set K, the predicate chooses the empty set

from among K.No recourse to

1 17

partially defined functions is necessary. As a special case of this use, we consider the treatment of the downscale polar opposite adjectives A°, like small (= big°), in case the background set from which a subset is to be selected consists of nothing but objects which are exactly as A ( N ) for some common noun N. In the f i r s t place, we want to avoid, that a sentence like "this object is big and small" can ever be true - not even in the case of a background set in which all objects are equally big, and in the second place we can in this way express the asymmetry which exists between A and A° polar opposite adjectives: the use of equally big in a and b are equally big does not imply that a and b are big - they may be very small. But the use of equally small implies smallness in some other background set. We will return to this topic. In other cases, the chosen set w i l l just consist of tomatoes, e.g. if we have a set sonsisting of oranges, bananas and tomatoes, then "tomato" will single out the set consisting of the latter. In still other cases "tomato" can be used to pick out sets of objects which, relative to some set K, could be considered to be the tomatoes in that set. K, but which we would not be w i l l i n g to call "tomatoes" in general, so e.g. if K would be a set of a r t i f i c a l f r u i t s , made of plastic. The point of all this is, that even non-complex common nouns are used in a relativized way, i.e. comparatively. Now a complex common noun, like "big tomato" is interpreted by a function similar to the one which interprets "tomato" i t s e l f . F(big tomato) = F ( b i g ) ( F ( t o m a t o ) ) picks out the subset of the big tomatoes within any subset K of the universe. In particular, F ( b i g ) can be applied to F(big tomato), and F ( b i g ) ( F ( b i g ) ( F ( t o m a t o ) ) ) will single out the big ones among the big tomatoes in any set K, the very big tomatoes in K, one might say. In this way a hierarchy of tomatoes in terms of bigness can be arrived at. Let us now have a closer look at our conditions. CO. says, that any common noun picks as a value a subset of the set K which serves as its argument in each case. As a t r i v i a l

118

consequence, let

Z = F(tomato) and K = U, then CO. reads:"every

tomato is a thing" (actually, everything is a thing, not surprisingly)\ Sticking to examples in terms of tomatoes, and interpreting Q+ as F ( b i g ) and Q~ as F(big°) ( = F ( s m a l l ) ) , C I . says that anything which is a big tomato in K is not a small tomato in K (and vice versa) even if K consists of nothing but big tomatoes, or, to put d i f f e r e n t l y , and more correctly, if all

it

tomatoes in K are equally

big. Sometimes it

is not possible to use a common noun - with or

without adjectives - to make a proper selection within a set of objects: if we are confronted with a set of tomatoes of absolutely equal size (as far as can be determined under the circumstances), then the common noun "big tomato" cannot be used to single out a subset of these in any sensible way. The same holds for the polar opposite "small tomato". On the other hand, if we have a set of objects to which the common noun "tomato" cannot be applied at

all

- a set o f , e . g . screws - then again "big tomato" cannot be used in any sensible way to single out a subset - but the reason for this is d i f f e r e n t from the f i r s t case: In the f i r s t case each object of the set is as much of a big tomato as each other object in a positive sense - since we have nothing but equally big tomatoes. In the second case each element of the set is as much of a big tomato as any other element in a negative sense - since any element in the set is as far removed from being a tomato

(let

alone a big tomato) as any other element. In the second case too the polar opposite, "small tomato" can be no more p r o f i t a b l y employed than in the f i r s t case. The possibilities discussed above are expressed in CI. and C I I . and C I I I . : In term of tomatoes, C I . states that no big tomato is a small tomato, and conversely. Moreover: if all are equally big,

tomatoes in a set K

then with respect to K, the noun "small tomato"

does not apply. C I I . states, that if Q + ( Z ) is not applicable to K, in the way in which "big tomato" cannot be used to make a choice within a set of screws, then Q ~ ( Z ) is not so applicable

119 either. Notice the following consequence of C I I . : there is a small tomato in any set K

only if there is a big tomato in K.

C I I I . states that if Q ~ ( Z ) nouns are not applicable to K there are two cases: either all

objects

not applicable to K at all.

in K are equally Q + ( Z ) , or Q + ( Z )

CIV. expresses what it

is

means for a

noun Z (the interpretation of a n o u n ) , whether composite or not, not to be applicable to an individual v: in that case v does not belong to any set

singled out by Z in any subset of the universe.

(Contrast this with the case where Z({v}) = ( ) , which, by saying that v is as much of a Z as v, implicitly states that Z can at least be applied to v ) . Suppose now that the set

K on which the f u n c t i o n

applied consists of j u s t two elements,

v

and

Q"*"(Z)

is

w . In this case

we have a means to d e f i n e the comparative by means of the positive. We will f i r s t give a provisional d e f i n i t i o n in the metalanguage, in order to f a c i l i t a t e the discussion of the remaining conditions. I n t u i t i v e l y one might read

Q+(Z)({v,w}) = ( v j

is the Q one of the pair {v,w} of Z ' s " or as "v is

(279)

1)

the more Q Z of

Def.

w is

a more Q Z than v

2)

w is a less Q Z than v

3)

w and v are

4)

as "v

equally Q Z ' s

w is no more a Q Z than v

Q+(Z)({w,v}) =

iff

+

iff

Q (Z)({w,v}) = iff

iff

{w} {v}

Q+ ( Z ) ({w, v j ) = +

Q ( Z ) ( { w , v } ) = ff

{w,v}

26)

In 1) and 2) w ^ v. If we disregard natural language, then clearly ( 2 7 9 ) could be framed in terms of "-" equally well as in terms of "+". However, as hinted at above people's i n t u i t i o n very often seems to be, that polar opposites of adjectives do not behave completely symmetrically in expressions like those introduced in ( 2 7 9 ) . Let us consider inferences 4), 4a) and 4b) of Ä q v i s t ' s list, repeated here for convenience:

120

(262)

4.

John is a more intelligent student than Mary. John is a less stupid student than Mary.

4a. John is a more intelligent student than Mary. John is an intelligent student and so is Mary. 4b. John is a less stupid student than Mary. John is a stupid student and so is Mary. to these examples we add

(280) 1. John and Mary are equally i n t e l l i g e n t . John and Mary are equally stupid. 2. John and Mary are equally stupid. John ist

stupid and so is Mary.

3. John and Mary are equally stupid. John and Mary are equally intelligent. 4. John and Mary are equally i n t e l l i g e n t . John is intelligent and so is Mary. According to Aqvist (1979, p. 2 1 ) ( 2 6 2 , 4 ) is valid. Äqvist is well aware that many people w i l l not agree with this, but claims that the feeling of uneasiness evoced by 4) is caused by the assumption that not only 4) is valid, but 4a) and 4b) as well. It would then follow, that John and Mary are both intelligent and stupid. My feeling is,

that in many cases - also in the case of the pair

"intelligent - stupid" - the negative pole is more marked than the positive one, in that its

use carries more of an assumption or

presupposition of a placement of an object or objects on the negative end of a scale than the positive pole does. E . g . I think that 4a) is not valid, but that 4b) might very well be. At least I would i n f e r from the premiss in 4b) that John is far b r i l l i a n t , and that Mary is stupid indeed.

from

121

Likewise, my intuition is, is,

that (280,1) is not valid, but ( 2 8 0 , 2 )

and that ( 2 8 0 , 3 ) is^ valid, but ( 2 8 0 , 4 ) is not.

If we wish to express this asymmetry, it

can easily be done - of

course this does not settle the matter whether the asymmetry should be expressed for all

polar p a i r s , or just for some, and if

so, for which pairs. In any case, one possible d e f i n i t i o n would

be: (281) D e f i n i t i o n w is

a more Q° Z than v

iff v e Q ° ( z ) ( U ) and Q ° ( Z ) ( { w , v } ) = (w}.

w is a less Q°Z than v i f f w € Q ° ( Z ) ( U ) and Q°(Z)({w,v}) = (vj. w and v are equally Q° Z ' s i f f w C Q ° ( Z ) ( U ) or v d Q ° ( Z ) ( U ) and

w is no more a Q°Z than v

iff

Q ° ( Z ) ( { w , v } ) = gf Variants of (281) can easily be formulated - one might e.g.

wish

to d e f i n e that v or w are not Q ° ( Z ) in the most general sense, i.e. It

in Q ° ( Z ) ( U ) , but at least for some K C U , such that ( w , v ] c K . is easily seen, that for any w e l l , w and w are equally Q Z ' s , or

w is no more QZ than w, which is as it

should be. It also follows

immediately from ( 2 7 9 ) , that if w is a more QZ than v, v is a less QZ than w ( i . e .

disregarding the actual behaviour of many polar

opposites in naturel language as possibly described in d e f . ( 2 8 1 ) ) , Suppose that in CV. K = U, and Z = Q + ( Z I ) , where Q+ = F ( b i g ) and Ζ χ = F ( t o m a t o ) . Then CV. reads: "All objects are equally big

122

tomatoes if and only if for no pair of objects {w,v}, w or v is the bigger (equivalently: the b i g ) tomato of the two. Under the same interpretation of K and Z , C V I . can be read as follows: "All objects are equally far removed from being big tomatoes of and only if for any pair {w,v} of objects, w is no more of a big tomato than v". Notice the contrapositions of CV. and CVI.: Under the given interpretation they say, respectively, that there are objects which are not big tomatoes iff there are two objects which are not equally big tomatoes (or no tomatoes at a l l ) , and that there are objects which are big tomatoes or equally big tomatoes iff there are two objects one of which is a bigger tomato than the other, or which are equally big tomatoes. This may lead us to accept inference 13 of Aqvist's list as valid assuming that "dangerous number" is applicable at all: If 13 is the most dangerous number, than it is more dangerous than other numbers, and therefore there are dangerous numbers - and 13 will certainly be one of them. It does of course not follow, that dangerous numbers are dangerous things. Similarly, it follows that there are large numbers if some one number is larger than some other. Finally, notice that in CV. and CVI. and Z could have been just F ( t o m a t o ) ( o r F ( N ) for any basic common noun N ) . Then CV. says, that within K all objects are as much of a tomato iff "tomato" is applicable to each pair £w,v} of K, and in no pair {w,v} one object is more of a tomato than the other. And CVI. states, that "tomato" is not applicable to K as a whole, iff it is not applicable to any pair {w,v}(and therefore not applicable to any individual if w= v). C V I I . expresses e.g. that if v is a big tomato, then it is a tomato. This condition seems to hold for most adjectives, but there is a number of adjectives for which it does not hold. Examples would be fake, alleged, f u t u r e or late (of dead people). Below we will give a special condition for the socalled privative adjectives, of which fake is an example. A consequence of CVII. is

CVIl'.: Z ( K ) = /Zf-»Q + ( Z

the

following:

123

which says, that if Z is not applicable to K, than any m o d i f i cation Q + ( Z ) of Z is not so applicable either. It may be sensible to include CVII. among the generally valid conditions, i.e. even for those adjectives for which CVI. does not hold. I hesitate to do so, however, because there might be languages, with adjectives having the same kind of e f f e c t as the English p r e f i x un-, such that Q " * " ( Z ) ( K ) ? 0, although Z ( K ) = 0(1 know of no examples). Suppose that in C V I I I . K = U. Then our interpretation in terms of big tomatoes would lead to the following reading: If v and w are objects and v is a big tomato and w is not a big tomato, then v is the big tomato in the pair {v,wj. Let us look at some other possible readings: according to the formation rules for L, Z could be F ( 0 ) ( = 1 ) , and K could be F ( t o m a t o ) ( U ) . In this case C V I I I . says, that if v and w are tomatoes, and if v belongs to e.g. the big objects in K and w does not belong to the big objects in K then v is the big object in the pair {v,wj. Alternatively we could let Z stand for e.g. F ( g r e e k ) ( F ( 0 ) ) , K still being F ( t o m a t o ) ( U ) . C V I I I . would then read: if v and w are tomatoes and v belongs to the big greek objects in K and w does not belong to the big greek objects in K, then v is the big greek object in the pair {v,w}. In CIX. let again K = U, Z = F(tomato) and Q+ = F ( b i g ) . Then CIX. reads: if v is the big tomato in {v,w} and v and w are objects, then if w is a big tomato, then v is a big tomato. Another possibility: let Z = F ( g r e e k ) ( I ) and K = F ( t o m a t o ) ( U ) ? then read : if v is the big greek object in {v,w}, and v and w are tomatoes, then if w is a big greekt object in the set of tomatoes, then v is a big greek object in the set of tomatoes (i.e: a big greek tomato). Intuitively, holds for an that v and w and w is not.

in its most general form, neither CVIII. nor CIX. adjective like clever. In the case of C V I I I . : Suppose are politicians and that v is a clever politician, Then it does not follow that v is the more clever

124

object in {v,w}. E . g . apart from being a politician, w might be a mathematician as well, and so clever as such, that actually he, w, is the clever object in {v,w}. Concerning CIX. we can say that conversely from the fact that v is a more clever object than w in {v,w} and the fact that v and w are politicians and, moreover, w is a clever politician, it does not follow that v is a clever politician as well. The fact that v is a more clever object than w, i.e. that v is more clever than w "generally speaking" does not imply that v is more clever than w in every respect, e.g. as a politician^?) . From CVIII. we can easily prove two formulae, which we will call CX. and CX' . respectively, which can be used in proofs of transitivity: CX:

( A v ) (Aw) ( ( Q + ( Z ) ( { v , w > ) = {v,w} & v e K & w € K ) —*· ( v € Q + ( Z ) ( K ) «- w € Q + ( Z ) ( K ) ) )

CX': ( A v ) ( A w ) ( ( Q + ( Z ) ( { v , w » = V & veK (v€Q+(Z)(K)«*-»> w e Q + ( Z ) ( K ) ) ) .

& w ) =

0.

Suppose 3a) holds, and suppose that χ ^ Q + ( Ζ ) ( { χ , y , ζ } ) , then y X Q + ( Z ) ( { x , y , z » - for if not, then by C I X . χ e Q + ( Z ) ( { x , y , z } ) - , and likewise z / Q + ( Z ) ( { x , y , z} ) . So Q + ( Z ) ( { x , y , z } ) = # and by CV.: Q + ( Z ) ( { x , y } ) = f t , contradicting 1 ) . T h e r e f o r e : χ e Q + ( Z ) ( { x , y , z j ) . But then by C I X . z e Q + ( Z ) ( { x , y , z } ) . Now either y e Q + ( Z ) ( { x , y , z } ) - then Q + ( Z ) ( { x , y , z } ) = i x , y , z ) , and f r o m CV. Q + ( Z ) ( { y , z } ) = {y,z} ( contradicting

2) -,

or y ^ Q + ( Z ) ({x, y, z} )

and from C V I I . : Q + ( Z ) ( { y , z } ) = {ζ}, likewise contradicting 2). Therefore, we must drop 3 a ) , and assume 3b): The assumption that χ fiQ+(Z ) ( {x,y, z} ) leads to a contradiction in a way similar to the contradiction obtained b e f o r e . So let χ e Q + ( Z ) ( { x . y . z } ) ; .then by C V I I I . z € Q + ( Z ) ( { x , y , z » r then by C I X . y e Q + ( Z ) ( { x , y , z } ) , so Q + ( Z ) ( { x , y , z } ) = {x,y,z}, and by CV. Q + ( Z ) ( { y , z } ) = {y,z},

contradicting 2).

So f i n a l l y , we

have to assume 3c): We have seen, that if χ ^Q+(Z)({x,y,z}),

then

+

Q ( Z ) ( { x , y , z } ) = f i , and by CVI. Q + ( Z ) ( { y , z } ) = ) = {y}. Contrail i c t i n g 1). L i k e w i s e , z ^Q+(Z) ( {x,y, z ] ). Therefore Q + ( Z ) ( { x , y , z } ) = 6 , and by C V I . : Q + ( Z ) ( { x , y } ) = ^ , contradicting 1). So x € . Q + C Z ) ( { - < , X , z } ) and by C V I I I . : y € Q + { Z ) ( { x , y , z } ) , and si m i l a r l y z € Q + ( Z ) ( { x , y , z } ) . Therefore Q + ( Z ) ( { x , y , z } ) = { x , y , z } , and by C V . : Q + ( Z ) ( { x , z } ) = { κ , ζ ] . contradicting the assumption. Suppose 3b) holds: We must assume that x Q + ( Z ) ( { x , y , z } ) (see the case above). Then y Q + ( Z ) ( { x , y , z } ) and z Q + ( Z ) ( { x , y , z } ) . So Q + ( Z ) ( { x , y , z } ) = ( x , y . z j ani by CV.: Q + ( Z ) ( { x , z } ) - {x,z} ; contradicting the assumption. Suppose 3c) holds: If x ^Q + (Z ) ( { x , y , z) ) , then y ^ Q + ( Z ) ( { x , y , z}) and z Q + ( Z ) ( { x , y , z } ) , b u t then Q + ( Z ) ( { x , y } ) = U (by C V I . ) contradicting 1). So x £ Q + ( Z ) ( [x,y,z] ) , therefore y e Q + ( Z ) ( { x , y , z } ) , and z e. Q + ( Z ) ( ( x , y , z}) . So, as before, Q + ( Z ) ( { x , y , z } ) = {x,y,z}, and by CVI. Q + ( Z ) ( { x , z } ) = {x,z}, contradicting the assumption. This concludes the case of ( 2 8 3 ) .

127

For ( 2 8 4 ) : Suppose ( 2 8 4 ) is false then 1) Q + ( Z ) ( { x , y } ) = 0 2) Q + ( Z ) ( { y , z } ) = 6 3a) Q + ( Z ) ( { x , z } ) = ( x ) , or

3b) Q + ( z ) ( { x , z } ) = {x,y}, or 3c) Q + ( Z ) ( { x , z } ) = {z}. Suppose 3a) holds: Q + ( Z ) ( { x , y , z ] ) / ff, for if not, then, by CVI ., Q + ( Z ) ( { x , z } ) = $ , contradicting the assumption. Suppose x£ Q + ( Z ) ( { x , y , z } ) , then it must be the case that y j z i Q + ( Z ) ( { x , y , z } ) , for if not, by C V I I I . : 0 + ( Ζ ) ( { x , y } ) = {y\, contradicting 1). By the same reasoning: z &Q+ ( Z ) ( { x , y , z } ) . So Q + ( Z ) ({x,y,z} ) = jzf, contradicting the above. Therefore χ € Q*(Z) ( {"x, y, z}). By C V I I I . : ye Q + ( Z ) ({x,y,z} ). Again by CVIII. I z e Q + ( Z ) ( { x , y , z } ) . So Q + ( Z ) ( { x , y , z } ) = {x,y,z}; but this, by CV. contradicts 1) and 2 ) . This concludes the case of ( 2 8 4 ) . In a equally simple way two related formulae can be proved, using C V I I I . and CIX. (285) ( Q + ( Z ) ( { x , y } ) = {x,y}& Q + ( Z ) ( { y , z } ) = {z}) Q + ( Z ) ( { x , z } ) = {z}. (286) ( Q + ( Z ) ( { x , y > ) = Si & Q + ( Z ) ( { y , z } ) = {z} ) Q + ( Z ) ( { x , z } ) = {z}.

>

*

128

T r a n s i t i v i t y holds in general for adjectives like big, but not for adjectives like clever; If χ is a bigger tomato than y, and y is a bigger tomato than ζ , then χ is a bigger tomato than ζ . And if χ and y are equally big tomatoes and y and ζ are equally big tomatoes, then χ and ζ are equally big tomatoes. Moreover, when χ is bigger than y (whatever χ and y are) and y is bigger than ζ , then χ is bigger than ζ . And if χ and y are equally big (whatever χ and y a r e ) and y and ζ are equally big, then χ and ζ are equally big. This last kind of inference seems not to hold for clever: it may be that χ is more clever than y and y is more clever than ζ , but χ and ζ cannot be compared in cleverness, and likewise if χ and y are equally clever and y and ζ are equally clever, χ and ζ may be incomparable. But of course if χ and ζ are comparable, then t r a n s i t i v i t y holds. Therefore we can give weaker versions of CVI1I. and C I X . , C V I I I . and CIX. respectively, which are supposed to hold in general and which are like C V I I I . and CIX. but for one proviso: CVIII.

- like C V I I I . - and the common noun of which Ζ is the interpretation does not contain any occurrence of 0 .

CIX.

- like C I X . - and the common noun of which Ζ is the interpretation does not contain any occurrence of Ο .

From C V I I I . * a corresponding weaker version CX. of CX. follows, and from C V I I I . * and CIX.* we obtain a corresponding restricted form of transitivity.

129

VII.

Sub-Classes of Adjectives and their Properties Let us now discuss the properties of some sub-classes of the

adjectives in more detail. In some cases we will introduce extra conditions. Kamp's classification of adjectives (Kamp, 1975, p. 125) has been taken over and extended somewhat by Aqvist (1979, p.

22ff.).

We take Äqvist's list as our point of departure and extend it somewhat in our t u r n .

a.

P r i v a t i v e Adjectives An example is f a k e . Evidently, a fake tomato is not a tomato,

which means that C V I I . does not hold for privative adjectives. If we compare two objects, and want to establish which one is

the

fake tomato of the two, we must make sure that the chosen object lies completely outside the realm of tomatoes. This leads to condition CPRIV.: CPRIV.:

(Av)(v

€. Q + ( Z ) ( K )

>v

€.U/Z(U))

where Q4" is F ( A ) for any p r i v a t i v e a d j e c t i v e A , and U / Z ( U ) is the relative complement of Z ( U ) with respect to U . Notice that it

is a consequence of CPRIV. that there can be no fake

things.

b.

A f f i r m a t i v e Adjectives Example: genuine. A genuine tomato is a tomato. The great majority of the adjectives seems to be a f f i r m a t i v e ;

condition C V I I . is the condition we propose for these adjectives.

c.

Linear Adjectives These are the adjectives that induce a linear ordering on

subsets of

U . Examples are tall or hot (as expressions of mea-

130

sures ( K l e i n , 1979, p. 13 f . ) ) · As a special condition we propose CLIN.: CLIN: ( Q + ( Z ) ( K ) / +

j* & K'

Z ( K ) -*Q+ ( Ζ ) (Κ' ) / ff)

CT

&

+

( Q ( Z ) ( K ) Χ/ί - * Q ( I ) ( K ) X 0 ) where

Q+

is

are subsets of

F ( A ) for any linear adjective

A, and

K

and

K'

U.

We also assume that apart from the general conditions, C V I I I . and CIX.

hold for linear adjectives (there may exist privative linear

adjectives, although I don't know any example). This then leads to t r a n s i t i v i t y for is a more QZ, is a less QZ, are equally Q Z ' s and is no more a QZ than. From the d e f i n i t i o n for more and less i r r e f l e x i v i t y and asymmetry follow. CLIN ensures, that for any K such that Q + ( Z ) singles out a nonempty subset in K, the ordering induced by Q"*"(Z) is complete on

K. Finally CLIN ensures, that if of objects,

it

is possible to single out a set

determined by a common noun Z within a set +

linear dimension Q"*",then Q

K along

the

can also be applied to objects,

without any f u r t h e r specification (this is expressed by the identity function I) within K. This means, that any two objects, which possess the dimension Q + , can be compared as things according to Q + , since a consequence of the second conjunct of CLIN is: ( Q + ( Z ) ( K ) X JZT & Q + ( Z ' ) ( K ' ) X 0) - » Q + ( I ) ( K ^ K' ) X (?. (by C V I . )

To give an example: if Q+ stands for

"hot", expressing

temperature, then we can compare burning candlesticks and stars as hot objects - where the stars are very much hotter objects than the candlesticks, although, as everyone knows, burning candlesticks are very hot.

131

d.

Strongly Predicative Adjectives

I use the term strongly predicative adjectives for adjectives like fourlegged, which I want to distinguish from adjectives like red, which I will call weakly predicative adjectives. Notice that in English neither fourlegged nor red are accompanied by a non-basic polar opposite 2 9 ). On the other hand can red be sensibly used in the comparative, whereas fourlegged cannot. Any fourlegged animal is a fourlegged being. If we have a pair of animals v,w and one of them, say w , is the fourlegged one of the two, then w simply is a fourlegged animal and v is not, and therefore w is a fourlegged being and v is not. On the other hand, if v,w is a pair of unripe tomatoes and w is slightly more red than v , it makes sense to determine w as the red one of the two, without thereby implying that w is a red object, nor that v is an absolutely non-red object. However, it seems reasonable to assume that w would not be determined as being more red than v if w would show no trace of redness whatsoever. This means, that if w is a more red tomato than ν , it is not necessarily the case that w f i g u r e s among the red objects, but it is at least not an absolutely non-red object. For the strongly predicative adjectives we propose CSPRED.: CSPRED.:

(Av)((v eQ+(Z)(K)—v v eQ+(I)(U)) & ( ( v e Q + ( I ) ( U ) & v e Z (K) —* v e Q + ( Z ) ( K ) ) )

It is now easily seen, that any set Z ( K ) is completely partitioned by Q + ( Z ) and Q ~ ( Z ) , except for the exceptional case where Q + t Z H K ) = ff. In this case we do not need Q~ anyway. Suppose then, that there is a v such that v e Z ( K ) , v ^ Q + ( Z ) ( K ) and v £ Q ~ ( Z ) ( K ) . Let z e Q + ( Z ) ( K ) . If we also assume, that CVIII: holds for strongly predicative adjectives (an assumption which seems reasonable), it follows that v € Q ~ ( Z ) ( ( v , z ) ) (by CO, CI, C I I I , and C-f/-). From CSPRED and C+/~ we obtain v e Q ~ ( I ) ( U ) , and from CSPRED and the assumption that v c Z ( K ) it follows that v e Q ~ ( Z ) ( K ) (via C + / - ) , contrary to hypothesis. In particular, if Z = I and K = U, the universe U will be completely partitioned by Q + ( I ) and Q ~ ( D · This means that strongly predicative adjectives do not need a polar opposite, nor

132 do they need a comparative: saying that v is a fourlegged 0 elephant amounts to the same thing as saying that v is a non-fourlegged elephant, and saying that v is more of a fourlegged elephant than w amounts to saying that v is a fourlegged being and an elephant, and w is either not an elephant or not fourlegged (leaving presuppositions aside for the moment): strongly predicative adjectives are not vague.

e.

Weakly Predicative Adjectives

A f t e r what has been said in the previous paragraph, we will introduce our proposed condition for the weakly predicative adjective right away: CWPRED.:

( A v ) ( ( v € Q + ( Z ) ( K ) -* v ^ Q ~ ( I ) ( U ) ) & (v e Q + ( I ) ( U ) & v € Z (K) -> v e Q + ( Z ) ( K ) ) )

This condition has a consequence which is reminiscent of the consequence we derived for the strongly predicative adjective "fourlegged", where we saw that all fourlegged animals were equally fourlegged, and all non-fourlegged animals were equally nonfourlegged. In this way, one could say that "fourlegged N" ( f o r any common noun N) defines a maximum and a minimum on any set on which it (or rather its denotation) is d e f i n e d : no fourlegged N can be more fourlegged than any other fourlegged Ν , and no nonfourlegged N can be less fourlegged than any other non-fourlegged N . We have also seen that a set K is exhausted by the fourlegged and the non-fourlegged N ' s . The weakly predicative adjectives also introduce a minimum and a maximum in this sense, but only in the universe U i t s e l f , with respect to common nouns of the form A ( O ) , where A is a weakly predicative adjective: Suppose w e Q + ( I ) ( U ) . Then for any v , such that ν β Q + ( I ) ( U ) and ν X w: Q+ ( I ) ( { v , w } ) = {v,w}. For suppose not: Let Q + ( I ) ( { v , w } ) = (v). Then, by C O . - C I I I . : Q ~ ( l ) ( { v , w } ) = ( w ) , and by CWPRED. and C+/-i w # O + ( I ) ( U ) , contrary to hypothesis. Let Q + ( I ) ({v, w} ) = {»} . Then, by the same reasoning, v ^Q+(I)(U), contrary to hypothesis. Finally, let Q + ( D ({v.wj )- 0. By CVI: Q + ( I ) ( { w } ) = i

133 +

+

and Q ( I ) ( { v } ) = 0. By CIV: w ^ Q + ( I ) ( U ) and v / Q ( I ) ( U ) , a contradiction. In the same way we prove that if w e Q ~ ( l ) ( { v , w } ) and v e Q ~ ( I ) ( { v , w } ) , then Q ~ ( I ) ( { v , w } ) = {v,w}. This means that in the set of e.g. red objects (assuming that red is weakly predicative) no object can be more red than any other object, and in the set of absolutely non-red objects no object can be lese red than any other. This situation is quite d i f f e r e n t from the situation we encountered in the case of e.g. "big" - in the set of objects which are big relative to the u n i v e r s e , some objects may be bigger than others. However, in contrast to the strongly predicative adjectives, Q + ( I ) and Q ~ ( I ) do not necessarily exhaustively partition U (where Q+ and Q~ are F ( A ) and F(A°) respectively for any weakly predicative adjective A ) . The fact that within the set of objects, i.e. relative to the universe, no red object can be more red than any other object, does not mean that within proper sub-sets K of U no element that is red with respect to K can be more red than another which is also red with respect to K . We can easily prove the following: if within any set Κ , ν is a more red Z ( K ) than w , then w does not belong to the set of objects which are absolutely red with respect to the universe. Let v 6 Q + ( Z ) ( K ) and = {v}. Then, by C O . , C I I I . ,

w £ Q + ( Z ) ( K ) , and let Q+(Z)({v,w}) Q ~ ( Z ) ( { v , w } ) - {n} , and by CWPRED.

w ^ Q + ( I ) ( U ) . The fact that Q + ( I ) ( U ) and Q ~ ( I ) ( U ) do not necessarily exhaustively p a r t i t i o n U if Q+ and Q~" translate weakly predicative adjectives is responsible for the fact that indeed v is a more red N than w does not mean the same thing as v is a red N and w is not a red Ν , as ( m u t a t i s mutandis) in the case of "fourlegged". 3 0 )

f.

Strongly Extensional Adjectives

A division of extensional adjectives into strongly extensional adjectives and weakly extensional· adjectives is made by Aqvist (1979, p. 2 4 ) . In order to preserve a certain terminological uniformity in this paper, I will call strongly extensional· those adjectives which Aqvist calls weakly extensional and weakly extensional will be his strongly extensional adjectives. The analogy

134

with my strongly and weakly predicative adjectives will become clear in a moment, when the extra conditions will be given. Strongly extensional adjectives are those adjectives for which inference 12} would hold. Clearly s k i l f u l is not one of them ( i t is not weakly extensional e i t h e r ) . An example of a strongly extensional adjective that is not weakly extensional might be f a t ; If all pygmys are human beings, then it is not necessarily the case that all fat pygmys are fat human beings. But if all and only cobblers are darts players, then the fat cobblers are the fat darts players. The condition I propose corresponds to Aqvist's condition for his weakly extensional adjectives (op. c., loc. c . ) : CSEXT.:

( A v ) ( v €. Z ^ K ) «—» V € Z 2 ( K ) ) —>

(v e Q + ( Z i ) ( K ) . M

or

M

[A] '9([N] '9)({g(d),g(e)}) = Fat most as

IX.

(A,N) (d,e)]M'9 = 0

otherwise.

An Axiom System If we want to formulate an axiom system for

L

and meaning

postulates for the several classes of adjectives we have been dis cussing above, it will be practical to introduce another neutral expression into

L , the neutral adjective

have a corresponding polar opposite

A° .

A , which does not F(A)

will be the

identity function on the domain of possible denotations of common nouns, i.e.

the f u n c t i o n

Ia

such that

Ia(Z) = Z

where Z is a

possible denotation for common nouns. The axiom system for

L

consists of

I.

The usual axioms and rules for q u a n t i f i e d predicate logic

II.

All substitution instances of the following scheraate, where N

is any common noun,

A

any a d j e c t i v e and P, R any predi-

cates : 1)

(Ax)T(x)

2)

( A x ) ( 0 ( P ) ( x ) «-» P ( x ) )

3)

( A x ) ( A ( N ) ( P ) ( x ) *-» N { P ) ( x ) )

4)

( A x ) ( N ( P ) ( x ) —> P ( x ) )

5)

( A x ) ( A ( N ) ( P ) ( y ) -» -A°(N) (P ) ( y ) )

6)

< A x ) ( - A ( N ) ( P ) { x ) ) —» ( A x ) ( - A ° ( N ) ( P ) { x )

7) (Ax)(-A°(N)(P)(x))-» ((Ax)(P(x)

A(N)(P)(x))v

8)

( A x ) ( ( ( A y ) ( P ( y ) P ( x ) ) -» ( A y ) ( A z ) ( ( P ( y ) & P ( z ) ) -> a s + ( A , N ) ( y , z )

10)

( A x ) ( - N ( P ) ( x ) ) —» ( A y ) ( A z ) ( ( P ( y ) fi. P ( z ) ) -> as' ( A , N ) ( y , z ) )

142 III.

Rule R°: If

in any axiom an adjective

A

where by its corresponding polar adjective

is replaced everyA° (if

it has

o n e ) , then the resulting formula is an axiom. As meaning postulates we w i l l have the following schemata: MP1)

(Ax)((Ey)(N(P)(y)

_*

(A(N)(P)(x) _

N(P)(x))

(MP1. corresponds to C I V . for the a f f i r m a t i v e MP2)

MP3)

(Ax)(Ay)((P(x)

adjectives)

& P ( y ) & A ( N ) ( P ) ( x ) & - A ( N ) ( P ) ( y ) ) _»

more

(A,N)(x,y))

(MP2.

corresponds to C V . )

( A x ) ( A y ) ( ( m o r e ( A , N ) ( x , y ) & P ( x ) & P ( y ) ) - » ( A ( N ) ( P ) ( y ) -> A(N)(P)(x))) (MP3. corresponds to C V I . )

MP4)

(Ax)(A(N)(P)(x)

-> - N ( P ) ( x ) )

(MP4. corresponds to CPRIV. for the p r i v a t i v e MP5)

(Ax)(Ay)((A(N)(P)(x)

&

A(N)(P)(y)

more(A,N)(x,y)

&

(A(N!)(P)(X)

v

adjectives)

—A ( a s + ( A , N ) ( x , y )

less(Α,Ν)(x,y)))

s, A ( N 2 ) ( P ) ( y ) —> ( a s + ( A , o ) ( x , y )

more(A,O)(x,y)

ν

v

ν

less(A,O)(x,y))))

(MP5. corresponds to C L I N . ) MP6)

(Ax)((A(N)(P)(x) &

N(P)(x)

—>

-* A ( 0 ) ( P ) ( x ) )

&

(A(O)(P)(x)

A(N)(P)(x)))

(MP6. corresponds to CSPRED. for the strongly predicative adjectives) MP7)

( A x ) ( ( A ( N ) ( P ) ( x ) —* - A ° ( N ) ( P ) ( x ) ) &

N(P)(x)

& (A(O)(P)(x)

—> A ( N ) ( P ) ( x ) ) )

(MP7. corresponds to CWPRED. for the weakly predicative adjectives)

143

MP8)

(Ax)((N1)(P)(x)

«-» N 2 ( P ) ( x ) )

—>

A(N2)(P)(x»)

(MP8. corresponds to CSEXT. for the strongly extensional adjectives)

MP9)

( A X ) ( ( N ! ( P ) ( X ) -» Ν 2 ( Ρ ) ( χ ) ) —> A(N2)(P)(x)))

(MP9. corresponds to CWEXT . for the weakly extensional adjectives) ( A x ) ( A y ) ( a s + ( A , N ) ( x , y ) —» ( ( m o r e ( A , N ) ( x , y )

MP10)

A m e r i c a n ( s h i p ) ( T ) ( d ) ) Therefore, by Modus Ponens:

(316)

A m e r i c a n ( b i g ( s h i p ) ) ( T ) ( d ) —» A m e r i c a n ( s h i p ) ( T ) ( d ) Moreover,

from the assumption that American is a f f i r m a t i v e ,

it follows that (317)

A m e r i c a n ( b i g ( s h i p ) ) ( T ) ( d ) —> b i g ( s h i p ) ( T ) ( d )

145

Therefore, from the assumption that A m e r i c a n ( b i g ( s h i p ) ) ( T ) ( d ) by MP from (316) and (317), (318) follows: (318)

i.e.

American(ship)(T)(d)

& big(ship)(T)(d)

Conversely, assume that (318) that (319):

is true and that (312) is not,

(319) - A m e r i c a n ( b i g ( s h i p ) ) ( T ) ( d ) In the paragraph on strongly predicative adjectives we have proved, that if b i g ( s h i p ) ( T ) ( d ) , then if - American ( b i g ( s h i p ) ) (T) ( d ) , then Aaierican°(big(ship)) (T) (d) . Again from the assumption that American is strongly predicative, it follows that (320)

American°(0(T))(d) However, under the same assumption it

(321)

follows from (318) that

American ( 0 ( T ) ) ( d )

which leads to a contradiction, via axiom 5). Therefore American big ship is equivalent to the conjunction reduced form American ship and big ship, but is lacking the proper conjunction reduced f o r m .

XI.

Some Further Remarks on the Comparative

It is not our objective in this essay to treat the comparative in final detail, but we would like to show, that the proposed semantics for the operator more can be taken as a basis for the semantics of cases of comparative sentences which are more complex than the simple "John is a bigger man than Peter".

146 In Klein ( 1 9 7 9 ) , p. 6 0 f f . , sentences of the form ( 3 2 2 ) are discussed: (322)

NPi is more ADJ]^ than N?2 is _

ADJ2

e.g. (323)

Mary is more clever than Bill is _ The dash in ( 3 2 2 ) ,

stupid

( 3 2 3 ) indicates that it

is assumed that

sentences like these contain a gap, as has been argued by Bresnan (1976). Notice that no degree m o d i f i e r can occur in front of the adjective in the subordinate clause of ( 3 2 0 ) : (324)

Mary is more clever than Bill is so/ that/very stupid ( K l e i n , op. c., Moreover,

it

p. 59)

seems that for many speakers contraction of the

copula preceding stupid is unacceptable, which might be interpreted as showing that the sentence contains a deletion or movement site at this point: (325) (326)

If Mary is clever, then B i l l ' s stupid Mary is more clever than B i l l ' s stupid In view of this hypothesis,

( 3 2 3 ) receives the following syn-

tactical analysis for its main adjective phrase: (327)

AP LAP more c l e v e r j

than

UP r Spe c(A)

£|

[ s B i l l is

ΓΑ

stu

P i d Jjj]

This is not the place to reproduce Klein's formalism in f u l l detail, but we can make our point in a more informal way as well. Notice that ( 3 2 7 ) contains the lexically u n f i l l e d node Lspec(A) £J · This node w i l l be translated into intensional logic as a variable, ranging over functions which serve as possible

147

interpretations of degree m o d i f i e r s . Informally, then, the conditions for the truth of ( 3 2 3 ) can be formulated as follows ( K l e i n , op. c., p. 61): (328)

( 3 2 3 ) is true in a context c iff there is some Lspec(A) ° such that Mary is ot clever is true at c , while Bill is of stupid is false at ted as a degree m o d i f i e r .

c , where h 2 & [a] M '9 e ( h x ( [ A 1 ] M , g ) ) ( J c N l ] M ' g ) ( U ) & [b] M '9 € ( h 2 ( [ A 2 ] M , g ) ) ( [CN2]M,g)(U)) To give an example: ["more (clever (woman) , r i c h ( m a n ) ) (Mary, J o h n ) J M ' g = 1

(361)

iff ( E h 1 ) ( E h 2 ) (hi £ DH & h 2 e. DH & h x M

M

> h2 & M

[Mary] ' 9 c h, ( [clever] ' 9 ) ( [woman] ' 9) (U) & [john] M '9 e h 2 ( f r i c h ] M ' 9 ) ( [man] M '9) ( U ) ) As a consequence of the d e f i n i t i o n of h^ ^ h 2 , we have that if χ e h 2 ( A ) ( N ) ( U ) , then not χ £ h x ( A ) ( N ) ( U ) + . Likewise, if x e h ! ( A ) ( N ) ( U ) , then not χ e h 2 ( A ) ( N ) ( U ) . We do not have, however: if or:

χ h 2 & pohn] M '9 £ n i ( [rJ^] M '9)( [man] M '9) (U) & [Peter] M ' 9 £ h 2 ( [rich] M '9)( [man] M > 9) (U) ) We can now prove, that under the given semantics ( 3 6 4 ) is true if and only if (365)

( 3 6 5 ) is true:

more (rich, man) (John, Peter)

which is the formula which was originally introduced to represent simple cases of comparative sentences, in this case "John is a richer man than Peter". In general we can prove, that a formula of the form ( 3 6 6 )

(366) is true iff (367)

( 3 6 7 ) is true:

more(ADJ 1 ,CN 1 ) ( a , b ) There is one proviso here: for a general proof concerning all

adjectives, we must apply t r a n s i t i v i t y . Therefore as a condition we must stipulate that in

CN^

there is no occurrence of

Ο .

Then we can use the weak version of t r a n s i t i v i t y which can be proved from C V I I I . and CVIX.

154 Now suppose f i r s t , that [ m o r e ( r i c h ( m a n ) , r i c h ( m a n ) ) { John, Peter )] M ' 9 = 1

(368)

Then there are

hj^

and

h2

M

with

h^ >· h2

M

rJohn] 'g € h!( [ r i c h j M / g ) ( f m a n ] ' 9 ) ( U ) M

M

and

and

M

peter] ' 9 €. h 2 ( [rich] '9)( [ m a n J ' 9 ) ( U ) It then follows directly from the d e f i n i t i o n of h

l >

h

2

that

[rich] M '9

( [man]

M

' 9 ) ( [John] M '9,

[peter] M '9 ) = [John] M ' 9 .

Conversely, suppose that ( 3 6 5 ) ( m o r e ( r i c h , man) (John, P e t e r ) ) is true, i.e.

that

M

[rich] '9( [ m a n ] M ' 9 ) ( [John] M '9, [p e ter] M '9 ) = Let,

for any individual

C DH , and for

(369)

be the f u n c t i o n such that

and any

N f. DCN :

h^

be the f u n c t i o n such that

(h 2 d ( A ) ) ( N ) ( U ) = ( x : x X d & A ( N ) ( { x , d } ) = {d}}

Then

h^^ )> h^ . For let

adjective and

Ν

A be the interpretation of any

of any common noun (under the restriction men-

tioned above). Let for

A €. DA

h1^

(h!d ( A ) ) ( N ) ( U ) = { x : A ( N ) ( l x , d } ) = ί Similarly let

(370)

any

d ,

[john] M '9 .

any individuals

χ e (h1,^ (A) ) (N) ( U ) χ

and

x e {x:(A(N)({x,d}) = (xftand

y . Then y£

and

y e (h 2 d (A) ) (N) ( U )

χ = d , or

(x:x /

d & A ( N ) ({x, d > ) = {d}} .

155

F i r s t , we have to consider two special cases: A) If A ( N ) ( { d } ) = X , then ( h 2 d ( A ) ) ( N ) ( U ) = β and the conditions for satisfied.

n

"*"d^ n ^d

to

hold are t r i v i a l l y

B) If ( A ) ( N ) ( { x , d ) ) = {x,d} for every x, then ( h 1 d ( A ) ) ( N ) ( U ) = {d} , and (h2d(A))(N)(U) =

fS

and what has been said under A) holds in this case as well. The remaining cases are the following: Case 1. Case 2.

χ = d. Then x / d. Then

weak t r a n s i t i v i t y : Under all

A ( N ) ( { y , x } ) = {x} = {d} A ( N ) ( { x , d > ) = {x} and A ( N ) ( { y , d > ) = {d} A ( N ) ( { x , y } ) = {x}

then, by

circumstances h 1 d (A) ( N ) ( U ) Π η 2 ά ( Α ) (Ν) (U) = &

So, if A ( N ) ( ' [John]M'9, [Peter] M '9 ) = [John] M '9 (where A interprets an adjective and N an O-less common n o u n ) , then [john] M 'g C h 1 Q T ^ n ] M , g ( A ) ) ( N ) ( U ) and [Peter] M > 9 € (h 2 pohn] M . g ( A ) ) (N) (U) . and hi jj^-j M , g > h 2 ^^ M , g

156 Actually, we can do better than this. We have seen that in Seuren (1973) as well as in Klein (1979) a sentence like "John is a richer man than Peter" is explained as something like "John is rich to an extent e and Peter is not rich to that extent". The negation in the second conjunct is supposed to explain the negative polarity e f f e c t s ,

the "ne-expletif" and other items discussed

above. We have also seen, that if then

h^ >

h2

and

χ e. h2 (A) (N) (U) ,

χ $ί h 1 ( A ) ( N ) (U) , but in general the converse doesn't hold.

To introduce negation in the than-clause of the comparative, we d e f i n e an extent in the following way:

(371) D e f . A function

h € DH

A € D^,N €. U£jj

is an extent iff

and

x,y € U

the

for

all

following holds:

If A ( N ) ( { x , y } ) X^T, then if χ € ( h ( A ) ) ( N ) ( U ) & y X ( h ( A ) ) ( N ) ( U ) , then A ( N ) ( { x , y > ) = {x} . It

is easily seen that according to (371) the denotation of

very is an e x t e n t , but the denotation of f a i r l y i s n ' t . We also see,

that if we change the f u n c t i o n

hd

of ( 3 6 9 ) in the following

way: (372)

for all

A, N,

( h d ( A ) ) ( N ) ( U ) = {x:x e U & A ( N ) ( { x , d } ) = (x>

v

A ( N ) ( { x , d } ) = {x,d>} then

hd

is an extent.

(hd(A))(N)(U)

w i l l be called the extent

to which d is A ( N ) . Now we can m o d i f y D e f . ( 3 4 7 ) for

(373) D e f . Let DHE C. DH

be the set

more t

of extents in

DJJ . Then

M

[more (A x ( C N ] _ ) , A 2 (CN 2 ) ) (a,b)] · 9 = 1 iff M M ( E h ) ( h C DHE & [a] «9 £ ( h ( [ A ! j ' 9 ) ) ( [ C N ] ] M ' 9 ) ( U ) M

[b] '9 ^

M

M

(h( [A2] '9))( [ C N 2 ] ' 9 ) ( U ) )

&

157

The proof of the equivalence of m o r e ( A D J i , C N j ) ( a , b ) the semantics given for more o r i g i n a l l y and of

with

m o r e ( A D J i ( C N i ) , ( A D J i ( C N i ) ) ( a , b ) under the newly given semantics can be carried out as before, by an additional use of formula (283) for t r a n s i t i v i t y . Apart f r o m the notion of extent, we can introduce the notion of degree;

(374) D e f . D

degree =

(h: h € DH 5, ( A A i ) (ANi.) ( A x ) ( A y ) (U £ h(A1)(N1)(U) &

y fi M A i M N i M U ) ) —*

A 1 ( N 1 ) ( t x , y } ) = {x,y> &

( E A 2 ) ( E N 2 ) ( x €- h ( A 2 ) ( N 2 ) ( U )

—> ( A N 3 ) (x t

N3(U)

->

χ £ h(A2)(N3)(U)))}

The degrees are those m o d i f i e r s , that model expressions like three f e e t . D e f . ( 3 7 4 ) is intended to capture the properties of degrees exemplified by the following: 1)

If χ pygmy,

is a three feet tall pygmy and y is a three feet tall then χ and y are equally tall pygmy's.

2)

If χ is a three feet tall pygmy and χ is a basketball player, then χ is a three feet tall basketball player. In p a r t i c u l a r , by taking 0 for Ν , χ is a three feet tall being.

We can order ^degree by means of >· and £ir , but it is clear that the ordering so induced will not be total. E . g . if the adjective to be modified is warm and the degree is three f e e t , then three feet warm should be undefined. We can express this property by stipulating that, for any N c D ^ j j : ( Γχ f e e t l M « 9 ( [ w a r m ] M » 9 ) ) ( N ) ( U ) = 0. The proper degree for warm will of course be in degrees of temperature (of whatever scale). But χ feet and y_ degrees C° are incomparable. We can, however, distinguish the notion of dimension:

158

(375) D e f . Any subclass &dim of ^degree i s a dimension iff there is an A, A e DA J such that for all h €. all N £ D C N : if A ( N ) ( U ) X fS , then ( h ( A ) ) ( N ) ( U ) X and for all h

and ϊΐ€·Όάπι: ~^>^

or

h >

h

or

The Description of Change We have seen that one of the functions of the English progressive seems to be the expression of protracted change, and we have reported on several e f f o r t s to d e f i n e a notion of BECOMING, non of which we considered to be very s a t i s f a c t o r y . We have also seen, that change involves comparison: If John grows, then he is taller than he was. In general - if something changes, it has more ( l e s s ) of a certain property than it had before. Conversely - if something has more ( l e s s ) of a certain property than it had bef o r e , then it has changed, or is ( s t i l l ) changing. Thus we might try to combine the treatment of comparison which we proposed above, with Russell's remark on change (1903, p. 4 6 9 f . ) : "Change is the d i f f e r e n c e in respect to truth or falsehood, between a proposition concerning an entity and a time

Τ ,

and a proposition concerning the same entity and another time

T',

provided that the two propositions d i f f e r only by the fact that occurs in the one, where

T'

Τ

occurs in the other." It seems that

what Russell had in mind, were propositions containing numerical measures, like ( 3 7 6 ) (376)

John is six foot tall at

T

For surely, if ( 3 7 6 ) is true, and the same sentence, with T' (= T) replacing T is false, then a change has taken place. But although

( 3 7 7 ) and ( 3 7 8 ) can both be t r u e , a change can neverthe-

less have taken (takes/will

take) place:

(377)

John is a tall man at

T

(378)

John is a tall man at

T'

159

simply because at T' John is even taller than at T . However, if we take John as he was at Τ , and compare him with John as he was at T' (I say was for the sake of s i m p l i c i t y ) , and if we restrict the CN tall man to this pair of "stages" of John - just as we did before comparing John with Peter -, then we see that with respect to the pair "John at T" and "John at T ' " we must say that "John at T'" is tall, and "John at T" is not. What, then, does the expression "concerning the same entity" in Russell's def i n i t i o n mean? If John is a six foot tall entity at T and a not six foot tall entity at Τ' , how can these two entities be the same? A solution for this problem will be to construe "a proposition concerning an entity ..." as a proposition concerning an individual concept and its extension at a moment T and the same individual concept and its extension at T' . In Hoepelman (1979) I have pointed out that in such a construction it is quite unint u i t i v e to stipulate that the individual concepts which serve as the interpretations of proper names, should be constant concepts, i.e. functions which at each index have the same entity as their value, as is done in e . g . Montague (1974, p. 2 6 3 ) . For suppose that the denotation of "John" is a certain physical entity - the same physical entity - at each index. As a physical entity it has a certain length - six foot, say. And six foot is the length it has at each index, or else it would not be the same entity everywhere. How are we now to describe that John grows? If John grows, then at a certain moment Τ , he is an element of the set of six foot tall entities, and at a later moment T' he is an element of the set of, say, seven foot tall entities. But as a constant individual concept, the extension remains the identically six foot tall physical object. To avoid a contradiction, we have to assume that the expressions "six foot tall" and "seven foot tall" have changed their meanings in time, and that at T' "seven foot tall" denotes, what "six foot tall" denoted at T . This, of course, is highly undesirable.33)

160

X I I . 1.

The Combination of the Tree-System for Action Sentences with the System for Adjectives

A f t e r the remarks made in the beginning of this chapter,

it

will come as no s u r p r i s e , that in order to describe change, we will introduce i n d i v i d u a l concepts as denotations for i n d i v i d u a l constants and variables. We w i l l not introduce intensions of predicates or of other elements of the language, since these are not needed for the points we want to make h e r e . The notion of a tree can be directly taken over from chapter V 1 ( 2 0 2 ) of Part I. For the convenience of the reader we repeat it here:

(202)

Def. A tree is

a structure

T = ( T , Q , W , < , —^ ) , where

I.

T

is a nonempty set of moments

II.

Q

is a nonempty set of s i t u a t i o n s

III.

Q Γ^ T = 0

IV.

W x ( w ( t ) )*T Q ~ ( Z ) ( K ) ) )

CII'.

Q+(Z)(K) = 0 - » Q - ( Z ) ( K ) = 0

GUI'.

Q - ( Z ) ( K ) = 0 - > ( Q + ( Z ) ( K ) = 0 v Q + ( Z ) ( K ) = K)

CIV'.

(Ax)(Aw)(At)(Z({v(w(t))>) = 0

CV'.

( Z ( K ) = K) «-» ( A x ) (Ay) ( A w ) ( A w ' ) ( A t ) (At' ) (x(w(t))

—»

K & y(w'(t')) «K

Z(fx(w(t))f

—»

y(w'(f ))}) = (x(w(t)), y(w'(t

( Z ( K ) = 0) *-» ( A x ) (Ay) ( A w ) ( A w ' ) ( A t ) ( A t ' )

CVI'.

(x(w(t))€ K & y(w'(t')) € K Z ( { x ( w ( t ) ) , y ( w ' ( f ) ) } ) = 0) C V I I I * ' . ( A x ) (Ay) ( A w ) ( A w ' ) ( A t ) ( A t ' ) ( x ( w ( t ) ) € K & y ( w ' ( t ' ) ) €K & x ( w ( t ) ) € Q + ( Z ) ( K ) & y(w'(t'))XQ+(z)(K))->Q+(z)({x(w(t)),y(w'( fx(w(t))} ) CIX*' .

( A x ) (Ay) ( A w ) (Aw' ) ( A t ) ( A t ' )

x ( w ( t » € K & y ( w ' ( t ' ) ) t K)

—>

( y ( w ' ( t ' ) ) £ Q + ( Z ) ( K ) - > x ( w ( t ) ) e Q+ N.B.:

In C V I I I * ' . and C I X * ' . we assume that

Ζ = F(N)

for

an

N in which 0 does not occur. C O ' . - C V I ' . as well as C V I I I * ' . and C I X * ' . are assumed to hold for all adjectives. The following conditions are assumed to hold for most, but not for all

adjectives

(we r e f e r to our

discussions in previous chapters): CVII'. ( A x ) ( ( A w ) ( A t ) ( x ( w ( t ) ) € Q + ( Z ) ( K ) -» x ( w ( t ) ) € Z ( K ) ) ( C V I I * . is the condition for the a f f i r m a t i v e adjectives) C V I I I ' . - Like C V I I I * ' . , without the r e s t r i c t i o n on Ζ CIX'.

- Like C I X * ' . , without the restriction on Ζ It

is obvious seen that the proofs we have given in Chapter

V I . of Part II can be given again, e.g. for t r a n s i t i v i t y : replace

166

any v , w , and z occurring in the old proofs by v ( w ( t ) ) , y ( w ' ( t ' ) ) , z ( w ' * ( f " ) ) respectively, and the proof will change into a proof from the conditions newly given. The conditions for the more restricted groups of adjectives can be adapted in the same way. We will b r i e f l y list them: CPRIV'.

(privative adjectives) (Ax)(Aw)(At)(x(w(t))

CLIN'.

€

Q + ( Z ) ( K ) -» x ( w ( t ) )

£ D/Z(D) )

( l i n e a r adjectives) - remains unaltered -

CSPRED'. (strongly predicative adjectives) (Ax)(Aw)(At)((x(w(t)) & (x(w(t)) x(w(t))

e

e

€

Q + ( I ) ( D ) & x ( w ( t ) ) £ Z ( K ) —·· +

Q (Z

CWPRED'. (weakly predicative

adjectives)

(Ax)(Aw)(At)((x(w(t)) & (x(w(t)) e x(w(t)) CSEXT'.

£

Q + ( Z ) ( K ) —> x ( w ( t ) ) c

e

Q + ( Z ) ( K ) -» x ( w ( t ) ) X Q - ( I

Q+(I)(D) & x(w(t))

£

Z ( K ) —*

+

Q (Z)(K)))

( s t r o n g l y extensional a d j e c t i v e s ) (Ax)(Aw)(At)(x(w(t))

e

Zi(K)

< t & t < t

2

S , (At3)

(t! < t 3 < t —» j-AjM,w,t,g ( ^ N ] M ' w ' t ' g ) ( { g ( d ) ( w ( t 3 ) ) , g ( d ) ( w ( t ) ) } ) =

{ g ( d ) ( w ( t ) ) } & ( A t 4 ) ( t < t 4 < t 2 —* ,w,t,g ([N]M'w't'9)(tg(d)(w(t4)),g(d)(w(t))}) = = 0

otherwise

Correspondingly, we have for less ; (388) D e f .

ΓΑ l e s s ( A > N ) ( d ) l M ' w ' t > g = 1 ( E t 1 ) ( E t 2 ) ( t l f 2 «· T & tl )) = 0

otherwise

170 In Part I, Chapter I l l . b , we put forward the hypothesis that one of the functions of the progressive is the expression of a change going on during a certain period. Now definitions (387) and (388) give us the possibility to express this: if we have a language containing operators expressing that a state of a f f a i r s takes place during a certain period, e.g. Kamp's

S^

and

U , as

employed in Hoepelman (1979), then we can say that a change in A , N of d is going on during a period I in world w iff for all moments t' of I we have Γ Δ m o r e ( A , N ) ( d ) ] M , w , t ' . g = ι or [ A l e s s ( A , N ) ( d ) ] M ' w ' t > ' 9 = 1. In the f i r s t case each moment t ' of I is preceded by an open interval of moments t ' ' at which d is l e s s ( A , N ) than at t' , and followed by an open interval of moments t ' ' ' , at which d

is

more(A,N)

than at

t' , i.e.

no moment

t'

of

I

can be

a moment which is the last one of a period of change. In this respect ( 3 8 7 ) is in accordance with the assumption concerning the progressive, that PROG(VP) is true of an argument (arguments) at a point or at an interval iff the point is an element of an open interval during which VP is true of its a r g u m e n t ( s ) , or the interval is a non-terminal part of an open interval at which VP holds of its a r g u m e n t ( s ) - so e.g. in Montague (1968), Scott ( 1 9 7 0 ) , Bennett & Partee ( 1 9 7 2 ) . We have already pointed out, that a condition like this, correct as it may be, cannot be enough, for it f a i l s to explain why only certain verbs appear in the progressive form. Moreover, it has been noticed by Dowty (1977, 1979), that the assumption that the progressive denotes an interior point or an interior interval (relative to an i n t e r v a l ) leads to problems in the explanation of cases like ( 3 8 9 ) , ( 3 9 0 ) (Dowty, 1979, p. 150): (389)

John was crossing the street when he was run over by a truck

(390)

John was watching television when he f e l l asleep As a way out, Dowty introduces "inertia worlds" as alterna-

tives to the actual world, in which those things go on, which would "normally" have taken place in the actual world, were it not

171

that something had prevented this. Of course this solution is not without problems, as Dowty himself ( 1 4 8 f f . ) and Vlach (1981, p. 285, 286) notice. I believe that we can circumvent this problem if we modify the semantics for "becoming" ( i . e . our Δ-operator ) , in such a way, that a moment of time t can also be a moment of d ' s becoming m o r e ( A , N ) , when t is a last moment of d's in- or decreasing in respect ( A , N ) . (Compare this with the graph of an increasing function f , which is broken off at a point χ . We can still say, that at χ the f u n c t i o n f increases, when at χ the left derivative of the function f is defined and greater than zero, although the right derivative is not d e f i n e d . ) For our treatment of accomplishment verbs (see Part I . , Chapter I I I . c . 2 ) this means that we must consider the f i r s t moment at which the result of the action connected with the verb is obtained, to be the last moment at which the change towards this result takes place. Now for our new d e f i n i t i o n s : (391) D e f . [Amore(A,N)(d)]M'w't'g = 1

*-^

( E t j ^ U - L e τ & tj_ < t & ( A t 2 ) ( t 1 ^ t 2 JV|M,w,t,g ( j - N ] M ' w ' t ' g ) ( { g ( d

(392) D e f . [A l e 8 B ( A , N ) ( d ) ] M ' w ' t * g - 1

( E t 1 ) ( t 1 € T & t < t x & ( A t 2 ) ( t < t 2 < tj_ |-AjM,w,t,g ( [ N ] M ' w ' t ' g ) ( { g ( d ) ( w ( t 2 ) ) ,

(393) D e f . rA(A,N)(d)]M'W't,g = ι ''

w

'

t

'g = 1

«_ or

From the d e f i n i t i o n s given for more and less and from conditions C O ' . - G U I ' . , it is immediately clear that

172

rAmore(A,N)(d)]M'w't'g = l

(394)

ΓΔ l e s s ( A ° , N ) ( d ) j [A less (A, N ) (d)]

M

w t

> ' 'g = l

M w

fc

' ' »g = l

M w

) (d)J ' ' g '

fc

and

= l

Notice also, that (under the assumption that the universe consists of not only fourlegged objects) it for the strongly predicative adjectives, predicative adjective expression

that for any strongly

N

and any individual

d , we have

[Amore(B,N)(dr|M'w't'g - 1

(395) only if

t

is the last moment of an interval which is closed to

the right by

t

and such that at any two moments

of this interval,

g(d)(w(t'))

suppose there is a [B]

B , any common

follows from CSPRED' .

M w t

' ' 'g([Nj

M

t' < t

is a s + ( B , N )

as

t'

and

t''

g ( d ) ( w ( t " ) ) . For

such that ( 1 ) :

w t

- ' 'g)((g(d)(w(t')),g(d)(w(t))}) =

and there is a t ' ' with t < t ' ' , such that ( 2 ) : |-BjM,w,t,g ( [ N ] M , w , t f g ) ( ^

then from CSPRED'. it would follow that ( 3 ) : g ( d ) ( w ( t ) ) €. and in combination with C O ' . - G U I ' . , and g(d)(w(t)) €

[B°]

M

C+/-, that ( 4 ) :

'W,t,g(I)(D)

an impossibility. To give an example: according to the semantics there is only one kind of moments at which an elephant can become a fourlegged elephant - those moments at which the animal has four legs for the f i r s t time a f t e r a period of not having four legs. But it

is impossible that an elephant becomes fourlegged during an

interval . 37) The semantics for WANT will be given in the next d e f i n i t i o n (396):

173

(396)

Def.

If £5 expression:

is a closed sentence of

[WANT(S,a)'[ M ' w ' t 'g = 1

I/ , and

a

an i n d i v i d u a l

iff

(Ew'Xw' € supg(a)(w(t))(*w*t) & [SJM'w''fc'g = 1 ) (Aw")(w"c i n f a w t ( * w * t ) _»-sM,w",g,t = 0)

&

As before, in this d e f i n i t i o n w*t denotes the set of worlds w' such that for all t' < t: w ( t ' ) = w ' ( t ' ) .

all

Finally we come to the d e f i n i t i o n for the semantics of DO. We can take the weak d e f i n i t i o n ( 2 2 3 ) of Part I, Chapter V . l . c , without modifications. For the sake of convenience we repeat ( 2 2 3 ) here: (223)

[DOP(a)]M'w't'g = 1 iff M w fc 1) [ P ( a ) ] ' > ' 9 = 1 & 2) ( A w ' ) ( ( w ' c *w*t & [WANT(- P ( a ) , a ) ] M ' w > ' t > g = 1) &

3)

-* [- P ( a ) ] M ' w ' ' t > g = 1) ( E w " ) ( w " € *w*t & [P_Uj;] M ·"''·*^ = 0)

Let us d e f i n e that at

w(t) a

W(-S, t , a )

to be the set of worlds

w , such

wants that not S :

Def.

(397)

W(-S,t,a) =

-[w c W : [WANT (-S , a)] M ' w ' * '9 = 1 }

Now we can slightly reformulate ( 2 2 3 ) : Def. [DOPU_)J M 'W't,g = ι

(398)

iff

M w>t>g

&

1) 2)

fL P ( a )ml l ' = 1 ( A w ' ) ( ( w ' e *w*t & w' e W ( - P ( a ) , t , a » -- > ' w > ' f c ' g = 0)

&

3)

( E w " ) ( w " c *w*t &

[ Ρ ( 8 ) ] Μ » Μ ^ ' 9 = 0)

Suppose that Ρ is O-less. Then, as a consequence of ( 3 9 8 ) 1) and 2), by condition C V I I I * ' . we have ( 3 9 9 ) :

174

(399)

If

[ b o p U f j M ' W ' t ' S = 1 , then

( A w ' ) ( ( w ' € *w*t & w ' e W ( - P ( a ) , t , a ) ) -> [ P ] M , w , t , g ( { g ( a ) ( w ' ( t ) ) , g ( a ) ( w ( t ) ) } ) = ( g ( a ) ( w ( t ) )} )

of

Now compare ( 3 9 9 ) with ( 3 9 1 ) , the d e f i n i t i o n of the semantics Amore. (391) is a condition on t , and ( 3 9 9 ) is a condition

on w, but the s t r u c t u r e of these conditions is the same: In one direction (391) says, that if

Γ Δ more (A, N ) (d)"[ M ' w > t ' 9 = 1 ,

t

such that for all moments

is element of an interval,

t"

then of

this interval preceding t , the stage of d at w ( t ) is A ( N ) as compared to the stage of d at w ( t ' ) . And ( 3 9 9 ) says, that if [ D O P ( a ) ] M ' w ' t ' 9 = 1 , then w belongs to a set of worlds *w*t such that for all elements w ' of *w*t , s a t i s f y i n g the condition of belonging to W ( - P ( a ) , t , a ) it is the case that the stage of a at w ( t ) is P as compared to the stage of a at w ' ( t ) . A situation like this, we call a modal change at a moment t , and we give the following d e f i n i t i o n : (400)

Def.

(EW')(W C W &(Aw')((w' ([N]

e

*w*t & w' e W)

-·>

M w t

' ' '9)((g(d)(w(t)),g(d)(w'(t))}) {g(d)(w(t))}) ν

We see that ( 3 9 3 ) and ( 4 0 0 ) can be considered as cases of something more general, for which we want to introduce the locution of an "unstable situation". (401)

Def.

Let w ( t ) be a situation; for w e W and Any set of situations S ( w ( t j ) , such that s

t e T .

(w(t))

C {s € Q : ( E t ' ) ( t ' < t & s = w ( t ' ) ) v ( E w ' ) ( w ' c *w*t & s = w' ( t ) } and s a t i s f y i n g the condition ( A s ) ( A t ' ) ( ( s = w ( t ' ) & t' < t & s e S ( w ( t ) ) ) -» ( A s ' ) ( ( E t " ) ( ( t ' < t" < t S, s' = w ( t " ) ) -» s' £ S ( w ( t ) ) ) >

175 is called a l e f t surrounding of w ( t ) .

(402)

Def. [UNST(A,N) (ES(w(t)))((As)(s

€ S

( w ( t ) ) —* ( [ A ] M , w , g , t ( | - N J M , w , t , g )

({g(d)(s),g(d)(w(t))}) = (As)(s€ S

(403)

(g(d)(w(t))})

v

—>

w t

Def. A situation respect

w(t)

(A,N)

is unstable for an individual

in a model

Μ ,

d

in

iff

'W't'g = 1 It follows, that any situation M w fc

[A(A,N)(d)] ' '

'g = 1

w(t) ,

such that

M w fc

or [POP(d)] ' ' · 1 = 1 ,

is unstable.

Now we are in a position to formulate in a more precise way the hypothesis concerning the semantics of the progressive in English which we announced at the beginning of this essay. Hypothesis;

Let

VP be a one-place verb-phrase, and let

d

be an individual expression. Then (404)

[PROG(VP(d) ) j M , w , t . g

=

!

iff

there is an interval

I cT

which may be open or closed to the right and which is open to the l e f t , stable for

such that for all d

in respect

t' e

VP , and

I »

w(t' )

is un-

t C I .

This hypothesis u n i f i e s the explanation of the occurrence of the progressive with verb-phrases expressing temporal change as well as with verb-phrases expressing agency. But surely there are more uses of the progressive, which are not captured by the hypothesis, as long as the d e f i n i t i o n of "unstable situation" is not enlarged. However, the examples I found in the literature indicate that it

might at least be worthwhile to go on searching in this

direction. We have discussed sentences like "Your beerglass is sitting on the edge of the table" - which is correct -, and "The statue of George Washington is standing in the park" - which is

176

not. Here, too, one intuitively feels that a judgement on the stability of a situation is given. Something similar seems to be the case if Dowty says on the futurate progressive (Dowty, 1979, p. 156): "The tenseless f u t u r e implies a greater degree of certainty of pre-deterraination, than the f u t u r a t e progressive", and gives examples like (405)

The Rosenbergs die tomorrow, although the President may grant them a pardon

(406)

The Rosenbergs are dying tomorrow, although the President may grant them a pardon

May be a notion of "epistemical stability" has to be introduced for cases like these.

XIII.

Coda; A Final Remark on the Logical Form of Comparatives

We have introduced a formula of the form m o r e ( A , N ) ( a , b ) to render simple comparative sentences like "John is a taller man than Peter". We have also seen, that this formula is not usable if we want to render more complex cases of comparatives, like "Mary is a more clever woman, than John is a rich man". For cases like these, we proposed the formation rule cum semantics ( 3 5 9 ) , ( 3 7 3 ) , which, for the sake of convenience we repeat here: (359)

Let ΟΝΊ and CN^ be common nouns, A^ and £2 adjectives and a.»b individual expressions. Then m o r e ( A i ( C N T ) , A ^ ( C N ^ ) ) ( a , b ) is an expression of L .

(373)

rmoretAT(ΟΝΊ),A?(CN?))(a,b)]M'9 = 1

iff

M

( E h ) ( h c DHE & [a] -g £ (h( [ A ^ M ' g j J i f C N j M ' 19 ) ( U ) & [b] M '9 ^ ( h ( [ A 2 ] M , g ) ) ( | - C N 2 ] M , g ) ( u ) )

Now ( 3 5 9 ) and ( 3 7 3 ) are not satisfactory either, if one wants to express even more complicated cases of the comparative, e.g. where tenses are involved, like ( 4 0 7 ) :

177

(407)

John will be a taller man than he was

I have been assured by native speakers of French, that in cases like ( 4 0 7 ) in French the "ne-expletif " is obligatory: (408)

Jean est plus grand qu'il n ' e t a i t (John is taller than he w a s ) Other cases are comparatives involving subordinate clauses,

like (409)

John is a taller man than Peter said he was

Notice that ( 4 0 7 ) is ambiguous. In one reading it says, that at some point t in the f u t u r e , it is the case that John is taller than he was in the past of t . But there is also a reading, which says, that at some point t in the f u t u r e , it is the case that John is taller than he was at some point in the past of the present moment. This ambiguity cannot be expressed if the comparative is analysed as a relation between an adjective phrase and a sentence, as in Klein (1979, p. 6 0 f f . ) , where e.g. (410)

Mary is more clever than Bill is stupid

is rendered as (411)

(more ( clever ) ( JC ( ~ stupid

{ Bill ^ )) (Mary)

with apparently no possibility to introduce a tense -, or other sentential operator in the f i r s t adjectival phrase more (clever) . In order to account for the ambiguity, we propose the following form: (412)

more( \h \x S^MA) (N) ( x ) ) , \h \x S 2 ( - h ( A ) (N) (x) ) ) ( a , b )

where 5η ( h ( A ) ( N ) ( x ) ) , S ^ ( - h ( A ) ( N ) ( x ) ) are sentences containing subphrases of the form h ( A ) ( N ) ( x ) and -h(A) (Ν) ( χ ) , respec-

178 tively,

h

variable,

is a variable ranging over extents, A any adjective and

Ν

χ

an individual

any common noun.

Semantically we will have: (413)

[more( Xh λ χ Β χ ( h ( A ) ( Ν ) ( χ ) ) , Xh λχ S 2 ( - h ( A ) ( N ) ( x ) ) )(a,'b)]M'v't'