# Negation as Maximal Distance in Discourse Space Theory

p. 351-378

## Résumé

This paper proposes that natural-language negation is best understood as a discourse phenomenon. It is also proposed that discourse can be treated in a geometrical rather than a propositional framework. This perhaps surprising notion is based on the prevalence of metaphorically spatial meanings in language and in the description of language. In this context, standard geometry is a natural tool. In particular, Discourse Space Theory (DST) uses a vector space of three dimensions, one of which is epistemically modal and incorporates counterfactuality as a location distal to the speaker or subject, who is located at the deictic centre, i.e. the origin of the coordinate system. Counterfactual concepts are interpreted in this model as the distal point on a modal scale. It is then shown how this model can efficiently represent and integrate a variety of well known types of negation. An implied affirmative ‘background’ can be included in a simple fashion. Within the sentence, constituent negation is easy to handle in terms of the location of referents in the abstract discourse space. The inaccessibility of anaphoric antecedents in negated clauses is also implicitly accounted for in the geometric approach. Strikingly, the DST geometry can provide an elegant representation of the relationship between negated and non-negated counterfactual sentences. Finally, the capacity of DST to deal with a classic problem concerning presupposition is outlined. It is suggested, in conclusion, that these examples demonstrate the validity and potential of this framework, which applies simple coordinate geometry to the abstract concepts of ‘space’ that underlie many linguistically encoded concepts

## Plan détaillé

## Texte intégral

1The proposal in this paper is twofold: first, it is proposed that natural-language negation is best understood a discourse phenomenon; second, and perhaps more controversially, it is proposed to treat it in a geometric rather than a prepositional framework. The geometric approach that is outlined here is based on standard vector geometry, because such a formalism provides a natural system for manipulating concepts of “distance” and “direction”— concepts that are widely used in an informal way in cognitive linguistics. The question is whether negation can be handled in such a framework. Sections 1 and 2 of this paper lay the groundwork of Discourse Space Theory and section 3 utilises its inherent properties to explore the representation of negation in discourse.

# 1. FORMAL APPROACHES TO NEGATION

## 1.1. Propositional negation

2Negation in propositional calculus is a binary operation on propositions. More precisely, it is a function NEG or—which assigns the value true (T) or false (F) to a proposition (p). Here p is a variable which can take a value Τ or F (just as X might take the value 1 or 0, say):

3NEG: NEG (T) = F and NEG (F) = Τ

4or:

5(p = T) = (p = F).

6There are no degrees of negation in propositional calculus. The set theory definition of negation as complementarity is of course also binary.

7The value of p can be regarded as arbitrary if one's main concern is the definition of In formal semantics, the assignment of Τ or F is supposed to depend on truth with respect to the state of a possible world. Its assignment function is defined as a relation between p and a possible world. Here we do not need to go into all the possible objections that have been raised against the “possible world” notion. The binary logical operation described above is probably only a small part of what humans do when they do negation. If we assume that they do in fact engage with something like the binary negation operator, we have also to assume that they are engaged in cognitive activity, and once one starts to consider cognitive activity, then negation emerges as inextricably linked with other types of operation related to judgements about what is and what is not true or real for the mind/brain engaged in that activity. Linguistic expressions of numerous kinds are clear indicators of underlying cognitive actions and states, and the linguistic expressions linked with linguistic negation are many and subtle.

8There are some limitations on classical negation that are well known (cf. Jackendoff 1972: 254). That is, formal negation is unable to describe certain phenomena in natural language that by any reasonable criteria must be counted as negation. For example, sentence focus is an intrinsic part of discourse meaning. The syntax of human languages makes it possible to linguistically negate constituents of sentences (*mutatis mutandis,* to negate arguments of propositions). Where sentence focus interacts with linguistic negation, we have a phenomenon that is not only to do with relative salience but also to do with specific aspects of states of affairs in some world. In the following examples, italicisation stands for sentence stress:

(1) John didn't murder the president. (It was not John who murdered the president)

(2) John didn't murder the president. (It was not the president John murdered)

(3) John didn't murder the last president. (It was not the president who was the last president whom John murdered)

9Though Kadmon says that constituent negation is not a truthconditional effect (Kadmon 2001: 259), this might be contested. Such sentences concern the value of argument variables in a proposition. It is surely the case that in (1), for example, the normal reading includes (a) that some person x murdered the president, and (b) x does not equal John. The fact that (a) is conventionally described as a presupposition does not alter the fact that the presupposition is part of the sentence's meaning for any hearer and that the presupposition is a proposition subject to truth conditions. Truth conditions in the usual sense are not of course part of the present account, though they* are* if they are understood as a cogniser's/speaker's assumptions about what is true or real for them. Whichever way one views truth conditions, it is important to be able to account for constituent negation. One clue as to how to do it (and this is the way it is done in DST) is to note that a similar effect occurs with modal expressions (cf. Rooth 1996, reported by Kadmon 2001: 259):

(4) John probably murdered the president with a cleaver.

(5) It was the president John might have murdered with a cleaver.

10Although the effect is similar in that the modal expression scopes over a constituent, there is a semantic difference between* not, probably* and* might* that needs to be accounted for—otherwise that is, than simply using the lexical items themselves as predicates or operators in a logical notation. One would thus think such phenomena are important for logicians and formal semanticists. Yet there is no formal account of such phenomena.

## 1.2. Negation in formal pragmatics and Kamp's DRT

11Kamp and Reyle's DRT (Kamp and Reyle 1993) is an extension of truth-conditional semantics designed originally to deal with the problem of anaphora between sentences (i. e. in discourse), especially cases where anaphors appear infelicitously to link up with antecedents in negative clauses, conditional clauses, etc. Kamp's theory depends on diagrams, though these do have formal definitions. The diagrams use nested boxes. Discourse variables (x, y, z, etc.) are inserted into the “universe” of a primary or “matrix” box, the discourse representation structure (DRS). The variables correspond to what discourse participants (speakers) assert as facts, i.e. as having existential status. The variables are arguments of predicates that give them semantic content: the NP* a dog* is represented as (x, dog (x)). Then the rest of the sentence is also predicated of x: e. g. (bark (x)). There are correspondences here with the approach outlined in the present paper (DST, not to be confused with DRT) in so far as discourse referents in DST are also inserted into a space (geometrically defined, however, unlike DRT boxes).

12In Kampian DRT, a matrix box may contain subordinate boxes corresponding to denied facts, doubtful facts, temporary assumptions,* if-then* sentences, etc. There may be several degrees of embedding. However, there is no way of differentiating between a negation box, a doubt box, a* might-possibly-be-the-case* box and so on, otherwise, that is, than using the graphical conventions for designating logical negation, modal-logic possibility, etc. That is to say, in DRT negation boxes, possibility boxes, etc. can all appear at the same level of hierarchical embedding. In Discourse Space Theory, however, these non-assertoric spaces are “embedded” in a base space (cf. “matrix” box in DRT) in a principled scalar fashion derived from the notion of “distance” as used in cognitive linguistics.

13One of the strengths of Kamp's approach is that it can solve the problem of capturing the anaphoric regularities in the examples of the following type:

(6) Rodney owns a Rolls. He does notlike it.

(7)? Rodney does not own a Rolls. He likes t.

14In informal terms, Kampian DRT includes a stipulation that anaphors cannot find antecedents in subordinate DRSs that are not “accessible”. Negated sentences are boxes nested inside matrix boxes. In (6), “it” can refer “upward” to the referent “Rolls” in the matrix box. But in (7), “it” cannot refer “downwards” to the referent “Rolls” inside the subordinate negation box. The accessibility rule applies to a number of other cases, such as conditionals. This account is couched in topological terms, and the notion of “accessibility” is suggestive in the cognitive context, though such a line is not overtly pursued by Kamp. The DST account outlined in section 3.3 below aims to deal with the same problem.1

## 1.3. Mental Space Theory

15The most developed cognitive theory with relevance to negation of various kinds is Fauconnier's Mental Space Theory (Fauconnier [1985] 1994), which has been developed by other linguists investigating related issues (Sweetser 1996, Dancygier 2002, Dancygier and Sweetser 1996). Mental spaces are discrete cognitive entities that are established during the processing of discourse. The latter correspond to the processor's interpretation of a speaker's representation of what is real and what is unreal or spatially or temporally displaced. Mental spaces are set up in response to particular linguistic expressions (“space builders”), such as “in 1066”, “in Beijing”, “in* War and Peace”.* Space builders also include conditional constructions* (if),* modal expressions, counterfactual desiderative expressions, negation. There is a base or “parent” space that represents the speaker's reality space. Referents can occur inside more and can be connected across spaces. Some such links handle anaphoric connections.

16Both Kamp's Discourse Representation Theory and Fauconnier's Mental Space Theory are important for DST: there may be formal equivalences. However, the big difference is that DST makes deictic discourse centring intrinsic to the model and to all utterances: it is able to do this because of the geometric coordinate and vector space theory that underlies it.

# 2. DISCOURSE SPACE THEORY (DST)

17Discourse Space Theory is a theory of discourse processing that is fundamentally deictic. It uses geometrical notation, specifically coordinate systems and simple vector spaces, on the grounds (a) that these provide already well defined mathematical notations and (b) that there may be neuro-cognitve motivation for modelling certain mental processes in the geometrical framework (cf. Gallistel 1990, Gärdenfors 2004, O'Keefe 2003). The basis for this approach to discourse comes from Cognitive Linguistics, especially its emphasis on spatial representations and conceptual derivatives of spatial representation. DST incorporates notions of “distance” and “direction” (the ingredients for defining vectors mathematically), which have been informally invoked by cognitive linguists in order to explain a number of language and discourse phenomena (cf. especially Langacker 1991). It also owes something both to Discourse Representation Theory and to Mental Space Theory. However, the most general aim of DST is to capture the “situatedness” of human discourse.

## 2.1. Discourse Space Models: coordinate systems

18In its current form, DST uses geometric diagrams (discourse space models or DSMs) involving only three abstractly defined dimensions. DSMs aim to homologically model the conceptual structures constructed by human discourse processors in response to linguistic input. DSMs have three axes defined, as shown in Figure 1:

19The origin is the deictic centre attributed to S: we can think of “S” as standing for Speaker, but also Self, or Subject. The vertical axis (s) is derived from three-dimensional spatial representations, preserving only direction (toward or away from the deictic centre) and distance from deictic centre. The s-axis can stand also for “discourse distance”, e. g. foregrounding/backgrounding (figureground) effects and viewpoint, which are derived from visual perception-cognition.2 As is well known, evidence from many languages indicate that spatial concepts commonly map onto temporal ones. The t-axis in DSMs is also defined in terms of spatial direction and distance, as indicated in Figure 2. The most important axis for the investigation of negation is the third axis: the “modal” axis (m-axis).

20The m-axis in discourse space represents the claim that all utterances involve epistemic (and other modal) evaluations on the part of S, ranging from what is believed (or claimed) to be true, real, and “right” (in all senses). Including a modal scalar axis involves recognising a theory of modality, primarily with respect to epistemic expressions that draws on spatial concepts, especially the concept of “distance”. Figure 3 shows how DST models the symmetry between the different axes of the discourse space:

21Several authors have approached modal meanings in this way, notably Frawley (1992: 384-436), who views tense as vectorial, without giving a formal definition, and defines modality in terms that are essentially vectorial, though he does not use the term in that context. Moreover, for Frawley, negation itself is modal—and this is also how it is treated in DST. Similarly, Langacker (1991: 246) proposes that the modals can be described as contrasting with one another because they situate the process at varying distances from the speaker's position at immediate known reality.

22Fleischman 1989, Sweetser 1996, Dancygier and Sweetser 1996 also adopt the notion of “distance” to make sense of degrees of conceptual differences on a* realis-irrealis* scale. The character of the m-axis is complex and is necessarily simplified here to the following intuitive gradations:

true > probable > possible > improbable > false.

23DST thus treats S as “located” at the origin of a coordinate system representing discourse space. The proximate region around this point includes what is spatially close to S, ordering in terms of discourse referents, utterance time and temporal “closeness”, combined with the greatest degree of epistemic certainty from S's viewpoint. If S is using utterances that express less than full commitment to the truth of some proposition they are treated in DST as more “distant” from the origin in the base (i. e. S's) coordinate system. DST works on the hypothesis that negation is maximally distant to S on the m-axis. That is, various linguistic expressions (negative particles, counterfactuals) give rise to conceptualisations that are maximally distant from S in S's discourse space. In what follows, this point and the plane in which it lies is simply labelled “neg”.

24The adoption of the geometrical formalism has automatic benefits. One is the use of coordinates to “locate” discourse referents and track anaphoric relations across different parts of the space defined by the coordinate system. Further automatic benefits come along with the elementary vector geometry. A standard operation that is important for present purposes is transformation on coordinate systems.3 Transformations in this sense are mappings which shift a copy of the base axes to other positions within the initial coordinate system. Translation of axes, parallel to the base system, is important as will be seen. In addition, we shall investigate the idea that certain kinds of negation are rotations as well as translations of the base coordinate system.

## 2.2. Discourse referents and predicates

25In its current state of development, DST, like other theories of discourse uses lexical labels for referents and for predicates. Discourse referents (*Napoleon, the mountain,* etc.) are simply labelled with English words. The vectors used for relations between referents* *(*travelled, climbed,* etc.) are labelled in the same way. Discourse referents may be unlabelled: this is necessary to account for certain discourse phenomena, including negation. It is not clear at the present time whether properties* *(*blue, round, high,* etc.) should or can be represented in the DSM in the same way that relations such as* go, put, see,* are.

## 2.3. What negation is not in DST

26Negation is not the mathematical symbol “–”: this symbol is used for negative vectors. Negative vectors are not non-vectors, but vectors whose direction is reversed. This is an extremely important concept in DST because it corresponds to direction of physical motion or direction of application of physical force, and because these physical concepts have counterparts in the human conceptualisations encoded in linguistic expressions—e. g. motion, location, causation, agency, viewpoint, and other relationships.

27Nor is the mathematical symbol “0”, “zero” (or English “nought”) what is meant by negation. In fact, in DST, 0 is the reverse. The origin, denoted by positional 0, is defined as the cognitive centre of S, the speaker (or cognitive subject). As argued above, spatial, temporal and modal proximity converge on epistemic certainty or reality for the speaker. In this sense, 0 is plenitude rather than emptiness.

# 3. NEGATION IN DISCOURSE SPACE THEORY

## 3.1. The denied background

28Many writers have noted that negation implies something asserted (cf. Bergson 1909, Horn 1989). How can we capture the notion that negative sentences somehow give rise to cognitive representations in which a “background” assertion is denied? We can think of this phenomenon as follows. An assertion that is true in one system (say, another speaker's mind) is not true with respect to another system (say, the mind of the speaker modelled in a DSM). The former is “contained” in the latter. DST uses coordinate transformation, a standard mathematical concept, to capture this cognitive double vision. The negative particle (and the same is true, as we shall see for modal expressions and conditionals) trigger a copying of S's* realis* axis to the distal end of S's m-axis—the locations which, as argued above, can be thought of as corresponding to counterfactuality (maximally* irrealis)* with respect to S's base coordinate system.

29Consider the simple negative sentence:

(8) Bill didn't paint the house.

30Simple logical negation does not capture the cognitive complexity. DST attempts to do so in the Discourse Space Model (DSM) given as Figure 4. In Figure 4, the denied background is shown as a set of translated coordinates (dashed lines), whose origin has coordinates on the base (i. e. S's) coordinates at S k somewhat distal to S's deictic centre (origin), with t-axis the same as S's, but with the m-axis coordinate* maximally distant* to S's origin. The coordinates for the origin of the embedded coordinate system can be stated as < s k, 0, neg >. Thus, the denied proposition is true with respect to the embedded coordinate system, but simultaneously false with respect to the base (S's) coordinate system—which is what we want to show. The event is in the past relative both to S and the embedded projection.

31What about the discourse referents? We have a coordinate, sk, on S's s-axis that locates the embedded origin of the embedded coordinate system. This we might call S', but it remains otherwise unlabelled.4 The discourse referents Bill and the house are real for S, and hence have coordinate m = 0; they project onto the coordinates of S', where they are m'= 0, i. e. also real for S', as we require. But the vector labelled paint has the value m = 0 (i. e. real) in the system of S', while it has m = neg (i. e. not real) in the base system of S, which is also what we require.

32The DST approach thus automatically makes sense of the notion that negated sentences have a positive “background”. The background is the shifted set of coordinates, which is distal to the speaker's reality plane.

## 3.2. Constituent negation and focus

33The problem of constituent focus in relation to negation (see above section REF) can be handled in a relatively simple fashion because of the three dimensions of DSMs. We will continue to assume that the basic geometry of negation in a DSM is as in Figure 4 above in order to explore how the framework might handle the following sentences:

(9) Rodney doesn't own a Rolls.

(10) Rodney doesn't own a Rolls.

(11) Rodney doesn't own a Rolls.

34In Figure 5, from S's viewpoint, Rodney is real and Rodney's owning is real (both have m = 0). It is also real for S that Rodney owns some object (also m = 0). The object owned is real for S, but its properties are unspecified.5 For the translated coordinate system (the one that belongs to S'), however, the Rolls is real, while it is simultaneously maximally* irrealis* (coordinate m = neg) for S. The discourse referent* Rodney* has coordinates also on the embedded axes, as is required, as well as a vector to the owned entity.6 It is possible to capture these effects of negation because DST allows for coordinate transformation, treats discourse referents and their property labels as separate items, and automatically allows for their coordinates to correspond in different parts of the space. This corresponds to the variables and “conditions” in Kamp's discourse theory, but in DST all variables, or discourse referents, are intrinsically “located” relative to the speaker's, or embedded speaker's, deictic centre.

35Sentences (10) and (11) can be handled in a similar fashion, as shown in Figure 6:

36In sentence (10), somebody owns a Rolls, but from S's viewpoint, the identification of this referent with Rodney is denied: hence, Rodney appears in S's negation plane, while simultaneously appearing in the* realis* plane for the embedded the coordinate system of S'. In sentence (11), Rodney and a Rolls are realis for S, and have corresponding coordinates for the embedded axes. But the identification of the relation (vector) between them is different. For the Speaker, there is a relationship but S's sentence does not name this relation. In the embedded speaker's system, the relation is truly named* own,* and simultaneously denied in the base system of S.

## 3.3. Negation and anaphora

37In this section we consider sentences of a type often considered in the literature (e.g. Kamp 1993: 99-108), exemplified here in (12) and (13):

(12) Rodney owns a Rolls. He doesn't like it.

(13) Rodney doesn't own a Rolls. * He likes it.

38Figure 7 shows the DSMs for these sentences. In the second sentence of (12), there does not seem to be a sense in which S is denying an assertion by some S'. Rather, it appears to be in the base epistemic space of S. In the case of (13), however, as we have claimed above (section 4.1), the first sentence appears to deny a background assertion that requires the appearance of shifted axes. In the DSM for (12), coordinates and property labels are first given in S's discourse space for* Rodney* and* Rolls* at m = 0. In the same plane (the* realis* plane for S), there is also a vector showing the direction and (relative) length for* own.**7* The pronouns* he* and* it* are the arguments of a proposition whose predicate is the vector* like*-a predicate that is counterfactual, i. e. denied, from S's viewpoint, hence located at the distal end of the S's m-axis. The coordinate system of the discourse space ensures that the coordinates for* he* and* it* in S's* irrealis* plane are the same as for* Rodney* and* Rolls,* respectively, in S's* realis* plane.

39Consider now the discourse space as affected by the linguistic input of sentences (13) in the second diagram in Figure 7. The first sentence of (13), at least in my reading, does seem to be denying a prior assertion. Read with unmarked intonation contour it has stress on* doesn't.* The effect appears to be that while* Rodney* is real in S's system,* own a Rolls* is not. Thus* own a Rolls* is shown in the DSM as simultaneously* realis* in the discourse space of some prior speaker S'and* irrealis* in the base system of speaker S. That is, the axis system of S'is shifted, within the discourse space of S to the distal end of S's m axis. The real Rodney (so to speak) has a labelled coordinate in S's system, and a projected coordinate in the system of S'. The Rolls is not real, and is labelled only in the denied system.

40What we can see from this is that the coordinates for the labelled* realis* discourse referents can project in the direction of the negation plane distal to S (the case for (12)), whereas the labelled* Irrealis* referent cannot project “backwards” to S's reality from out of the negative plane. So far, this corresponds more or less to the accounts given by Kamp and Reyle and Jackendoff. There are, however, two ways in which the present account perhaps could go a little further towards a more detailed and principled account.

41First, the m-axis is scalar, in the fashion suggested in section 3.1. This inherent feature of DST enables us to make sense of the point made by Jackendoff (2002: 401), namely that the “may of possibility”, the modal future tense auxiliary* will,* and questions (which inherently do not make assertions), have an effect on the possibilities of anaphoric reference. Similar points are made in Fauconnnier (1994) and are implicit in Kamp and Ryle (1993). In DST these observations are implicit in the claim that the m-axis is a scale from epistemic certainty to counterfactuality, relative to the viewpoint of S. The anaphoric phenomena in question are illustrated below:

Rodney probably owns a Rolls. * He likes it

Rodney possibly owns a Rolls. * He likes it

Rodney may own a Rolls. * He likes it

Rodney might not own a Rolls. * He likes it

Rodney probably doesn't own a Rolls. * He likes it

Rodney cannot possibly own a Rolls. * He likes it.

42These examples show that we cannot map “back”* from* epistemic points on m>0 to the* realis* plane of S. It also seems to be the case that we cannot map back from the maximally distal negation plane* to* any point closer to m<neg, as suggested by the following:

Rodney doesn't own a Rolls. * He possibly doesn't like it

Rodney doesn't own a Rolls. * He probably doesn't like it

Rodney doesn't own a Rolls. * He mightnot like it

Rodney doesn't own a Rolls. * He may like it

Rodney doesn't own a Rolls. * Possibly he likes t.

Rodney doesn't own a Rolls. * Probably he likes it

43Second, the DST account follows automatically, it might be argued, from the axioms of the vector space mathematics. It is certainly not necessary to introduce ad hoc devices such as Jackendoff s “grounding arrow”, while directionality and distance, in terms of which the DST account is couched, are inherent, indeed necessary elements of the vector space formalism.

44There is a further possible way of deriving the unidirectionality of anaphoric reference from the geometry, which I only sketch here. To this end, we first have to argue that the shifted axis system is in fact not three-dimensional but two-dimensional. The grounds for this might be established by investigating further examples such as (14) and (15):

(14) * Rodney does not might own a Rolls.

(15) Rodney might not own a Rolls.

45Sentence (15) can be seen as involving a point on S's m-axis that is beyond the epistemic mid-point, as are expressions like* possibly doesn't, probably doesn't, definitely doesn't,* etc. which tend towards* doesn't.* That is,* might* is relative to S's system, not to the denied embedded system. However, sentence (14), if it were possible, would require an embedded system such that the* might* concept is simultaneously true for the embedded system of S'but negated in the base system of S. But it is not a possible sentence. So we could provisionally conclude that the system embedded at the negative point for S is actually a planar vector space not a three-dimensional one. Now, if this is assumed, the result may well fall out from elementary linear algebra, though we can only engage in handwaving at this point. The shifting of axes is a linear transformation, and certain linear transformations have the property of “losing” a dimension. In such cases, one cannot map back into the initial domain. Given the points made above about the epistemic scale of the m-axis, this argument may generalise to any plane in S's space with m > 0. Summarising this line of thought, we could propose that negation is a function that maps from the three dimensional space of S into a two-dimension negation plane.

46One could also search for a cognitive explanation. It is not helpful for human survival to map questionable representations of reality into real ones upon which one wants to be able to act safely. The converse, however, is not so risky. Viewing negation in this light—that is, as located the end point on a spectrum of epistemic distancing—leads us to another grammatical construction that is closely bound up with the human ability to entertain the unreal.

## 3.4. Counterfactual conditionals

47Conditional constructions, like the negative and epistemic distancing expressions touched upon above, seem to have the job of decoupling mental representations triggered by language input from the current environment that a human cogniser accepts as real and true (on “decoupling” see Cosmides and Tooby, 2000). Like the epistemic gradient of the modal expressions, conditional sentences can be modelled in DST in terms of distance from S's deictic centre. One of the remarkable things about conditional constructions is that, in certain forms marked by the word* if* and specific tense choices, a syntactically positive clause can be negative (and conversely). That is, a clause can be negative without even containing a negative particle.

48Writers on conditional sentences agree (e. g. Fleischman 1989, Sweetser 1996, Sweetser and Dancygier 1997, Michaelis 1998) that conditional sentences involve degrees of epistemic “distancing”.8 The spatial metaphors suggest that the geometric approach of DST might be appropriate.

49*If* constitutes a function that translates a copy of S's coordinate system away from S's reality plane to various points on S's m-axis. How far it shifts depends on the tense forms together with verb meaning and contextual factors, as discussed in the previous section. The second set of axes is a “new reality” space, similar to Fauconnier's mental space, except that “distance” and “direction” and deictic centring are built into the fundamental formalism of DST. Within the new set of axes, propositions that are dependent on* if* are represented. They are simultaneously “iffy” with respect to S's initial coordinate system, but “real” within the new system. That is, all representations stimulated by conditional sentences are relative to 0', the origin of the shifted axes. In the new space, anaphors find their antecedents and further embedded coordinate systems (“spaces”) can be set up.

50Examples (16) to (20) represent the scalar possibilities of conditional sentences:

(16) If John goes to the party, he will see Sarah.

(17) If John went to the party, he would see Sarah.

(18) If you looked younger, you'd get a promotion (Michaelis 1998: 146)

(19) If John had gone to the party, he would have seen Sarah.

(20) If John had had a free day tomorrow, he would have gone to the party.

51Figure 8 models two examples that give rise to embedded coordinate systems whose origins are at different points on the epistemic m-axis: (16) is relatively close to and (17) relatively farther from the base system of S. The morphological tense of the verbs in (16) and (17) is not at issue here; what the DSM represents is conceptual structures evoked by what have become conventional meanings signalled by particular tense forms in the//construction together with situational factors that we do not need to specify here.

52The coordinates for John, Sarah and the party are labelled in S's base coordinate system, but the vectors representing* go* and* see* are located relatively to S's axes. Thus in (16), for example,* John goes to the party* would be “real” (i.e. at m'= 0) in the if-space but only “likely” relative to S's reality plane; in (17) the vector would again be “real” within its own embedded coordinate system, but further still from S on S's m-axis. (For clarity the vectors are not shown in Figure 8)

53We can also see that the new set of embedded//-axes is a full 3-dimensional system. The reason for this is clear from the modal possibilities that are relative to the conditional* if* illustrated in (16'):

(16') If John goes to the party, he will/might/probably will... will not see Sarah.

54Conditional sentences whose apodisis includes any point along the epistemic scale that we have postulated for the m-axis are grammatical. Negation in either the protasis or apodosis, or both, also has to be included. By way of illustration of one of the myriad possibilities, Figure 9 shows how axis shift would deal with one case, the case in which there is negation in the apodosis of a conditional sentence:

55The speaker's axes shift to the appropriate point on* m;* John, the party and Sarah remain at the coordinates on s at m = 0, but the relationships and negations of them are relativised to the//-point on the shifted m axis. The proposition of the protasis corresponding to* John goes to the party* is represented by a vector located relative to the embedded axes. The proposition of the apodosis,* he will not see Sarah,* is represented by a vector that is in the negative plane relative to the same embedded axes. In general, the proposition of the apodosis is represented as located at a time relatively later than that of the protasis. In the same fashion, we can model sentences like (16') but which have a modal expression in the apodosis instead of the negative particle—for example:* If John goes to the party, he will probably see Sarah.* In this case, the new axes would be situated as in Figure 10, but the vector for* John see Sarah* will have a coordinate on the m-axis closer to the embedded origin.

56Interpreting the protasis of (17) as expressing a less likely eventuality than that of (16), the DSM would depict the origin of the embedded coordinate system as further away from the origin of the base system than is the case for (16), and the same remarks concerning the use of the embedded m-axis apply.

57The geometric operation we are using here is cognitively motivated. The new set of axes is relative to S's reality plane, i. e. the base coordinate system. But the new set of axes provides a discourse space—always anchored to an epistemically questionable point on m—in which* irrealis* discourses can be set up.

58Let us consider now the counterfactual representations arising in (18), (19) and (20). Assuming contexts in which these three examples have their full counterfactual reading, the differences between them (with respect to the question at issue) is one of time placement relative to S's coordinate system. The embedded systems each have coordinates m = neg (i. e. maximally distant from S); the t-and s-axes coincide with those of S. This means that the vectors occurring in the DSMs for (18), (19) and (20) all have coordinates m = neg, but for (18) t = 0, for (19) t < 0 and for (20) t > 0. Figure 10 proposes one way in which we might construct a DSM for (19):

Figure 10. Counterfactuals: First approximation.

59(19) If John had gone to the party, he would have seen Sarah

60As before, the embedded m-axis is required for cases where the verb tense of the apodosis evokes epistemic distancing of one kind and another. However, Figure 10 may not be optimal. For we also need to take account of the fact that (19) illustrates polarity reversal. Let us consider its counterpart in (21):

(21) If John had not gone to the party, he would not have seen Sarah.

61One's experience of interpreting such sentences, in context and out of context, is that they produce two simultaneous representations: one in which John did in fact go to the party (the* real is* representation) and one in which he did not (the counterfactual one). Pragmatically, these language-induced cognitions presumably have many functions. For example, it might be desirable to justify the doing of some action by evoking the consequences of not doing it—in which case, what is required is to set up an imagined contrary world and assert consequences that are plausible. The simultaneity phenomenon here is crucial. It surely cannot be captured in classical logical form. Yet it is surely a distinctive feature of human cognition, and perhaps uniquely supported by linguistic manipulation. What is needed is a formal system that can somehow naturally define a homologous structure.

62Throughout our discussion of conditionals we have used the concept of transformation of axes, an elementary mathematical operation on coordinate systems. In particular we have utilised a form of translation of axes. But other transformations can be defined. The one that fits our needs for representing negative counterfactual sentences is a reflection of the base coordinate system to the distal (negative) about the plane bisecting the mid-point on the m-axis, as shown in Figure 11.

Figure 11. Counterfactuals: second approximation.

63(21) If John had not gone to the party, he would not have seen Sarah

64Figure 11 assumes that the discourse referents are first defined in S's reality plane, i.e. as proper names and definite phrases they presuppose existing antecedents. The vectors* go* and* see* lie in the negation plane of the embedded* if-system* and in S's reality plane in S's system. In other words, from the point of view, so to speak, of the//-sentence, the events of going and seeing did not happen, but from S's point of view they did happen.

65Sentence (19) can of course be represented in the same diagram, by locating the vectors at m'= 0, which means that from the viewpoint of the//-sentence the event did happen* (John had gone),* but simultaneously, in S's perspective, it did not. In general, for counterfactual conditional sentences we need the translation-plusrotation transformation of the base coordinates. We can also see that polarity reversal, as instanced by counterfactual conditionals, can be defined in the present framework as a symmetrical reflection. It is highly likely that this holds for other types of negative polarity expressions. The scales examined in detail by Horn (1989: 231) all have the characteristic that two entailed series of expressions are mirror images of one another, both series being superimposed on a linear scale—which is the property that is incorporated in the DST spaces.

## 3.5. Metalinguistic negation

66In discussing the now well known problem of negation and presupposition, illustrated in (22) and (23) below, Kadmon (2001: 145-150) seems to come to the conclusion (p. 149) that it is necessary to distinguish between logical and metalinguistic negation:

(22) The king of France is not bald

.(23) The king of France is not bald—there is no king of France!

67In fact, in order to describe the latter, Kadmon has to refer to “utterances” and “the speaker”. However, formal pragmatics is wedded to truth conditional semantics. This means that a wedge is driven between logical negation, which has a logical formulation, and metalinguistic negation, for which Kadmon offers no formal account. So we end up not only with no formal account of metalinguistic negation but with no unified account of the two types. The following is a cautious proposal to resolve this problem.

68As already said, DST treats all negation as metarepresentation, along with other conceptualisations on m>0. The type of negation that Horn, Kadmon and others call “metalinguistic” is in fact an epiphenomenon of the discursive nature of language. If the disembodied, decontextualised examples that have been analysed since Russell are recognised as utterances in context (and maybe also if it is recognised that the design of language, the language module itself, incorporates the situated usability of language), then what is usually called metalinguistic negation re-emerges as discursive negation. This means that the infamous sentence (22) is treated in DST as an utterance in a ghost dialogue—essentially what was proposed in section 3.1. above.

69Because DST models mental representations relative to a speaker's cognition of the world, truth-conditional semantics does not get in the way of explaining the meaning of (22). The DSM simply models the presupposition as part of the speaker's realis plane. The problem of capturing (23) in a DSM is more complicated, but it can be done using the same principles, thus offering a unified theory of both “logical” and metalinguistic negation. Let us start with the geometric DSM for (23) (Figure 12).

70We take it that specific features of the syntax and intonation of (23) have a particular function in a discourse continuum. In Fauconnier's terms, these features (e. g. the negative form is no and the stress on is) are “space-builders”. Further, these features make it clear that (23) is part of an ongoing discourse, say, a dialogue with another speaker, S', who has uttered, say, “the king of France is bald”. (Note that all utterances are implicitly situated in discourse in DST.) S'is located relatively distal to S on S's s-axis. In DST terms, the features mentioned set up a second coordinate system, whose origin has a coordinate for S'on s, but is located at the most distal, i. e. counterfactual position on the m-axis, while t it is aligned for both S and S'. In other words, S “s truth centre is counterfactual with respect to S's coordinates. France is presupposed as real in S's coordinate system: this fact is captured in the DSM by its coordinates, which have m = 0. Although it is not crucial to the present argument, it should be explained that Figure 12 treats” king of'in a localist fashion: the vector joining France and king “locates” king with respect to the landmark France (cf. ownership). The coordinates for “king” are at m = 0 for S'(i. e. France truly has a king, for S') but are counterfactual for S, as required. I have not here indicated the property ascription “is bald”, but the point at issue is not affected.9

Figure 12. Metalinguistic negation.

71(23) The king of France is not bald—there isn't a king of France!

72The advantages of this approach are as follows. First, we can capture some of the traditional observations concerning (23). The most obvious of these is that (23) stands for S's judgement about another judgement (cf. Bergson, 1909: 312): i.e. it is metarepresentational. It is often said that the presupposition, viz. the proposition* there currently exists a king of France,* “disappears” in contexts such as (23). In fact, it does not “disappear”; this metaphorical notion is inappropriate. The proposition remains, but is categorised as counterfactual in S's discourse space (coordinate system). Second, we can capture some detail that earlier accounts neglect, namely, the fact that while* king* has two conflicting interpretations in the complex DSM,* France* does not.

73Second, the axis shift operation comes with the mathematical apparatus, and is naturally motivated: it is a type of viewpoint shift. It is this inherent property of DST that enables it to provide a unified account of “logical” negation and “metalinguistic” negation. The latter is induced by transformation of S's coordinates. The notion of the “disappearance” of presuppositions, which appears in the traditional truth-conditional accounts, actually does not arise.

# 4. CONCLUSION

74The aim of this paper has been to seek a formal model of discourse that can,* inter alia,* incorporate negation. The fundamental claim has been that spatial cognition is re-represented in temporal and epistemic dimensions. If this is so, then it makes sense to utilise a geometric model, specifically a vector space that stands for abstract discourse space. This paper has tried to show that Discourse Space Theory can model negation as a distal epistemic region in discourse space. It has also tried to demonstrate that certain standard properties and operations of vector geometry provide a naturalistic account of certain types of negation in discourse. Such geometric formalisms may indeed be natural in the sense that they correspond with the nervous system's basic method of building on cognition of the physical environment.

## Bibliographie

Des DOI sont automatiquement ajoutés aux références bibliographiques par Bilbo, l’outil d’annotation bibliographique d’OpenEdition. Ces références bibliographiques peuvent être téléchargées dans les formats APA, Chicago et MLA.

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*English Language and Linguistics*(Vols. 6, Issues 2, pp. 347-377). Cambridge University Press (CUP). https://doi.org/10.1017/s1360674302000278

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## Notes de bas de page

1 Jackendoff (2002: 315, 394, 402) also claims affinity with Kamp's DRT and has a conceptualist version of non-accessibility.

2 DSMs model* discourse* space, not full conceptualisation of physical space. A complete model of discourse conceptualisation would of course include the three physical spatial dimensions.

3 Axis translations are also central to understanding the viewpoint phenomena commented on extensively by Langacker and others. Viewpoint is essentially a spatial phenomenon that can be modelled by vectors and coordinate transformations.

4 In actual discourse, S would probably have an actual S'in mind, whose positive utterance the negated sentence is denying.

5 In fact, the sentence would probably be understood as meaning that Rodney owns a car, or at least a vehicle, of some sort, so the discourse referent in question could be considered to carry at least (he property [+ car] or something of the sort.

6 There is one problematic detail which I do not attempt to resolve here. The effect of translating the axes means that the coordinates on the embedded s-axis are relatively closer to the embedded 0', the deictic centre of the speaker whose representation is denied.

7 Length of vectors needs to be defined in some conceptually relevant fashion. Possession is defined in localist terms in DST. Direction is given by the vector whose tail is at the landmark (the owner) and whose head is at the object owned. Distance requires further discussion but a case can be made for working with unit vectors defined in terms of peripersonal space.

8 However, Michaelis (1998) challenges Fleischman's (1989) view that the protasis of English conditionals map onto a* realis-irrealis* scale according to the tense of the verbs. In Michaelis's analysis the protases of counterfactual conditionals and hypothetical conditionals do not give rise to gradations of a single epistemic scale. Rather, degree of likelihood and counterfactuality are viewed as distinct but overlapping semantic categories. In terms of the general argument outlined here, the two past tenses (past perfect and past) can both refer to counterfactual (relative to S) representations up to the present Past, however, can refer to a medial-epistemic possibility in the future, a lesser degree of epistemic certainty than is expressed by the present. Michaelis's point about the semantic duality of past is therefore maintained. However, we do not abandon the notion of an epistemic scale that has counterfactual representations triggered by certain conditional sentences at its distal maximum relative to S.

9 It could be represented by another vector, pointing from* king* to the property* bald.*

## Auteur

**Paul A. Chilton**

Received his first degree and doctorate from the University of Oxford. He is currently professor of Linguistics at the University of East Anglia. His varied publications include work on French Renaissance literature and translation from French, as well as work in the analysis of political discourse and in cognitive linguistics. He is now working on spatial representation and the use of abstract spaces in a discourse-based model of grammatical constructions

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