6.1 Counterfactuality

Counterfactuals have been intensively studied and puzzled over by philosophers of language and science in the last half century, going back at least to David Hume and J. S. Mill. Philosophers are preoccupied with the conditions under which if-sentences in English, which do not assert truth, would in fact be true. Nelson Goodman (Reference Goodman1947) famously discusses the example ‘if the match had been scratched, it would have lighted’. The problem that can arise with such a sentence emerges if one focuses on the causal relation between the if-clause and the main clause. For example, to convince oneself that scratching the match would cause its lighting, other conditions have to be assumed: e.g. that the match is dry, that there is sufficient oxygen in the appropriate region of space, etc. The question is how many other things are needed to guarantee the truth of the sentence and how they are all to be computed. The work of David Lewis (especially Lewis Reference Lewis1973) sought to resolve such questions. Lewis's proposals build on the important notion of possible worlds and ‘closeness’ of different possible worlds to actuality. A much more general way to state the puzzle might be: counterfactuals are by definition not true, so how can they be used, when truth and factuality are required? An even more general way of stating the puzzle might be: given the empirical fact that humans do use counterfactual utterances, amongst which counterfactual if-sentences are perhaps prototypical, how, why, and for what purposes, do humans use them? This has been the approach of psychologists (e.g. Johnson-Laird Reference Johnson-Laird1983, Byrne Reference Byrne2007). It has also been the approach more recently of cognitive science and cognitive linguistics. Fauconnier and Turner (Reference Fauconnier and Turner2002) propose a theory of the way counterfactual thinking works, in language and outside of language, as well as demonstrating the countless manifestations of counterfactual thinking and communicating in science, philosophy and everyday life. Here, however, I focus on a more narrowly linguistic question: how can we analytically describe the fact that counterfactual conceptualisations – in effect thinking that something is not the case – spring from a sentence like Goodman's ‘if the match had been scratched, it would have lighted’ which has no explicit negative word like not in it? This may seem a less important question than the philosophical ones, and maybe that is the case. Nonetheless, language is the sine qua non of philosophy and it is rational and even important to ask how it works. Moreover, there are plainly ways in which the approaches converge in common concern.

Counterfactual thinking is in fact activated and communicated by many different linguistic forms but counterfactual if-sentences are of particular interest because they are relatively complex and variable linguistic structures that facilitate not only thinking and communicating about things and events that do not exist or have not taken place, or do not yet exist or have not yet taken place, they are also, as philosophers have been above all aware, instruments for thinking about causal relations. In the realm of social psychology they are, in their contexts, a means to setting up an intersubjectively shared mental space, on which joint attention is focused, for the pursuit of practical reasoning about the past as well as about future action. Fauconnier and Turner (Reference Fauconnier and Turner2002: 221, for example) describe such uses minutely in terms of Blending Theory. What Blending Theory seeks to tell us about is the integration of different representations of the world prompted by particular sentences, including counterfactual if-sentences, e.g. ‘if President Clinton were the Titanic, the iceberg would sink’. But it does not tell us how the specific grammatical construction, the if-sentence that can give rise to the blending of different spaces can have the form it has. The account I will outline below shows that the seemingly paradoxical form of counterfactual conditionals follows naturally from the fundamental premises in DST concerning the fundamental deictic space. In the Clinton–Titanic example, the blending of factuality and counterfactuality comes from the conceptual frames associated not only with the grammatical frame but also with the conceptual frames associated with lexical items Clinton, Titanic, iceberg and sink. Blending Theory demonstrates over many cases how this works. DST shows something in a way more meagre: the basic structure of counterfactual conceptualisation itself and its dependence on abstract geometric transformations.

6.2 If-sentences and counterfactual conceptions

Grammatical constructions have meaning. If-sentences (conditional sentences) in general are sometimes said to be ‘counterfactual’. This is true to the extent that there is a range of ways in which it is possible for a proposition to be ‘counterfactual’, ranging from assertions of probability, through possibility, to negation of some other proposition. Conditional sentences, that is two-clause sentences consisting of an if-clause (the antecedent clause or traditionally protasis) and a consequent clause (traditionally apodosis, syntactically the main clause), yield mental representations that lie somewhere along the m-axis. They may be thought of as having the potential to express degrees of irreality, ranging from the not-quite-certain about the near future to that which is counter to fact. In speaking of these degrees, I am not thinking of absolute truth but the reality as intended to be communicated by a speaker to a hearer, given the conventional structures of the language they are using and the range of conceptualisations that can conventionally be associated with these structures. This is not to say that particular forms, say of the English conditional constructions, are uniquely associated with particular conceptualisations. It is well known that the particular meanings that conditional sentences can give rise to depend on a number of factors – the meaning of if, the tense of the verb, the semantics of the particular verb, expected intentions of the speaker and other contextual factors. The tenses of verbs in conditional sentences do not simply denote times; they are often only indirectly related to times to which a speaker may wish to refer. These points concerning the meaning of tenses in conditional constructions have been well made by, among other scholars, Fauconnier (Reference Fauconnier1994), Dancygier (Reference Dancygier1998), Dancygier and Sweetser (Reference Dancygier and Sweetser2005) and Byrne (Reference Byrne2007).¹

The past perfect can be used in a conditional sentence to refer to the speaker's present, past or future. One can say, for instance, if John had been here now, he would have been/would be astonished but one can also say if John had been here last week, he would have been astonished. And one can say if John had been coming tomorrow, we would all have been very pleased. A particular time adverbial can then determine how the time reference relative to the interlocutors is interpreted. The use of were, and the simple past tense, also allows variation in time reference, since we can say if John were here now and if John were here next week – although one cannot use this form to refer to the past (?if John were here last week). The simple past tense form likewise allows varying time reference: if John came yesterday, if John arrived at this moment, and if John turned up next week are simple past tense forms referring to different times in the speaker's reference frame.

The degree of certainty or uncertainty itself is not wholly predictable from the tense form, though there is a tendency for past tense and past perfect tense forms, denoting relatively more ‘remote’ past times, to correlate with more ‘remote’ epistemic interpretations of the verb meaning in the antecedent clause of conditional constructions. Thus, for example, in making sense of If John knew the answer, we'd be pleased, the hearer has to judge whether the speaker intends a counterfactual to be understood or whether John may perhaps know the answer. However, in saying if John had known the answer, we'd have been pleased, it is more difficult – though not impossible – to imagine a context in which, the speaker intends to communicate anything other than a counterfactual, i.e. that John did (or perhaps does) not know the answer. Even in the case of the subjunctive form were, contextual expectations of the interlocutors determine the epistemic interpretation. It is clear that if pigs had wings, they'd fly is counterfactual, given our knowledge of pigs. But in if that man were a doctor, he'd know what to do, the interpretation depends on what is known or not known about that man. Furthermore, if a past perfect conditional, for example, is embedded inside a context that itself is understood as counterfactual, then the embedded space is not necessarily read as counterfactual: If John had gone to the party he would have met Sarah. If he had found her attractive, he would have flirted with her. The second sentence can be understood as a possibility rather than an assertion that he did not flirt with her – the scenario of John at the party is set up in the first sentence and in the second sentence a possible happening in that scenario is mooted. A similar case would be John came home at midnight. If he had been to the party, someone saw him there. Taken as a whole the two sentences are counterfactual, are in a counterfactual ‘space’, but the second is not counterfactual with respect to the first (cf. analyses of such contexts by Dancygier and Sweetser Reference Dancygier and Sweetser2005: 73–6).

However, it is important to observe that the varying tense forms in if-constructions do not allow any interpretation but steer the construing hearer in limited directions. Dancygier and Sweetser (Reference Dancygier and Sweetser2005: 76–7) note that English speakers, despite the crucial role of context, nonetheless have a strong sense that decontextualised past perfect – e.g. if John had gone to the partypresupposes the counterfactual status of the propositional content of the sentence, an unuttered sentence John did not go to the party. A counterfactual reading may be said to be the default reading for the past perfect tense form in an if-construction. In the case of the simple past tense form, a default epistemic distancing effect is also found. For example, if John came to the party, he would meet Sarah is naturally understood as a non-committal assertion of a possible event in the future. Degrees of past-ness apparently, in the scope of if, get interpreted in terms of degree of epistemic certainty, unless contextual factors apply. The correlation of temporal distance in the past with epistemic distance looks like a natural cognitive association.

The use of the ‘distance’ metaphor in cognitive accounts is different from Lewis's, though perhaps ultimately connected cognitively. Lewis (Reference Lewis1973), following Stalnaker (Reference Stalnaker and Rescher1968), was working with the theory of possible worlds and speaks of the ‘closest possible world’ in the sense of ‘most similar possible world’ to the actual world except that the antecedent of the if-sentence is true in it. This notion is then held to deal with the problem of when counterfactual sentences are true: they are true if antecedent and consequent are true in such a world. This theory and its subsequent developments are an extraordinarily rich achievement. Its general approach is, however, different in nature and purpose from the cognitive-linguistic approach. The philosophers' problem with counterfactuals arises from seeking to translate them into truth-conditional bi-valued propositions and further treating the relation between antecedent and consequent parts as a causal relation that has to be matched with a plausible account of causation. The question we are posing here is simpler. Counterfactual expressions in human languages give rise to conceptualisations: what kind of conceptualisations are they? What are their properties? What do they tell us about the human mind?²

If there is an overlap between the possible-worlds approach in philosophy and the mental-space approach in cognitive semantics it is not in terms of true or false or degrees of similarity. Rather, the scalar notions developed in cognitive linguistics concern cognitive scale, specifically a scale of epistemic judgement concerning the speaker's assessment of degrees of likelihood or real-ness. For linguists investigating conditional sentences the notion irrealis concerns kinds of conceptualisation not ‘possible worlds’ in the sense of truth-conditional semantics. The term ‘distance’ (e.g. Fleischmann Reference Fleischmann1989) is a spatial metaphor for talking about degrees of epistemic scalarity. Linguists working with ‘mental spaces’ (Fauconnier Reference Fauconnier1994) are also working with a quasi-spatial scale according to which some conceptualisations, phenomenologically, ‘feel closer’, some ‘feel more remote’ (Sweetser Reference Sweetser1990, Reference Sweetser, Fauconnier and Sweetser1996, Langacker Reference Langacker1991, Werth Reference Werth, Athanasiadou and Dirven1997a, Reference Werth, Nuyts and Pederson1997b, Reference Werth1999, Dancygier Reference Dancygier1998, Reference Dancygier2002, Dancygier and Sweetser Reference Dancygier and Sweetser2005; see also Lyons Reference Lyons1977: 718–19, 796 n.1). An important assumption is that types of conditional sentence can be arranged, at least intuitively, in a scalar fashion in terms of their epistemic distance from the speaker. Counterfactual conditionals could, then, be defined as the furthest ‘possible world’ (‘mental space’ in cognitivist terminology) relative to the ‘position’ of the speaker. An alternative, but not inconsistent, definition of counterfactuality in cognitive semantics is ‘forced incompatibility between spaces’ (Fauconnier Reference Fauconnier1994: 109, Fauconnier and Turner Reference Fauconnier and Turner2002: 230), which can be understood as factual inconsistency, in the speaker's consciousness, between one mental space and another. What also seems to be essential to counterfactual conceptualisation is the way speakers (and thinkers) entertain two incompatible conceptualisations apparently simultaneously (cf. also Byrne Reference Byrne2007: 34–40 on ‘dual-possibility ideas’).³ Blending theory (Fauconnier and Turner Reference Fauconnier and Turner2002) gives an account of simultaneous conceptualisation of incompatible spaces in terms of blending theory. The model I propose below takes account of these insights, with the aim of showing how the geometric properties of the deictic space as set up in DST can integrate the sense of contrariness and the simultaneity that is incorporated in the lexico-grammatical details of conditional sentences interpreted as counterfactuals.

In the exploration below, it is assumed that if is a cognitive operator that combines with the tense of the verb in the antecedent clause, the semantics of the verb used, interlocutors' expectations and other contextual factors. Without speculating too far into the semantics of if, we can say that is inherently modalising: any expression in its scope is going to be some ‘distance’ along the m-axis. Such linguistic events give rise to, and express, the conceptualisation of things, processes and states of affairs that are non-actual in S's deictic space. We shall consider conceptualisations of the unreal at different points along the m-axis, ranging from degrees of possibility to the main object of concern, counterfactuality. The overarching concern with respect to the latter is why certain grammatical forms that have no negative marker give rise to negative – in the sense of counterfactual – conceptualisations.

6.3 Tense in the modal mirror

In Chapter 2 I discussed the relationship between time and modality. There is clearly a very close relationship cognitively, though it seems to me inadvisable to argue for the reduction of tense concepts to modal ones. Indeed, the fundamental structure of the deictic space that DST postulates has time and modality as independent cognitive dimensions. Nonetheless, it is also implicit in the deictic space framework that there are correspondences between its three axes (and underlying that framework a correspondence with spatial experience). These relationships are made explicit in Chapter 2, Figure 2.8. In looking at the tense forms of verbs in if-clauses there is, despite context-influenced variations, a tendency for temporal ‘distance’ encoded in English past- and present-tense forms to correlate with epistemic ‘distance’. This apparent fact is already inherent in the fundamental DST model. Moreover, to focus just on the t- and m-axis correspondence, this correspondence is a geometric reflection. Consider Figure 6.1, which is a partial view of the usual DST diagram, showing the plane defined in t and m – ‘looking down on it’, so to speak.⁴ The line r is a mirror line that maps points on the t-axis onto corresponding points on the m-axis; the line s is also a mirror line. The distances are scalar and relative to the origin – the point 0 that is S's immediate consciousness of time and certainty. The tenses correspond to progressively distanced modal meanings, at least in prototypical and decontextualised cases of if-sentences.

Figure 6.1 Reflection of time onto modality

Figure 6.1 is not intended as a DSM of a particular construction; rather, it is a schematic model of the conceptual (non-linguistic or pre-linguistic) relationship between temporal and epistemic experience. Linguistic forms such as tense affixes schematically, conventionally and by default denote subjectively represented times relative to S, and to one another; these are labelled on the temporal axis. The m-axis is also a non- or pre-linguistic scale – a cognitive gradation of ‘realness’. Both t and m represent ‘distance’ from S, and both reflect also onto the attentional distance d-axis, all three being grounded in spatial perception. The crucial point for the present discussion is the mapping from t onto m: relative temporal distance is mapped onto relative epistemic distance, all in conceptual pre-linguistic space.

In exploring how this cognitive reflection shows up in basic decontextualised if-constructions I refer to the following:

Past perfect maps to the irrealis extreme on the m-axis, as illustrated in sentence (3), which presupposes the counterfactuality of the proposition in the if-clause: John did not go to the party. Simple past maps to an epistemic midpoint on the m-axis – neither wholly certain not wholly uncertain – as suggested by sentence (2): John may or may not go to the party. As it happens, these past tense mappings can be modelled geometrically as a reflection through the theoretical mirror line labelled r. The main point here is the default modal reflection from the tense form. The intended time reference in sentences like (2) and (3), when they are uttered, is another question.

Do future tense forms also reflect onto the epistemic axis? This is only partially the case but is worth noting. The future tense form expressed in English by the auxiliary will can be used with modal meaning, as in that will be the postman. This construction is somewhat restricted but its existence is nonetheless evidence of the general mapping of time concepts onto epistemic ones. In this ‘putative’ use of the future tense form, it appears that S communicates relatively high epistemic certainty in respect of a process or state, and that this is possible because future time reference by way of future tense morphology is mapped onto epistemic distance. Contrary to what is sometimes argued, despite the fact that future events are inherently uncertain, speakers of languages nonetheless take one another to be confident in referring to future events (see Dancygier Reference Dancygier1998: 45–6), though a variety of tense forms reflect the background uncertainty. Geometrically, it turns out that this modal use of future tense forms is consistent with the fundamental design of the deictic space. The t-axis consists in fact of two half-lines, the one being a reflection of the other through the deictic point of origin. If we then postulate a point on +t that is future relative to S and can be denoted by the will form, then we can see that it maps by geometric reflection onto the modal axis. Taking future tense forms, in particular the English will form, as denoting confident prediction of the future, it is plausible to see it as mapping reflectively onto a relatively ‘close’ region on the epistemic axis.⁵ The mirror line r in Figure 6.1 is a geometric way of modelling this kind of conceptual ‘backshifting’ of the future tense forms.

Continuing to interpret the implications of Figure 6.1, we are left with questions concerning the reflection mapping of present tense forms. The point at the origin – i.e. S's now and point of epistemic ‘realness’ – reflects onto itself. For present progressive tense (ing forms) one should include all those points falling within the peripersonal (temporal and epistemic) region. In the case of statives such as know, simple present can refer to either time zero or to the future, as indicated in (1c): if John knows the answer can be followed by he is writing it down at this moment or by he will write it down. In the case of non-statives the only tense form in English that can refer to now in if-sentences is the present progressive ing form (called ‘presencing’ operator in Chapter 4), which extends over peripersonal time. It reflects therefore onto what we might call ‘peripersonal modality’, and can be understood as high epistemic certainty. Sentence (1b) seems to express more subjective epistemic certainty than (2), which has the more distal simple past tense, and (1b) is indeed readable as ‘assuming that John is going to the party…’. The modal reflex of the progressive form might be the maximal degree of epistemic certainty possible under the scope of the semantics of if.

This is not quite the end of the story. Sentence (1b) is most likely to be understood as referring to the relative future, but the present progressive in if-clauses can also refer to the present ongoing event, e.g. if John is walking to the party, he's getting wet in this rain refers to the speaker's peripersonal present time. One cannot use the simple present for this meaning: if John walks to the party, he is getting wet in this rain. Outside of context, in if-sentences, present progressive for non-statives and simple present for statives can refer to either ongoing now or to some future time. What is of interest is that non-statives can be in simple present tense form in if-sentences, provided they refer to the future. Can we account for this? As argued in Chapters 4, simple present is connected to ‘instants’ but also to temporally unbounded, or ‘timeless’, categorical concepts, contrasting with the ‘presence’ of events conceptualised via the progressive tense forms. If the conceptual contrast with ing meanings that I am suggesting is plausible, then it may be that, under the if-operator, the more abstract simple present denotes an abstract certainty but not the ‘present’ concrete certainty of factual experience expressed by the ing forms. What might the epistemic reflexes of simple present be? The simple present in (1a)If John goes to the party seems to indicate a relatively high degree of epistemic certainty in the future: not so epistemically ‘distanced’ as the simple past in (2) and not so epistemically certain or ‘close’ as the present progressive referring either to present or to future in (1b). The fact remains that simple present with non-statives under if can only refer to the future, as is the case in ordinary declarative sentences. In addition, abstract category conceptualisation associated with simple present are compatible with the abstraction implicit in the semantics of if.

Finally, one might wonder why the regular will-future is not possible in the if-clause? The answer has to do with natural cognitive consistency and the semantics of this future tense form. Such forms, as noted above, are in fact taken in context to indicate confident prediction of future events: it may be that this is a degree of confidence incompatible with the abstract hypotheticality inherent in the meaning of if. The English will form is, unlike for example the Romance future tenses, based on the concepts of volition and intentionality. Consequently, the primary meaning of if John will go to the party seems to be focused on John's willingness rather than on the speaker's epistemic assessment of the event's likelihood.

Once granted that past tense forms define scaled regions on the m-axis, these regions, marking epistemic distance relative to S, are available for any time reference, however such time reference is achieved linguistically or pragmatically. This tense reflection phenomenon does seem, in languages generally, to establish a conventional default pattern. This does not mean that tense forms map automatically, when they are used in practice in particular sentences, onto the epistemic m-axis. What I take it to mean is that the cognitive experience of ‘temporal distance’ is available to be reflected onto the cognitive experience of ‘modal distance’. This is a non-linguistic process that provides a resource for linguistic structure. It is a stable mapping that remains in background cognitive structure. Linguistic tense forms are influenced by it and can draw upon it, but they are not univocally determined by it.

6.4 The geometry of if-sentences

In this section I turn to the main concern of this chapter. How can we model, using the DST framework, the epistemic distancing effects associated with the different degrees and kinds of epistemic ‘distance’ associated with different tense forms in the protasis of if-constructions? Unlike the preceding section, this section is concerned now with the DSMs of particular sentences rather than the cognitive mapping of tense for meanings to the epistemic scale.

The conjunction if is a cognitive operator that transforms the basic coordinate system, S's deictic space R, by translating a copy R′ to various points on S's m-axis. How far it shifts depends on the tense forms, together with verb meaning and contextual factors. The second set of axes is a ‘new reality’ space, similar to a Fauconnier mental space (cf. also Dancygier and Sweetser Reference Dancygier and Sweetser2005), except that ‘distance’, ‘direction’ and deictic centring are already built into the fundamental structure of DST. Within the new set of axes, R′, propositions that are dependent on if are represented. They are thus governed by if with respect to S's initial coordinate system, but simultaneously ‘real’ within the new system. That is, all representations activated by conditional sentences are relative to 0′, the origin of the shifted axes, and the locus of S's conceptualising consciousness. In the new space, anaphors find their antecedents – ‘trans-world identity’ – across the base deictic space and the translated copy. The discourse entities referred to in if-sentences may be real in R, in which case they are labelled on the d-axis in R. They may also, however, be non-real, in which case they are labelled only on the d′-axis in R′. This would be the case, for example, in a sentence such as If John goes to a party, he will meet people; the same can be true for de dicto meanings in general. The time axis is aligned in both R and R′. While t represents conceptual time in DST, it is supervenient on metaphysical time and cannot be out of joint in the sense that there are two separate time realities that do not correspond. The actual time references that are intended and understood on the basis of tense forms in both protasis and apodosis of if-sentences are represented within the coordinate system of R′.

Taking examples (1a) to (3), simplified and repeated below, as representing something of the graded epistemic possibilities of conditional sentences, Figure 6.1 models two examples that give rise to embedded coordinate systems whose origins are at different points on the epistemic m-axis: (1) is relatively close to, (2) relatively farther from the base system of S. What the DSM represents is conceptual structures evoked by what have become conventional meanings signalled by particular tense forms in the if-construction together with situational factors.

In Figure 6.2a below, the coordinates for John, Sarah and party are labelled in S's base coordinate system, but the vectors representing go and see are located in the translated coordinate system determined by the verb form, together with contextual and pragmatic factors. Because of the coordinate system, the real labelled entities transfer automatically from R to R′. Figure 6.1 represents the copied and translated coordinate system R′ for sentence (1) and its relative position on the m-axis in S's base system R. In the diagram this is an approximate impressionistic system that is intended to correspond with the relative epistemic ‘closeness’ of events represented under the if operator by the simple present – as discussed in the previous section. The verbs in conditional sentences have the event represented in the protasis temporally prior to that represented in the apodosis (consequent clause). However, both clauses are subsumed under the semantic effects of if. The events in the two clauses are part of separate mental representation, a quasi-reality or ‘world’ in which entities that are real in S's base reality are represented as relating to one another in quasi-real ways (modelled as vectors) with real-time relationships to one another. As also discussed in the previous section, simple present if-clauses refer to times later than S's now. The exact intended temporal ‘location’ is a matter of contextual pragmatics, so any diagram can only position the event vectors arbitrarily, while maintaining temporal relationships.⁶

Figure 6.2a Conditional sentence (1) present tense

Figure 6.2b differs from Figure 6.2a only with respect to the epistemic positioning of $$ within S's base reality space: its origin on the m-axis is epistemically more ‘distant’ than $$ in Figure 6.2a. Time reference is still to the future. The diagram shows the origin of $$ at the epistemic midpoint. This is of course not a precise metric and contextualised reactions to (2) may vary. However, the theoretical positioning at the midpoint is supported by the difference between If John goes to the party and If John went to the party, where speakers usually report that the latter feels more ‘distanced’ than the former. In addition, one may note that a paraphrase of the past tense version, If John went to the party, can be John might go to the party and John might not go to the party, where the speaker's judgement as to possibility is divided. Figure 6.2b, then, is an attempt to model one of the salient cognitive effects of the tense form in the first clause of (2), viz. the simple past, modalised under if, in the way discussed in the preceding Section 6.3.⁷ It is of course possible to use (2) in reference to a past event relative to S, depending on contextual influences. In such a case, the vectors go and meet in the $$ coordinates would be positioned on the past time axis −t. Whether a past-time interpretation affects the sense of epistemic distance is a matter for consideration but perhaps cannot be determined theoretically.

Figure 6.2b Conditional sentence (2) past tense

The essential point is that epistemically distanced mental representations occur in response to tense forms combined with pragmatic contextual factors. In the DST account, which in many respects follows Dancygier and Sweetser (Reference Dancygier and Sweetser2005), R′ space can glide along the epistemic axis. It may be indeed that epistemic adjustment is in actual processing a matter of variable on-line adjustment.

Within R′ the main clause (apodosis) of if-sentences can be modalised. This includes negation, producing complete conceptual counterfactuality. This is modelled by the positioning of the event vector on m′. The if-clause itself (protasis) cannot normally be modalised, but can of course be negated, as in (6):

The reason for the oddity of such sentences is presumably that if is already epistemically modal; conversely, such examples are evidence that if is indeed an epistemic operator. The auxiliary will, which serves in English to form the future tense is, however, possible in the protasis, as noted earlier and the reason for this is presumably that will is not epistemic.

To return to the main clause of if-sentences, modalisation and negation in the main clause, illustrated in examples (7) and (8) below, can be modelled in a natural way in the framework developed above.

Figure 6.3 outlines a DSM for (7). The meet vector is located in R′ at the midpoint of the m′-axis and at some time t_i later than t_j, the time point of go. The sentence (8) is as modelled in Figure 6.3: the meet vector is at the distal (irrealis) limit of m′ in R′.

Figure 6.3 Modalised apodosis: sentence (7)

Modalisation in the main clause is entirely relative to R′, which in turn is relative to R. So modalisation in the main clause is, from S's viewpoint, a second layer of modalisation – modalisation within modalisation. Further conceptual nesting can occur. For instance, in (9) the negation operator is itself modalised:

The obvious and perhaps only way to model such a case in DST is to add a further epistemic space R″, with origin at m′ in R′, such that the event meet is modelled by a vector located at m″. We might speak here of first-order (under if) and second-order (under might) conceptual models.

If-sentences like (1) and (2) – and there are many similar possibilities – can be modelled in DST geometry by translation transformations that set up an irrealis space that is inconsistent with respect to the initial space that S takes as reality. These transformations are geometric translations that glide along the m-axis to a point of epistemic distance that satisfies S's epistemic processing, given the if operator, tense forms and contextual factors. These epistemically translated coordinate systems have an important property in DST that we will come across in later chapters. They are not aligned with the base system R. Specifically, the modal axis m′ in R′ is not aligned with m in R; that is, points on the m-axis are not aligned perpendicularly with equivalent points on the m′-axis. The d- and t-axes are aligned, however.⁸ This is as one might expect. The spaces R and R′ are modally distinct: by definition the world of if is separate from what is the real world for S. For counterfactual sentences like (3), however, the case is significantly different.

6.5 Through the looking glass: counterfactual if-sentences

The significant difference between the if-sentences examined so far and sentences of the type illustrated below (including (3), which is repeated) is that all of the latter presuppose the falsity (from S's viewpoint) of the expressed assertions:

Sentence (3) presupposes that John did not go to the party and he did not meet Sarah. In an imaginary world where he did, however, he met Sarah. In each clause the polarity of the expressed meaning is reversed in the presupposed clause. This is the case for (10), (11) and (12). This effect arises even when there is no context. The default time reference out of context appears to be past relative S but, as noted earlier, can be shifted to ongoing present and to future relative to S. In the case of the if-sentences with present and simple past tenses, discussed in the previous section, the reversed presupposition is heavily dependent on contextual assumptions: e.g. if John was a doctor [and we know he isn't], he could prescribe a pill. It is possible, however, and such a conceptualisation would be modelled in the same way as we shall now do for the construction exemplified in (3), which appears automatically interpreted counterfactually, and to which I restrict the term ‘counterfactual’.

What sort of DSM should be constructed to model a sentence such as (3)? A first approximation would be to assume that there is an embedded if-space, R′, as before, as shown in Figure 6.4.

Figure 6.4 Counterfactual sentence (3): first approximation

In this first approximation we note that in the counterfactual if-construction, the protasis needs to be true in one space but simultaneously false – it being understood that by ‘true’ and ‘false’ is meant S's estimation of realis and irrealis polarities. This can be achieved by locating the origin 0′ of R′ at the epistemic limit of the m-axis in R. Thus Figure 6.4 would give us a configuration in which, from S's point of view, John is at the party in R′ (if John had been at the party), but not at it in the presupposition (or better, in S's base epistemic state). Otherwise put: the negated if-clause is simultaneously negative in R (S's real world) and positive in R′ (in the if-space). However, for any sentence with a negated protasis, as in (11) and (12), 0′ would have to be located at the epistemic limit m′ in R′, and this would fail to model the kind of epistemic double vision that counterfactual if-sentences induce. Furthermore, the entire modal axis would theoretically have to be understood as ‘outside’ the reality space R of S. While such a configuration is appropriate for some constructions (cf. figures in preceding section 6.4), and modal non-alignment is appropriate as we have seen for certain types of if-sentence – those communicating some degree of epistemic distancing short of counterfactuality – it is inadequate for counterfactual if-sentences.

It seems that the two representations (the expressed one in the sentence and the presupposed one) involved in understanding sentences like (3) are simultaneous. Phenomenologically, they seem to be experienced as simultaneous. Moreover, polarity reversal is systematic in examples like (3), (10), (11) and (12). What we need in order to model these properties of counterfactual sentences is in fact a reflection transformation, as shown for (3) and (12) in Figure 6.5. Strictly speaking this configuration is a glide reflection: a reflection in the mirror perpendicular to m and passing though its limit point, combined with a translation of the reflected axes such that 0′ coincides with the limit of m and the limit of m′ coincides with 0. The reflection of the space R′ interlocks with the base space R.

Figure 6.5 Examples (3) and (12)

Figure 6.5 shows the polarised correspondences apparent in (3) to (12). In sentence (3) the vector representing go in the protasis, as well as the vector representing meet in the apodosis, lie in the realis plane in R′ – and simultaneously in the irrealis plane of R, the plane lying at the end of the m-axis in the base space R. Thus in the diagram S knows that John did not go to the party but raises at the same time a mental representation according to which he did go. Conversely, for sentence (12), Figure 6.3 models the go and the meet vector as lying in the irrealis plane of R′ and simultaneously in the realis plane of R′. So S knows John did go but activates a simultaneous representation of his not going. In other words, for sentences (3) and (12), what Figure 6.5 seeks to model are counterfactual states of mind communicated by those sentences.

Counterfactual sentences of this type can have epistemic modals in the apodosis: e.g. If John had gone to the party, he might have seen Sarah. The model is simply as for (3), with the meet vector positioned at the epistemic midpoint in the if-space R′. The meet vector then has the same position in both R and R′. This is what is needed, since the meaning we have for the apodosis in both spaces is one of possibility.

This geometrical account thus enables us to model the conceptual effects produced by combining the semantics of if with the semantics of the most distal tense (past perfect). We do not have to introduce any more machinery for this: the reflection transformation is given by geometry and is needed for modelling other linguistic constructions. Some of these are related to mental-state verbs such as wishing, imagining and supposing, as in the boy wished he had not stolen the apple, if only I were a millionaire!, what if she had left the door open that night?, let us suppose that the boson had been found fifty years ago. As will be seen in Chapter 10, the counterfactual mirror appears also in deontic modal constructions – e.g. the boy should have apologised for stealing the apple – where the sense of the auxiliary only makes sense against a simultaneous background reflecting something that did not happen.

6.6 Concluding reflections

A mind engaging in counterfactual thinking, including by means of language, must have two mental representations: one that is epistemically real and simultaneously one that is counter to it – this is what the DSMs model explicitly; in other models, even in mental space and conceptual blending theories, which are the closest to the present account, I believe this dual mental vision is not explicitly modelled as such. It also seems to me that counterfactual conditionals proper are of a different kind, not just a different degree of epistemic distance. Nonetheless, the introduction of ‘distance’ by Dancygier and Sweetser (Reference Dancygier and Sweetser2005) has been pivotal to the development of the theory of counterfactuals in natural human language, pointing toward the crucial role of spatial conceptualisation in linguistic meaning.

The exploration of geometric modelling in this chapter, based on the postulate of the three-dimensional deictic space, seems able to bring together a number of the frequently mentioned features of conditional and counterfactual grammatical constructions. The idea of a geometrical transformation on the basic deictic space gives us a way of modelling a secondary imaginary (irrealis or ‘counterfactual’ in a broad sense) space. Because this secondary space is a copy of the first, it ‘inherits’ its structure because it is positioned within it in such a way that referent entities are given coordinates that give them ‘trans-world identity’. The phenomenon of polarity occurs in a number of linguistic constructions but is most dramatic in counterfactual if-sentences, and the geometric reflection modelling handles this in a natural way as a consequence of general geometric principles. Most importantly, the reflection transformation captures the intuition that counterfactual and factual occur simultaneously.

Counterfactuals proper are special. Other conditionals that are epistemically distanced are ‘parallel’ worlds that do not map isometrically onto the whole of R. The parallel worlds of distanced conditionals are best characterised as ‘alternative realities’ – they are ‘positive’ in the sense that a potential (possible) event or relation is imagined, whereas in counterfactuals proper, the reality is counter, it is polarity-reversed. Some specific relations between entities are changed that hold in the subject speaker's reality are changed in his simultaneously imagined reality. This seems to me to be cognitively different from entertaining alternative realities, as in conditional sentences that are not strictly counterfactual ones; it is not surprising that models of counterfactuals in DST turn out to have their own special kind of isometry. It also makes the present account different from (though still indebted to) mental-space models (Fauconnier and Turner Reference Fauconnier and Turner2002) and from ‘epistemic distance’ models (Dancygier and Sweetser Reference Dancygier and Sweetser2005), as well as from Lewis's ‘closest possible world’ theory (Lewis Reference Lewis1973). It is also perhaps what makes them useful to the human mind. Because they retain referent identity and the universal time-frame they make reasoning, retrospection and prospection possible. Curiously, they are tools of reflection in a broader sense. They are essential to highly abstract thinking yet rooted in embodied spatial experience and the cognitive phenomenon of deixis. Finally, however, if the reflection model of counterfactuals has any plausibility, we might notice that they are actually self-reflective: thinking in counterfactuals is a self-contained conceptual activity. This makes them different again from the structures that we shall examine in the next chapter, which can be related to a large degree to the imagining of the minds of others.