Tense and continuity

BARRYTAYLOR

TENSE AND CONTINUITY’

ABSTRA~. The paper proposes a formal account of Aristotle’s trichotomy of verbs, in terms of properties of their continuous tensings, into S(‘state’)-verbs, K(‘kinesis’)-verbs, and E- (‘energ&z’)-verbs. Within a Fregean tense framework in which predicates are relativized to times, an account of the continuous tenses is presented and a preliminary account of the trichotomy devised, which permits an ihuminatmg analogy to be drawn between the temporal properties of E- and K-verbs and the spatial properties of stuffs and substances. This analogy is drawn upon in constructing a sophisticated version of the preliminary theory accommodating more of the hnguistic data.

0. PRELIMINARIES

This paper is intended as a contribution to Davidson’s semantic programme. We Davidsonians believe’ that a semantic theory for a language L should take the form of a recursive assignment of truth-conditions to the sentences of L, of the sort which Tarski taught us how to construct for the special case of first-order languages. Faced with the task of constructing such a semantic theory for English, we posit the existence of a vaguely first-order formal language, which I shall call ‘Base English’. Each surface sentence of English we assume to receive a paraphrase (called a ‘base paraphrase’) in Base English; and we further suppose that base paraphrases can be linked to their surface realizations by the meaning-preserving transformations of a Choms- kyian syntax. Under these assumptions, the task of producing a semantics for English reduces to that of constructing a semantic account for Base English; and the assumption that Base English is ‘vaguely first-order’ means that we can rely at least in part on Tarski’s methods in discharging the latter task. But of course, it is far from obvious that all of the complexities of surface English can be adequately paraphrased into a strictly first-order framework.The interesting question for the semanticist is, accordingly, how far Base English will need to deviate from outright first-order structure in order to accommodate the subtleties of surface English, and what modifications to Tarskian methods these deviations entail.

’ Preliminary drafts of this paper benefited greatly from discussions with my doctoraf super- visor, Michael Dummett.

* See for example Davidson, D., ‘Truth and Meaning’, Synthese 17 (1967), 304-323, and ‘Semantics for Naturaf Languages’, in Visentini et al. (eds.), Lingruzggi n&u societa e nella tecnicu (Edizioni di Communita, Milan 1970).

Linguistics and Philosophy 1 (1977) 199-220. All Rights Reseroed. tipyright @ 1977 by D. Reidel Publishing Company, Dordrecht, Holland.

200 BARRY TAYLOR

In investigating this question, one should bear in mind that base paraphrases are not Quinean explications. Our aim in framing a base paraphrase for an English sentence is simply to display it in a form congenial to the operation of a recursive truth-theory; and this purpose is quite compatible with the survival of unclear concepts from the surface in the formal base paraphrase. Indeed, since the semantic properties of the surface sentence are to be identified with those of its base paraphrase, it is arguable that the concepts deployed in the surface sentence cug/rr to re-emerge in a base paraphrase; and, though the vagueness of the criteria of identity for concepts makes this no firm general test for the adequacy of base paraphrases, it should give us pause before e.g. invoking heavy set theoretic resources in framing base paraphrases for simple English sentences, no matter what clarity we may thereby achieve. ’

In this paper, I want to offer an account of the continuous tenses within this Davidsonian framework, and to defend my account by showing how it enables us to understand and develop an Aristotelian classification of verbs in terms of properties displayed in their continuous tenses. My account of the continuous tenses will however presuppose a certain treatment of the simpler tenses, and my next section will be devoted to outlining this.

1. THESIMPLE AND PERFECT TENSES

Many philosophers, faced with any demand for formal elucidation of the temporal locutions, reach for heavy Quinean artillery, proposing the use of a paraphrasing language whose variables range, not over the familiar three- dimensional items of commonsense, but over unchanging four-dimensional entities extended in time as well as in space. From my preliminary remarks, however, it will be clear that, whatever the merits of this approach in devising for purposes of science or metaphysics a canonical deGur?crr from ordinary English, the radical conceptual reorientation involved in translat- ing into such a formalism renders the resulting paraphrases unsuitable as &e paraphrases for English sentences.

Fortunately, contemporary wisdom provides a choice of two less radical ways for fitting the subtleties of tense into a first-order framework. One method is to augment first-order apparatus by adding ?ense oper&crs: indexical sentential operators used to generate tensed renditions out of tenseless first-order formulae. The other way - Frege’s - is to require that ostensibly n-place predicates of surface English capable of bearing tense

’ Cf. Davidson’s distinction between ‘analysis’ and ‘logical form’ (e.g. as drawn in ‘Causal Relations’, Journal ~~PMw@~ LXIV (1967), 69 l-703); and the ‘third condition’ imposed on semantic theory in ‘Semantics for NaturaI Languages’, (op. cit., p. 179).

TENSE AND CONTINUITY 201

inflexions shall be represented by (n + l)-place predicates in first-order paraphrases, with the extra place reserved for occupation by a singular-term for a time; tense is then to be accommodated by means of a system of predicates expressive of relations between times, along with indexical constants reproducing the surface ‘now’ and ‘then’.

Although most attention in recent years has been focussed on the first of these alternatives, for present purposes Frege’s strategy has a number of advantages over its rival. One of these is that it involves no special complication of the structure of Base English to accommodate tense; for what it proposes is devious paraphrase of tensed sentences into a first-order base language with indexical constants, and it is generally conceded by semanti- cists on independent grounds that such constants will need to be included in Base English, and the modifications entailed for Tarskian semantics have been fairly extensively researched.’ A further point is that part of our present interest lies in exploring the rationale of Aristotle’s classification of verbs in terms of their tense-properties; and this makes more attractive a starting-point which treats tense as a feature of thepredicutes which underlie verbs (as Frege’s account does) rather than as a feature attaching to whole sentences (as the tense-operator strategy makes it appear).

For these reasons, Frege’s approach is the one I shall adopt. To implement it, we need to suppose Base English to contain various temporal resources, beyond the indexicals ‘now’ and ‘then’. First, variables of Base English must be construed as ranging over a domain embracing times. Apparently times of two sorts will have to be acknowledged, for indivisible temporal moments are required by any sophisticated account of time, whilst on the other hand many apparent singular terms of English (e.g. ‘Thursday afternoon’) refer not to moments, but to longer temporal periods. It is natural to suppose Base English equipped with the means of marking this distinction, in the shape of the predicates ‘Mom (t)’ (‘t is a moment’) and ‘Per (t)’ (‘t is a period’). The base formalism will also need predicates to express rekztions between times: I write ‘t< t” to mean ‘t is earlier than t”, ‘td’ to mean ‘t falls properly (wlzolly) within f”, and ‘td’ to mean ‘ffalls within t” (i.e. t either is t’, or else falls properly within t’).

Clearly, the notions just invoked, and the existential assumptions of the theory of times they presuppose, cry out for an axiomatic articulation; moreover, for an articulation which will shun the natural use of set-theory, lest the paraphrases we offer of simple tensed English sentences be sus- pected of commitment to powerful conceptual resources rendering them unsuitable as candidates for base paraphrases. This debt to formal propriety

’ See e.g. Weinstein, S., ‘Truth and Demonstratives’ A&s 8 (1974), 179-184.

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I discharge in the Appendix, where I offer an axiomatic theory of (dense, continuous) time taking as primitives just the relations of identity and temporal precedence. My official account of temporal notions is that given by the formalism there; but to follow the bulk of this paper, an intuitive construal should suffice.

One matter remains to be clarified before we can turn to actually proposing Fregean base paraphrases of tensed English sentences, and this is the intuitive content of Frege’s basic proposal that predicates should be relativized to times. The obvious suggestion is to suppose that the time to which a predicate is relativized gives the time ut which the predicate is said to be instantiated, so that e.g. a sentence like

(1) Hirsute (es, t)

should be ‘read’ as

0) Esau is hirsute at t.

However, the proprieties of English render such a construal unattractive, since the preposition ‘at’ is happily followed only by a singular term for a moment, whereas, as will emerge in the next section, it is sometimes necessary to suppose that predicates relate individuals to periods. I therefore propose for (1) the more neutral reading.

(3) t is a time of Esau’s being hirsute

which, even if less than idiomatic, gives enough grasp on the meaning of the formalism to enable the theory to get under way.

By now it will be clear how sentences involving verbs in the simple tenses are to be accorded Fregean base paraphrases. Thus the simple present as exempliGed in

(4) Esau is hirsute

(verb tensed) goes over as

6) Hirsute (es, now),

the simple past of

(6) Esau was hirsute

is paraphrased as

(7) (3 t)(t < now & Hirsute(es, t))

whilst the simple future in

@I Esau will be hirsute


is rendered as

(9) (3 t)(now -c t 4% Hirsute(e.s, r)).

It is less clear how the perfect tenses are to be accommodated. The obvious move on the present perfect, exemplified in

00) Esau. has been hirsute

is to assign to it the base paraphrase (7); but this obvious ploy, though I believe it to be correct, generates a puzzle. For in assigning to the perfect- tensed (10) the same base paraphrases as the simple-tensed (6), we commit ourselves to treating the two sentences as semantically equivalent; yet some differences between the two must be accounted for, since they are non- interchangeable in standard English speech.

A solution to the problem can however be found by invoking the apparatus developed by Reichenbach in his celebrated discussion of tense. ’ Reichenbach distinguishes three times associated with each utterance of a tensed sentence, viz.

(i) the point of speech, i.e. the time at which the utterance is made,

(ii) the point of f/re event, i.e. the time at which the speaker asserts the event (or state) described in the sentence to occur (or obtain),

(iii) the point of reference, i.e. the temporal standpoint from which the speaker invites his audience to consider the occurrence of the event (or the obtaining of the state).

In uttering a sentence cast in a simple tense, Reichenbach goes on to claim, a speaker’s reference-point is to be taken as identical with the point of the event, which in turn he asserts to be simultaneous with, prior to, or subsequent upon the point of speech according as his utterance is in the present, past, or future simple tense. A systematic account of the relation- ship between the perfect and simple tenses then emerges: the temporal points associated with an utterance of a perfect-tensed sentence are to be supposed related as far as possible in the same way as they are in an utterance of a sentence in the corresponding simple tense, subject to the overriding requirement that the point of reference is to be her than the point of the event. The relations indicated between the temporal points associated with utterances of sentences in simple and perfect tenses may therefore be represented under obvious diagrammatic conventions as follows (where the points marked ‘S’, ‘E’, and ‘R’ represent respectively the

’ In $51 of his Elementi of Symbolic Logic (Macmillan, New York 1947).

204 BARRY TAYLOR

points of speech, event, and reference):

Simple Perfect

Present ‘- S,R,E E S, IX

Past m’ I% IX S E R S

Future w e S IT IX S E R

This account of Reichenbach’s enables us to maintain the semantic equivalence of (6) and (lo), and to consign their differences to the pragmatic component of linguistic theory. For in uttering either (6) or (lo), it reveals, a speaker asserts the same thing - that Esau was hirsute at some earlier time; but in choosing the simple-tensed form of words (6) he directs the attention of his audience to the earlier time as his reference-point, whilst in selecting the perfect-tensed (10) it is rather to the present that attention is drawn. Moreover, his articulation of the relations holding between temporal points associated with utterances of the other perfect tenses suggests obvious proposals for their treatment; thus the Pust pe$ect exemplified in

(11) Esau had been hirsute

and the future petfect of

02) Esau will have been hirsute

will receive the base paraphrases

031 (3 t)(t -C then & then -C now & Hirsute (es, t))

and-

041 (3 t)(now < t & t -C then & Hirsute (es, t))

where, on each occasion of utterance, the demonstrative ‘then’ serves to indicate the reference-point of the speaker.


2. CONTINUOUSTENSES AND ARISTOTLE~STRICHOTOMY

The stage is now set for the main business of this paper: an account of the continuous tenses, and of Aristotle’s classification of verbs of natural languages in terms of properties of their continuous tenses. Aristotle’s classification6 is a trichotomy, which would distinguish amongst the verb- phrases of English:

(a) S (‘state’)-verbs, e.g. ‘is hirsute’, ‘is red’, ‘is soluble’, ‘is taller than (Caesar)‘, ‘loves (Beatrice)‘, ‘understands (Godel’s proof)‘. The distinguish- ing characteristic of S-verbs is that they do not occur in genuine continuous tenses in standard English speech.

(b) E (‘errergeiu’)-verbs, e.g. ‘chuckles’, ‘talks’, ‘falls’, ‘blushes’, ‘moves’, ‘strokes (the dog)‘, ‘ponders’. E-verbs are distinguished by the fact that, for E-verbs V, the present continuous ‘x is V-ing’ entails the perfect ‘X has V-ed’.

(c) K (‘kinesis’)-verbs, e.g. ‘stabs (Caesar)‘, ‘discovers (America)‘, ‘builds (the house)‘, ‘polishes (the boot)‘, ‘grows up’. K-verbs are distinguished by possessing a contrary property to that characteristic of E-verbs, a fact commonly expressed by saying that for K-verbs, ‘x is V-ing’ entails ‘X has not V-ed’; though it is clear that this slogan cannot be taken as a precise expression of the property in question. For the account of the present perfect given in the last section permits distinction of two senses in which the phrase ‘X has not V-ed’ may be taken, depending on the scope of the negation; that is, we may construe the phrase as meaning (roughly) either

(9 - (3 t)(t < now & t is a time of x’s V-ing) or (ii) (3 t)(f < now & - (tis a time of x’s V-ing)).

And yet under neither of these construals does the common slogan accu- rately express a property which can be taken as definitive of K-verbs. ‘Brutus is stabbing Caesar’ does not entail ‘Brutus has not stabbed Caesar’ in sense (i) of the latter expression, since the current stabbing may be Brutus’ second for the morning. And for E-verbs us well as K-verbs it is true that ‘x is V-ing’ entails ‘x has not V-ed’ in sense (ii). No time need be wasted, however, in a search for refinements of vernacular formulations avoiding these difficulties, provided a formal account of the properties of E-verbs underlying their characteristic entailments can be found. For then we may hope a candidate for a contrary property to be attributed to K-verbs will emerge, and the common slogan, despite its imperfections, should be suggestive enough to permit evaluation of any such candidate.

’ The main references in Aristotle are Met 06 (1048b, 18-35); De. An. 417 a30-b2; hk. Eda. x4.

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To approach the account I want to give of these distinctions and the continuous tenses giving rise to them, consider the case of Rod, a hirsute barman, who pulls a pint, taking all the time in some period P to do so, and chuckles the while. Then it is reasonable to say, of any moment m within P, that nr is a time of Rod’s being hirsute; indeed, Pcounts as a time of Rod’s being hirsute, it seems, just because each moment within P is such a time. On the other hand, although at each moment m within P it is true to say that Rod is c/nA&zg and is puZling u pint, it is plausible to hold that no moment within P can be a time of Rod’s chuckling or of his pulling a pint; for both pulling pints and chuckling take time in a way in which being hirsute does not. These remarks suggest that the continuous tenses may be construed as functioning so as to mark the presence of a time t (typically a moment) which, though not itself a time of application of the tensed verb, occurs within a more inclusive time which is a period of the verb’s application. This construal explains both why continuous tensings of K- and E-verbs can be true at times which are not themselves times of application of the verb, and also why S-verbs should lack continuous tenses - for every time within a period of application of such a verb itself being a time of its application, there is no place for tenses designed to register the existence of times of non- application of the verb within broader periods of its application. ’

These informal views are elucidated by the following symbolism. I assume that verb-phrases in each of the categories of S-, K-, and E-verbs will be represented in the base by use of a predicate constant in the corresponding category, to be defined below as the theory is developed. Where P)’ is the j-th n-place predicate constant of the base formalism, let c be the result of filling its first n - 1 argument places by distinct variables (say, the first n - 1 variables in some specified standard ordering). ’ The formal upshot of the above remarks on S-verbs is then that Py should count as an S-predicate iff it meets

PostuZute 1 Per (t) + (qt ti (Vt’)(Mom (t’) & t’c t . + . qt’))

’ In emphasizing the importance of temporal periods in accounting for the continuous tenses, my account shares a number of features with recent (independent) work by David Dowty, Linguistics and Philosophy 1, 45-77; however, our differences of framework and subsidiary assumptions quickly lead to wide divergences.

s In using this convention in the sequel, I assume that ‘x’ and ‘y ’ are the first two variables in the standard ordering; and also that the insertion of parentheses around terms following predicates is not required by the formation-rules of the base in constructing atomic formulae, but is merely a convenient punctuation device to aid in construing the formalism.


so that, for example the predicate Hirsute is an S-predicate on the assumption that it fits

Posthte lu Per (f) + (Hirsute (x, t) w (Vt)(IvIom (t’) & t’ z t . + . Hirsute (x, r’))).

Again, the thesis that mere moments do not suffice as times of application for E- and K-verbs issues in the requirement that where Py is an E- or K-predicate, it should meet

Poshhte 2 Vyf + Per(f)

so that, e.g., the E-predicate Chuckle and the K-predicate Stab will respectively fit

Postulute 2u Chuckle (x, t) + Per (f)

Postulute 2b Stab (x, y, t) *Per(t).

Finally, the analysis of the continuous tenses can be expressed in intuitive symbolism by means of the schema

Cont Py -ingxr . . . xn-lt .s- - vyt 6% (3 f’)(hx’ & vy).

(The left-hand side may be read here in quasi-English as ‘x1 is Py - ingx2... x~-~ at f’.) Combining this account of the continuous with the paraphrases of the last section, therefore, we will assign respectively to

WI Brutus is stabbing Caesar (16) Brutus was stabbing Caesar (17) Brutus will be stabbing Caesar

the base paraphrases

-Stab (b, c, now) & (3 t’)(now r t’ & Stab ((b, c, f’))

(3 t)(t < now & -Stab (b, c, t) & (3 t’)(trt’ & Stab (b, c, f’)))

(3 t)(now <t & -Stab (b, c, t) & (3 t’)Wt’ & Stab (b, c, t’))).

The informal explanation given for the failure of S-verbs to occur in the continuous tenses is now reflected in the formalism by the fact that a sentence like

Esau is being hirsute

208 BARRY TAYLOR

receiving the base paraphrase

cm -Hirsute (es, now) & (3 r’)(now W’ & Hirsute (es, f’))

is trivially false in virtue of Postulate la, which entails by the logic of times that for any x, r, and t’,

or’ & Hirsute (x, t’) + Hirsute (x, t).

Postulate 2 expresses a feature common to both K- and E-verbs, and we now turn to the problem of describing the further properties of E-verbs needed to account for their characteristic entailment. Let us say that a temporal period t is qe~-~r~rzre~ (OF(t)) iff it contains within it no earliest moment; and dually, that a period t is open-ended (OE(r)) iff it contains within it no latest moment. Then the following two further assumptions need to be made to accommodate the peculiarities of E-verbs:

(a) that every period of application of an E-predicate falls within (though not necessarily properly within) an open-fronted temporal period which is itself a period of its application; and

(b) that every period falling within a period of application of an E- predicate is itself a period of its application.

The first of these assumptions is a weakened version of a thesis argued for by Aristotle (P/ry&s 236a, 7-28), ULZ. (in effect) that ull periods of application of both E-predicates and K-predicates are open-fronted. Aristotle’s arguments, however, hardly establish the point; his best one proceeds from the premise of the existence of a latest moment within any period immediately preceding a period of application of a K- or E-predicate and then argues from the density of the moments to the open-frontedness of that period itself, but no reason is given for accepting the crucial premise. The weaker assumption (a) is, however, all that is needed here, and I see no bar to accepting it even without subsidiary argument if it is needed to accommodate the data. (In particular, I do not think it is refuted empirically by examples of the sort advanced in discussion by Christopher Peacocke, whose alarm-clock rings at the laudable hour of 7 a.m. -for, though ‘ring’ (as said of alarm-clocks) is certainly an E-verb, there is no reason to classify 7 a.m. as the first moment of the period of the alarm-clock’s ringing, rather than as the lower bound of that period, hence possibly as the last moment of the preceding period.)

The second of my assumptions is open to more serious objection, as the next section will reveal; but for the moment let it be accepted in the same spirit, as a working hypothesis to accommodate the data. Combining both assumptions with Postulate 2, we are left with the view that E-predicates PT


will be those meeting

Postulute 3 Vyt + Per (t) & (El t’)(OF(t’) & t!zt’ & vt’)

& (Vt”)(t”W & Per (t’? + qt’y

so that, e.g., the E-predicate Fall will operate under

Postzdute 3u Fall (x, t) + Per (t) 8~ (B t’)(OF(t’) & t&t’ & Fall (x, t’))

& (Vt”)(t”ct & Per (t”) + Fall (x, t”)).

Granted the further plausible supposition that the demonstrative ‘now’ always serves to indicate a moment (designating e.g. on each occasion of its use the least upper bound of the time taken for its utterance), this postulate combines with our analyses of the tenses and intuitive properties of times to explain the characteristic entailment of E-verbs. For suppose e.g. that x is falling, i.e. by Cont that there is a t such that now Et and Fall (x, t). By Postulate 3u, t is open-fronted; from this and the density of moments there are accordingly moments m amd m’ within t such that m < m’ < now. So there is a period t’ stretching from m to m’ which is within t and earlier than now; by the former property and Postulate 3u, Fall (x, t’); hence by our account of the simple perfect, x has fallen. Since moreover this informal reasoning can be reconstructed as a formal deduction within the theory of the Appendix, we have thus apparently arrived at a rigorous account of the idiosyncrasies of E-verbs.

Further, this account seems to yield, as was hoped, a plausible candidate explicating that contrary property of K-verbs which is suggestively if impre- cisely captured in the standard slogan that for such verbs V, ‘x is V-ing’ entails ‘x has not V-ed’. For it appears that such a candidate emerges if we assume that K-predicates possess a property contrary to that supposed above as assumption (b) to hold good of E-verbs - thus that for any K-predicate, no period within a period of its application should itself be such a period. Combining this assumption with Postulate 2, the K-predicates Py therefore emerge as those meeting

Poshdute 4 VjY + Per (t) & (Vt’)(t’ r t + - vt’)

so that the predicate Stab, for example, will fit

Postzdute 4u Stab (x, y, t) + Per (t)

& (Vt’)(t’ E t + -Stab (x, y, t’)).

On this view, accordingly, ‘x is stabbing y’ will entail a statement which we may read as ‘x has not yet stabbed y during this period of his stabbing’ -

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which, though hardly vernacular, is reasonably construed as the truth which the common slogan tries to capture. ’

Here, then, is an account of the continuous tenses, and of the Aristotelian trichotomy of verbs in terms of the properties they display in these tenses. But considerations to be advanced in the next section reveal that the latter account needs some complication and modification.

3. COMPLICATIONS BY SPATIAL ANALOGY

The account of E-verbs and K-verbs given in the last section permits of an illuminating spatial analogy, which at the same time deepens the under- standing and reveals the limitation of the preceding formalism.

Suppose that a lump of gold were completely Iromogeneozq so that every three-dimensional area wholly within it was itself a lump of gold. (Real gold is, we know thanks to molecular theory, not homogeneous; but before that theory was developed, it was perfectly reasonable to hold that it was.) Then there would be no lower limit on the spatial extent of a lump of gold-though of course some lower limit would operate on the size of the samples which normal criteria could identify m gold, and for samples lower than this minimum we could know, without recourse to sophisticated scientific tests, that an item was a lump of gold only by knowing that it was (or had been) part of a larger lump to which the standard tests could be applied. But even within a homogeneous lump of gold, there is no lump of gold at a pint; for lumps of gold, even homogeneous gold, must occupy some area, however small, of three-dimensional space.

A homogeneous stuff such as gold (on the present assumption) differs instructively from a substance (e.g. a table) in the matter of space occupancy. For while no point of space within a table is occupied by a table, in the same way that no point within a homogeneous lump of gold is occupied by a lump of gold, in general no (three-dimensional or other) space within a table is

’ My postulates do, however, validate the entailment from ‘x was V-ing’ to ‘some time is a time of x’s V-ing’;for K-verbs V; and many people have objected to this, on the ground that e.g. Coleridge was writing ‘Kubla Khan’ at the time t of the Porlock person’s untimely arrival, despite the fact that the interruption ensured that no time was to be a time of his writing the poem. (Assume, for the example, that ‘Kubla Khan’ is the whole of Coleridge’s projected work, not just its successfully completed fragment.) Very briefly, my response is to deny that Coleridge really was writing ‘K.K.’ at t (since t did not fall within a period of his writing the poem); rather he was at t doing something which would /uzue ken writing ‘K.K.’ had the arrival not occurred (i.e. t falls within a period t’ which would have been a time of Coleridge’s writing ‘K.K.’ had the Porlock person not arrived). If all purported counterexamples to the entailment can be met with this count&factual construal, my postulates stand; though a systematic treatment of the counterinstances must await a Davidsonian analysis of the subjunctive conditional.


occupied by a table, whereas er~?ry three-dimensional area within a homogeneous lump of gold is occupied by a lump of gold. For a form of words to encapsulate the difference, we may say that a homogeneous stuff fills, whereas a substance &IM&r, the space it occupies.

Now, taking ‘falls’ and ‘stabs’ as paradigms respectively of E- and K-verbs, the views of the last section can be summed up as the theses that fallingfiUs time as a homogeneous stuff fills space, whereas stabbing delimits time as a substance delimits space. Thus no moment is a time of falling or of stabbing, just as no point is a place occupied by a lump of gold or a table; for falling and stabbing take time, just as both stuffs and substances take up three-dimensional space. Further, just as in general no spatial area within a table is itself an area occupied by a table, so the last section holds that no period within a period of stabbing is itself such a period; and just as euery three-dimensional spatial area within a lump of homogeneous gold is itself such a lump of gold, so every period within a period of falling is itself a period of falling.

But once it is seen that this is indeed the spatial analogy pre-supposed by the account of the last section, doubts arise about its aptness in all cases. For it is clear that, even continuing to waive the claims of molecular theory and to maintain the homogeneity of stuffs such as gold, various heterogeneous stuffs ought also to be recognized, Fruit-cake will serve as an example. Division of a lump of fruit-cake will produce a lump of fruit-cake only until a sample of some minimal size is reached; a mere sultana does not in itself constitute a lump of fruit-cake, and can at best be a part of such a lump. There is indeed no obvious requirement that there should be some ubsolute lower limit on the size of a lump of fruit-cake, as there presumably is on the size of the samples our normal criteria can identify as such lumps, since there is no obvious conceptual bar to supposing that a shrinking machine might miniaturize a sample beyond any limit which might be imposed. But even within a sample of miniaturized fruit-cake there will be areas (e.g. one occupied by a miniature sultana) which are not themselves occupied by a lump of fruit-cake; and, failing theoretical grounds to the contrary, it is natural to suppose that the minimal lumps of fruit-cake within a sample which is on the everyday scale are those of just sufficient size to be recognized as fruit-cake by our normal criteria. The contrast of heterogeneous fruit-cake with homogeneous gold therefore emerges sharply in that whereas any lump of normal-scale fruit-cake must in principle be directly recognizable as such by means of our normal criteria, there may (as already noted) be genuine samples of gold which, themselves of insufficient size to be recognized directly, can be told as such by our standard criteria only indirectly via the knowledge that they form (or have formed) part of a larger lump to which these tests can be applied.

212 BARRYTAYLOR

Now doubts concerning the universal applicability of the account given of E-verbs in the last section arise because in some cases it is more apt to compare the structure of an E-verb’s period of application with the space- occupancy of a heterogeneous rather than a homogeneous stuff. True, there ure cases where the analogy with homogeneous stuffs is appropriate: even a microsecond within a period of falling is plausibly reckoned as itself genuinely a period of falling, even though it can be told as such by means of normal empirical criteria only indirectly, via the knowledge that it does indeed come within some wider period long enough for those criteria to be applied. ‘Falls’ we may therefore classify as a /romogeneousE-verb, grouping it with e.g. ‘moves’, ‘ponders’, and ‘blushes’. But an example like ‘chuckles’ provides a case more naturally conceived on the analogy of a heterogeneous stuff, since any sounds emitted in a microsecond during a period of chuckling (at the normal rate) hardly constitute chuckling themselves, but rather appear to stand to chuckling as a sultana might stand to fruit-cake, viz. as at best falling within some period of chuckling though themselves occupying a time too short to constitute such a period. So, though there is no need to be committed to a lower limit on the length of a possible period of chuckling (since the rate of chuckling can conceivably be increased just as fruit-cake can be miniaturized), within any period of chuckling there will be minimal periods of chuckling, and it is natural to identify the minimal periods of a chuckling carried out at the normal rate with those which everyday empirical criteria can identify as such. These afhnities between the way chuckling fills time and a heterogeneous stuff fills space suggest we should classify ‘chuckles’ as a /heterogeneous E-verb, grouping it with e.g. ‘giggles’, ‘talks’, ‘walks’, and ‘strokes (the dog)‘; and the account of the last section, whilst now seen to provide in Postulate 3 an explication of the logical properties of Jromogene - ous E-verbs, requires a supplementary explication of the properties of the important class of heterogeneous E-verbs thus distinguished.

The obvious way to obtain such an explication is to construct a temporal analogue of a condition definitive of the way heterogeneous stuffs fill the space they occupy; and in the light of the foregoing discussion, I propose the following as such a condition. Where S is a heterogeneous stuff (e.g. fruit-cake), let an S-pluce (ureu) be a place (area) occupied by a lump of S; then any heterogeneous stuff S should neet the SputiulHeferogeneify Cbndi- tion :

(9 (ii)

any S-place is a (three-dimensional) S-area any S-area A falls (spatially) within (though perhaps not properly within) some area A’ such that (a) A’ is S-maximal (i.e. an S-area falling properly within no further S-area)


(b) some area within A’ is S-minimal (i.e. an S-area within which no further S-area falls) (c) any area within A’ is an S-area iff there is within it some S-minimal area.

To obtain a formal temporal analogue of this condition, it is convenient first to introduce some abbreviatory notation; so I defme

03 t is maximal w.r.t. a set z of times

(Max (t, z)) . ti . t E z & -(3 t’)(tct’& t’ c z)

02) t is minimal w.r.t. a set z of times

(Min (t, z)). * . t E z & - (3 t’)(t’ct 8~ t’ E z)

and further abbreviate by the definition-schemata

PI Max”: (t) +B Max (t, {t’/ Vyt’})

WI Minq (t) ++Min (t, it’\ qt’}).

Then the heterogeneous E-predicates Py emerge as those meeting

Postdate 5 vt + Per (t) 8~

(3 a)(Max”y (a) & %a & (3 b)(Min”y (6) &L La) &

(Vc)(cca + (Vyc - (3 ~)(Min~; (b) & 66~))))

so that, e.g., the heterogeneous E-predicate Chuckle (or Ch, for brevity) operates under

Postzdate 5a Ch (x, t) . + Per (t) &

(!l a)(MaxG~xj (a) & t G a & (3 b)(MinacXj (b) & b L a) &

(Vc)(c = a + (Ch(x, c) ~(3 b)(MinatXJ (b) & b E c))))

where by the definition-schemata [I] and [II] we have

Me~,(xj 0) -h&ix tc 0+Xx, ON

Minor,tXj (t) H. Min (t, {t’]Ch(x, t’)}).

An obvious difficulty in accepting this as an explication of the logical properties of heterogeneous E-verbs is that it fails to provide for the characteristic entailment, shared by all E-verbs, from ‘x is V-ing to ‘X has V-ed’; for, e.g. Postulate 5a allows that there should be a first minimal period within any period of X’S chuckling, hence that there should be some absolutely first period which is a time of x’s chuckling - and at any time within that period it will on the present elucidation of the tenses be true that

214 BARRY TAYLOR

x is chuckling, but false that he has chuckled. This difficulty is endemic to the conception of heterogeneous E-verbs at temporal analogues of heterogeneous stuffs, and cannot be avoided by modifying or augmenting the postulate which gives rise to it. But there is no cause for undue concern, provided the natural assumption be made that the minimal periods of chuckling within a piece of normal-rate chuckling are the least times of chuckling so discernible by normal empirical criteria. For then it will at least remain true that no speaker will be in a position warrantably fc assert that x is chuckling until, some minimal period of chuckling having passed and been recognized, it is true that x has chuckled; so although on the present view it must be denied that there is a genuine entailment from ‘x is V-ing’ to ‘x has V-ed’ for heterogeneous E-verbs, at least it is clear why it should have seemed plausible for theorists to have held that there is.

The distinction of two classes. of E-verbs has its ramifications for the theory of K-verbs as put forward in the last section. For the strategy pursued there assumed that there would be some one property characteristic of E-verbs, an explication of which would enable identification of an opposed property which could be taken as characteristic of K-verbs; but now it has emerged that the assumption underlying this strategy is false, and that the account of the last section should be seen as opposing K-verbs, not to E-verbs generally, but more particularly to homogeneous E-verbs. So, without denying the prima facie plausibility of the earlier explication, the possibility arises of an alternative and perhaps more satisfactory theory which will oppose K-verbs rather to the heterogeneous E-verbs.

The claims of spatial analogy (such as they are) point towards this second alternative. In comparing the structure of a K-verb’s period of application (as construed in the last section) with the way a substance delimits space, on the ground that e.g. a table generally occupies a space no area within which is itself occupied by a table, the need to restrict attention to the standard cases by means of the hedge in ‘generally’ indicates the imperfection of the analogy. For tables can be so constructed as to enable a number to slot together to form a table; and so an accurate account of the delimitation of space by substances must allow that a substance of a given kind can occupy an area itself containing areas occupied by substances of that kind. Of course, it is still .true that not every area within an area occupied by a substance can be occupied by a substance of similar kind - like heterogeneous stuffs, any substance will contain minimal areas of occupancy by substances of the same kind. But some areas falling properly within a substance will nevertheless fail to be themselves occupied by a substance of the given kind, even if containing within them one of these minimal areas. So substances are really to be naturally contrasted with heterogeneous stuffs in


the matter of space-occupancy, just as on the currently canvassed suggestion K-verbs are to be opposed to heterogeneous E-verbs in their temporal properties; so an attractive feature of that suggestion is that it seems to allow for a fuller vindication of the spatial analogy suggested even by its com- petitor.

For a detailed implementation of these ideas, we observe that (where S is now a &r&tce-k!&, and an S-place (area) is a place (area) occupied by an instance of S) a condition definitive of the way substances delineate space can be obtained from the Spatial Heterogeneity Condition by substituting for clause (ii) (c) the requirement

w for any S-minimal area A” properly within A’, there is an area within which A” falls, and which falls within A’, but which is not an S-area.

By constructing a temporal analogue of the condition thus obtained, we therefore arrive at a view of K-verbs which at once opposes them to heterogeneous E-verbs and maintains a full analogy with the space- occupancy of substances, VLZ. the view that the K-predicates Py are those meeting

Postufute 6 Vj’t . + . Per (t) &

(3 u)(Maxv; (u) & tmz & (B b)(Min”; (b) &L bm) &L

(Vc)(Minvy (c) & CQ + (3 d)(ccd & dru & - V/‘d))).

This condition is, one notes, weaker than that imposed on K-predicates by Postulate 4 in the earlier account, in that any predicate meeting Postulate 4 meets Postulate 6, but not vice versa. The question which arises is therefore whether (apart from the dubious desire to maintain the spatial analogy point for point) there is any need to construe K-predicates according to this new and laxer standard, rather than to cleave to the simplicity of the older story.

Surprisingly enough, such a need does arise out of complexities, ignored in the last section, connected with the continuous tenses of quantified sentences containing K-verbs. A sentence like

(23) John is polishing all (the) boots

can be true (as I shall say) consecz&eZy and not just simzhzneously ; that is, it can be true at any time during a performance in which John systematically works his way through a heap of boots, polishing each in turn, and does not require that any time of his polishing one boot should be a time of his polishing any other (a feat presumably impossible, assuming John to have the normal complement of limbs). Now the truth of (23) on the present

216 BARRY TAYLOR

account of the continuous tenses requires that there is a time 2 such that now w and

04) (Vy)(Boot(y) + Polish& y, g)).

Hence we must suppose that K-predicates so relate times to individuals that there is such a time, even when (23) is consecutively true. This is ensured by the supposition that the K-predicates PT meet a further requirement, namely that we have (for each i such that 1 s i c n).

P~st~hte 7 (Vxi)(A(xi) + (Zl f’)(t’ct & qr’)) &

(Vt”)(Mom (t”) & f”ct +

(3 xi){3 t”‘)(f’~t”’ & A (Xi) & qt”‘))

a + e (Vxi)(A(xi) + Vyt)

where xi is the i-th variable in the standard ordering and A(Xi) is any formula containing xi free. (Recall, in construing this formalism, that xi also by our syntactic conventions occurs as the i-th free variable in q.) For as an instance of this schema we have

Postzdute 7~2 (Vy)(Boot (y) + (3 l’)(t’W & Polish (x, y, t’))) C%

(VC)(Mom (t”) & ?“cl+

(Zl y)(Zl f”‘)(l’W” & Boot (y) & Polish (x, y, t”‘)))

+ . (Vy)(Boot (y) + Polish (x, y, t))

which ensures that even when John’s polishing of the boots is consecutive, there will be a time - viz. the time stretching from his applying the first lick of polish to the first boot, up to his laying aside of the last - which meets the condition imposed in (24) (assuming, as seems reasonable, that we are prepared to count any moment in a pause between his laying aside the n-th boot and his first brush-stroke on the (n + 1)th as belonging either to the time of his polishing the n-th boot, or to that of his polishing the (n + 1)th).

Postulate 7, needed for these reasons, is of course straightforwardly incompatible with the old account of K-verbs as enshrined in Postulate 4; for the very reason it has been invoked is to ensure the existence of a time 2 such that


in circumstances where t is not a minimal period of John’s polishing &, i.e. for some time t’W

Polish G, bi, f’)

in contradiction to Postulate 4. This possibility is, however, allowed by Postulate 6, whose laxer conditions appear indeed to describe quite suc- cinctly the structure of the more complex periods of application of K-verbs to which we are now committed. The latter postulate should accordingly be preferred in the final reckoning; K-predicates are thus to be construed as contrasting with heterogeneous rather than homogeneous E-predicates, and the spatial analogy with substances is vindicated in the fuller detail.

Some puzzles connected with Postulate 7, however, survive this decision. One of these is that if the time t of John’s whole consecutive boot-polishing performance is to count as a time of his polishing any boot bj, then by the favoured account of the continuous tenses, at uny moment within t it is true that John is polishing boot bi - even e.g. at a moment when he has laid & aside as finished, and begun applying polish to boot bi+r. This may seem just wrong; for (Case 1) no obseruer of John’s performance can legitimately assert that he is polishing boot bi when he has manifestly moved on to boot bi+l. On the other hand, not all the linguistic evidence is hostile to the present consequence. Thus (Case 2) a hotel-guest, having left his boots out for cleaning the previous night, may enquire of a maid as to their where- abouts, and be informed that John is currently cleaning them; and for the maid’s answer to be honest, it is not necessary that John actually be operating on the boots in question, but will suffice that they be part of the heap of guests’ boots he is currently working his way through as part of his duties.

The solution to this difficulty lies in accepting the consequence of the proposed semantic account that the assertions of both observer and maid are literally true, but appealing to pragmatic conventions governing assertion to contrast the illegitimacy of the former with the acceptability of the latter. For there is a general duty of the speakers of a language (based on the mutual interest of all concerned) not to mislead their audience, hence to base their assertions on the most specific information to which they have access. So we may posit a Principle of Strongest Assertion governing the conditions under which speakers are to assert continuous-tensed sentences containing K- verbs V, dictating (roughly) that a speaker should not assert that x is V-ing merely on the basis of a belief that the current moment falls within some norm -minimal period of x’s V-ing, if he also has grounds for believing that the current moment does not fall within a minimal period of x’s V-ing. The observer of John’s polishing is, accordingly, not entitled to assert that John is

218 BARRY TAYLOR

polishing boot 6i on the basis of his belief that the current moment falls within a time f of John’s polishing of every boot (even though the sentence thereby asserted would be literally true), since the testimony of his senses informs him that the current moment falls within a minimal period of John’s polishing of boot ~i+~ but not of boot bi ; whereas the maid of Case 2, who presumably lacks any knowledge of the minimal periods within the nonmini- ma1 time r of John’s polishing every boot, is thereby entitled to use her less specific belief about the wider period in framing a continuous-tensed assertion which is not just literally true, but sanctioned by the conventions of assertion.

Appeal to this pragmatic principle also enables us to see how the common slogan that, for K-verbs V, ‘x is V-ing’ entails ‘X has not V-ed’, is to be interpreted on the present account of K-verbs. It can indeed no longer be claimed, as it was in the last section, that the insight underlying this slogan is that whenever the current moment falls within a time f of x’s V-ing, no time earlier than the current moment and within ? is a time of x’s V-ing; for as we have seen, we must now allow that if John polishes the boots consecutively, a time ti of his polishing boot bi can precede the current moment and both nevertheless fall within a time t which, as a time of John’s polishing every boot, is a time of his polishing boot bi. Nevertheless, it remains true that a totally informed observer, operating in accordance with the Principle of Strongest Assertion, will not usserf that x is V-ing unless the current moment falls within some minimul time of x’s V-ing, in which no time earlier than the current moment can be a time of x’s V-ing; and it seems reasonable to offer this in explication of the standard slogan.

I accordingly claim for my account of the continuous tenses the virtue of permitting a systematic account of Aristotle’s intuitive trichotomy. The official account of that trichotomy with which I have concluded is a fairly far-reaching revision, through spatial analogy, of the first attempt of the last section: whilst S-verbs are still to be explicated by positing underlying S-predicates meeting the old Postulate 1, E-verbs are now divided into the homogeneous and the heterogeneous, with Postulate 3 explicating the properties of predicates underlying verbs of the former class, and Postulate 5 those of predicates underlying verbs of the latter; and K-verbs are to be those whose underlying predicates meet Postulate 6 rather than Postulate 4, with Postulate 7 as an additional condition governing at least some of them. It will be noted that of the prima facie entailments used to draw up the original classification, only that of homogeneous E-verbs has been vindicated as genuine. But appeal to pragmatics enables the ground for the supposed entailment in other cases to be understood; and it is commonplace for a philosophical theory to refine the intuitions which give it rise.


A final suggestion. My account discovers and exploits a systematic analogy between K- and E-verbs and the sortal and mass terms we use to individuate space-occupying entities (substances and stuffs); perhaps, then, it can be used as the basis of a theory of time-occupying entities (events). I think this is correct, but cannot defend the suggestion here.

APPENDIX: A THEORY OF TIMES

Rmitiues: identity and temporal precedence. Defined predicates are intro- duced as follows (read ‘Lin (f, t’)’ as ‘f and t’ are linearly related’; ‘01 (f, f’)’ as ‘f overlaps with t”; and ‘Str (t, x, y)’ as ‘f stretches from x to y’):

Lin (t, f’) - f-c f’ v f = f v f’< f.

01 (f, f’) * -Lin (f, f’) &

(3x)(x<r&-x <f’. v .r<x&-f’<x).

rcr’ M 01 (f, f’) & - 01 (t’, t).

fGf’-f~f’ v f = f’.

Mom (t) e - (3 f’) 01 (f’, f).

Per (f)e(3 f’)(Mom (f’) 8z f’rf).

Str (t, x, y)@xW & ycf & (b/2)(2 -Cf+z -Cx. &

. f-cz-+y~z).

OF (t)*Per (t) & (Vt’)(Mom (t’) & f’cf

+ (3 f’y(f”=f & f”K f’)).

OE (t) *Per (t) & (Vf’)(Mom (f’) & f’cf

+ (3 f’9(f”Cf & 2’4 f”)).

As uxioms based on a conception of time as stretching continuously in both directions, take the following:

0-W Per (f) v Mom (f).

643 (vx)(x-cf~x~f’.&.f-cx~f’-cx)~f=f’

(A3) r<r’+-f’<f.

644) f < tr & f’ < r”+ f < f#.

(A5) -(3 t)(Vf’)(Mom (t’) & f # f’+ f’< f).

w3 -(g f)(Vf’)(Mom (f’) & r # f’+ f -c f’).

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(A7) (Mom (t) v Mom (t’)) & t <t’ . + . (3 f)(t -c t” & P-C f).

MN x<y+(ilr)Str(f,x,y).

OW (3 f)(w’)(r’m -u -c f’ & r’ -c b)

(AlO) ~<~‘&OE(~)&OF(~‘)~(~u)(f<u&u<~‘).

University of Melbourne

Tense and continuity

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