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Semantics of Pictorial Space

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Abstract

A semantics of pictorial representation should provide an account of how pictorial signs are associated with the contents they express. Unlike the familiar semantics of spoken languages, this problem has a distinctively spatial cast for depiction. Pictures themselves are two-dimensional artifacts, and their contents take the form of pictorial spaces, perspectival arrangements of objects and properties in three dimensions. A basic challenge is to explain how pictures are associated with the particular pictorial spaces they express. Inspiration here comes from recent proposals that analyze depiction in terms of geometrical projection. In this essay, I will argue that, for a central class of pictures, the projection-based theory of depiction provides the best explanation for how pictures express pictorial spaces, while rival perceptual and resemblance theories fall short. Since the composition of pictorial space is itself the basis for all other aspects of pictorial content, the proposal provides a natural foundation for further pictorial semantics.

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Notes

  1. The artist was probably Dieudonnée Lancelot. See: https://commons.wikimedia.org/wiki/File:Barthtimbuktu.jpg.

  2. A caveat for the case of photography: because the process by which photographs are produced is to such a high degree mechanical, reasonable questions have been raised about whether photographs should count as representations at all (Scruton 1981). Here I will proceed on the assumption that photographs are representations: they are pictures, they express contents, and their contents encode conditions of accuracy. But the inclusion of photographs is not central to my theoretical aims.

  3. This use of “semantics” doesn’t distinguish pictorial semantics from pragmatics, instead referring to the general interpretive framework for pictures, whatever it may be.

  4. Prominent analyses of pictorial representation which focus on, or culminate in conditions on “depiction-as” include Abell 2009, 217; Goodman 1968, 26-31; Hopkins 1998 76-77; Lopes 1996, 151-153; and Walton 1973, 312-15. Meanwhile, literature on the psychology of pictorial perception has followed the holistic concerns with pictorial space advocated here. See footnote 5.

  5. Note that the term “pictorial space” has often been employed by psychologists to refer to the space that one perceives when looking at a picture (e.g. Rogers 1995; Ittelson 1996; Koenderink and van Doorn 2003; Cutting 2003; DeLoache et al. 2003; Vishwanath et al. 2005). Here I use it to describe the form of a picture’s content, independent, at least in principle, from the perceptual content that the picture elicits from viewers.

  6. See Peacocke (1992, 62) for analogous remarks about perceptual space.

  7. This definition is informal, but will suffice for present purposes. See Greenberg (2020) for a more in-depth discussion.

  8. Another group of authors, including (Peacocke 1987; Budd 1996), and (Hopkins 1998), enlist projection within the framework of the resemblance theory of projection; I address these theories in Section 6.

  9. There are other aspects of pictorial content, in particular those that reflect pictorial reference to individuals, which do depend on causal connections between a picture and elements of the context in which it is created.

  10. The following presentation of the idea of projection draws from Dubery and Willats (1972) and Sedgwick (1980), Willats (1997, ch. 2), and Greenberg (2013) among others.

  11. The projection lines in the diagram below are only a sample of the full array.

  12. Note that the temporal location of the source and plane must be the same. Also, the picture plane itself must define at least up-down and left-right orientations; front and back can typically be defined by the relative position of the projection source.

  13. The notion of general viewpoint introduced in the last section can be defined as the pair of an abstract projection source and picture plane, specified only in terms of their positions relative to one another.

  14. Willats (1997, 4-20) organizes methods of projection into “projection systems” (≈ my “projection condition”) and “denotation systems” (≈ my “marking condition”). See Durand (2002) for an extension and critical discussion of this approach.

  15. Many of the illustrative pictures in this essay also include “blank space”, unfilled white areas around the central figure that extend to the border of the page. In principle, such blank space may be thought of as outside the picture plane, not the product of projection, or it may be analyzed as the projection of deep space or void. In the second case, a theory may treat blank space as a different kind of mark than white (though the difference may not be visible). The issue awaits further discussion. I gloss over such complications in the text.

  16. With some translation, the Projection Principle is comparable to the various projection-based proposals in the literature. (i) Kulvicki: Kulvicki posits a basic layer of pictorial content, called bare bones content: “whatever scenes could have resulted in a particular [linear perspective picture] via a perspective projection count as parts of the [bare bones] content of the picture” (Kulvicki 2006, 59). He further holds that bare bones content constrains a picture’s overall content. The result is equivalent to the Projection Principle for content. However, in order to preserve the property of “transparency,” Kulvicki takes a non-standard view of pictorial structure, with the result that Kulvicki’s principal is logically weaker than the Projection Principle. Here I believe the Projection Principle captures intuitive distinctions in content that Kulvicki’s principle misses; see footnote 33 and the Appendix for a detailed discussion.

    (ii) Hyman: the Projection Principle is comparable to the Occlusion Shape Principle proposed in Hyman 2006, ch. 5. The occlusion shape of an object is, roughly, the shape optically projected from an object to an intersecting plane along a given “light of sight.” The Principle is then formulated as follows: if a part P of a picture depicts an object O, then “the occlusion shape of O and the shape of P must be identical” (Hyman 2006, 81). Assuming that the sum of objects a picture depicts always determine a specific set of scenes, and reading “depicts an object” as accurately depicts an object, the principle expresses a constraint very much like the Projection Principle for accuracy, at least with respect to overall shape. In the end, however, Hyman’s principle is tied to an optical interpretation of projection, which significantly narrow its range relative to the Projection Principle; see footnote 32.

    (iii) Greenberg: for Greenberg (2013), “accurate depiction” of a scene may be understood in the present nomenclature simply as accuracy at a scene; Greenberg terms this scene the “referent” of a picture; I understand this simply to be the index relative to which a picture is evaluated. Also see the discussion later in this section on Greenberg’s view that projection is sufficient for accuracy.

    (iv) Howell: a final proposal not mentioned in the text is that of Howell (1974, §3), who defines a picture’s picture space as a type of spatial array for which the picture could be a projection. However, because Howell’s goal is to provide a formal semantics for sentences which attribute content to pictures, rather than for pictures themselves, the semantic properties of pictures are never fully set out.

  17. To state the principle formally, let ⟦PS, c denote the content of a picture P, relative to a system S and context c. And let projS(⋅) be a projection function from scenes to pictures, for the method of projection characteristic of S. According to the Projecting Principle, given a system S and context c, for any picture P in S: ∀wv : if 〈w, v〉 ∈ ⟦PS, c then projS(w, v) = P. Equivalently: \(\llbracket P \rrbracket _{S,c} \subseteq \{ \langle w,v \rangle ~ | ~ proj_{S}(w,v) = P \}\).

  18. In the sense of relative accuracy I have in mind, all pictures with contents have accuracy-conditions, even if only some such pictures function to be accurate. Imperatival pictures, for example, do not function to be accurate, but they still have the same kind of spatial content and accuracy conditions as any other picture.

  19. Formally: given a system S and context c, then for any picture P in S, and any world w, viewpoint v: if P is accurateS, c at 〈w, v〉 then projS(w, v) = P.

  20. For example, see Greenberg (2018, 867-70, 893-94) for an account of singular and generic content which is compatible with a scene-based approach to a picture’s overall content.

  21. The need for levels of content beyond the skeletal are vividly demonstrated by the case of ambiguous figures (Voltolini 2015, 42-43). Voltolini further argues that the relevant perceptual capacities which help flesh-out ambiguous figures also operate at the level of recognition of the pictorial vehicle itself (pp. 127-31). Though I have provisionally treated pictures as syntactically unstructured here, I welcome the more structured account that these observations suggest.

  22. The Projection Principle determines a unique perspectival feature map for every picture, on the assumption that, in the content of each picture, the internal structure of the general viewpoint— the relationship between projection source and plane— is held fixed across scenes. This assumption accords with intuition, because there does not seem to be ambiguity about the directional structure of space in a given picture, even when there is ambiguity about depth and shape. If projection source and picture plane are allowed to vary with respect to one another among scenes within the content of a single picture, then the Projection Principle associates each picture with a set of perspectival feature maps, rather than a unique map.

  23. The dependence of high-level content on skeletal content is asymmetrical. In general, for any given bit of high-level content C (which may be only a part of a picture’s total content), it appears to be impossible for a picture to express C without also expressing some skeletal content. The reverse is not true, because there are genuine skeletal contents– those expressed, for example, by fields of static or indecipherable marks– that lack any substantive high-level content.

  24. Linear perspective is often simply called “perspective.” But it should be distinguished from curvilinear perspective, the sort of image produced by a fisheye lens.

  25. The same point can be made with respect to the difference between systems of black and white line drawing and systems of color line drawing. Relative to a system of black and white line drawing, Picture J might depict a solid of any color or shade— its content is indeterminate in this respect. Relative to a system of color line drawing, it must depict a white solid on a white background.

  26. This isn’t always the case. There are systems of depiction which involve more than one method of projection. Generally such systems fall under the rubric of “impure projection,” discussed in Section 7.

  27. Hagen (1986, 142-43) offers an alternative analysis, according to which the projection source of parallel projection is actually a point, but it is located at infinite distance from the scene. But this analysis is unwieldy to implement in geometrical terms; and it undermines any finite attributions of depth that might figure in a picture’s fleshed-out content.

  28. The familiar examples of axonometric, isometric, and elevation projections are all species of parallel projection, derived in part by varying the relationship between projection source and picture plane. Still other methods of projection can be defined by varying the structures of the projection source, picture plane, projection lines, and their relative relationships.

    In the text I discuss methods of projection where the projection lines are themselves straight lines. But in a prominent alternative class of projections, lines of projection instead closely recapitulate the behavior of light rays, reflecting and refracting in the environment before reaching the picture plane. Such optical methods of projection are the norm in realist painting, computer animation, and, of necessity, photography. The construction of images in modern 3D animation uses a method of rendering known as ray tracing— essentially tracing the complex path that light would realistically take in the scene.

  29. As an anonymous reviewer points out, topological (or quasi-topological) maps, such as subway maps, as well as circuit diagrams, do not fit this mold. Still, they may be derived from projections by abstracting away from metric properties; I’m inclined to think of these as a blend of pictorial and diagrammatic representations, exploiting interpretive competencies from both visual and geographic cognitive modalities.

  30. There has been extensive research on the marking conditions at work in line drawing: see e.g. Kennedy (1974, chs. 7-8), Willats (1997, ch. 5), Palmer (1999, §5.5.7), DeCarlo et al (2003). For discussion of color and other “optical” methods see Willats (1997, ch. 6), who shows that even photographs can be conceived as the products of projection. Most photographs exploit a projective geometry approximately like linear perspective with ray-tracing, and an optical system of marking loosely analogous to the “color-to-color” method illustrated in the text.

  31. In terms of the geometry of projection lines, optical methods are characterized both by a perspectival organization of projection lines, and by a ray-tracing approach to their propagation through space. Systems can be optical with respect to geometry, but not markings, and vice versa.

  32. Hyman (2006) writes as if projected shape (for Hyman, “occlusion shape”) can always be defined relative to a point-sized projection source (p. 76), with projection lines that follow the behavior of light (p. 77). In addition, since Hyman’s Occlusion Shape Principle is modeled after optical projection to the eye, it doesn’t distinguish the elements of picture plane and projection source, which in the present framework make up a viewpoint. As a consequence, the principle fails for systems of anamorphosis, which involve projections where the projection source and picture plane are positioned at an acute angle with respect to one another. Since the Occlusion Shape Principle is invalid here, Hyman (2006, 93-98) is compelled to introduce an additional principle for the special case of anamorphosis. By contrast, the Projection Principle handles anamorphosis as a yet another system of depiction, albeit one which imposes distinctive constraints on viewpoint.

  33. To be precise, Kulvicki holds that for a system to be pictorial, it must exhibit “transparency” or near enough; see ch. 3, especially p. 76. Intuitively, transparency requires that a picture of a picture have the same syntactic structure as the original. In Kulvicki’s system, this amounts to the claim that pictures in a given system satisfy their own skeletal content. (See the Appendix for a more detailed discussion.) Suffice to note for now that perspective and parallel projections are transparent, while curvilinear systems are not (Greenberg 2013). Whether this marks a contrast with the Projection Principle depends on the scope of Kulvicki’s “near enough” clause. More vivid challenges are marking conditions used in edge- and topography-based line drawing. Such systems are stark violations of transparency, but standard cases for the Projection Principle.

  34. To spell out the analogy: the retinal surface is analogous to the picture plane; the focal point defined by the lens of the eye is analogous to the projection source; and the ray-like behavior of light which links the scene to patterns of activity in the retina resembles the straight lines of projection.

  35. The sense in which such laws are “assumed” by the visual system is an open question; it is unlikely that they are actually represented, but they may well be encoded in the normal operation of the system itself (Johnson 2020, 9-13). Either way, the inverse problem is intractable unless assumptions about the projective behavior of light are in some way respected.

  36. Communication, in the relevant sense, involves inter-personal coordination of information, and isn’t limited to the transfer of information. Thus an architectural plan might facilitate coordination among a group of builders discussing the best course of action, even when all parties were antecedently familiar with the plan. As an anonymous reviewer suggests, points parallel to those made in the text could also be made for intra-personal coordination. The stability and power of the visual system makes it possible for a single individual to exploit pictorial signs to extend their access to visual information over time, and in doing so, their capacities for visual reasoning.

  37. The theoretical situation is akin to one familiar within the philosophy of language. Various formal-logical definitions of the concept of language are available, but many of these formal systems are so abstract, complex, or bizarre that they resist the moniker “language” in any straightforward sense. Theorists have tended to respond (e.g. Chomsky 1965, Lewis 1975) by identifying some subset of these formal systems as the humanly usable languages.

  38. My proposal here bears some affinities to that of Kulvicki (2006, 76), since we both identify systems of depiction in terms of their relative proximity to a core class. But Kulvicki defines that class in terms of the formal property of transparency, while I define mine in terms of visual cognition. (See footnote 33 and the Appendix for more on transparency.) The two ideas come apart because systems can vary their proximity to transparency, without varying in their proximity to visual cognition, and vice versa. Systems like edge- and topography-based line drawing, for example, strongly violate transparency, but are closely related to visual cognition.

  39. This taxonomy is not meant to be exhaustive or exclusive. For example, Voltolini’s (2015, 16-22, 164-67) recent proposal weaves together elements from all three traditions.

  40. Hyman (2006, 111) describes his projection theory as “the defensible residue of the resemblance theory of depiction.”

  41. In the latter case, isomorphism is understood as a kind of abstract structural “match” between two spatially defined systems. Despite its apparent abstraction, isomorphism of this kind is just another form of similarity— similarity with respect to abstract, relational features.

  42. There are other kinds of account which might also be called “resemblance theories,” but which which cast the concept of similarity in quite different roles. One “meta-semantic” theory holds that for a system to be pictorial (or perhaps, iconic), similar pictures must express similar contents— hence the space of possible pictures is similar or isomorphic to the space of possible contents. (Thanks to John Carriero and Matthew Fulkerson for suggesting this idea.) The Projection Principle is, as far as I can tell, fully compatible with this proposal.

  43. The distinction between resemblance- and projection-based theories is sometimes clouded because projective invariants are superficially analogous to the properties held fixed in a resemblance theory. To be sure, projection does entail certain projective invariants— properties of a scene that always survive projection to the picture plane. But, in general, the relation of projection cannot be equated with that of sharing projective invariants, because nearly every form of projection imposes systematic differences between its relata that go beyond invariants. In addition, projective invariants do not even imply shared properties between pictures and scenes. For example, the property of line straightness is invariant under linear perspective projection: if an edge is straight in the scene, its projection will also be straight. Yet the reverse is not true. For there can be straight lines in the picture which are not projected from straight edges in the scene, but instead from curved edges rotated perpendicularly to the picture plane. So while line straightness is an invariant under perspective projection, it is not in general a property shared by a picture and its content. Thus projective invariants provide little help for resemblance theories.

  44. This idea is expressed differently by the different authors. Peacocke (1987) writes of shapes of regions in the visual field (p. 385). Budd (1996) uses the idea of two-dimensional aspects of the visual field (pp.158-159), understood as a two-dimensional abstraction of three-dimensional perceptual space. And Hopkins (1998, ch. 3) defines outline shape as the solid angle subtended by the object relative to a viewpoint. This is not quite the same as a shape on the visual field, but the difference is inconsequential, since every visual field shape determines a unique outline shape, and vice versa.

  45. Note that the relevant experience of similarity is between the outline shape of P and the outline shape of an F— not between P (simpliciter) and the outline shape of an F.

  46. In fact, Hopkins has various ways of reaching this conclusion. One is to allow that the features of the picture plane which are the subject of experienced resemblance are only those which are especially salient; and for some systems, only rough shapes, not fine-grained spatial detail, is salient. The result is that experienced resemblance may be to a table or even a cube, but not one with narrowly defined spatial dimensions (Hopkins 2003, 153). A second tack is to allow that, under certain circumstances, the actual content of a picture can be less determinate than experienced resemblance (Hopkins (1998, ch. 6). Hopkins(2003, 155) indicates that this is his preferred approach to the problem raised by parallel projection.

  47. Hopkins’ conclusion here is not that pictures in the parallel system are entirely indeterminate or devoid of substantive content. A parallel picture like M may be experienced as resembling a cube in outline shape; but it cannot be experienced as resembling anything more specific in outline shape— a cube of certain dimensions, or a cube in a certain orientation, for example. An experience of resemblance to one of these more determinate situations would require a more precise match in outline shape. So the perceptual resemblance theorist can allow that, for pictures in parallel projection, their high-level content may be relatively fleshed-out, even though there would still be considerable indeterminacy about low-level spatial content.

  48. By systems of “indeterminate geometry,” I have in mind especially those which apply aspects of known pure projection systems, but inconsistently. This is characteristic of late medieval European depictions of space, prior to the codification of perspective during the Renaissance.

  49. That said, many forms of caricature do blend visual exaggeration with stylization, in which case the comments on stylization below would apply.

  50. A different diagnosis holds that such images are “distortions” of their subjects, which is to say, they are systematically and grossly inaccurate (cf. Voltolini 2015, 63-64). This would bring the analysis of stylization in line with my analysis of caricature, and consequently obviate the theoretical challenge facing the projective analysis. Yet stylization seems to me too ubiquitous, and its uses too often sincere and factual, to dismiss with a widespread error-theory. Stylized images are used to record fact in a way that caricatures are not. For better or worse, I think that projection semantics is stuck with the problem of stylization.

  51. That said, the schemas adopted in stylized depiction are not arbitrary. Instead, they are loosely derived from projections from stereotyped viewpoints. The bubble-figure in the crosswalk sign illustrates this principle, for the overall organization of legs, arms, torso, and head correspond loosely to the positions those objects would have in a pure projection from a side view. (A projection from above or below would have a rather different layout.) In general, in a stylized system, the schema for a type of object X is derived from the pure projection(s) of a standard instance of X from some canonical viewpoint (or viewpoints). See Hagen (1986, ch. 6) and Willats (1997, 212-214) for discussion of canonical viewpoints.

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Acknowledgments

This paper has grown out of years of discussion with dozens of friends, colleagues, teachers, students, and audiences. I owe far too many personal and intellectual debts to enumerate here. Parts of this research were supported by NSF IGERT DGE 0549115.

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Appendix: Bare bones content and the Projection Principle

Appendix: Bare bones content and the Projection Principle

In this appendix I argue that, in the context of Kulvicki’s (2006, ch. 3) claims about pictorial structure, the constraint he offers on pictorial content is significantly weaker than the Projection Principle.

To begin, I restate the Projection Principle within the framework of possible-world semantics. Where S is a system of depiction and c is a context, ⟦⋅⟧S, c is an interpretation function mapping pictures to sets of scenes, understood as 〈world, viewpoint〉 pairs. For a system S, projS(⋅) is the characteristic projection function of S, mapping scenes to pictures. Then according to the Projection Principle, given a system S and context c, for any picture P in S:

$$ \forall w \forall v:\ \text{if}\ \langle w, v \rangle \in \llbracket P \rrbracket_{S,c}\ \text{then}\ proj_{S}(w,v) = P $$
(1)

Next, Kulvicki posits a basic layer of pictorial content, called bare bones content, which in turn is defined in terms of projection: “whatever scenes could have resulted in a particular [linear perspective picture] via a perspective projection count as parts of the [bare bones] content of the picture Kulvicki (2006, 59).” This formulation, in fact, expresses only the sufficiency of projection for bare bones content, but the necessity clause is clearly implied as well. So we can think of the bare bones content of a picture as the set of all scenes which project to the picture, relative to a sytem. Letting BS(ϕ) be the bare bones content of ϕ, relative to system S, we can render Kulvicki’s definition within a scene-based semantics, as follows:

$$ B_{S}(P) = \{ \langle w, v \rangle ~|~ proj_{S}(w,v) = P \} $$
(2)

Kulvicki further holds that bare bones content constrains a picture’s overall content. That is:

$$ \llbracket P \rrbracket_{S,c}\subseteq B_{S}(P) $$
(3)

Put together, these entail:

$$ \llbracket P \rrbracket_{S,c}\subseteq \{ \langle w, v \rangle ~|~ proj_{S}(w,v) = P \} $$
(4)

which is equivalent to the Projection Principle.

However, Kulvicki is motivated to preserve a property of systems he terms transparency (Kulvicki 2006, ch. 3). Intuitively, transparency is preserved when a picture of a picture is (syntactically) the same as the original. Formally: a system S is transparent iff, for any representations R and R from S, if R satisfies the content of R, then R and R are of the same syntactic type (p. 53). Kulvicki seeks to establish that core systems, including linear perspective, satisfy this constraint. Yet he acknowledges that the relevant level of content is not plausibly the full, “fleshed-out” content of a picture (p. 52). But it cannot even be its bare bones content, as defined above, because if P is projected from P relative to an oblique viewpoint, P will satisfy P’s bare bones content, but the two have manifestly different metrical structures (pp. 53-55).

To solve this problem, and save transparency, Kulvicki proposes to revise the operative notion of pictorial syntax. Rather than defining pictorial syntax metrically, as is standard, he holds that pictorial syntax consists only of projective invariants. Since pictorial syntax is the only feature of a picture which determines bare bones content, it now follows that all pictures which share projective invariants have the same bare bones content (Kulvicki 2006, 57).

To compare the resulting idea with the Projection Principle, I’ll translate Kulvicki’s proposal back into the framework of this essay in which pictures are treated metrically. Kulvicki’s (final notion of) bare bones content can be defined as the set of all scenes which project to something which shares projective invariants with the target picture. Since projections of a picture always share projective invariants with it, I can restate the final definition of bare bones content, now B, as follows. Here I’ll allow that a picture P defines a world wP which contains only that picture.

$$ B^{*}_{S}(P) = \{ \langle w, v \rangle ~|~ \exists v': proj_{S}(w,v) = proj_{S}(w_{P},v') \} $$
(5)

The resulting constraint on content can now be set side by side the Projection Principle:

$$\text{if}\ \langle w, v \rangle \in \llbracket P \rrbracket_{S,c}\ \text{then}\ \exists v': proj_{S}(w,v) = proj_{S}(w_{P}, v')\qquad \text{(Kulvicki's Constraint)} $$
(6)
$$ {\kern-.4pc}\text{if}\ \langle w, v \rangle \in \llbracket P \rrbracket_{S,c}\ \text{then}\ proj_{S}(w,v) = P{\kern5.8pc} \text{(Projection Principle)} $$
(7)

The difference here is significant. Consider the diagram below in which pictures B and C are projected from A, thus share projective invariants with it:

figure p

Intuitively, A, B, and C have different spatial content. The Projection Principle delivers this result: since each image can be projected from scenes which the others could not, they do not have the same skeletal content, hence they do not have the same content. The content of A, for example, is compatible with scenes containing a cube, but this is not the case for the contents of B or C.

But Kulvicki’s Constraint cannot capture this claim. Let L be the system of perspective projection. Perspective projection from a plane is reversible, modulo issues of scale; so if there is some v such that \(proj_{L}(w_{P}, v)=P'\), then there is some v such that \(proj_{L}(w_{P'},v')=P\). As a consequence, Kulvicki’s Constraint puts the same extensional requirement on each of A, B, and C. This is to be expected, given the background theory: if all three have the same pictorial structure, and structure determines bare bones content, then Kulvicki’s Constraint can’t distinguish between them. Thus the Projection Principle issues in a more determinate constraint that Kulvicki’s Constraint.

While the semantic differences between A, B, and C are in principle compatible with Kulvicki’s view at the level of what he calls “fleshed-out content,” only the Projection Principle actually delivers the relevant result. Since neither Kulvicki nor I have offered a detailed account of how fleshed-out content is determined, the Projection Principle goes farther, given the theoretical resources available, towards explaining these differences in a systematic manner.

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Greenberg, G. Semantics of Pictorial Space. Rev.Phil.Psych. 12, 847–887 (2021). https://doi.org/10.1007/s13164-020-00513-6

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