Predicative subject matter

Abstract

The notions of subject matter and aboutness have been objects of considerable attention among philosophers over the last few years. Current theories of subject matter take sentences to be the primary bearers of subject matter: “sentences have aboutness properties if anything has” (Yablo, Aboutness, Princeton University Press, 2014). However, some subsentential expressions can also be thought of as being about something. Moreover, it appears that the subject matters of sentences depend in a systematic way on the aboutness properties of their subsentential components. In this paper, we focus on the question of what predicates are about. We provide an account of predicative subject matter in which subject matters are assigned to predicates in a natural way, and which can be smoothly integrated with some existing accounts of sentential subject matter. We also argue that the notion of predicative subject matter is a worthy object of study, both within the current debate on subject matter and in its own right.

1 Introduction

In recent years, philosophers and logicians have been devoting considerable effort to elucidating the elusive notions of subject matter (SM) and aboutness (Fine, 2016, 2020; Yablo, 2014; Hawke, 2018; Berto, 2018; Moltmann, 2018; Plebani & Spolaore, 2021). Yablo introduces the notion of aboutness in this way:

Aboutness is the relation that meaningful items bear to whatever it is that they are on or of or that they address or concern (2014, 1)

Aboutness, then, is the relation between meaningful items and their SM. But which meaningful items? The standard answer is sentences:

One [referee] asked, reasonably enough, what subject-matters are “of”, on my account. I think the answer is sentences in context, as suggested originally by Kaplan (Yablo, 2014, ix)

sentences have aboutness properties, if anything does (Yablo, 2014, 1)

Accordingly, many existing treatments of aboutness (see, e.g., Yablo, 2014; Fine, 2016; Hawke, 2018) assign a SM only to sentences. This is particularly clear in the case of truthmaker semantics, where the SM of a complex sentence depends on the SMs of its subsentences, but the assignment of SM to atomic sentences is taken as primitive, and no SM is assigned to subsentential expressions.¹

However, the fact that sentences are about something does not entail that sentences are the only items endowed with aboutness properties. On the one hand, items that are larger than sentences should be assigned a SM, too: “[b]ooks are on topics” (Yablo, 2014, 1), and e-mails have a subject (Osorio-Kupferblum p.c.). On the other hand, items that are smaller than sentences also seem to have aboutness properties. In this paper, we focus on the aboutness of predicates, that is, on predicative SM.²

The view that predicates are about something is plausible. To appreciate this point, contrast them with a class of expressions that are generally taken to lack aboutness properties: Boolean connectives (Berto, 2022). At an intuitive level, it makes little sense to ask what a conjunction is about, or whether it is about the same thing as a disjunction. The situation with predicates looks very different: there is nothing absurd in asking what a predicate like “red” is about, or whether it is about the same thing as “round”. In fact, our approach to predicative SM comes with plausible and systematic answers to these kinds of questions. Moreover, the fact that predicative SM has been largely ignored in the literature so far does not depend on a principled opposition to assigning a subject matter to predicates but, rather, on broadly methodological considerations. At an initial stage in the formal study of aboutness, it makes perfect sense to operate within simplified semantic frameworks, and restrict attention to the sentential level. However, this is not a reason for neglecting the SM of subsentential expressions.

The key reason to be interested in predicative SM is that there seems to be a regular connection between the SM of sentences and the predicates they contain. “The table is red” and “The table is round” have different aboutness properties, and a natural explanation of this fact is that “red” and “round” are about different topics. An attractive feature of our proposal is that, as we shall see, it smoothly combines with two influential approaches to sentential SM (Lewis, 1988a, 1988b; Plebani & Spolaore, 2021) in a way that accounts for this connection.

At a first approximation, in our view, the SMs of predicates are aspects of things (sometimes we use “respect” instead of “aspect” when it improves readability). Examples of aspects of things are weight, color, size, shape, velocity, but also mood, emotion, and so on (we use boldface to indicate aspects). Weight is the respect in which two things are alike when they weigh exactly the same, color is the respect in which two things are alike when their color is exactly the same, and so on. Let us stress that our view is different from, and alternative to, a conception of predicative SM that might sound at least initially plausible, that is, simply identifying the SMs of predicates with the properties they express. (On the difference between aspects and properties, see below, p. 5).

There are different possible formal approaches to aspects of things.³ In this paper, we take for granted a Lewis-inspired (1988a, 1988b), possible-worlds treatment of aspects, which makes it easy to combine and compare our proposal with extant theories of subject matter.

Aspects can be considered at different levels of resolution. When we say (in English) that two things have the same color, we can mean that they have the same basic color (e.g., they are either both red, or blue, or green ...), or that they have the same color shade (they are either both purple, or navy blue, or mint green ...), or even that they have exactly the same color. These are different claims, and the difference is important. Basic color is, intuitively, a coarser-grained version of color shade, and both are coarser-grained versions of color. We shall speak of different categories of an aspect to mean coarser-grained versions of that aspect. Thus, basic color and color shade are categories of color, weight in kilos and weight in grams are categories of weight, and so on.

In our full-fledged proposal, we distinguish between broad and narrow predicative subject matter: the broad SMs of predicates are aspects of things and their narrow SMs are categories of those aspects. Thus, for instance, “red” and “green” have color as their broad SM and basic color as narrow SM. We assume that each language comes with a stock of basic aspects and categories.

The idea that predicates are about aspects/categories of things is based on independently plausible considerations. Here are some of them.

• Predicates are naturally divided into families: we tend to put “red” and “green” in one family, “round” and “square” in a different family, “large” and “small” in yet another family, and so on. An explanation of this division is in terms of subject matter: we put two predicates in the same family when they are about the same aspect or category. We classify “red” with “green” because they are both about the (basic) color of things, “round” with “square” because they are both about the shape of things, and so on.

• There is wide evidence that children learn the meaning of adjectives based on a previous ability to keep track of and categorize the corresponding aspect—that is, the ability to partition things based on identity or closeness in that aspect. For instance, children start keeping track of and categorizing color—that is, grouping objects and surfaces based on chromatic identity or closeness—very early in their development, and prelinguistic chromatic categories form the basis for the subsequent learning of color predicates, at least in the sense that color predicates are learned faster and easier when they fit into these prelinguistic categories (see, e.g., Forbes & Plunkett, 2020 for an overview of findings in early color word acquisition). Similarly, the ability to keep track of and categorize shapes plays a key role in lexical development and, more specifically, in the acquisition of shape predicates (see, e.g., Gershkoff-Stowe & Smith, 2004; Verdine et al., 2016).

• In genuine cases of disagreement, people appear to disagree about the same subject matter. Delia claims that the table is round, and Tabitha replies that it is oval. They clearly disagree, and they appear to be speaking about the same thing. Indeed, their disagreement is so clear and straightforward precisely because they are speaking about the same thing. But what are they speaking, and disagreeing, about? Intuitively, they disagree on the shape of the table.⁴

• Philosophers and semanticists are interested in the subject matters of sentences, in addition to their truth conditions, because they think that the SMs of sentences can do explanatory work which cannot be done by appeal to standard semantic values alone (Yablo, 2014, p. 2). More generally, one can hold that the meaning of any expression is the result of two factors: its (standard) semantic value, i.e., its contribution to the truth conditions of the sentences that contain it, and its subject matter. (This is the key assumption behind so-called two-component semantics; see, e.g., Berto, 2022, ch. 2 for an overview). But there is no explanatory bonus if an expression’s SM is simply a function of its semantic value. Thus, for at least some expressions, SMs ought to be independent of semantic values. Our proposal clearly obeys this desideratum when it comes to predicates: the standard semantic values of (unary) predicates are properties, and in general, properties do not determine the corresponding aspects/categories.

Here is the plan of the paper. Section 2 specifies the notion of aspect of things and defines a few interesting relations between aspects; Section 3 clarifies the distinction between aspects and categories; Section 4 discusses the interplay between predicative and sentential subject matters; Section 5 concludes.

2 Aspects of things

We often say that two objects are identical in certain ways but different in others. ${{\square }}$ and ■ are identical in shape but different in color. The ways in which two things can be identical or different, in this sense, are what we call aspects of things. Thus, there is a respect, shape, in which ${\square }$ and ■ are identical, and another respect, color, in which they are different.

Now let us ask: in what circumstances would we agree that two things are identical, or different, in some specific respect? A natural answer is that they are identical (different) in a specific respect when they (do not) share a certain property. The answer might suggest that aspects are just properties. But this suggestion cannot be correct. Aspects are related to properties but are not identical to properties.

To see why, suppose that Delia and Tabitha share the property of being happy. If so, we can say that they are identical in their mood. But clearly, the aspect mood is not the same thing as the property of being happy, for two people can be in the same mood without being happy. They can be both angry, or sad, or melancholic. In fact, there is no specific mood property that two individuals must share in order to be identical in their mood. Rather, there is a family of mood properties, and being in the same mood is sharing one or another of these properties. What holds for the aspect mood holds for any other aspect.

For any aspect $\textbf{s}$, there is a corresponding equivalence relation, namely, the relation $\equiv _\textbf{s}$ of being identical in respect $\textbf{s}$ ($\textbf{s}$-equivalence), and it is safe to simply identify each aspect with the corresponding equivalence relation (see Lewis, 1988a, 1988b). Now we can summarize the above discussion by saying that, for any aspect $\textbf{s}$, there is no specific $\textbf{s}$-property $\mathcal {P}$ such that two objects must both possess $\mathcal {P}$ in order to be $\textbf{s}$-equivalent. Rather, there is a family of two or more incompatible $\textbf{s}$-properties $\mathcal {P}_1,\mathcal {P}_2\dots$ such that two objects are $\textbf{s}$-equivalent precisely when they share one of those properties. In other words, for any such family of $\textbf{s}$-properties $\mathcal {P}_1,\mathcal {P}_2\dots ,$ the following principles hold:

• $\lnot \exists n \forall x \forall y (x \equiv _\textbf{s}y \leftrightarrow (\mathcal {P}_n x \wedge \mathcal {P}_n y))$

• $\forall x \forall y (x \equiv _\textbf{s}y \leftrightarrow \exists n(\mathcal {P}_n x \wedge \mathcal {P}_n y))$

It is important to consider the possibility that some object has none of the properties that are relevant to an aspect, for some aspects do not sensibly apply to all kinds of things. It makes no sense to ask about the color of a neutron or the spin of an armchair. We shall familiarly speak of the domain of an aspect to mean the set of objects the aspect sensibly applies to.

Aspects induce partitions of their domains. Consider a famous Fregean example. An aspect of lines is their direction. Lines have the same direction when they are parallel. Therefore, the aspect direction can be also identified with the partition of the set of lines that assigns two lines to the same cell if and only if they are parallel. Similarly, an aspect of cars is their color. The aspect color can be identified with the partition in which cars are in the same cell if and only if their color is the same.

2.1 Aspects in intension

We have just said that aspects can be identified with equivalence relations between objects (or, alternatively, with the corresponding partitions). In an intensional framework, however, the same object can have different properties in different possible worlds. A natural way to take this possibility into account is to equate aspects with equivalence relations between pairs (o, w), where o is an object and w is a possible world. Color, for instance, can be seen as the equivalence relation $\equiv _\textbf{color}$ such that $(o,w)\equiv _\textbf{color}(o',w')$ when o has the same color in w as $o'$ in $w'$. For the sake of readability, sometimes we will speak simply of “objects” to refer to object-world pairs.⁵

We shall speak of the domain of an aspect (in intension) $\textbf{s}$ to indicate the set of object-world pairs (o, w) such that $\textbf{s}$ sensibly apples to o, as it is in w. The set of pairs (o, w) that are not in the domain of an aspect $\textbf{s}$ is called the garbage set of the aspect. For instance, and arguably, any pair (o, w) such that o is a neutron is in the garbage set of color. For the sake of formal convenience, we assume that, if two pairs (o, w) and $(o',w')$ are both in the garbage set of an aspect $\textbf{s}$, then $(o,w)\equiv _\textbf{s}(o',w')$. In other words, we identify any aspect $\textbf{s}$ with an equivalence relation that is defined for all object-world pairs, including those belonging to the garbage set of that aspect.

We stress that we make this choice just to keep things as formally simple as possible. An analogy with Lewis’s account of subject matter can be helpful here. According to Lewis (1988a, 161), two worlds are “alike with respect to” the SM the 17th century if and only if either the 17th century of one world is a perfect duplicate of the the 17th century of the other, or they both lack a 17th century. Similarly, in our account of aspects, two objects are alike with respect to the aspect mass if they have the same mass or if they both lack mass. Objects that belong to the garbage set of an aspect are vacuously alike in that respect: 4 and 17 have the same mass in the sense that they have no mass at all. All this is compatible with maintaining that we cannot sensibly talk about the mass of a number or the 17th century of worlds that don’t have a 17th century. In fact, the reason why we introduce the notion of garbage set is precisely to single out the case in which two objects are vacuously alike in a certain respect as a special case.⁶

It is possible to show that there is a one-one correspondence between aspects (in intension) and families of incompatible properties in intension ($\mathcal {P}_1,\mathcal {P}_2\dots$), understood as functions from worlds to sets of objects (i.e., to extensions of monadic predicates). The reason is simple: an equivalence relation between object-world pairs induces a partition of the domain of object-world pairs, and there is a one-one correspondence between the cells of this partition and properties in intension. Consider, for instance, the two-cell partition such that (o, w) is in cell A if and only if o is red in w, and (o, w) is in cell B otherwise. Clearly, cell A determines a function that assigns, to each possible world, the set of objects that are red in that world—that is, the property red. And it works also the other way around: given a property in intension, we can define a class of object-world pairs. For instance, given the property red, we can define a class of object-world pairs such that (o, w) belongs to the class if and only if o is red in w. Clearly, this class coincides with the above cell A. Thus, aspects in intension can also be seen as families of properties.

2.2 Aspects and their parts

Lewis (1988a) defines a variety of quasi-mereological relations between sentential subject matters (more on sentential SMs in § 4). Within the framework introduced thus far, it is immediate to extend Lewis’s definitions to predicative subject matters (aspects of things). Let us start with the relation of parthood (where $x,y\dots$ vary over object-world pairs):

Definition 1

(Parthood) $\textbf{s}'$ is part of $\textbf{s}\Leftrightarrow$ for any x, y, if $x\equiv _\textbf{s}y$, then $x\equiv _{\textbf{s}'} y$.

Intuitively, Definition 1 says that aspect ${\textbf{s}^\prime}$ is part of aspect $\textbf{s}$ when $\textbf{s}$-equivalence entails ${\textbf{s}^\prime}$-equivalence. For instance, upper body features is part of body features because, if two people have exactly the same body features in certain possible worlds, then they also have the same upper body features in those worlds. If we think of aspects as partitions, we can also say, equivalently, that $\textbf{s}^\prime$ is part of $\textbf{s}$ if and only if each cell of $\textbf{s}^\prime$ is a union of one or more cells of $\textbf{s}$. Intuitively, the cells of a part are “ larger” than those of the whole—or, if you prefer, the whole aspect is more discriminating than its parts.

Another interesting relation between aspects is orthogonality:

Definition 2

(Orthogonality) $\textbf{s}$ is orthogonal to $\textbf{s}' \Leftrightarrow$ for any x, y, there exists z such that $x\equiv _{\textbf{s}} z$ and $z \equiv _{\textbf{s}'} y$.

If we think of aspects as partitions, we can also say that $\textbf{s}$ is orthogonal to $\textbf{s}'$ if and only if every $\textbf{s}$-cell intersects every $\textbf{s}'$-cell. Intuitively, orthogonal aspects are metaphysically independent. For instance, arguably, color is orthogonal to shape, for the color of a thing is metaphysically independent of its shape: every color is compatible with every shape.⁷

In this framework, the notion of the composition (sum) of two aspects (qua equivalence relations) can be defined as their intersection: $\textbf{s}{}+\textbf{s}' =_{\mathrm {\,def}} \textbf{s}\cap \textbf{s}'$. For instance, shape + color is the respect in which (o, w) is identical to $(o',w')$ if and only if o in w has both the same color and the same shape as $o'$ in $w'$. In other words, shape + color is the largest equivalence relation that entails both shape- and color-equivalence. Clearly, by Definition 1, both shape and color are part of shape + color.

Complex aspects such as shape + color raise a potentially tricky question: how are we to define the garbage set of $\textbf{s}+\textbf{s}'$ in terms of the garbage sets of $\textbf{s}$ and $\textbf{s}$ $'$? Intuitively, the garbage set of an aspect collects all the objects that are undefined relative to that aspect. The garbage set of mass collects all the objects that lack mass and that of parity, all the objects that are neither odd nor even. Now, it is natural to assume that if an object is undefined relative to an aspect $\textbf{s}$, then it is also undefined relative to the sum of $\textbf{s}$ with other aspects. If natural numbers are undefined relative to mass, then they are also undefined relative to ${\textbf {mass}}+{\textbf {parity}}$, although they are defined relative to parity. If so, then the garbage set of a complex aspect $\textbf{s}+\textbf{s}'$ is naturally defined as the union of the garbage sets of $\textbf{s}$ and $\textbf{s}$ $'$.

Two consequences of this definition are worth noting. First, if $\textbf{s}$ and $\textbf{s}$ $'$ are endowed with complementary garbage sets, then any object is in the garbage set of $\textbf{s}+\textbf{s}'$. Thus, for instance, if any object endowed with mass lacks parity, the complex aspect mass $+$ parity does not sensibly apply to any object. This result is in harmony with the literature on the logic of nonsense (see, e.g., Halldén, 1949; Goddard & Routley, 1973). Second, the relation between aspects and garbage sets turns out to be different in the case of simple and complex aspects. In the case of simple aspects, the garbage set is bound to coincide with a cell of the corresponding partition. In contrast, the garbage set of a complex aspect might be the union of two or more cells of the corresponding partition.⁸

Many ordinary aspects of things can be analyzed into component aspects or “dimensions” (see, e.g., Gärdenfors, 2004: § 1.5 and 2014: § 2.1). For instance, color can be analyzed as the composition of three sub-aspects, namely hue, brightness and saturation. Something similar holds for sound (which can be analyzed as pitch + loudness + timbre), taste (salt + bitter + sweet + sour + umami), and emotion (arousal + value).

2.3 Aspects and the decomposition of identity

Just as hue, brightness, and saturation are dimensions of color, any aspect can be seen as one dimension of the most discriminating aspect of all, identity, the aspect $\textbf{s}$ such that, for any individuals x, y, $x \equiv _{\textbf{s}} y \leftrightarrow x=y$. Seeing aspects as dimensions of identity helps to understand what aspects are for.

Suppose you are a secret agent, and you want to identify an enemy spy. The question (subject matter) you are dealing with is Who is the spy?. A strategy to tackle a difficult problem is to break it down into several simpler problems, in such a way that combining the solutions of the simpler problems yields a solution to the original, harder problem. A similar idea applies to questions. In order to address the question Who is the spy?, it might be useful to break it down into several easier questions, in such a way that combining the answers to the easier questions yields an answer to the harder one. Questions can be identified with equivalence relations or the corresponding partitions (see § 4.3), so the idea of breaking down a question into several simpler ones is the idea to decompose an equivalence relation into several equivalence relations that are parts of the original one (in the sense specified in the previous section). In the case at hand, you might want to break down the question Who is the spy? into easier questions, each corresponding to an aspect of the spy (e.g., height, body type, eye color, accent and so on).

3 Aspects and categories

Many aspects correspond to extremely fine-grained partitions of their domain. For two objects to be equivalent in their mass aspect, for instance, they must be endowed with exactly the same amount of mass: any minimal difference, even on the order of thousandths of a gram, will break the equivalence. From a metaphysical viewpoint, this is all well and good. However, such extremely fine-grained partitions are not well suited for representing how finite agents organize information.

By a category of an aspect we mean, roughly, a coarser-grained version of that aspect. Consider mass. We can categorize objects based on their mass in different ways. For instance, we can count objects as equivalent in mass when they have the same mass rounded to the nearest gram. This mass-based categorization corresponds to a category of mass, let us call it mass in grams. Other categories of mass can be either finer-grained (e.g., mass in milligrams) or coarser-grained (e.g., the two-class category heavier than 1 kg|lighter than 1 kg).⁹ We assume that each category is a category of exactly one aspect.

Just like aspects, categories can be conceived with reference to a specific domain of objects or in intension. Officially, categories (in intension), like aspects, are equivalence relations on, or partitions of, the domain of object-world pairs. However, for the ease of exposition, we shall often speak of categories as partitions of a given domain of objects.

Aspects and categories correspond to two different ways to split up and organize information. Consider again the example from the previous section. You are a secret agent, and you want to identify an enemy spy. The question (subject matter) you are dealing with is Who is the spy?. You start by breaking it down into easier questions, corresponding to the spy’s relevant aspects (height, body type ...). But once you have done this, it is still important to decide how to categorize those aspects. How are you going to split up the aspect height? There are different, more or less fine-grained categories you may choose (e.g., compare short|average|tall with height in inches). But regardless of the choice you will actually make, some choice has to be made, for you can only specify the height of the spy at some level of granularity. See Table 1 for other examples of aspects and corresponding categories.

Table 1 Aspects and categories

3.1 Formal constraints on categories

The notion of a category can be made more precise by imposing formal constraints on categories and their relations with aspects. Due to space limitations, here we shall introduce only two minimal necessary conditions on the categories of an aspect $\textbf{s}$, and we simply mention other two possible, more specific constraints.

The first condition is parthood: the categories of an aspect are parts of the aspect.

Parthood.:

$\textbf{c}$ is a category of an aspect $\textbf{s}$ only if $\textbf{c}$ is part of $\textbf{s}$.

This is a very natural requirement: if two objects are equivalent in an aspect (say, mass), they must be equivalent also in any category of that aspect (mass in grams, mass in pounds, and so on).

Not all the parts of an aspect are, intuitively, categories (coarser-grained versions) of that aspect. Left ear features is part of head features but, plausibly, it is best thought of as a restricted version of head features, for its domain is smaller: left ear features does not apply to entities that do have a head but have no ears. The role of our second condition is that of distinguishing the categories of an aspect $\textbf{s}$ from the parts of $\textbf{s}$ that come with restrictions on its domain:

Domain identity.:

$\textbf{c}$ is a category of an aspect $\textbf{s}$ only if $\textbf{c}$ and $\textbf{s}$ share the same domain.

The conditions introduced thus far concern all aspects. There are two other conditions that are worth considering, as they help understand the intended relation between aspects and categories, although they only concern aspects endowed with specific features. As mentioned above, some aspects are complex in the sense that they can be analyzed as the sum of different dimensions—for instance, color as hue + brightness + saturation. The first specific constraint on categories is based on the idea that, if we categorize a complex aspect, then we also get a categorization of its dimensions:

Downward-analysis.:

If $\textbf{s}$ is a complex aspect that can be analyzed into coordinates ${\textbf {x}}_{1},\dots ,{\textbf {x}}_{n}$ (with $n>1$), and $\textbf{c}$ is a category of $\textbf{s}$, then $\textbf{c}$ can be analyzed into categories of ${\textbf {x}}_{1},\dots ,{\textbf {x}}_{n}$.

For instance, basic color is a category of color also because we can break it down into coarser-grained versions of hue, brightness, and saturation.

Finally, many aspects are (say) structured in the sense that they come with an associated ternary relation of comparative closeness or betweenness on their domains. For instance, Kansas City lies in between New York City and Los Angeles in terms of distance, while Los Angeles lies in between New York City and Kansas City in terms of population. Intuitively, y lies in between x and z in $\textbf{s}$ ($B_\textbf{s}(xyz)$ in symbols) when y is strictly closer to x and z in $\textbf{s}$ than x is to z.¹⁰ Now, and this is our last condition on categories, it is very natural to require that the categories of a structured aspect $\textbf{s}$ do not jeopardize $\textbf{s}$’s underlying betweenness relation. For instance, if $\textbf{c}$ is category of population and New York City counts as c-equivalent to Kansas City, then we want New York City to count as c-equivalent to Los Angeles, too. This natural condition can be laid down as follows:

Betweenness.:

For any structured aspect $\textbf{s}$ and any category $\textbf{c}$ of $\textbf{s}$, we have that $B_\textbf{s}(xyz)$ and $x\equiv _\textbf{c}z$ entail $x\equiv _\textbf{c}y$.¹¹

4 The interplay between predicative and sentential subject matter

4.1 Existing accounts of subject matter

Extant theories of subject matter present a recursive procedure to assign SMs to sentences. A typical example is truthmaker semantics (see, e.g., Yablo, 2014; Fine, 2020; see also above, footnote 1), where the positive SM of a sentence is identified with the set of its truthmakers, the negative SM of a sentence is identified with the set of its falsemakers, and the overall SM of a sentence is identified with some combination of the positive and negative SM. The assignment of SMs to sentences starts with a pair of functions that assign truth(false)makers to atomic sentences, and then a set of rules determines how the truth(false)makers of a complex sentence depend on the truth(false)makers of its immediate subsentences. The function that assigns a SM to the atomic formulas is completely unconstrained (see also above, footnote 1).

An exception is Berto (2022), which assigns a SM to individual constants and sentences and defines the SM of a n-ary predicate as a function from n-tuples of SMs to SMs. This sits well with the idea that SMs should be compositional: the SM of an atomic sentence should result from applying the SM of its predicate to the SM of its individual constant(s). However, merely saying that SMs are compositional in this sense tells very little about the nature of SMs. In contrast, our account provides a natural characterization of the SM of predicates that also explains why the SM of a predicate determines a function from objects (which can be taken as the SMs of names) to the SM of sentences.

Theories of sentential SM pivot around two key issues. The first is the typology issue: the issue of specifying what kind of things sentential SMs are. The second is the pairing issue: the issue of specifying how each sentence ends up having the SM(s) it has. An important bonus of our proposal is that it smoothly combines with extant approaches to both issues. More specifically, our proposal combines with Lewis’s (1988a, 1988b) theory of sentential SM to yield a compositional approach to the typology issue, and with the recent neo-Lewisian proposal in Plebani and Spolaore (2021) to yield a compositional approach to the pairing issue.

4.2 The subject matters of predicates

For most purposes, it is natural to identify the SM of a predicate with a category. Think of a color predicate like “red”. It expresses the property of being red, which is a rather coarse-grained property: two objects can both be red without having exactly the same color: one might be magenta, or purple, and the other might be scarlet, or crimson. Therefore, “red” does not correspond to a cell in the partition induced by the aspect color. Rather, it corresponds to a cell in the partition induced by the category basic color. Or if you prefer, red is one of the properties in the category basic color, while it is too coarse grained to belong to the aspect color.

There are, however, considerations that might pull in a different direction. Different languages categorize the color continuum in different ways. For instance, the Welsh basic color predicate “gwyrrd” is close in meaning to the English “green,” but it is generally used to indicate relatively brighter or fresher greens (see, e.g., Geeraerts, 2009, 130). Thus, English and Welsh have different underlying basic color categories. However, it might be natural to say that “gwyrrd” and “green” are both about the same aspect of reality, viz., color.

Let us say that the narrow SM of a predicate is its category, while the broad SM is the aspect corresponding to that category. Relatedly, we distinguish between two notions of relevance between predicates, namely narrow and broad relevance: predicates p and q are narrowly (broadly) relevant to each other if they have the same narrow (broad) SM. As we shall see, the notion of relevance helps connect the subject matter of a sentence to the subject matter of the predicates it contains,¹²

4.3 From predicative to sentential subject matter: the typology issue

We have said above that our proposal naturally combines with Lewis’s theory of sentential SM to yield a compositional solution to the typology issue (the issue of specifying what kind of objects sentential SMs are). In Lewis’s approach, a sentential SM is an aspect of worlds, conceived as an equivalence relation on the set W of all possible worlds (viz., the logical space), or the corresponding partition. For instance, Delia’s mood is the equivalence relation two worlds are in when Delia is in the same mood in both.

In semantics, it is common to identify a question with the set of all possible complete answers to a certain interrogative sentence.¹³ If we take propositions to be sets of possible worlds, then also Lewisian SMs can be conceived as questions. For instance, Delia’s mood is the set of all possible answers to What is Delia’s mood?. The idea that SMs are strictly connected to questions is only natural. It is the reason many of us tell students to focus on research questions when they are in search of a topic for their dissertation. Lewis’s account of SM vindicates this natural idea.

Lewisian sentential SMs are aspects of worlds, while in our proposal, predicative SMs are aspects of things, that is, of inhabitants of possible worlds. Now, it is easy to see that aspects of things induce functions from objects to aspects of worlds: we can map each aspect of things $\textbf{s}$ with a function $f_\textbf{s}$ such that, for any object o, $f_\textbf{s}(o)=\{(w,w'):(o,w)\equiv _\textbf{s}(o,w')\}$. For instance, if o is Delia and $\textbf{s}$ is mood, then $f_\textbf{s}(o)$ is the equivalence relation two worlds are in when Delia has the same mood in both, that is, it is the aspect Delia’s mood. Thus, intuitively, aspects of things combine with individual objects to yield aspects of worlds, just like properties combine with objects to yield propositions.

To summarize, aspects of things induce functions from individual objects to aspects of worlds. As a result, if we specify an object and an aspect of things (the kind of entity we assign to predicates as their SMs), we immediately get an aspect of worlds (the kind of entities Lewis assigns to sentences as their SMs).

Our account is designed to assign a SM to simple predicates. But what about complex predicates, constructed out of primitive predicates (Stalnaker, 1977)? We leave a full treatment to future work, and here we limit ourselves to outlining a natural approach to complex predicates obtained by lambda astraction from other predicates and Boolean connectives. The key ideas behind this approach are the same underlying negation transparency and junctive transparency in the sense of Berto (2022) (see also Parry, 1933; Ferguson, 2023). In a nutshell, given a predicate P, the SM of $\lambda x( \lnot Px)$ is the same as that of P, and given predicates P and Q, the SM of $\lambda x (P x \wedge Q x)$ and $\lambda x (P x \vee Q x)$ is the sum of the SMs of P and Q, i.e., the intersection of the corresponding equivalence relations (see above, p. 9). Hence, two objects will be equivalent with respect to the SM of $\lambda x ({x\;is\;green} \wedge {x\;is\;round})$, i.e., color $+$ shape, if and only if they have the same color and the same shape. This construction yields the result that the subject matter of $\lambda x (P x \wedge Q x)\,a$ is the same as that assigned by Plebani and Spolaore (2021) to $Pa \wedge Qa$ (assuming that the atomic sentences have the right focus profile; see the next subsection). Note that under this approach, the SM of $\lambda x ({red} \, x \wedge \lnot {red} \, x)$ is identical to that of “red,” i.e., color, and different from that of $\lambda x ({round} \, x \wedge \lnot {round} \, x)$, i.e., shape.¹⁴

4.4 From predicative to sentential subject matter: the pairing issue

In this section, we shall briefly explain how our account combines with an extant theory of sentential SM (Plebani & Spolaore, 2021) to yield a compositional solution to the pairing issue. Plebani and Spolaore (2021) explain how to deal with atomic sentences with different focus profiles and how to extend the account to the case of complex sentences. For simplicity’s sake, here we will only deal with atomic sentences with a specific kind of focus profile.

Solving the pairing issue requires finding a principled way to assign SMs to sentences. We follow Plebani and Spolaore (2021) in taking as our starting point Lewis’s (1988b) approach to this issue. Lewis defines an important relation between sentences and SMs: a sentence $\phi$ is entirely about SM $\textbf{s}$ iff $\phi$ has the same truth value in $\textbf{s}$-equivalent worlds. Lewis’s proposal seems to go in the right direction. For instance, it correctly predicts that “Delia is happy” is about Delia’s mood, for in all possible worlds where Delia is in the same mood the sentence has the same truth-value; by the same token, it predicts that “Delia is happy” is not about Delia’s height.

However, a sentence is entirely about (in Lewis’s sense) a lot of SMs. At one extreme, every sentence $\phi$ is entirely about what Plebani and Spolaore (2021) call its binary SM: the two-cell partition where one cell is the set of worlds where $\phi$ is true and the other cell is the set of worlds where $\phi$ is not true. At the other extreme, every sentence is entirely about the SM whose cells are singletons of possible worlds. As Plebani and Spolaore (2021) argue, neither of these extremes is a plausible candidate for the role of SM of $\phi$. This raises the problem of how to choose, among the many SMs in between these two extremes, the one that should be assigned to $\phi$ as its SM.

One way out of this impasse is to think of SMs as questions and, given a sentence $\phi$, look for a question Q such that $\phi$ is a perfectly natural answer to Q. Now, the same sentence can be heard as an answer to different questions, depending on the focus profile of the sentence. “The table is red” is naturally heard as an answer to the question Which objects are red?, “The table is red” is naturally heard as an answer to Is the table red or not?, and “The table is red” is naturally heard as an answer to What is the color of the table?

Let us restrict attention to the last case. What is the equivalence relation that corresponds to the question What is the color of the table? Plebani and Spolaore (2021) use boldface to signal focus, formalize “The table is red” as “$\textbf{P}a$,” and assign to sentences of this form the following equivalence relation on worlds as SM:

$$\begin{aligned} \{ (w, w'): \text{for all the predicates $X$ relevant to $P$, }\, w \vDash Xa \leftrightarrow w' \vDash Xa\} \end{aligned}$$

This definition captures the idea that the SM of “The table is red” is the color of the table. However, here Plebani and Spolaore (2021) appeal to a notion of relevance among predicates that they do not define. We are now in the position to define this notion: predicates P, $P'$ are relevant to each other exactly when they have the same SM. More precisely, we can distinguish between two relations of relevance, as explained above, and let predicates $P,P'$ be narrowly/broadly relevant to each other when they have the same narrow/broad SM. As a result, for instance, two worlds are equivalent with respect to the broad SM the color of the table when the table has the same color in both. Hence, the broad SM of the sentence “The table is red,” i.e., the color of the table, is connected to the broad SM of “red,” i.e., color, and similarly for all the other atomic sentences, as desired.

Let us elaborate on the connection between the SM of an atomic sentence and the SM of the (monadic) predicate it contains. As we said above, the SM of a predicate induces a function from objects to Lewisian SMs. The pleasing result is that the SM that Plebani and Spolaore (2021) assign to $\textbf{P}a$ is precisely the output of the function induced by the SM we assign to P when applied to a. Let f be the function induced by P’s SM: two worlds are equivalent with respect to f(a) exactly when a is identical in the relevant respect in those worlds, i.e., exactly when the same relevant predicates are true of a in those worlds, i.e., exactly when the worlds are equivalent with respect to the SM that Plebani and Spolaore (2021) assign to $\textbf{P}a$. Hence, there is a very tight connection between the SM of an atomic sentence and the SM of its predicate: the predicate’s SM induces a function that takes as input the referent of the singular term to which the predicate applies and delivers as output the sentence’s SM.

If we identify the SM of a singular term with its referent, then we obtain as a consequence of our characterization what Berto (2022) simply imposes as the defining feature of a predicate’s subject matter: the SM of a predicate corresponds to a function from singular-term level SMs to sentence-level SMs.¹⁵

Before wrapping up, let us spend a few words on how to extend our account of SM to quantified expressions. This is an important, if difficult, topic that we intend to address in future work. Here we limit ourself to some preliminary remarks. Plebani and Spolaore (2021, 616) assume transparency for quantifiers: formulae of the forms $\forall v\phi$ and $\exists v\phi$ have the same SM as their immediate subformula $\phi$. This approach works reasonably well within their framework but, at least in part, because they stipulate (somewhat artificially) that two worlds are alike with respect to the subject matter of atomic formulae (with the right focus profile) only if they have the same domain (615). If one wants to get essentially the same result without resorting to this stipulation, one can assign to quantifiers the Lewisian SM what there is, that is, the equivalence relation two worlds are in when the same objects exist in both ($\{(w, w'): D_w = D_{w'}\}$). As a result, for instance, the SM of a formula like $\forall x\, red (x)$ turns out to be color + what there is, that is, the equivalence relation two words are in when the same objects exist and have the same color in both.

5 Conclusions

We have presented an account of subject matter that satisfies two desiderata: (i) assign a SM to predicates; (ii) explain the connection between the SM of a sentence and the SM of the predicates it contains. In our view, predicates are about aspects of things. The aboutness of predicates has both cognitive and semantic significance, and it impacts the way we group predicates into families. Sentences are about aspects of worlds. The aspect of worlds an atomic sentence is about is obtained by applying the aspect of things a predicate is about to a specific object. This procedure yields the result that each sentence is entirely about its SM in Lewis’s (1988b) sense (see Plebani & Spolaore, 2021, 615–616 for a proof of this claim).

We also introduced a distinction between an aspect of things and a category of that aspect, i.e., the way it is categorized in certain linguistic practises. Sometimes it is convenient to identify the SM of a predicate with a category. In a sense, the English “green” and the Welsh “gwyrrd” are about different SMs, different ways of categorizing the color aspect (see above, p. 15). But sometimes it is convenient to identify the SM of predicate with an aspect: if two objects are indistinguishable color-wise, then one of them is in the extension of “green” (“gwyrrd”) if and only if the other is. “Green” and “ gwyrrd” have the same loose SM in the sense that they are both color predicates. Whether to identify the SM of a predicate with a category or an aspect might be a contextual matter, even though the default option should be to take the SM of a predicate as a category.

The resulting integrated account of SM is a form of two-component semantics in Berto’s (2022) sense. The idea of a two-component semantics was originally presented with reference to sentences: there are pairs of sentences with the same truth conditions, but different SMs, and there are pairs of sentences with the same SM, but different truth conditions. In our account, something analogous holds for predicates. Predicates like “yellow” and “blue” have the same SM (color), but different intensions. Similarly, the SM of a predicate is strictly connected to the system of alternatives with which it is contrasted. To borrow an example from Yalcin (2018, 14–15), even if “water” has the same intension as “H$_2$O,” it is natural to think they have different SMs, and thus different full contents. For when we use the former “we (inter alia) locate [water] amongst the beverages,” while with the latter we embed it “in some part of the subject matter chemistry” (15). As a consequence, the full content of a predicate is partly determined by the contents of other predicates relevantly related with it. This idea is in harmony with a variety of traditions in both lexical and cognitive semantics.¹⁶

Our investigation of predicative subject matter has borne fruit: the notion of aspect, and the cognate notion of category, can play an important role in semantics, connecting recent work on subject matter with other semantic traditions. In light of this, we think that predicative subject matter should be regarded as an interesting object of study, both within the current debate on subject matter and in its own right.