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Resemblance and Representation

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Ben Blumson

9. Perspective and Possibility

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1Depictions, like thoughts and sentences, distinguish different ways things might be; the Mona Lisa, for example, represents Lisa by distinguishing between the various possible ways which Lisa might have looked. This suggests analysing the states of affairs depictions represent in terms of possible worlds: the state of affairs represented by the Mona Lisa, for example, may be analysed as the set of possible worlds in which Lisa’s appearance is as it portrays. This chapter argues that analysing states of affairs in terms of possible worlds addresses the lacuna in the last chapter in a way which is consonant with the platitude that depiction is mediated by resemblance.

2The first section introduces the analysis of states of affairs as sets of possible worlds. The next three sections address three problems for this analysis. The second addresses the problem posed by pictures in perspective. The third addresses the problem posed by pictures of metaphysical or a posteriori impossibilities. The fourth addresses the problem posed by pictures of logical or a priori impossibilities. All three problems require revision to the analysis of states of affairs as sets of possible worlds, but the revisions are consonant with both a strong analogy between depiction and description and the platitude that depiction is mediated by resemblance.

9.1 The possible worlds analysis of content

3A possible world is a consistent and complete way things might be. The actual world, for example, is one consistent and complete way things might be: it includes not just the earth, but also other planets, solar systems, galaxies, intergalactic space, and anything that actually exists. Other possible worlds include different planets and galaxies, but all of them are complete: there is no possible world which leaves any question undecided, since it is impossible for things not to be one way or another. And all of them are consistent: no possible world answers any question inconsistently, since inconsistencies are impossibilities.

4The main application of possible worlds is in the analysis of necessity and possibility: necessity may be analysed as truth in every possible world and possibility may be analysed as truth in some possible world. It is necessary that twice two is four, for example, because twice two is four in every possible world. And it’s possible that there are alien species, for example, because there are alien species in some possible worlds. Similarly, it’s impossible that twice two is five because there’s no possible world in which twice two is five and it’s not necessary that there are alien species because there are not alien species in every possible world.

5Since possible worlds are consistent and complete the rules of classical logic hold within them, so this analysis has the advantage of reducing an ill understood phenomenon – modal logic – to a well understood one – classical logic. It explains, for example, how it is necessary that there either will or will not be a sea battle tomorrow, without being necessary that there will be or necessary that there won’t be a sea battle tomorrow, because – since possible worlds are complete – every possible world is one in which there either is or isn’t a sea battle tomorrow, even though some possible worlds do have sea battles tomorrow and other possible worlds don’t.

6A state of affairs can be analysed as a set of possible worlds (Lewis, 1986, 185). The state of affairs of grass’ being green, for example, is the set of possible worlds in which grass is green and the state of affairs of Lisa’s smiling is the set of possible worlds in which Lisa smiles. The state of affairs of a horse’s grazing is the set of possible worlds in which a horse is grazing: since different horses graze in different possible worlds, the state of affairs of a horse’s grazing need not be the state of affairs of any particular horse’s grazing. Similarly, the state of affairs of Santa’s laughing can be analysed as the set of possible worlds in which Santa laughs: the puzzle of Santa’s inexistence is overcome by postulating his existence in other possible worlds.

7One clarification. The analysis of states of affairs in terms of possible worlds requires no particular assumptions about their nature or existence. Possible worlds may, for example, be concrete entities like the actual world, abstract entities akin to numbers and sets, non-existent objects, or merely useful fictions. If it’s correct that there’s no distinction between what exists and what there is, then if there are possible worlds they must exist. But if there is a distinction between what exists and what there is, then it’s possible to agree with the conclusions of this chapter, but not the last, by construing possible worlds and their constituents as Meinongian non-existent objects.

9.2 Centred properties and possible worlds

8Many pictures represent what they do from a particular viewpoint or perspective: profile portraits, for example, represent their subjects from their sides, rather than their fronts. The analysis of depictive content simply in terms of sets of possible worlds, as Jeff Ross (1997, 73-97) shows, is unable to accommodate perspective pictures. To illustrate the point, Ross (1997, 73) uses two pictures: whereas the first picture (Fig. 1) depicts a white sphere in front of a black sphere, the second picture (Fig. 2) depicts the same spheres from the opposite direction, with the black sphere in front of the white sphere.

Fig. 1 A white sphere in front of a black sphere. From Jeff Ross (1997), Semantics of Media (Dordrecht: Kluwer), p. 73.

Fig. 2 A black sphere in front of a white sphere. From Jeff Ross (1997), Semantics of Media (Dordrecht: Kluwer), p. 73.

9There is an obvious difference in content between the two pictures, but there is no difference in the set of possible worlds they represent. Every possible world in which the white sphere is in front of the black sphere is also a possible world in which, from another perspective, the black sphere is in front of the white sphere. And every possible world in which the black sphere is in front of the white sphere is also a possible world in which, from another perspective, the white sphere is in front of the black sphere. So every possible world in which the first picture is accurate is a possible world in which, from another perspective, the second picture is accurate: the two pictures represent the same set of possible worlds, but differ in content.

10This is an important difficulty for analysing the contents of depictions as sets of possible worlds. However, as Ross (1997, 75-76) points out, the problem is not unique to depictive representation. The analysis of the content of thought and language in terms of possible worlds, for example, also has to be revised to accommodate beliefs and sentences with egocentric content. The rest of this section explains the solution to the problem of accounting for egocentric content in general, the application of that solution to depiction and then the consonance of that solution with the platitude that depiction is mediated by resemblance.

11Consider the example, due to Lewis (1979, 139), of two gods. One lives on the tallest mountain and throws down mana; the other lives on the coldest mountain and throws down thunderbolts. Both gods know everything about which possible world they inhabit: for example, both know they inhabit a possible world with two gods, one on the tallest and one on the coldest mountain. However, neither god is omniscient, because both gods are ignorant about which god they are: neither god knows whether he is the god on the tallest mountain who throws down mana or the god on the coldest mountain who throws down thunderbolts.

12Because the gods already know which possible world they inhabit, their lack of knowledge cannot be analysed as ignorance about which world is actual; rather, their ignorance is about their location within the world. The solution to the problem is to analyse contents as sets of centred possible worlds or, in other words, ordered triples of locations, times and possible worlds. When the gods know the time and everything about their world, the content of their knowledge can be characterised as a set of two triples. The world and time coordinate of each triple is the same, but the location coordinate is different: one triple’s location coordinate is the tallest mountain whereas the other’s is the coldest mountain.

13The difference in content between the picture of the white sphere in front of the black sphere and the black sphere in front of the white sphere can be analysed as a difference in the location coordinates of the centred possible worlds in the sets they represent. The possible worlds coordinates of the centred possible worlds in the sets represented by both pictures are the same, but the location coordinates differ: the location coordinates of ordered triples in the set represented by the first pictures are locations to which the white sphere is closer, whereas the location coordinates of the ordered triples in the set represented by the second are locations to which the black sphere is closer.

14Two clarifications. First, Lewis and Ross (1997, 75-83) characterise egocentric belief in terms of the self-ascription of properties rather than triplets of locations, time and worlds. The later proposal was originally suggested by Quine (1969); Lewis (1979, 147) argues the two views turn out to be equivalent. Ordered triples of individuals, times and worlds are often used rather than triples of worlds, times and locations: I prefer locations to individuals because I prefer to leave open the question of whether or not the point of view in some pictures is inhabited by an individual (Walton (1990, 337-348) and Currie (1995, 170-179) discuss this issue).

15Second, Figure 3 and Figure 4 show that accommodating every example of difference in perspective requires centred possible worlds to be defined as ordered quadruples of locations, orientations, times and possible worlds. Whereas Figure 3 depicts a sphere to the left, Figure 4 depicts the same sphere to the right. They both depict the sphere from the same location: the difference in content is produced by a difference in the orientation of the viewer, as illustrated by Figure 5, which shows from above the orientation of the viewer in Figure 3, and Figure 6, which shows from above the orientation of the viewer in Figure 4.

Fig. 3 A black sphere to the left.

Fig. 4 A black sphere to the right.

Fig. 5 A black sphere to the left from above.

Fig. 6 A black sphere to the right from above.

16Similar examples show that the contents of thoughts and sentences must also be defined as ordered quadruples of locations, orientations, time and possible worlds: a change in orientation can change the truth of the sentence ‘the sphere is to the left’, for example, even while location, time and possible world are kept constant. Alternatively, if centred possible worlds are defined as ordered triples of individuals, times and possible worlds, then orientations may be dispensed with (in the analysis of linguistic, mental and depictive content) on the grounds that orientation and location may both be determined by the direction in which the individual is faced at the relevant time and possible world.

17Analysing what depictions represent in terms of sets of ordered quadruples of orientations, locations, times and possible worlds, instead of as sets of possible worlds simpliciter, is not as completely straightforward as it is for the case of egocentric beliefs. Egocentric beliefs represent that the believer is located at the centre of one of the centred possible worlds corresponding to their beliefs; in the case of pictures, this would suggest that what is depicted is the onlooker instead of, as it should be, what they look on. Perspective pictures do not represent that people are located at certain points of view, but the way things look to people when they are seen from those points of view (Ross, 1997, 85).

18Depictive representation is not representation of the properties of viewers, but the representation of the properties of depicted objects. To solve the problem of how things can be represented by pictures from particular points of view, properties have to be found which the things have from some points of view but not others. In the case of the pictures of the black and white spheres, a distinction must be drawn between the white sphere having the property of being in the foreground and the black sphere having the property of being in the foreground. I will argue in the rest of this section that the problem can be resolved by accepting a slight and intuitive revision of the standard analysis of properties in terms of possible worlds.

19The analysis of properties in terms of possible worlds is motivated by a problem for an extremely naïve theory of properties: the analysis of properties as the sets of objects that possess them. The property of red, for example, is analysed as the set of red things. The problem with this analysis is that it cannot distinguish between properties which happen to be possessed by all the same things. The property of being a creature with a heart differs from the property of being a creature with a kidney, for example, but the analysis of properties in terms of sets cannot distinguish them, because it so happens that the set of creatures with a heart and the set of creatures with a kidney are the same.

20This problem can be resolved by analysing properties as functions from possible worlds to extensions, which take each possible world to the set of things possessing the property at that world. So the property of being green, for example, is analysed as the function which takes the actual world to the set of things which are actually green and other possible worlds to the sets of things which are green in those worlds. This analysis can distinguish between the properties of being a creature with a heart and being a creature with a kidney, for example, because the functions associated with each property take possible worlds in which not all creatures with hearts have kidneys to distinct sets.

21Since the set of possible worlds in which the black sphere is in the foreground is the same as the set of possible worlds in which, from a different perspective, the white sphere is in the foreground, a function from possible worlds to the set of things that are in the foreground in that world cannot provide a property the representation of which can distinguish between the two pictures. More generally, if properties are functions from possible worlds to extensions, it is not possible to distinguish between different properties that things may have from different perspectives within the same possible worlds.

22This problem can be resolved by analysing properties as functions from centred possible worlds, rather than possible worlds simpliciter, to extensions, which take each ordered quadruple of orientation, time, location and possible world to the set of things possessing the property in question relative to that orientation, time location and world (Egan, 2006a, 509-513). The property of being in the foreground, for example, can be analysed as a function which takes centred possible worlds in which the white sphere is closer to the location coordinate to extensions including the white sphere and centred possible worlds in which the black sphere is closer to the location coordinate to extensions including the black sphere.

23So perspective pictures can be accommodated by taking the content of a depiction to be the set of centred possible worlds in which the depicted objects have the depicted properties, where those properties are functions from centred possible worlds to extensions, rather than functions from possible worlds simpliciter to extensions. The content of the picture of the white sphere in front of the black sphere, for example, is the set of centred possible worlds relative to which the white sphere has the centred property of being in front of the black sphere; the content of the picture of the black sphere in front of the white sphere is the set of centred possible worlds relative to which the black sphere has the centred property of being in front of the white sphere.

24Originally, introducing centred possible worlds may have seemed to be problematic for the claim that depictions resemble what they represent. But the introduction of the corresponding centred properties reveals some attractive features of the combination of a resemblance theory of depiction with the claim that the contents of depictions are sets of centred possible worlds. The natural suggestion is that pictures from particular points of view resemble what they represent because they share centred properties with what they represent. Depictions resemble what they represent because they possess centred properties relative to the intended positions of the viewer which the represented object possesses relative to the points of view represented by the picture.

25Take, for example, anamorphic pictures, which appear to resemble what they represent only if viewed from certain angles. This can be explained because the property in which the anamorphic picture resembles what it represents will be one that it has only relative to the unusual position from which the picture has to be perceived. In Holbein’s famous detail of the skull, for example, the shape of the details appears to be the same as a shape of a skull only relative to a viewpoint far off-centre. So the recognition of centred properties is important for understanding both the contents and the representational features of depictions, as well as the kinds of respects in which depictions resemble what they represent.

26One clarification. To accommodate the point that some pictures, such as anamorphic pictures, only resemble what they represent relative to a certain time and place, the analysis has to be revised as follows:

(26) An object depicts a state of affairs if and only if it is intended that if the object reaches an audience of a certain type at a certain time, place and orientation then:
a. the object’s having a property relative to that time, place and orientation resembles that state of affairs in respect of being states of affairs of something’s having that property relative to a time, place and orientation
b. the audience recognises the object’s having a property relative to that time, place and orientation resembles that state of affairs in that respect
c. the audience infers at least in part from the fact that the object’s having a property relative to that time, place and orientation resembles that state of affairs in that respect that it is intended:
d. that the object induce an attitude or an action directed towards that state of affairs in the audience
e. this effect be induced by means of providing a reason
f. and the audience recognise intentions (a)-(f).

27So, for example, Holbein’s detail depicts a skull because if the detail is seen by an audience at the extreme right of the picture, then the detail’s having the property of appearing to be a certain shape at the extreme right of the picture resembles the state of affairs of a skull’s having that shape in respect of both being states of affairs of something’s appearing to have that shape (see Greenberg (2013, 261-263) for further discussion of centred worlds and properties in the analysis of depiction).

9.3 The two-dimensional analysis of content

28The introduction of centred possible worlds shows how perspective pictures, even though they require an important revision, can be easily accommodated within the spirit of the possible worlds framework. Impossible pictures, on the other hand, seem more threatening: the possibility of pictures of impossibilities is an obvious difficulty for the analysis of what pictures are of in terms of possibilities. Similar difficulties arise for pictures of necessities and pictures with distinct but necessarily connected subjects. The problems divide into two kinds: depictions of metaphysical or a posteriori and of logical or a priori impossibilities and necessities.

29Inexistence and identity are the two main sources of a posteriori necessities and impossibilities. The inexistence of unicorns, for example, is known only empirically. But it’s empirical knowledge of a necessary truth, since it follows from the fact that unicorns do not exist that unicorns could not have existed, because although there are numerous possible but inexistent horned horse-like species, no one of these species is uniquely entitled to be identified with the unicorn (Kripke, 1980, 156-158). It follows that depictions of unicorns – and other inexistents whose essential properties are unspecified – are depictions of impossibilities. Their content can’t be analysed in terms of the sets of possible worlds in which they are accurate, since they aren’t accurate in any.

30Similarly, it was an empirical and a posteriori discovery that Hesperus – the brightest star in the evening – is Phosphorus – the brightest star in the morning. Nevertheless, it was an a posteriori discovery of a necessary truth, since it follows from the identity of Hesperus and Phosphorus that Hesperus and Phosphorus are necessarily identical (Kripke, 1980, 97-105). Imagine a star chart produced before the discovery that Hesperus is Phosphorus, which depicts them as simultaneously possessing different locations. The analysis of depictive content in terms of sets of possible worlds predicts that the content of this depiction – and all other depictions of a posteriori impossibilities – is the empty set, since there are no possible worlds in which the depiction is accurate.

31One way to escape this problem would be to argue that the chart does not really represent the impossibility of Hesperus and Phosphorus possessing different locations, but merely represents the possibility of the brightest star in the evening possessing a different location from the brightest star in the morning. Similarly, one may argue that pictures of unicorns do not represent any particular impossible species, but merely the general possibility of the existence of horse-like animals with horns. In general, apparent depictions of impossible states of affairs concerning particulars can be reconstrued as depictions of possible states of affairs concerning generalities.

32But this strategy obscures an important distinction. There is an important difference in content, for example, between a depiction of Pegasus flying and a depiction of a flying horse – but no horse in particular – which closely resembles Pegasus. Similarly, there is an important difference in content between depictions of stars with certain properties, and of particular stars with which we are familiar: depicting Hesperus is different from merely depicting a star – but no particular star – which rises in the evening. The strategy of arguing that apparent depictions of a posteriori impossible states of affairs concerning particulars are really depictions of possible states of affairs concerning generalities obscures these distinctions.

33A better solution is to argue that all depictions of a posteriori impossibilities do depict the empty set, but under different modes of presentation. Although depictions of unicorns and depictions of werewolves, for example, both depict the same impossible state of affairs, they do so under a different mode of presentation: the former represents the impossible state of affairs under a mode of presentation involving horns and horse-like features, whereas the latter represents the impossible state of affairs under a mode of presentation involving teeth and wolf-like features. The rest of this section clarifies this proposal and reconciles it with the platitude that depiction is mediated by resemblance.

34Prima facie, the introduction of modes of presentation into the analysis of depictive content is problematic for the platitude that depiction is mediated by resemblance. Depictions of a posteriori impossibilities, for example, seem to resemble neither the impossible states of affairs they depict nor the abstract modes of presentation under which those states of affairs are represented in any relevant respect. Moreover, what a mode of presentation is needs further explanation. Both these problems can be resolved by the two-dimensional theory of modes of presentation, which is naturally reconcilable with the platitude that depiction is mediated by resemblance.

35Depictions of a posteriori impossibilities might, if things had turned out differently, have been depictions of possibilities. If, for example, it had turned out that the brightest star in the morning and the brightest star in the evening were distinct, then the chart which represents Hesperus and Phosphorus as having different locations would have represented the genuine possibility of two other possible stars having different locations. Similarly, in other possible worlds in which a horned horse-like species does exist, depictions of unicorns might have depicted the possibility of members of that species appearing in a certain way.

36Just as the states of affairs depictions represent may be analysed as sets of possible worlds, the various states of affairs depictions might have represented may be characterised by two-dimensional intensions: functions from possible worlds to the set of possible worlds which a depiction would represent if that world were actual. The various states of affairs which the chart which represents Hesperus and Phosphorus as having different locations might have represented, for example, is characterised by a function which takes the actual world to the empty set, but possible worlds in which the stars appearing in the morning and evening are distinct to sets of possible worlds in which those stars have the locations shown by the map.

37A two-dimensional intension determines which states of affairs a depiction represents in each possible world, but they also determine one further state of affairs: the diagonal state of affairs is the set of possible worlds which the function takes to sets in which they’re included. To illustrate, suppose there are just two possible worlds: i and j. In i the brightest star in the evening is the brightest in the morning, but in j it is not. The rows represent the states of affairs the chart would depict in i and j:

i

j

i

0

0

j

0

1

38The diagonal from upper left to lower right represents the diagonal state of affairs: in this case, the set of possible worlds in which the brightest stars in the morning and evening are distinct.

39One clarification. In some possible worlds, what a picture would have depicted if that world were actual is completely irrelevant to its actual meaning (Schroeter, 2003). The chart representing that Hesperus and Phosphorus differ in location, for example, might have depicted sandwiches instead of stars, but sandwiches are irrelevant to its content. Resolving this problem requires either restricting the possible worlds involved to those compatible with the presuppositions of a representation’s perpetrator and audience (Stalnaker, 1978) or arguing a two-dimensional intension is part of a representation’s actual content, which reflects the perpetrator and audience’s understanding of how its truth depends on the facts (Chalmers, 2000; Jackson, 1998).

40Characterising the content of a depiction using a two-dimensional intension distinguishes between depictions of a posteriori impossibilities with different content, because different depictions of a posteriori impossibilities are associated with different diagonal states of affairs by their two-dimensional intensions. The diagonal state of affairs associated with the chart depicting Hesperus and Phosphorus, for example, is the set of possible worlds in which the brightest stars in the morning and evening are differently located, whereas the diagonal state of affairs of a picture depicting unicorns is the distinct state of affairs of there being a horned horse-like species, even though the horizontal state of affairs of both is the empty set.

41But characterising the content of a depiction using a two-dimensional intension also distinguishes between depictions of a posteriori impossibilities concerning particular and general states of affairs, because although depictions concerning particular and general states of affairs are associated with the same diagonal states of affairs by their two-dimensional intensions, they are associated with different horizontal states of affairs. A depiction of Pegasus grazing, for example, differs from a depiction of a winged horse, but no horse in particular, grazing, because the non-empty horizontal states of affairs associated with the former all concern particular horses, whereas the horizontal states of affairs associated with the latter do not.

42Originally, characterising the content of depiction in terms of modes of presentation might have seemed problematic for the platitude that depiction is mediated by resemblance. But characterising modes of presentation as diagonal states of affairs resolves this prima facie inconsonance, by showing that the state of affairs of the picture’s having a certain property may resemble the picture’s diagonal state of affairs in respect of both being states of affairs of something’s having a certain property. The state of affairs of Pegasus’ portrait’s being partly white, for example, resembles the diagonal state of affairs of a winged horse’s being partly white, because they are both states of affairs of something’s being partly white.

9.4 Structured intensions and impossible worlds

43The possibility of depicting logical or a priori impossibilities is more directly problematic for the analysis of depictive content in terms of possible worlds. The Penroses’(1958, 31) or Reutersvard’s triangle (see Fig. 7), for example, is a picture of an a priori, rather than a merely a posteriori, impossibility (see Mortensen (2010, 117-119) for a defence of the logical inconsistency of the impossible triangle). The depicted triangle does not exist in any possible world, nor would the picture have depicted an existent triangle if the world had turned out differently, so the content of the picture cannot be analysed as a set of possible worlds or two-dimensionally.

Fig. 7 An impossible triangle.

44Impossible pictures like the impossible triangle are a manifestation of a more general problem for the analysis of content in terms of possible worlds: just as it is possible to draw a priori impossibilities, it’s possible to believe a priori impossibilities: a person who believes the premises but disbelieves the conclusion of a deductive argument, for example, believes an a priori impossibility. This is problematic for the analysis of the content of belief in terms of possible worlds, because there are no possible worlds in which all their beliefs are true, nor any possible worlds in which, had things turned out differently, their beliefs would have represented something true.

45A natural solution to this problem is to argue that inconsistent beliefs divide into consistent partitions: although the beliefs of a person who believes the premises but disbelieves the conclusion of a deductive argument, for example, are logically inconsistent, that person’s beliefs divide into consistent partitions corresponding to each premise and the conclusion’s negation, and these partitions can be analysed in terms of sets of possible worlds (Stalnaker, 1984; Lewis, 1986, 34-35). Similarly, although the Penroses’ triangle is inconsistent, each part is consistent: covering any two of the picture’s sides with a piece of paper reveals that the remaining side depicts a consistent part of a triangle.

46But not every depiction of, or belief in, a logical impossibility can be treated in this way. Take, for example, a picture of a straight line captioned ‘square circle, side view’ (Sorensen, 2002, 343). The picture depicts a logical impossibility – a square circle – but not by consisting of individually consistent parts whose combination is inconsistent: every part of the straight line depicts a part of the square circle which is both straight and curved and therefore impossible. Since not even the parts of the straight line depicting a square circle depict possibilities, the inconsistent content of the whole cannot be analysed in terms of the consistent content of the parts.

47Side view depictions of square circles do not succeed in representing impossibilities explicitly; the depiction is successful only by representing an impossibility at an angle from which it is invisible. This phenomenon is widespread: depictions which are not composed of consistent, but jointly inconsistent, parts all seem incapable of representing impossibilities explicitly. This suggests an odd disanalogy between depictive and descriptive representation, since sentences seem straightforwardly capable of explicitly representing impossible states of affairs. It seems to follow that depictions differ from descriptions not merely in whether they are mediated by convention or resemblance, but in the kind of contents that they have.

48But this apparent disanalogy between depictive and descriptive content can be more neatly explained by the platitude that depiction is mediated by resemblance, which suggests that a picture depicts a property explicitly only if the depictive resembles the depicted states of affairs in respect of both being states of affairs of something’s having that property. So the Mona Lisa, for example, represents Lisa’s colour explicitly, because the Mona Lisa resembles Lisa in respect of colour, but represents that Lisa is smiling only implicitly, because the Mona Lisa does not resemble Lisa in respect of smiling.

49It follows from this characterisation of explicit depiction of properties that depictions cannot represent things’ having inconsistent properties explicitly, because depictions cannot resemble things in respect of having inconsistent properties. A straight line, for example, can resemble a square circle in respect of appearing like a straight line from the side, but a straight line cannot resemble a square circle in respect of being both square and circular, because it cannot be both square and circular. The straight line depicts that the square circle appears straight from the side explicitly, but depicts that it is both square and circular only implicitly.

50Even the implicit depiction of impossibilities is problematic for the analysis of depictive content in terms of possible worlds. According to the analysis, the content of a depiction is the set of – or a function from possible worlds to sets of – possible worlds in which it is accurate, but since impossibilities do not occur in any possible world, the analysis predicts that every depiction of an impossibility has the same content: the empty set or the constant function from every possible world to the empty set. Since neither a straight line depicting a square circle nor a straight line depicting a triangular hexagon, for example, are accurate in any possible world, both seem to represent the empty set, when in fact what they represent is obviously different.

51There are two options for dealing with this problem. The first option is to replace the analysis of depictive content in terms of sets of possible worlds with an analysis in terms of sets of worlds simpliciter, by allowing the inclusion of impossible – incomplete and inconsistent – as well as possible worlds (Malinas, 1991, 288). This would accommodate depictions of inconsistencies straightforwardly, since depictions of different inconsistencies would depict different sets of impossible worlds. The set of worlds represented by the straight-line depicting a square-circle, for example, would include impossible worlds in which there is a square-circle, whereas the set of worlds represented by the straight-line depicting a triangular hexagon would not.

52The introduction of impossible worlds can naturally accommodate the resemblance of depictions to what they represent, by enabling properties to be reanalysed as functions from ordered quadruples of locations, orientations, times and worlds – possible or impossible – simpliciter to extensions. A depiction which represents a triangle as equilateral but not equiangular, for example, represents the triangle as having a property characterised by a function which takes all possible worlds (and other coordinates) to sets of triangles which are both equilateral and equiangular, but which takes some impossible worlds (and other coordinates) to sets of triangles all of which are equilateral but some of which are not equiangular.

53However, the introduction of impossible worlds does have some costs. Because possible worlds are consistent and complete, the rules of classical logic hold within them, so that analyses in terms of possible worlds provide an analysis of an ill understood phenomenon in terms of an extremely well understood phenomenon. The possibility of depicting a horse, but no particular horse, for example, is explained by the possible worlds analysis without appealing to non-particular horses in other possible worlds, whereas the possibility of depicting an impossible horse is explained by the impossible worlds analysis only by appealing to impossible horses in the impossible worlds.

54The second option is to analyse the content of depictions by introducing more fine-grained contents than sets of possible worlds, such as ordered n-tuples of objects and properties (Soames, 1987; Malinas, 1991, 288). A line depicting a square circle on its side, for example, may depict the ordered quintuple of an object, the property of being square, the property of being circular and the centred property of being viewed from the side. Although ordered n-tuples of objects and properties are more fine-grained than sets of possible worlds, the proposal remains close to the analysis in terms of possible worlds, because each ordered n-tuple of objects and properties determines a set of possible worlds in which the relevant objects possess the relevant properties.

55Ordered tuples of objects and properties are fine-grained enough to distinguish between the content of all necessarily equivalent depictions only if it is possible to distinguish between the necessarily equivalent properties represented. The content of a depiction of an equiangular triangle can only differ from the content of a depiction of an equilateral triangle, for example, if it is possible to distinguish between the properties of equiangularity and equilaterality, since if equiangularity and equilaterality are the same property, there is no difference between ordered tuples of objects and properties containing equiangularity and those containing equilaterality.

56The solution is to introduce finer-grained properties in addition to states of affairs. Structured properties and relations are ordered tuples of unstructured properties and relations. So equiangularity, for example, can be analysed in terms of an ordered tuple containing the property of being an angle, whereas equilaterality can be analysed in terms of an ordered tuple containing the property of being a length: since the unstructured properties of being an angle and being a length are distinct, so are the structured properties of being equiangular and equilateral and structured states of affairs constituted by ordered tuples of objects and those properties (Lewis, 1986, 56-57).

57Since non-particulars do not exist, non-particular states of affairs cannot be analysed as ordered n-tuples of non-particulars and properties. Instead, non-particular states of affairs must be analysed as ordered n-tuples of properties and higher-order properties of properties (Soames, 1987, 224). The state of affairs of something’s grazing, for example, can be analysed as the ordered pair of the property of grazing, and the higher-order property of being instantiated. The state of affairs of a horse’s grazing can be analysed as the ordered triple of the property of being a horse, the property of grazing, the higher-order relation of being co-instantiated, and so on.

58Depictions of non-existents are also problematic for analysing depictive content in terms of fine-grained states of affairs. If Holmes does not exist for example, then the content of a depiction of Holmes smoking cannot be analysed as the ordered pair of Holmes and smoking, because there is no such ordered pair. In general, depictions of non-existents cannot be analysed in terms of ordered tuples of non-existent objects and properties, because there are no ordered tuples of non-existent objects and properties. The analysis of depictive contents as fine-grained states of affairs cannot distinguish between depictions of different non-existents with the same properties.

59The solution is to combine fine-grained states of affairs with the two-dimensional solution to the problem of depicting a posteriori impossibilities. Just as a function from centred possible worlds to sets of possible worlds characterises the different states of affairs a depiction might have represented, functions from centred possible worlds to particulars and properties characterise the different particulars and properties it might have represented. A structured two-dimensional intension is an ordered tuple of functions from centred possible worlds to properties and particulars (Chalmers, 2011b). The content of depictions can then be fully characterised by structured two-dimensional intensions.

60The structured two-dimensional intension of a depiction of Holmes smoking, for example, is the ordered pair of a function which takes centred possible worlds to detectives called ‘Holmes’, if anything, and a constant function which takes centred possible worlds to the property of smoking. The structured two-dimensional intension determines an ordinary two-dimensional intension, a function from centred possible worlds to the empty set or sets of possible worlds in which a detective is smoking, as well as the various structured horizontal states of affairs Holmes’ portrait might have represented and the structured diagonal state of affairs consisting of the ordered pair of the property of being a detective called ‘Holmes’ and the property of smoking.

9.5 Conclusion

61The application of possible world semantics to the contents of depiction requires revision to cope with depictions in perspective and depictions of metaphysical and logical impossibilities, but the revisions that are required are close to the spirit of the possible worlds framework and consonant with the platitude that depiction is mediated by resemblance. The success of these revisions in defending the application of possible world semantics to depiction suggests that similar strategies should be pursued for resolving the similar problems which arise for the analysis of the contents of thought, language and fiction in terms of possible worlds, rather than revisions which depart more radically from the spirit of the possible worlds framework.

62Whatever revisions need to be made – even if they depart markedly from the analyses discussed in this chapter – the examples discussed here suggest that most plausible analyses of depictive content will be consonant with the platitude that depictive representation is mediated by resemblance. The reason is that whatever complications are introduced into the analysis of depicted states of affairs, similar complications will need to be made to the analysis of properties, so that depictive will always resemble depicted states of affairs in respect of both being states of affairs of thing’s having certain properties. So whatever theory of depictive content is right, that theory seems likely to support the platitude that depiction is mediated by resemblance.

63Similarly, whatever revisions need to be made – even if they depart markedly from the analyses discussed in this chapter – the examples discussed here suggest a strong analogy between the content of depictive and other kinds of representation. The reason is that whatever complications are introduced into the analysis of depictive content, analogous complications will need to be made to the analysis of linguistic, mental and other kinds of representation. The few disanalogies – such as the inability of depictions to represent impossibilities explicitly – discovered in this chapter were either incidental or arose from the platitude that depiction is mediated by resemblance.

Table des illustrations

Légende Fig. 1 A white sphere in front of a black sphere. From Jeff Ross (1997), Semantics of Media (Dordrecht: Kluwer), p. 73.
URL http://books.openedition.org/obp/docannexe/image/2428/img-1.jpg
Fichier image/jpeg, 12k
Légende Fig. 2 A black sphere in front of a white sphere. From Jeff Ross (1997), Semantics of Media (Dordrecht: Kluwer), p. 73.
URL http://books.openedition.org/obp/docannexe/image/2428/img-2.jpg
Fichier image/jpeg, 12k
Légende Fig. 3 A black sphere to the left.
URL http://books.openedition.org/obp/docannexe/image/2428/img-3.jpg
Fichier image/jpeg, 12k
Légende Fig. 4 A black sphere to the right.
URL http://books.openedition.org/obp/docannexe/image/2428/img-4.jpg
Fichier image/jpeg, 16k
Légende Fig. 5 A black sphere to the left from above.
URL http://books.openedition.org/obp/docannexe/image/2428/img-5.jpg
Fichier image/jpeg, 12k
Légende Fig. 6 A black sphere to the right from above.
URL http://books.openedition.org/obp/docannexe/image/2428/img-6.jpg
Fichier image/jpeg, 12k
Légende Fig. 7 An impossible triangle.
URL http://books.openedition.org/obp/docannexe/image/2428/img-7.jpg
Fichier image/jpeg, 26k

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