AND FRENCH COGNITIVE ADVERBS - Cahiers de linguistique ...

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(2) Tout le monde a une passion, mais ae n'est pas la mime pour tout le monde. (Everybody ..... (15) is odd and the minuteness of the difference cannot cure it. To. (hopefully) gain ..... bonne (1)", SEMANTIKOS I 1, 1-28. :"ll 6tait une fois une ...
- 247 GAMES, FRAMES/ AND FRENCH COGNITIVE ADVERBS (i) Jacques Jayez Annemasse

INTRODUCTION This paper is the first part of a general exposition concerning the use of game-theoretical techniques in pragmatics. The emphasis will be on the logical side, while the second part will be designed to provide a computer-based approach. Anyhow the two parts can be read separately, and no particular connection will be assumed for understanding. Although it is impossible to track every influential work in a research project, I will mention briefly some probable "triggers". - first, there are seminal papers by ANSCOMBRE and DUCROT (see, e.g., Anscombre 1973, Anscombre 1975, Ducrot & al. 1980, Ducrot 1980). 1 am convinced that, basically, pragmatics need such studies and other areas in linguistics and psychology need them too. - Second, there is syntax : because I do

not have any proof of the

so-called derivative character of syntax, I do not dream of any linguistic universe without syntax, and I am rather careful (should I say "timid" ?) about pragmatic accounts of syntactic phenomena (see Jayez 1982b). - There is a second, independent, reason for

moderate use of prag-

matics : we are at a loss to offer a precise rating of how much conversational analysis matters to pragmatics. - Last there is cognitive psychology : here too pragmatics comes across stimulating ideas and safe warnings by advisory persons (see, e.g., Hall-Partee 1979, Peters & Saarinen 1982, Abelson 197 3, Abelson 1975, Bobrow & Winograd 1977, Bobrow & Winograd 1979, for introductory readings ; further references will be given in the second part).

I deliberately omitted diachronic studies, to which I will return in the sequel.

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1.1. Games in general Definition 1 : Let L be a first-order language and K a set of connectives. A game on (L,K) is a set G (L, K) such that : (a) each element of G (L, K.) is a triple of form (p , (k., A) , (P , J-

J

Jv

(k ,B)) where : (b) p. and P. are sets, x k {c) k. and k are sets of indexed connectives and quantifiers having at least one occurrence in A (for k.) or B (for k , ) , j 1 (d) A and B are sentences of L with connectives and quantifiers indexed according to their respective positions in the sentence, or expressions obtained from sentences of L by substituting variables or constants for subformulas in the sentences, (e) these variables and constants obey the following minimal requirements : . each wff of L is designated by a single constant, . the cardinal of the set of such variables and constants is "sufficient" (it is at least CARD (L)). Please note : it follows from (d) that A and B can contain both variables for wffs and wffs. To avoid unnecessary complexity, we shall use wffs autonymously (the constants for wffs are wffs themselves). Definition 2 : Let G(L,K) be a game on (L,K), a play of G(L,K) is a sequence m^,..., m (n an ordinal) such that : la) if i is a successor ordinal, each m

- I x l y J z ( x , y , z ) CG(L,K) . m

is such that :

= (x,y) . m = z) , or

- "dx ]y }z 3 VJZ J k J l ((x,Y,Z) CG(L,K) . Y = Su (y) . Z = Su (x,y) . m. = z ) , where Su i s a f u n c t i o n such t h a t : i n .Su > (P i , (k ,A)) = ( p i , S u h ( k ,A)) for any p ^ k ,A, .Su

(z)

Vi

can :

•change only k (still respecting (c) in def. 1 ) , •change A, correlating to it a more abstract form by use of variables and constants for wffs (see (d) and (e) above), and change k

accordingly, suppressing all elements no longer present

in the more abstract form, renumbering the new elements if necessary, and possibly making a new selection >

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(b) if i is a limit ordinal m

must be such that its form is not secured

as in (a), but can be determined only by considering an infinite number of m, ; (c) m o is (p , (k.,A)) for some p , k , and A. Please note : the existence and constructibility of indices h for Su h functions may prove incredibly difficult, so it is safer not to hazard any restriction. This couple of definitions is much more general than is currently tolerated in game-theoretical semantics ; so it requires some intuitive counterpart. The basic idea consists in regarding the "verification" of an expression as a succession of moves controlled by "teams". Suppose we start with an expression E : the game G(L,K) is a set of options among which we have to make a choice at each stage of the play. First we encounter n^ (the initial move) : m o means that the team p. tries to win E, focusing on the logical articulations the members of k. stand for ; the game associates to (p., (k ,E)) some legal move{s) ; when choosing one of these we indicate the next team concerned, the next relevant connectives, and the next expression at stake ; then these informations are compared again to the set of options, and so on. The standard game-theoretical approach is a special case of def. 1, namely a case in which there are severe restrictions on teams and determination of relevant connectives (quantifiers). Let me illustrate the last feature : a standard play always proceeds according to depth degrees, that is, it selects the connective or quantifier which ranks highest after the last connective or quantifier selected in the hierarchy of complexity. For instance, in a sentence like Vx ]y((P(x) . Q(x)) V R(x)), the successive k. will be {Vxf , }] y\,

\ . \, and j V } . For sure the rule is convenient, but I am

unable to offer any principled reason fort it. Consider dealing with quantifiers : are we to impose the left-to-right traditional order ? I have been arguing at some length in Jayez 1981b and Jayez 1982a that argumentative relations between sentences could be analyzed, to a certain extent, as relations between games correlated to these sentences (more accurately, to formal expressions "corresponding" to the sentences). This suggestion seems endangered if it draws too heavily upon standard gametheoretical rules. For consider :

250 -

(1) Tous les Chinois ont une voiture, mais a'est la mime pour le monde. (Every Chinese has a c a r , but i t ' s t h e same for everyone)

tout

(2) Tout le monde a une passion, mais ae n'est pas la mime pour tout le monde. (Everybody has a passion, but it's not the same for everybody) From now on I shall use the expression ARG.(x,y) to mean that x is an argument in favour of y j by and large, it is supposed to translate the French un argument

pour,

or, better, etre

presents

oomme un argument

etre

pour •, the

subscript i reminds us that there may be different formal structures connected with argumentation. In isolation Tous les Chinois or Tout

le monde a une passion

ont une

voiture

seem to be interpreted preferentially as

V3-forms. The differences between (1) and (2) are perfectly consistent with a standard game-theoretical approach. If we stick to Jayez 1981b and Jayez 1982b, we could admit the following points : - sentences like Tous les

Chinois

ont une voiture

a representation like Vx(P(x)_j 3y(Q (y) - given the mais,

have among others,

. R(x,y))),

the ARG. relation imposes only very weak restrictions,

- the general structure is : x

MAIS

ARG^ECx) ,z)

y ARG^Efy), i z)

E(x) and E(y) designate formal expressions correlated to x and y, z is a third formal expression ; ARG, controls the game-theoretical relation between E(x) (or E(y)) and z (or Iz). - In cases like (1) or (2) the ARG

relation can be (with deliberate

over-simplification) paraphrased by : "there is at least one choice of individuals which allows successful checking of E(x) and z (or E(y) and lz)". - Here we can do Ely) ~ z. Since a Vx 3y-form can be verified by situations in which y is the same for all x or is not the same for all x, we readily get an explanation for the possibility of (1) and (2). Obviously, we do not need any change in the standard processing of quantifiers and we may retain the left-to-right order. However in many structures the point is not so clear. When game-theoretical semantics is proof-oriented, it duly stresses on the notion of "winning strategy". Suppose a game has only two opposite teams, p. and p , we say that p

has a winning strategy for an initial expression E if p. can construct

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a verification of E no matter what the moves set up by p, could look like. This amounts to saying that, in any case, p

has a way out. When such

circumstances obtain, they raise the problem of equivalence : when proving predicate tautologies, the sequence of relevant connectives and quantifiers is partly irrelevant. Furthermore, there is something not really so clear with examples like (1) or (2) : what it a successful checking of V x(P{x)

>iy(Q(y) . R(x,y))) ? In game-theoretical semantics we have two

possible answers, viz a proof (a winner strategy) or a (partial) verification. In the verification the winner has been able to find two individuals (on a stricter interpretation, one would suffice), say a and b, such that P(a)—*(Q(b) . R(a,b)) is true. Unfortunately, it would be exactly the same story if the expression was |_x(P(x) »3y(Q(y) . R(x,y))). So it seems we must endorse the proof reading and translate the ARG. relation (in a standard game) by : "there is at least one winning strategy for E(x) which allows (at least) one successful verification of z" (similarly for E(y) and iz). Even at this highly informal level, we can understand the hierarchy : for a precise expression there is a (possibly null) set of winning strategies, each winning strategy is a set of verifications, each verification being of course a particular conclusive play. The reader must realize that the selection between all possible rules for games in general depends ultimately on truth-conditions } as long as different rules do not impinge on verification, they may be reputed as equivalent, and it is only fair to admit that game-theoretical approach is deeply thruthconditional.

1.2. Classical games The reader interested in game-theoretical semantics can find valuable explanations and discussions in Hintikka 1973, Hintikka 1979a, Hintikka 1979b, Hintikka 1979c, Rantala 1975, Rantala 1976, Karttunen 1976, Lorenzen 1967, Lorenzen 1982,Iorenz 1982, Inhetveen 1982, Gunther 1982, Dummett 1978, Tennant 1979. He must be warned that the theory itself is not so simple as it seems to be (the connection between games and intuitionistic logic is philosophically difficult and many mathematical aspects need seme training). Definition 3 : Let A be a wff, K(A) will designate the main connective of A and Q(A) its main (outermost) quantifier, or the null set 0 if A is atomic

. The standard

(1) Naturally, Q(A) is the null set if A has no main quantifier, and similarly for vK(A) (at

- 252 game-theoretical game G(L,| . ,V, T , [ ) is the smallest set which includes

l-J .» { I P , . (•1 ,i» l . | Ji ftp,, \.\,n [V] ,S l(p t . \\\ ,n iv] ,U l(p2, Jvj.B

• cj,»p 2 l ^K(m^t;^MU)) . m i l ) . c),iP e »i{Ktcftu{iHC))>u))j • c),ip*,UK R(y,x))) } . Recall that

R is a partial ordering ; then T

imposes the existence of MAX(A). Consider

M(A,T ) j since L has enough constants (i.e. constant symbols), any two models of M(A,T ) with the same diagram will be elementary equivalent (since they will be isomorphic) and the converse is true. So we can think of

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"... to have the same diagram as..." as an equivalence relation on M(A,T ), and the truth of T on (M(A,T ) boils down to positing a disjunction whose each member is the representative of an equivalence class generated by this equivalence relation. We could, to some extent, maintain that such a disjunction is the "meaning" of presque.,

if we

separate "meaning" and "content", or, better, if we understand that the "content" of an expression is not independent from the potentialities the expression offers. For sure, to say presque

x is not to assert a

(possibly infinite) disjunction, because the context may well compel us to do some selection thereby reducing the number of adequate models, but it is to endorse the general structure of this reducing process. There are obviously many ways of determining a correct model (or a class of such models), and elements like presque

control some aspects of this

determination interacting with many other "controllers". Thus our assertions are not determinations themselves but, so to speak, highly interactive programs for determination. This idea is reminiscent of the "instructional" approach of Anscombre-Ducrot and of the "cognitive behaviorism" advocated by Ballmer-Brennenstuhi (see Ballmer 1979, Ballmer & Brennenstuhl 1980, Brennenstuhl 1982). However, in contradistinction with some pragmatic approaches, I detect no major sin in truthconditional semantics and I am consequently not prepared to dismiss model-theoretic considerations. As soon as we accept this notion (or equivalent ones), we are not very far from games. The general use of presque

(and many other "cognitive markers", see Jayez 1981b) is to

constrain a determination process, but there is something more subtle about constraining, something (A) cannot capture under its present interpretation. Actually, (A) is based over a mixing up : presque

does

not operate just on degrees, but on significant degrees -, this is the reason why the quasi-contradiction pinpointed in (b) is only apparent : there is no real incompatibility between admitting that something can possess a property to a given degree and admitting that this degree is at least equal to any significant degree, because, of course, not every degree is significant. To adjust more properly the "movement" metaphor, we could nearly say : if the speaker asserts that a

is presque

P, he

commits himself to asserting that, for any significant degree x of P-ness, a

is P to a degree y at least equal to x. So I propose the new

following convention :

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-L is enriched by a monadic predicate S ("significant"), -T is : jvx( .S(x) ^.;y(R(x,y) . S(y)})} U jvxVy((S(x) , R(x,y>) >S(y))| U |ix(x - MAX(A) . Vy(( -z(R(y,z) V R(z,y))) -»R(y,x)))} Let us designate these three conditions by T

,T , and T , 1,1 1>1, i respectively. T is the repetition of the condition ensuring the existence of MAX(A). T excludes for S the possibility of decrea1, * sing, and provides a link with the ordering of intensities. T

corresponds to the so-called "movement". What is the relation between T

and the more traditional approaches which use predicates suche as

"...possesses the property... at a degree..."? T

is supposed to give

a general structure, according to which the choices of significant degrees can be made ; A is a set of degrees and some (or all) of them are significant but the "number" (cardinal) of such degrees depends on the interpretation function, I, of the model. We can draw from this a first sketchy image : if P had the power of selecting a model from M(A,T ) , p

could always pick up a significant element

( i.e. a member of A which verifies S) , no matter what model p

has

chosen. Note that in the worst case MAX(A) would be the significant element. Even if this account seems too abstract, it somehow embodies the essential "fuzziness" of presque uses presque

; after all, when a speaker

he does not know, in most cases, exactly what he says ;

he does not intend to call attention to a restricted class of degrees, let alone to a special degree : primarily, he rules out any suspicion about the substantiality of some property for a given object j even if the degree which verifies S is located very high by the opponent it can be found owing to the structure of T . d) Until now, I have tried to rephrase the scale metaphor and to give it a more accurate form. Nevertheless I have used a predicate, S, whose meaning and properties are far from obvious. When urged to provide some analysis of it, there is at least one explanation avoid : I would not answer that x is

that I would carefully

not significantly different from

y if their difference is so small that it can be neglected. For "presque" is not used to make a quantitative evaluation ; there are cases where a difference of O.OOOi does matter (for some unit), and cases where it is (almost) nothing, so we must give up the hope of constructing a

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measurement device which, when fed with numerical values, would return a degree of correctness for the use of "presque". To my mind, the meaning of S has to do with the holistic notions developped in modal logic (see Gabbay 1976, Gabbay 1981, Nute 1980 for recent expositions). Suppose that P(x) is "x is P" and RP(x,y) is "x is P at a degree y", if y is such that it represents a significant degree of P-ness (or, alternatively, such that the difference between y and the degree z is such that RP(x,z)

-P(x) is not significant), then the difference between

RP(x,y) and P(x) (i.e. RP(x,z), if the decisive degree) is itself not significant for some given "domain of significance", no matter "how large" the difference between y and z appears to be. It is plainly true that there is no domain of significance per se : to assert that a difference is of no import might be a matter of causation, of emotional consequences, and so on. it is likely that linguistic exchanges set up successive domains of significance. This last proposal need some qualification, however : I put quotes around "how large" because I do not think we can completely dispense with quantitative aspects ; from the fact that the truth (or falsity) of a given proposition has no significance in a given domain it does not follow that we have a right -o say "presque P" if P(x) is false. If a book is bright yellow and this color has no consequence on its content, I am not entitled to say "this book is almost red". This difficulty is related to fuzzification (see Lakoff 1972, Bellman & Zadeh 1977) . It is true that no quantitative absolute scale can be produced, but this does not implicate that any sort of contraint is off the point. Consider (15) :

(15) C'eet presque 0.0002, 0.0001 exactement.

(?)

(It's almost 0.0002, exactly O.OOOl) (15) is odd and the minuteness of the difference cannot cure it. To (hopefully) gain some understanding, let us define a distance function d on A ; by definition of "distance function", d has the following properties : -d(x,y) £ -d (x,y) • -d(x,y) = -d(x,z) ^

0, d(y,x), 0