§1 Analytic philosophers interested in depiction have focused for the most part on two problems: first, explaining how pictures represent; second, describing the distinctive kinds of artistic value pictures can possess, or the distinctive ways in which they can embody artistic values that extend more broadly across the arts. I shall discuss the first problem here. The main concepts I shall be concerned with are depiction, resemblance, sense and reference.

§2 My main aim is to reassess the traditional idea that representation in painting and sculpture depends on resemblances in form and colour between works of art and the objects they represent. The philosophical literature about representation has been dominated for fifty years – to its detriment, I shall argue – by the view that the traditional idea is wrong. I reject the so-called resemblance theory of representation, the theory that representational works of art are iconic signs, which Nelson Goodman attacked with devastating effect in Languages of Art.Footnote ¹ But there are other ways of making the traditional idea precise.

The question I shall consider first is whether resemblance is a relation. This may seem remote from the interests of philosophers of art, especially since we cannot expect to be able to answer it without first clarifying the idea of a relation itself. But it is where I believe we must begin.

According to logicians, if two or more names in a true sentence are replaced by variables, the term that results will normally express a relation, which obtains between the bearers of the names. For example, if we start with the sentence ‘Reggie and Ronnie are twins’, and replace the names with variables we get the predicate ‘x and y are twins’, which expresses a relation that can obtain between two siblings. Here are a couple more examples: ‘Paris is west of Moscow’ yields the predicate ‘x is west of y’, which expresses a spatial relation; ‘Scott is the author of Waverley’ yields ‘x is the author of y’, which expresses a causal relation.

However, the logical conception of a relation has two important limitations. First, it is too broad. For there are cases where – as logicians generally agree – following this procedure will not yield a predicate that expresses a relation. For example, ‘Brutus killed Caesar’ reports an action, but actions should not be assimilated to relations.Footnote ² And there are other cases that are controversial. For example, it is controversial whether identity-statements, such as ‘the morning star is the evening star’ and ‘Bronstein is Trotsky’, are about an especially intimate relation in which everything stands, and can only stand, to itself.

Second, the logical conception of a relation tells us where we can normally find a term expressing a relation in a sentence – it is the part we are left with when we delete the names – but it does not tell us anything more about what a relation is. This is rather like explaining that a father is the kind of thing Russians refer to with their middle names. If we want to know more than this, we need to refer to the traditional conception of a relation, which preceded the development of formal logic.

According to this conception, which modern formal logic complements, but does not supercede, a relation is a way in which one thing can stand to another thing, or several things can stand to one another. As a matter of fact, the last clause is untraditional, since the idea of a many-termed relation was only introduced in the second half of the nineteenth century.Footnote ³ But the definition in italics captures a conception of a relation which stems originally from Aristotle's Categories and on which the logical conception I have described is based. Here, for example, is Locke's definition, which confines relations to two terms, and also precludes things from standing in a relation to themselves:

And here is the definition in the current edition of the OED:

Like the definition in italics, the definition in the OED includes many-termed relations, and it allows a man to stand in a relation to himself, for example, being the one who shaves, or being the one who kills. For these are both ways in which one thing can stand to another thing. But being the same man as is not a way in which one thing can stand to another thing. Nor is it a way in which one thing is connected or associated with another thing. So it is not a relation, according to these definitions, whereas it looks like a relation according to the logical conception we began with, as I pointed out earlier. I shall not attempt to adjudicate between these views.

In the relation-stating sentences I mentioned earlier, the verb phrase that expresses the relation is flanked by proper names referring to the objects it relates: ‘Paris is west of Moscow’, ‘Scott is the author of Waverley’. These names can be replaced by descriptive terms such as ‘the capital of France’ or ‘the Laird of Abbotsford’. But, like proper names, these terms refer to particular places, people or things. Even the sentence ‘John knows a fireman’, which does not purport to identify John's acquaintance, implies that there is a particular fireman John knows, since one cannot know a fireman, but not any fireman in particular.

What about statements of resemblance? Do the verb ‘resembles’ and the verb-phrases ‘is like’, ‘looks like’, ‘sounds like’, etc., express relations? The answer, I suggest, is that sometimes they do and sometimes they do not. For example, in the sentence ‘Darwin looks like Socrates’ the verb-phrase expresses a relation, whereas in the sentence ‘Socrates looks like a satyr’ it does not.

Compare the verb ‘to be’. It is an elementary fact about English grammar that ‘to be’ is used both to express identity, as in ‘Bronstein is Trotsky’ and ‘Cicero is Tully’, and as a copular verb, as in ‘Cicero is a statesman’. As I mentioned earlier, it is controversial whether identity is a relation: Wittgenstein says that it isn't in the Tractatus, whereas Kripke says that it is. But if we assume that Kripke is right, that identity is a relation, the verb ‘is’ expresses a relation in the sentence ‘Cicero is Tully’ and the sentence ‘The morning star is the evening star’, whereas in the sentences ‘Cicero is a statesman’ and ‘The morning star is a planet’ it does not.

The verbs ‘resembles’, ‘is like’ and ‘looks like’ evidently have a similar dual use, that is, they are used both to express relations and as copular verbs. ‘x is like y’ is a regular two-place predicate, and if we replace the variables with names, the resulting sentence relates the individuals concerned: for example, ‘SoHo is like Hampstead’ relates part of New York and part of London. But the sentence ‘SoHo is like a village’ does not relate anything to anything. It is not comparable to ‘The morning star is the evening star’ or ‘Cicero is Tully’, but to ‘The morning star is a planet’ or ‘Cicero is a statesman’, the only difference being that it characterizes the place referred to by saying what it is like, rather than what it is. ‘SoHo looks like a village’ and ‘SoHo resembles a village’ are just the same.

It follows that although ‘SoHo resembles a village’ and ‘John knows a fireman’ are syntactically similar, they are logically different. If I tell you that John knows a fireman, you can ask me which fireman John knows, and if what I said is true, your question has an answer, even if I do not know what it is. In other words, we can add a namely-rider to the sentence ‘John knows a fireman’, for example, ‘… namely, the fireman who lives on Church Lane’, or ‘… namely, Jim’.Footnote ⁶ By contrast, if I tell you that SoHo resemble a village, and you ask me ‘Which village?’, I can say that I did not mean to imply that it resembles any village in particular, and there does not have to be a namely-rider we can add to the sentence, in order to prove that what I said was true. ‘SoHo resembles a village’ normally means that SoHo has some of the salient characteristics of a village, without any village in particular being involved.

So, is resemblance a relation? If the idea is that the verb ‘resembles’ sometimes expresses a relation, then it is true. If it is that ‘resembles’ always expresses a relation, then it is false.

§3 I shall turn now to the philosophy of art. According to Robert Hopkins, one of the principal objections to the view that pictures invariably resemble the scenes or objects they depict is that resemblance is a relation between two particulars, whereas depiction is not.

Hopkins's entry on depiction in the Routledge Encyclopedia of Philosophy contains a similar argument, but he adds the requirement that for two things to resemble one another both need to exist:

Catherine Abell follows Hopkins closely:

Interestingly, Hopkins does not appear to notice that his claim that ‘resemblance is a relation between two particulars’ contradicts the familiar idea that resemblance is a reflexive relation, which implies that true statements of resemblance may compare an object with itself. Even if this familiar idea is wrong, a person or place at one time can certainly resemble the same person or place at another time, and a kind of stuff in one place can resemble the same kind of stuff in another place. For example, the British Foreign Secretary William Hague famously looked the same at thirty-six as he looked at sixteen. Here we do not have two things, we have one thing at two different times. Again, Guinness in Tel-Aviv tastes the same as Guinness in Cork; and here we have one kind of stuff in two different places.

Neither Hopkins nor Abell says whether they believe that resemblances have to be between particular things that exist because this true of relations generally, or whether they think it is something special about resemblance. Be that as it may, it is not self-evident that when a statement of resemblance does relate two things, both need to exist in order for the statement to be true. And besides, as we have seen, true statements of resemblance do not always mention two particular things, to which the existence requirement can apply. I shall enlarge on these two points in turn.

When a statement of resemblance relates two things, must they both exist in order for the statement to be true? On the face of it, the answer is no. Reggie looks like Ronnie, Levin is similar in various ways to Tolstoy, Widmerpool resembles Malvolio, and Thor is like Zeus. At least these are things that we find it quite natural to say. It appears that a statement of resemblance can be true whether both or one or neither of the individuals concerned exists.

Here is a simple theory which explains this. The mark of a fictional character, as the disclaimer that sometimes appears at the front of novels or at the end of movies attests, is that any resemblance to any actual person is coincidental (at the time of writing); it is not that none is possible, or that none exists. (The qualification ‘at the time of writing’ is needed for the reason Oscar Wilde famously pointed out: life can imitate art.)

If this is right, we can still insist that someone who uses two names to make a true statement of resemblance refers to something with each of the two names, as long as we are prepared to acknowledge that it is possible to refer to things – such as fictional characters – that do not exist.Footnote ¹⁰ Arguably, what the referring use of a name requires is that the speaker be able to identify whom or what she is referring to, which is not ruled out where fictional characters are concerned. On the contrary, fictional characters can be identified quite easily. For instance, Widmerpool is the character in A Dance to the Music of Time who marries Pamela Flitton and Malvolio is the character in Twelfth Night who wears yellow stockings with cross gartering.

If this simple theory is right, when a statement of resemblance compares two things, neither of them needs to exist in order for the statement to be true. Of course, this theory is not universally accepted and statements of resemblance involving fictional characters have been interpreted in various ways. But we should not assume that the simple theory is false.

The second point I said I would enlarge on is that statements of resemblance do not always mention two things, to which the existence requirement could apply. The reason I gave is that while there are many statements of resemblance in which two specific persons, places or things are mentioned, for example, ‘SoHo is like Hampstead’, there are just as many in which only one is, for example, ‘SoHo is like a village’. Equally, there are many in which none are, for example, ‘A kibbutz is like a village’ and ‘Margarine is like butter’. It is puzzling that philosophers writing about the resemblance theory of depiction uniformly ignore these kinds of statements, and repeat the canard that resemblances are necessarily between particulars, or specific things.

As we have seen, the supposed fact that resemblances are necessarily between pairs of specific things, both of which exist, is thought to pose a problem for resemblance theories of depiction, because a picture may depict a man or a horse without depicting any man or horse in particular. Here is the passage from the Routledge Encyclopedia again:

One thing that is puzzling about this remark is that if a picture depicts a horse, but no horse in particular, it surely does not depict something that does not exist. On the contrary, it depicts something, a kind of animal, that does exist, unlike a picture of a centaur, for example. A kind of animal is not a particular animal, of course. But the question whether the particular horse it depicts exists does not arise, since ex hypothesis it does not depict any horse in particular. Stubbs's portrait of Whistlejack depicts a horse that exists, or existed when he painted it; whereas Rubens's painting Perseus and Andromeda depicts Pegasus, a mythical horse, which never existed. But when a picture depicts a horse, but no horse in particular, there is no particular horse about whose existence we can enquire. Hopkins appears to confuse pictures with generic content and pictures with fictional content. I shall return to this confusion later.

The passages by Hopkins I have quoted seems therefore to combine two errors: first, the idea that resemblances are necessarily between pairs of specific things, both of which exist; and second, the idea that genre pictures – pictures with generic content – invariably depict things that do not exist, that is, they depict things that do not exist whether they depict centaurs (which do not exist) or horses (which do). I have said more about the first error so far. But the second error is equally important, because it illustrates a failure to think clearly about the relationship between the concept of a picture with generic content and the concept of a picture that portrays an individual, which is of fundamental importance in the theory of art, as I shall argue in a moment.

Where does this leave the traditional idea that representation in the visual arts in general, and depiction in particular, depends on resemblances in form and colour between works of art and the things they represent?

It is true that the verb ‘depicts’ is sometimes used to express a relation, and sometimes not. For example, ‘It depicts a horse’, ‘… a bridge’, ‘ … a river’ can be read in either way. Read in the first, relation-involving way, the question ‘Which horse?’, ‘Which bridge?’, ‘Which river?’ has an answer, even if we do not know what it is, and the sentence can be continued with a namely-rider, ‘… namely, Whistlejack’, ‘… namely, the Rialto’, ‘… namely the Styx’. Read in the second, non-relation-involving way, the question ‘Which …?’ and the namely-rider are out of place. It is useful to mark this distinction clearly in the language we use to talk about pictures, and to a degree we do: the verb ‘portray’ is biased towards the relation, whereas ‘depict’ is not.

But as we have seen, the verb ‘resembles’ has exactly the same dual use. Hence, the statement that a picture (or part of one) resembles a horse does not imply that there is a particular horse that it resembles, and the statement that it resembles a satyr does not imply that satyrs exist. Satyrs have a distinctive appearance, which it is easy to describe, and if something has the same appearance as a satyr, then it resembles one. The fact that satyrs are mythical creatures does not prevent this from occurring. In Plato's Symposium, Alcibiades says that Socrates looks like a satyr. This may have been unkind, but it was not absurd.

§4 I said that the relationship between the concept of a picture with generic content and the concept of a picture that portrays an individual is of fundamental importance in the theory of depiction, but neither the phrase ‘picture with generic content’ nor the word ‘portrays’ has exactly the right meaning. Perhaps the simplest way to capture what I have in mind is to take a picture of a specific person, place or object, whether fictional or real, and to consider what we can call, for want of a better pair of terms, its reference and its sense. The words ‘reference’ and ‘sense’ are the normal translations of the terms ‘bedeutung’ and ‘sinn’, which Frege used to distinguish between the object that an expression stands for or designates, and the way in which the expression presents that object, the ‘mode of presentation’ as he called it.

Frege introduced the distinction between sense and reference to explain how identity statements can be informative, without being about words.Footnote ¹¹ Returning to the example I mentioned earlier, ‘The morning star is the evening star’ is not a statement about words, unlike the statement that the phrases ‘the morning star’ and ‘the evening star’ refer to the same object, because the first statement uses these two phrases, whereas the second one mentions them. And it is not merely an instance of a law of logic either, like the statement that morning star is the morning star. ‘The morning star is the evening star’ can state an astronomical discovery, Frege explained, because the phrases ‘the morning star’ and ‘the evening star’ have the same reference, but do not have the same sense. Here is another example. The expressions ‘2 × 3’ and ‘4 + 2’ both designate the number six, but the first expression presents it as the product of two and three, whereas the second presents it as the sum of four and two So again these expressions have the same reference, but do not have the same sense.

Similarly, two portraits of the same individual may present him as dark-haired and seated, wearing a black smock (Kramskoy's 1873 portrait of Tolstoy), or as grey-bearded and standing, wearing a white smock (Repin's 1901 portrait). The analogy between expressions in a language and pictures is not exact. But it is helpful to think of each of these two portraits as designating, or standing for, same the same individual as the other, while differing in its ‘mode of presentation’ – in other words, as having the same reference, but a different sense. And we can use the same distinction to think about two pictures of the same fictional person – for example, Michelangelo's fresco of the creation of Adam on the Sistine Chapel ceiling and Piero della Francesca's fresco of the death of Adam in the church of San Francesco in Arezzo – or the same place.

I said earlier that the verb ‘depicts’ is sometimes used to express a relation, and sometimes not, and that sentences like ‘It depicts a horse {… a bridge / … a river}’ can be read in either way. We can see now that this distinction corresponds to the distinction between sense and reference. In the relation-involving use of the verb, the use where the sentence can be continued with a namely-rider, depiction corresponds to reference; whereas in the non-relation-involving use of the verb it corresponds to sense. We can also see that the kind of picture Hopkins and Abell are concerned about, a picture that depicts a horse, but no horse in particular, is a picture that has a sense – as any intelligible figurative picture must – but no reference, like the phrase ‘the greatest integer’ or ‘the present King of France’. Henceforth, I shall use subscripts to distinguish between these two ways of using the verb ‘depicts’: ‘depicts_r’ for the use that corresponds to reference and ‘depicts_s’ for the use that corresponds to sense.

Together with the muddle about resemblance, the most important mistake philosophers have made about depiction is to confuse or amalgamate theories about the sense of works of art and theories about their reference, or to assume that a theory of depiction is first and foremost a theory of reference – as it were, a theory of the portrait – and that a theory of sense can be developed from it, rather as Wittgenstein's theory of meaning in the Tractatus was developed from his conception of a name. Each of the two most influential theories of depiction during the last fifty years, Richard Wollheim's and Nelson Goodman's, makes one of these mistakes. Wollheim makes the first, while Goodman makes the second.

Wollheim argues that a picture is a marked surface, which is designed to cause a distinctive kind of visual experience, which he calls ‘seeing-in'. Seeing-in, he explains, has two aspects or components:

This kind of experience is not caused by pictures alone. It can also occur, for example, when we look at a damp-stained wall. The element of intention or design is what distinguishes pictures from other marked surfaces that have the same kind of effect. Representation occurs when a standard of correctness is imposed on the natural capacity of seeing-in, and the standard of correctness is set by the artist's intention. Thus a picture represents a specific person or place, or an object of a given kind, if, and only if, the artist successfully intends the view to see that person or place or that kind of object in its surface.

There are several well-known objections to this theory.Footnote ¹³ What concerns us here is that Wollheim ignores the distinction between the sense and reference of a picture. He talks indifferently about seeing Henry VIII or Charles Laughton or a generic bison in a picture. But this is logically naïve. It is like failing to distinguish between the sense and reference of a phrase such as ‘the morning star’, as if the philosophy of language could make do with a single idea of meaning or signification that includes both.

One result of Wollheim's failure to acknowledge this distinction is his claim that the standard of correctness, which determines whether the viewer has correctly perceived what a picture represents, is set by the intentions of the artist. This is normally true of a picture's reference, but not its sense. Wittgenstein was surely right when he said in the Blue Book: ‘An obvious, and correct, answer to the question “What makes a portrait the portrait of so-and-so?” is that it is the intention.’Footnote ¹⁴ This is comparable to the question, What determines the reference of a proper name, in a particular instance of its use? For example, if I begin a letter with the phrase ‘Dear George’, what determines which of the myriad Georges in the world I am addressing? The answer is surely my intention.

But we cannot answer the corresponding question about sense in the same way, for there may be a difference between what a word or phrase I write or meant to utter means and what I meant to say. Similarly, a picture can depict_s a man in the uniform of a midshipman when the artist intended to depict_s a man in the uniform of an ensign, or it can depict_s a spruce when the artist intended to depict_s a larch. In both cases there are more general terms that apply to the depicted_s object and conform to the artist's intentions, such as ‘man’ and ‘tree’. But the divergence between intention and outcome remains, and this disproves the idea that an artist cannot produce a picture with unintended sense. As one might expect, inexpert artists are especially prone to do so. For example, most three-year-old children are no less capable of painting a picture that depicts_r their mothers than Rembrandt or Whistler was, but drawings by three-year-olds tend to depict_s arms growing out of heads.

Wollheim's error about the role of the intention is a result of his failure to distinguish between the sense and reference of a picture. He uses a single model to explain both what determines the reference of a portrait and what determines its sense or mode of presentation. But it is as elementary a mistake to overlook the difference between these questions as it would be to overlook the difference between the sense and reference of a descriptive phrase.

Goodman is a very different case. He does not overlook the distinction between sense and reference: he rejects it. The distinction he draws between a picture of a man and a man-picture is extensionally equivalent to the distinction between and picture that depicts_r a man and one that depicts_s a man, except that Goodman excludes the referring use of empty names. But the extreme form of nominalism he espouse reduces sense to reference, so we find the same failure to think about the sense of a picture, as opposed to reference, in his work. Thus his principal claim is that ‘denotation is the core of representation’ – denotation being a variety of reference – as if a portrait were the basic kind of picture.Footnote ¹⁵

Both of these approaches are disastrous, because representation by pictures depends on a systematic relationship between the shapes and colours on the surface of a picture and its sense that does not exist between the shapes and colours on the surface of a picture and its reference. So if we amalgamate sense and reference, or if we regard the question of how pictures refer as fundamental, we are bound to miss the basic mechanism that explains how pictures represent.

Thus, if we imagine hanging Whistler's portrait of his mother (Arrangement in Grey and Black No.1, 1871) next to a child's portrait of its mother drawn at the age of three – admittedly different artists and different mothers – it seems obvious that there isn't a systematic relationship between the shape and colour of part of the surface of a portrait and the shape and colour of the individual it portrays. But if we turn from the reference of a picture to its sense, the situation is quite different. The case of a free-standing sculpture is simpler but similar to the case of a picture, so that is where I shall begin. (The qualification ‘free-standing’ will be omitted in the discussion that follows.)

§5 Consider the part of Michelangelo's Bruges Madonna that represents Jesus's head. What is the reference of this part of the scuplture, and what is its sense? Its reference is Jesus's head, and it presents it as having various features: thick locks of hair, chubby cheeks, and so on. These features, we may say, comprise the sense or mode of presentation of this part of the sculpture. But the sense or mode of presentation of a sculpture expressed in the most general terms is simply an object or arrangement of objects with a specific shape. And except in the unusual case of an anamorphic sculpture, which is designed to be seen from an extremely eccentric point of view, this shape is evidently the shape of the part of the carved block itself. This applies to Michelangelo's Rome Pietà in exactly the same way. The Rome Pietà refers to the same two individuals as the Bruges Madonna, but the sense, or mode of presentation, of Jesus's head is different because the shape of the corresponding part of the sculpture is different, and Jesus's head is presented as an object with this shape.

It follows that the fundamental difference between representation in sculpture and representation in words is not a matter of reference, it is a matter of sense. The simple rule about sculpture is that what a sculpture represents has the same shape as the sculpture itself. Another way of making the same point would be to say that there is an exact resemblance in shape between a sculpture and what it represents. But by the phrase ‘what it represents’, I do not mean the sculpture's reference, I mean its sense. The rule does not relate two particulars, for example, the sculpture and the historical individual Jesus; it concerns a single particular, the sculpture, and its sense or mode of presentation. If resemblance were invariably a relation between two particulars, the rule would be incoherent; but as we have seen, this dogma about resemblance is a mistake.

The simple rule about sculpture should be obvious as soon as it is stated, so obvious that it seems trite. But notice that the rule is not conditional on the artist's intention; it does not involve a system of rules correlating symbols with the objects they refer to; it does not refer to any of the psychological states philosophers have postulated to explain how painting and sculpture represent; and it applies in exactly the same way to a Greek bronze figure of a generic horse from the Geometric period as it does to the Bruges Madonna or the Rome Pietà. The same is true of the equivalent rule for pictures, as we shall see.

I shall add four further observations, before discussing the rule for pictures. First, although the simple rule for sculpture can be stated in terms of resemblance it need not be. It is not a restatement of the theory that paintings and sculptures are iconic signs. That theory failed to distinguish between sense and reference, treated resemblance as a relation and was not limited to shape.

Second, the simple rule does not imply that a sculpture that represents Jesus resembles Jesus, or that a bronze figure of a horse resembles a horse. It says nothing about the reference of a sculpture, and nothing about its sense beyond its shape. It therefore combines naturally with the idea that reference is normally determined by the artist's intention, and with the idea that a viewer's ability to identify the sense or mode of presentation of a sculpture as a child's head or as a horse depends on her ability to recognize these kinds of objects by their shapes.

Third, the fact that the simple rule is not conditional on the artist's intention does not prevent the artist's intention from playing any role in the theory of representation apart from determining the reference of a work of art. The analogy with linguistic meaning suggests that it does play such a role. For acknowledging that the meaning of an utterance need not be the same as what the speaker meant to say is consistent with the idea that an utterance means nothing unless the speaker means something by the words he utters; and it is also consistent with the idea that a meaningful utterances cannot occur except against a background that includes the custom of making utterances with the intention of saying something. Both of these ideas can be transferred to the case of painting and sculpture in a straightforward way; but neither implies that an artist cannot produce a representational work of art with unintended sense, as Wollheim's theory implies.

Fourth, as I have indicated, the simple rule that what a sculpture represents has the same shape as the sculpture itself combines naturally with various ideas philosophers interested in representation have proposed. It combines easily with John Kulvicki's recent defence of the role of bare-bones content in the theory of representation as well as claims about the role of recognition in explaining how works of art represent, such as those advanced by Flint Schier in his book Deeper into Pictures and Dominic Lopes in Understanding Pictures;Footnote ¹⁶ it is consistent with various ideas about the relationship between the concept of representation and the concept of intention or design; and as we shall see, it suggests that the concept of occlusion shape (outline shape) plays a central role in the theory of depiction, as Robert Hopkins and the author of this article have both proposed.

The equivalent rule for pictures is less straightforward than the rule for sculpture, because sculptures represent objects with the same number of dimensions as they have themselves, whereas pictures normally represent three-dimensional objects on a two-dimensional (i.e. flat) surface, or on a surface whose curvature is slight enough to be ignored. But it is not difficult to see what the rule for pictures is, if we think about how their two-dimensionality affects their sense.

The principal point is that we cannot discover different aspects of an object represented in a picture by moving around it and studying it from different angles, as we can in the case of a free-standing sculpture. That is why Van Dyke sent Bernini a triple portrait of Charles I: one referent, three modes of presentation. So whereas the sense or mode of presentation of a sculpture, expressed in the most general terms, is an object or arrangement of objects with a specific shape, the sense or mode of presentation of a picture is an aspect or view of an object or arrangement of objects – or several aspects or views, in unusual cases such as this one – relative to a line (or lines) of sight.

Now if we want to formulate a shape-rule for pictures analogous to the simple rule for sculpture discussed above, we shall need to identify a two-dimensional shape-property that an aspect or view of an object represented in a picture invariably includes, whether it is a shadow, a rainbow, part of the sea or sky, or a medium-sized specimen of dry goods. It is not difficult to identify this property if we think about an object with a simple shape, such as a coin. Consider a circular coin viewed along an oblique line of sight. The coin will look circular to a viewer as long as she can see its orientation. But the two-dimensional cross-section of the cone of light the coin subtends to the viewer's eye will be elliptical, and this is also a visible property of the coin. It is especially salient when an object is backlit, and appears in silhouette. It has been called a perspectival shape or outline shape or occlusion shape, it is two-dimensional, and of course it is relative to a line of sight. It changes as the line of sight changes. But relative does not mean subjective. The shape of a cross-section of the cone of light an object subtends to the viewer's eye is not merely a feature of the viewer's experience. It belongs to optics, not psychology.

I said a moment ago that the sense or mode of presentation of a picture is an aspect or view of an object or arrangement of objects, relative to a line of sight. This means that a picture invaraibly depicts_s the occlusion shapes of objects. So suppose a picture depicts_s a circular coin with an elliptical occlusion shape. What shape would the corresponding region of the picture's surface have to be? With the same exception as we noted in the case of sculpture – that is, an anamorphic picture, designed to be seen from an extremely eccentric point of view – the answer of course is that it would have to be elliptical, and the surface of the coin would be foreshortened.

But it would be a mistake to think that the occlusion shape is only represented when an object is foreshortened. For example, the shield in a painting on a kylix attributed to the Foundry Painter (Munich 2640, ca. 490 BC) is among the earliest examples of foreshortening in Greek art; whereas the hoop in a painting on a krater attributed to the Berlin Painter (Louvre G175, ca. 500 BC) is not foreshortened. This is not because the Berlin painter did not represent the hoop's occlusion shape. It is just that the line of sight in this case is perpendicular to the hoop, and the hoop's occlusion shape is therefore a circle. So whereas the simple rule for sculpture is that what a sculpture represents has the same shape as the sculpture itself, the shape-rule for pictures is that the shape of a region on a picture's surface is the same as the occlusion shape of the object it represents. In other words, there is an exact resemblance between these shapes. This applies to pictures of objects such as shields and hoops, but it applies equally to pictures that represent a rainbow, the sea or the sky.

It is reasonable to suppose that the simple rule for sculpture has always been understood by sculptors and their public, even if it is too obvious to be stated or written down. By contrast, the shape-rule for pictures has always been implicit in artistic practice, but the concepts used to state it precisely and to explain the idea of occlusion shape were first developed in Greek geometry and widely disseminated – in Europe – only in the Renaissance.

I have defended the claim that the idea of occlusion shape plays an central role in the theory of depiction and stated the shape rule for pictures in several earlier publications, most fully in my book The Objective Eye. The approach I took there contrasted subjectivist theories of depiction, which seek to explain how pictures represent by defining the kind of experience they are designed to produce in viewers, with objectivist theories, which proceed without attempting to define the experience. (Objectivist theories stem from Plato, subjectivist theories from Descartes.) The simple rule for sculpture and the shape-rule for pictures belong in the objectivist camp.

The principal justification for subjectivism has always been the evident dissimilarity between a picture or sculpture and the objects it represents. This is the point Descartes seizes on. He writes:

The shape-rule for pictures addresses both of the arguments in this passage. The first is that ‘engravings represent to us bodies of varying relief and depth on a surface which is entirely flat’. But the only shape properties the shape-rule refers to are the occlusion shapes depicted_s in a picture, and occlusion shapes are two-dimensional. The second argument is that the rules of perspective may, for example, require an artist to represent a circle by means of an oval. But as we have seen the circular profile of a coin has an elliptical occlusion shape relative to an oblique line of sight. The dissimilarity between the physical shape of a circular coin and shape of the region on the surface of a picture that depicts_s a coin is consistent with the rule.

Another justification that is sometimes offered for subjectivism is that the objectivist emphasis on resemblance embodies a bias in favour of realistic, literal or accurate representation and a narrow and stultifying programme for artistic work.Footnote ¹⁸ But this is confused. One might as well argue that regarding a language as a system of semantic and syntactic conventions or rules embodies a bias in favour of conventional literature, or literature that follows rules – a bias, say, in favour of Rupert Brooke over T.S. Eliot or Arthur Conan Doyle over James Joyce. It should be obvious that this would be a gross misunderstanding. The rule that the gerund in English ends in ‘–ing’ does not limit the inventiveness of English writers, and it applies to Donne or Milton in exactly the same way as it does to lesser poets. Similarly, the simple rule for sculpture applies in exactly the same way to a geometric bronze figure of a horse, Donatello's Gattamelata and Marino Marini's L'angelo della Città, and the shape-rule for pictures applies to pictures irrespective of the style or tradition they belong to, their originality, or the artistic values they express. Both of these rules identify basic mechanisms of representation in the visual arts; they do not dictate or limit the forms artists create, the models they follow or the values they embody in their work.

However, the subjectivist position is not entirely without merit. For although the simple rule for sculpture and the shape-rule for pictures do not refer to viewers' experiences, their competence is limited in two ways. First, as we have noted more than once, the rules only provide an objective correlation – a correlation that is independent of the viewer's experience – between the shape of a sculpture or the shapes on the surface of a picture and the shape or occlusion shape of each object included in the sculpture's or the picture's sense. No specification of the sense or mode of presentation of a work of art beyond this can be ‘read off’ its non-representational properties in this way. Second, the parts of a picture that represent discrete objects or parts of a scene need to be distinguished from each other, and of course the rules cannot explain how this is done. Both of these limitations indicate ways in which psychological factors are essentially involved in defining the relationship between representational and non-representational properties of works of art.

As we have seen, there is a third limitation on the competence of the two rules, which not provide a gap that subjectivist ideas can fill. It is that they do not correlate the shape of a sculpture or the shapes on the surface of a picture with its reference. Here, as in the case of language, intentions and contextual factors are involved in complex ways that are difficult to summarize or survey. I shall not attempt that task here.

§6 I said earlier that it is sometimes alleged that philosophers who analyze the concept of depiction in terms of resemblance or occlusion (outline) shape express a bias in favour of realistic or literal representation and offer a stultifying programme for artistic work. Michael Podro makes this charge in his article ‘Literalism and Truthfulness in Painting’. These philosophers, he says, ‘treat depiction as a matter of mere visual representation’, they pursue ‘the project of approximating depiction to an abbreviated equivalence of ordinary environmental perception’, and ignore the ways in which pictures can ‘transform our experience of the subject’. ‘We need’, he adds, ‘to see how painting elaborates upon its underlying conditions as poetry elaborates on those of language.’Footnote ¹⁹

This is partly right and partly wrong. It is right to point out that defining the ‘underlying conditions’ is only part of the theory of art. The foundations are part of the structure, not the whole of it. Artists exploit the communicative possibilities inherent in the medium as such (i.e. its ‘underlying conditions’) with specific materials, tools and techniques, to communicate thoughts, feelings and perceptions in a work of art. Understanding art means understanding all three aspects of artistic activity, both in themselves and in relation to each other. But it is wrong to think that philosophical theories of depiction treat all pictures ‘as a mere matter of visual representation’, just as it would be wrong to think that linguistics treats literature as a mere litany of facts. In fact it is doubly wrong. It is wrong because philosophers need not mistake the part for the whole – and to my knowledge they have not done so. And it is also wrong because we cannot expect to understand how painting ‘elaborates upon its underlying conditions’ unless we know what these ‘underlying conditions’ are.

For example, I said earlier that expressed in the most general terms, the sense or mode of presentation of a picture is an aspect or view of an object or arrangement of objects, relative to a line (or lines) of sight. Several significant developments in the history of painting ‘elaborate upon this underlying condition’ in ingenious ways. First, novel views of objects – views associated with novel lines of sight – can be introduced by combining views along established lines of sight. For example, quasi-frontal views were composed at different times in the history of painting by combining two profiles or oblique views, so that the composite image divides along a vertical axis. This is how the Andokides Painter produced a frontal view of a wrestler's face around 515 BC (Berlin F2159), with his oddly pointed head, projecting ears and thick neck; and it is how Giotto produced a frontal view of a mourner in his Dormition of the Virgin (ca. 1310, Staatliche Museen zu Berlin, Gemaldegalerie), with his broad shoulders (figs. 1 and 2).

Figure 1.

Andokides Painter, Amphora, ca. 515 BC, detail.

Figure 2.

Giotto, Dormition of the Virgin, ca. 1310, detail.

Second, the lines of sight associated with distinct parts of a depicted scene can be coordinated or played off against each other. Thus, in an orthodox use of Renaissance perspective, the lines of sight associated with each part of the depicted scene are made to intersect, so that the entire scene is coordinated in relation to this implicit point of view. By contrast, in Masaccio's fresco of The Trinity (1425, S. Maria Novella, Florence), the architecture and the supporting figures are depicted as if seen from below, but the figures of the Father and the Son are depicted frontally, without any foreshortening at all.Footnote ²⁰

Third, the implicit line of sight can be associated with an implicit spectator. The idea of an implicit spectator was first used as a theoretical tool by Alois Riegl, in his analysis of Rembrandt's The Staalmeesters (1662, Rijksmuseum, Amsterdam). (Riegl credits the idea to Thoré-Bürger, who ‘correctly presumed the presence of an unseen party in the space of the viewer, with whom the syndics are negotiating.’Footnote ²¹) But the earliest examples are self-portraits, because here an implicit spectator can be introduced by accident, without being intended as a narrative device. For example, Dürer's drawing known as Self-Portrait with a Cushion (1493, Robert Lehman collection, Metropolitan Museum of Art, New York) shows the artist absorbed in the act of drawing himself, and so the view of the young man it depicts is necessarily represented as his own.

Of course other equally significant developments in the history of painting depend on other factors, and failing to understand in general terms how pictures represent is unlikely to impede the work of art historians interested in the impact of the Council of Trent or the supply of paint in tubes. But the inventions I have mentioned involve more abstract concepts, and the cost of misunderstanding them can be high. The theoretical debates about Renaissance perspective in the twentieth century are an embarrassing episode in art history for precisely this reason. Everyone understood that perspective is a geometrical system that enables artists to control the occlusion (outline) shapes and relative occlusion sizes of the objects represented in a picture, but misunderstandings about these properties inherited from philosophy and optics led art historians from Panofsky onwards into pointless controversy and needless confusion.Footnote ²²

In summary, the simple rule for sculpture and the shape-rule for pictures define part of the basic mechanism on which representation in the visual arts relies. (I have not discussed colour here. In The Objective Eye, I argue that analogous rules for colour can also be defined without referring to the experiences sculptures and paintings cause in viewers.) Alongside these rules, a comprehensive theory of representation in art will also refer to psychological factors, to the artist's intentions, to customs and conventions, and to other factors. But if we wish to adjudicate between the traditional view that representation in the visual arts depends on resemblances between works of art and the objects they represent and the theories of representation defended by Goodman and Wollheim and their followers, who reject this view, we are, I believe, bound to conclude that the traditional view is right. It was not eclipsed for fifty years because it is philosophically naïve or artistically stultifying, but because some exceedingly simple and well understood ideas in logic have been misunderstood or routinely ignored: first, resemblance is not invariably a relation between particulars; second, we need to unpack the general idea of representation and distinguish between sense and reference in order to understand how pictures represent.