Unity and Application

1.1. Application

A stack of plates stands to the individual plates in the stack in the same way that a pile of bricks stands to the individual bricks in the pile. Arguably, the stack of plates is not merely the plates that are stacked. The stack of plates is one thing whereas the plates are many things. The analogous point holds for the pile of bricks. What is this relation that the stack of plates bears to the plates and the pile of bricks bears to the bricks? The standard answer is that the stack of plates is the fusion of the plates and the pile of bricks is the fusion of the bricks. On this view, there is operation—fusion—such that applying it to the bricks gives you the pile, and applying it to the plates gives you the stack.⁴

A similar situation arises in the theory of propositions. The proposition that two is prime stands to two and being prime as the proposition that three is odd stands to three and being odd. That two is prime is not merely the plurality consisting of two and being prime: the proposition is one thing whereas two and being prime are two things. The analogous point holds for the proposition that three is odd. What is this relation that the proposition that two is prime bears to two and being prime and the proposition that three is odd bears to three and being odd? While there is no standard answer, it is natural to develop an answer by analogy with the above case. In particular, it is natural to postulate some operation such that applying it to two and the property of being prime is the proposition that two is prime and applying it to three and the property of being odd is the proposition that three is odd. On this way of thinking, the proposition that two is prime is identical to $App\text{(}f,x\text{)}$ where $App$ is an operation—which I’ll call application— $f$ is the property of being prime and $x$ is the number two.⁵

Generalizing from the above example, the operation of application should be understood so that every instance of the following schema comes out true.

• For any individual $x$ and property $f$, if $f$ is the property of being $F$, then

$App\text{(}f,x\text{)}$ is the proposition that $x$ is $F$.

An instance of this schema is a sentence that results from replacing the capital ‘$F$’ by a predicate (making appropriate adjustments for grammaticality). The schema is not a definition or analysis of $App$. But it provides us with nontrivial information about its “extension.” For example, we can infer from it that if $f$ is the property of being blue, then the proposition that $x$ is blue is identical to $App\text{(}f,x\text{)}$ for any individual $x$ (assuming standard disquotational reasoning).

We can further tighten our grip on application by analogizing it to function application. A function $f:A\to B$ can be applied to an element $a$ of $A$ to get an element $f\text{(}a\text{)}$ of $B$. For instance the successor function $s\text{:}\text{N}\to \text{N}$ can be applied to any number $n\in \text{N}$ to get its successor $s\text{(}n\text{)}=n+1\in \text{N}$. Hence the successor function is a function of type natural number to natural number that when applied to a natural number delivers its successor.⁶ Similarly, given a property of a certain type, say a property of individuals, and an individual, we can apply that property to the individual to get a proposition. So the property of being blue can be thought of as being a property of type individual to proposition that is such that, when applied to an individual $x$ delivers the proposition that $x$ is blue.⁷

The analogy is only partial. Property application is highly constrained in a way that function application is not. There is a function $f\text{\hspace{0.17em} : }\text{N}\to \text{N}$ that maps every number to 113. More generally, for any functional relation $R\subseteq \text{N}\times \text{N}$ there is a function whose graph is that relation. The analogous behavior plausibly fails for properties. Plausibly, there is no property whose application to any individual is the proposition that snow is white. Suppose we had in our language some predicate $F$ such that $⌜$the property of being $F\text{⌝}$ denoted this property. Then $\text{⌜}\forall xApp\text{(}f,x\text{)}=$ that $x$ is $F\text{⌝}$ would be true given the schema by which application was introduced. But then $\text{⌜}\forall x$ that snow is white = that $x$ is $F\text{⌝}$ would be true since $App\text{(}f,x\text{)}$ was stipulated to be the proposition that snow is white. But the sentence $\text{⌜}$that I am $F\text{⌝}$ does not express the same proposition as ‘that snow is white’ for any $F$ since the former is about me whereas the latter is not. It is not the case that for any functional relation $r$ between individuals and propositions, there is a property $f$ such that $x$ bears $r$ to $App\text{(}f,x\text{)}$.⁸

I think this gives us some grip on the notion of application. Some will demand an account of what it is for a proposition to be an application of a property to an individual. Application is not plausibly fusion. Let $f+x$ denote the fusion of $f$ and $x$. The proposal that $App\text{(}f,x\text{)}=f+x$ entails that the proposition that I am walking is the fusion of me and the property of walking. But this proposal has some counterintuitive consequences. For example, since parthood is transitive, it entails that all of my parts are parts of the proposition that I am walking. The issues become worse when we consider the applications of relations to relata. Consider an $n$-ary relation $R$ and its application $App\text{(}R,x_1,\dots ,x_n\text{)}$ to $n$ relata. The obvious generalization of the fusion theory is to define $App\text{(}R,x_1,\dots ,x_n\text{)}$ as $R+x_1+$ · · · + $x_n$. But this theory conflates the proposition that $x_1$ loves $x_2$ (the application of loving to $x_1$ and $x_2$ in that order) and the proposition that $x_2$ loves $x_1\text{(}$the application of loving to $x_2$ and $x_1$ in that order).

There is a response to these objections that solves both at once. Instead of defining $App\text{(}R,x_1,\dots ,x_n\text{)}$ to be the fusion $R+x_1+···+x_n$ of a relation and relata, we could define it to be the fusion $R+\text{⟨}{x}_1,\dots ,x_{n⟩}$ of a relation and a sequence of relata. When $R$ is unary the result is that the proposition that I am walking is the fusion of the property of walking and the one-termed sequence whose sole term is me. Since this sequence is plausibly a simple (or if one likes set theoretic reductions, its only parts are other sets), this avoids the transitivity argument and the “forgetting order” problem. But I do not see any particular reason to believe it unless one is already committed to using the fusion operation in one’s theory of the combinatorial features of propositions. This will become even more apparent when developing the theory of application; in many cases the fusion theory would only add unecessary complication to the theory.⁹

As I see things, there is no more basic operation in terms of which application can be defined. It is not function application: since it is a manner of combining things, its behavior is highly constrained in a way that function application is not. And it is not fusion: there doesn’t seem to be any notion of parthood standing to application as our ordinary notion of part stands to fusion. I suggest we take application as a primitive manner of combining elements and see where it gets us. This starting place for my theory is reminiscent of Mark Johnston’s (2006) account of the unity of the proposition. He states:

• we can identify the proposition that $a$ is $F$ as the predication of $F$-ness of $a$. We may think of that as a complex item built up from $F$-ness and $a$, by way of the relation of being predicated of. (2006: 684)

There are a couple of things to note here though. First, application is an operation rather than a relation. This will turn out to be important to the overall theory. The claim is not that that there is some relation, application, that holds between the $x$ and the property of being $F$ whenever the proposition that $x$ is $F$ exists. The claim is that the proposition is the result of applying $App$ to $x$ and $F$. Failure to distinguish between operations and relations I think has resulted in some confusion when addressing the problem of the unity of the proposition. The confusion is present in Johnston’s description of his own view since ‘the predication of’ syntactically combines with a singular term denoting a property to form a term. It thus denotes on operation. This conflicts with the second part of his quotation in which he characterizes it as a relation.

A second important thing to note: while our starting places may be similar, this doesn’t mean that our ending places are. As of yet I have merely explained my primitives, I have not yet stated what the theory governing them is. And it is the theory governing this primitive which is distinctive. The theory demonstrates the strength and perspicuity of the primitive. I will return to this point in Section 3.1.

1.2. Application and Algebraic Theories of Propositions

Philosophers in the algebraic tradition of theorizing about propositions have often made use of operations similar to application. As Bealer (1998: 10) says, it is a truism that “The proposition that Fx is the predication of the property $F$ of $x$.” This “predication” operation is essentially my “application” operation, though we’ll see that it plays rather different roles in our respective theories. In this section, I want to emphasize several ways in which the approach I develop differs from that in the algebraic tradition. Before doing this, it is important to point out one thing: the main point of this paper will be to explain various representational notions in terms of application. This is not something that algebraic theorists often do in their theories. So there is a certain sense in which the main parts of our respective theories are not in competition with one another. For instance one could attempt to develop a theory very much like the one I develop within, for example, Bealer’s approach if one preferred. But I think the framework I sketch here provides a better foundation for the theory that I go on to develop.

Algebraic theories of propositions often locate propositions as the 0-ary case of $n$-ary relations more broadly. The basic idea behind these theories, the reason why they are called algebraic, is that propositions, properties and relations fit into a certain kind of algebraic structure.¹⁰ The algebraic structure consists of the disjoint union of a family of domains, $D,R_0,R_1,$ … where $D$ is the family of individuals, $R_0$ the family of propositions, $R_1$ the family of properties and $R_n$ the family of $n$-ary relations for $n\ge 2$.¹¹ In addition, this disjoint union is equipped with a collection of (partial) operations. There is an operation ¬ (negation), for instance, that maps each proposition $p\in R_0$ to its negation ¬$p\in R_0$. Other Boolean operations are taken as primitive operations on propositions. There is also included an application operation, $App$, that takes an element $f\in R_1$ and elements $x$ in any domain, and produces an element $App\text{(}f,x\text{)}$ in $R_0$.¹²

On this approach, application is treated on a par with notions of conjunction, negation and other logical operations. Moreover, application is treated as a global notion: we can apply a given property $f$ not just to individuals, but also to properties and relations more generally. My preferred approach diverges from this approach in two ways. First, on the approach I prefer, application is treated as an operation, whereas the Boolean operations are treated as (higher-order) properties and relations. Thus the sole primitive operation of the theory is application. Second, the theory I prefer is typed, whereas authors working in the algebraic tradition tend to prefer untyped theories (like that sketched above). Here is a rough sketch of how this would go. The collection of types includes a basic type $e$ and for any finite sequence of types ${\sigma }_1$, …, ${\sigma }_n$, a derived type $\text{⟨}{\sigma }_1$, …, ${\sigma }_n\text{⟩}$. We then assign entities types as follows. Individuals are entities of type $e$. Anything that combines with things of type ${\sigma }_1$, …, ${\sigma }_n$, in that order, is of type $\text{⟨}{\sigma }_1$, …, ${\sigma }_n\text{⟩}$. Thus, for instance, propositions are of type ⟨⟩, properties of individuals are of type $\text{⟨}e\text{⟩}$, properties of propositions are of type ⟨⟨⟩⟩, and so on.

What do we mean by entities that combine with other entities in a given order? In my view this is where the notion of application comes in, and must be treated as a primitive operation (as opposed to a relation of some type). In particular we suppose that for any type $\text{⟨}{\sigma }_1$, …, ${\sigma }_n\text{⟩}$, there is an operation $App$_{$\text{⟨}\sigma 1,$ …,$\sigma n⟩$} such that $App$_{$\text{⟨}\sigma 1,$ …,$\sigma n⟩$}$\text{(}R,x_1,\dots ,x_n\text{)}$ is a proposition where $R$ is a relation of type $\text{⟨}{\tau }_1,\dots ,{\tau }_n\text{⟩}$ and $x_i$ is an entity of type ${\tau }_i$. Thus the application operation that we introduced in the first section of this paper is the application operation of type $\text{⟨}e\text{⟩}$: it “combines” properties and individuals to get propositions.

In this framework, conjunction and negation can be treated as themselves certain kinds of properties and relations between propositions. Let ∧ be the conjunction relation. Let $p$ be the proposition that snow is white and $q$ the proposition that grass is green. Then $App$_{⟨⟨⟩,⟨⟩⟩}(∧, $p,q\text{)}$ is the proposition that grass is green and snow is white. Quantifiers can similarly be treated as higher-order properties. Where $f$ is the property of being prime, and ∃ is the higher-order property of “being instantiated”, a property of type $\text{⟨}\text{⟨}e\text{⟩}\text{⟩}$, we can think of $App$_{$\text{⟨}\text{⟨}e\text{⟩}\text{⟩}$}(∃, $f\text{)}$ as the proposition that something is prime.

The framework is quite obviously inspired by models of typed lambda calculus. The move made in this paper is to treat application as “representationally significant”, in the sense of taking it to correspond to a real live operation out in the world; a theory of this operation, I will argue, can help us make progress with certain problems concerning the representational status of propositions (as well as other abstract objects). The view described provides something like an answer to the question: What is a proposition? The answer is: a propositions is an application of a relation to some relata. More precisely, a proposition is, for some types ${\tau }_1,\dots ,{\tau }_n$ and some relata $A_1,\dots ,A_n$ of types ${\tau }_1,\dots ,{\tau }_n$ respectively, an application of a relation $R$ of type $\text{⟨}{\tau }_1,\dots ,{\tau }_n\text{⟩}$ to $A_1,\dots ,A_n$ :

Fully comparing a typed view of this kind with the untyped view of Bealer is beyond the scope of this paper. The typed theory will mostly rest in the background in what follows since I will be primarily concerned with the operation $App$_{$\text{⟨}e\text{⟩}$} that takes properties of individuals and individuals to propositions. The reason for restricting my focus is that it is here that propositions make “contact” with the concrete world, as it were. There may be interesting things to say about higher-type entities, but for the purposes of this paper I will mostly ignore them.¹⁴

One might wonder whether this account is really a primitivist account of propositions: after all, haven’t we just defined propositions as the output of application? We have, but in explaining what application was, I made ineliminable use of the word ‘proposition’. Thus the theory does not provide anything like a reductive analysis of propositions. At least in one sense of ‘primitivist’, I take the theory to be a primitivist view. Nothing of great importance seems to rest on this fact though.

One might also object to describing this view as one that takes the operation of application as primitive: really the theory has posited a typed collection of application operations and so has posited many new primitive operations. One might even object on these grounds that the view should be rejected on the grounds of being “ideologically complex.”

I have two things to say in response. First, a typed collection of operations could easily be traded in for one single partially defined operation. The types then merely let us keep track of where the operation is defined. Second, just as ontological complexity is more a measure of how varied in kind items in one’s ontology are, as opposed to mere number of things, so too ideological complexity should be taken as a measure of how heterogeneous one’s ideology is, as opposed to merely the numbers of items included in one’s ideology. Since I’m inclined to regard all of the typed application operations as being of the same kind, I’m inclined to think that the theory is not very ideologically complex at all.

Contra some algebraic theories, I will not assume any strong decomposition principles for propositions like the following:

• Structure: If $App\text{(}f,x\text{)}=App\text{(}g,y\text{)}$ then $f=g$ and $x=y$.

It seems to me that allowing for a bit more freedom in the behavior of the operation $App$ can lead to some genuine explanatory advances; in the final section of this paper I will sketch one such case in particular. There is also a worry about inconsistency; given moderate resources, Structure seems to be inconsistent. For instance, fix a proposition $q$ and suppose that $f$ is the property of being a proposition $p$ such that for some property of propositions $h,p=Ap{p}_{\text{⟨⟩}}$$\text{(}h,q\text{)}$ and $p$ does not instantiate $h$. Consider the proposition $App\text{(}f,q\text{)}$. If $App\text{(}f,q\text{)}$ does not instantiate $f$, then for every property $h$ such that $App\text{(}f,q\text{)}=App\text{(}h,q\text{)},q$ instantiates $h$. Thus, since $App\text{(}f,q\text{)}=App\text{(}f,q\text{)},q$ instantiates $f$. So if $q$ doesn’t intantiate $f$, it does instantiate $f$. Classically, this entails that $q$ instantiates $f$. So for some $h,App\text{(}f,q\text{)}=App\text{(}h,q\text{)}$ and $q$ does not instantiate $h$. But if Structure were true, any such $h$ would have to be $f$, which we’ve already shown $q$ does not have; so Structure is false.¹⁵

One might worry that given the falsity of Structure there is no sense in which $App$ can be regarded as a way of combining things. But I don’t see why this should be so. Ordinary fusion is idempotent: the fusion of $x$ and $x$ is $x$. Thus even ordinary fusion wouldn’t satisfy something as strict as Structure. Since the theory I’ll develop explicitly denies that $App$ is fusion, we are free to posit even more radical failures of structure: for instance, we might allow for certain “freely absorbable” or “non-structure creating” properties: properties for which $App\text{(}f,-\text{)}$ is the identity map on objects, for instance. If there are explanatory gains for allowing such properties, we should. One possible example of such freely absorbable properties and relations is the relation corresponding to the operation of application. For instance, we might suppose that there is a relation $a$ of type $\text{⟨}\text{⟨}e\text{⟩}$, $e\text{⟩}$, the application relation, such that applying it to a property $f$ and individual $x$ is the same as applying $f$ to $x\text{:}$

These non-structure creating relations have some grounds for being called logical. Under that criterion, the application relation is itself a logical relation. We might then view the theory I will put forward as one that attempts to account for the representational features using only broadly logical resources. I will return to this point below. I now want to begin to develop a theory of the representational features of propositions within this framework.

2.1. Aboutness and Predication

Some of the key representational features of propositions are that propositions are about individuals and predicate properties of them. We can capture this fact with the following principles:

• Rigidity: Necessarily for any $f$ and $x$ and any proposition $p$, if $p=App\text{(}f,x\text{)}$ then necessarily if $f$ and $x$ exist $p=App\text{(}f,x\text{)}$.

• Aboutness: For a proposition $p$ to be about $x$ is for there to be some $f$ such that $p=App\text{(}f,x\text{)}$.

• Predication: For a proposition $p$ to predicate $f$ of an individual is for there to be some $x$ such that $p=App\text{(}f,x\text{)}$.

The principle Rigidity tells us that ‘$App\text{(}f,x\text{)}$’ is to be read as a rigid designator. Aboutness and Predication are proposed as analyses of a proposition’s representational features in terms of application. Together these principles entail some plausible facts concerning the representational features of propositions. For instance, the proposition that two is prime is necessarily about two. The theory provides the following explanation. The proposition that two is prime is the application of being prime to two (by the schema by which application was introduced). So by Rigidity it is necessarily the application of being prime to two. And so by existential generalization necessarily there is some property $f$ such that it is the application of $f$ to the number two. And finally by Aboutness it follows that necessarily it is about two. Similarly, since it is the application of being prime to two, it is, for some individual $x$, the application of being prime to $x$; hence by Predication it predicates the property of being prime (necessarily so by Rigidity).

It might be worth briefly mentioning how propositions expressed using definite descriptions fit into this theory. On my view the proposition that the present king of France is bald does not predicate baldness. One might worry that this means the view has lost contact with any pretheoretic notion of ‘predication’ since, surely on a pretheoretic sense, this proposition does predicate baldness.

In response I want to say two things. First, I’m not sure I have a grip on what it would mean for a proposition to predicate a property but not predicate that property of anything. To predicate, on my view, is to predicate of. If that’s right the pretheoretic data may be a bit murkier than the objection makes out. And second, I’m not really all that concerned with capturing all of the pretheoretic data: I reject the idea that one’s theory either aligns with the pretheoretic data or else it is revisionary. An abductive approach looks for the joint carving notions in the vicinity of the pretheoretic data without being held hostage to it. In the present case, I think there are good theoretical reasons for adopting my approach to predication rather than one that takes the proposition that the King of France is bald to predicate baldness. The basic reasons are just those that motivated Russell (1903) to treat propositions like the proposition that the present King of France is bald as being qualitative.¹⁶ The proposition is not used to pick something out, and predicate something of it. Rather the proposition is ultimately quantificational. One could follow Russell in taking it to express a more complicated proposition built up from universal and existential quantifiers. Or one could take ‘the’ to express a primitive higher-order relation, ι, and identify the proposition that the present king of France is bald as being identical to $Ap{p}_{\text{⟨}\text{⟨}e\text{⟩},\text{⟨}e\text{⟩}\text{⟩}}\text{(}\iota ,f,g\text{)}$ for some properties $f$ and $g$. On these sorts of views it is probably more accurate to say that the proposition that the present king of France is bald predicates^∗ the relation expressed by ‘the’ of the properties expressed by ‘present king of France’ and ‘is bald’, where “predicates^∗” is some higher-order analogue of first order predication. Expressing this idea on English is admittedly a bit difficult.

The principles proposed here bear some resemblance to the zipping theory briefly considered by Merricks (2015: 154):

[W]henever there is a state of affairs of an object’s being related by Zipping to a property, then that state of affairs essentially represents that object as having that property. Nothing explains why Zipping works this way. That is just how Zipping works.

But the theories are actually importantly different: they have a completely different form. Like predication in Johnston’s theory, Zipping is supposed to be a relation. The theory, schematically, says the following:

The proposition that $x$ is $F$ is about $x$ and predicates $F$ if and only if $x$ bears Zipping to the property of being $F$.

Notice though that the proposition that $x$ is $F$ is about $x$ and predicates $F$ whenever both $x$ and the property of being $F$ exist. Thus for all we have said about Zipping, it could simply be the relation of co-existence. The Zipping theory doesn’t offer us any account of what it is for a proposition to be about something or for it to predicate a property of something. The theory of application on the other hand, in particular the principles Aboutness and Predication, offers explicit analyses or metaphysical definitions of these relations. It does offer an account of what it is for a proposition to be about something. The analyses appeal to a new primitive. But that is not to bestow any mysterious powers on the primitive. The theory does not say that being an application of $f$ to x causes one to be about $x$. The theory is that being an application of some property to $x$ is just what it is to be about $x$. In this way the theory of application seems to me superior to the Zipping theory, even if superficially they look similar.¹⁷

One sort of objection the theory faces concerns propositions seemingly about non-existent entities. For instance, someone might want to hold the view that the proposition that Sherlock Holmes is a detective is about Sherlock Holmes, even though Sherlock Holmes does not exist. It’s worth pointing out though that this judgment is compatible with what the theory states thus far, though it does not entail it. By the schema by which application was introduced, we have

• That Sherlock Holmes is a detective = App(Being a detective, Sherlock)

It is tempting to then argue that if App(Being a detective, Sherlock) exists, Sherlock exists as well. But it is open to one to deny this. For example, those who deny Sherlock’s existence while affirming that Sherlock is a detective presumably will also affirm that the partner of Sherlock is Watson. But the locution ‘the partner of’ denotes on operation. We are thus going to need some sort of a theory of operations that permits them to be defined on “nonexistent arguments.” Thus the theory does not entail existentialism, the thesis that, schematically, $N$ exists whenever the proposition that $N$ is $F$ exists, though it is perhaps more natural in that setting.¹⁸

2.2. Predication, Truth and Instantiation

Predication bears an intimate relation to instantiation. A proposition that predicates $f$ of $x$ is necessarily true if and only if x instantiates $f$. What accounts for this fact? One reason that this question has proved difficult to answer is that authors have tended to look for explanations of a proposition’s truth in terms of which objects instantiate which properties. But once we have the operation of application in hand it becomes natural to reverse the order of explanation:

• Instantiation: For $x$ to instantiate $f$ is for $App\text{(}f,x\text{)}$ to be true.

The principle Instantiation is proposed as an analysis of instantiation in terms of truth. This reverses the traditional order of explanation. To many it will look like a hopelessly confused attempt to analyze the noumenon in terms of the phenomenon, or less grandiosely, to say how things are in terms of how they are represented to be. But the issues here are delicate. The proposed analysis is consistent with how things are being prior to how they are represented to be. Contrast the following two statements:

• (1)

  The proposition that two is prime is true because two is prime.

• (2)

  The proposition that two is prime is true because two has the property of being prime.

Ordinarily we might not distinguish these statements. But when doing metaphysics it is important that we recognize the coherence of the position that accepts (1) while rejecting (2). On the sort of view I am imagining, not only is the truth of propositions explained by how things are, but so too is the instantiation of properties. That is, in addition to accepting (1) and rejecting (2), this kind of theorist accepts (3):

• (3)

  Two has the property of being prime because two is prime.

This puts propositions and properties on equal footing by treating both as explanatorily posterior to how things are. Propositions are true or false whereas properties are true or false of things.¹⁹ And both truth and truth of are explained by some prior notion of how things are.

I prefer a different view. Instead of taking both properties and propositions to be explanatorily posterior to a prior notion of how things are, I think we should take both properties and relations to be constitutive of how things are: propositions correspond to distinctions in reality, not to distinctions in how reality is represented. Similarly, properties correspond to distinctions among individuals in reality, not to how individuals in reality are represented.²⁰ Less metaphorically, what it is for the proposition that two is prime to be true is for two to be prime, and what it is for two to have the property of being prime is for it to be prime.²¹ Several authors have recently argued that identifications like ‘for it to be the case that … is for it to be the case that … ’ obey analogous principles to ordinary identity predicates. In particular they obey the obvious analogues of transitivity and symmetry.²² If that is right then it immediately follows that for two to instantiate being prime is for the proposition that two is prime to be true. And this sort of argument generalizes. Schematically:

• P1 For the proposition that $x$ is $F$ to be true is for $x$ to be $F$.

  P2 For $x$ to instantiate being $F$ is for $x$ to be $F$.

  C Hence, for $x$ to instantiate being $F$ is for the proposition that $x$ is $F$ to be true.

This provides confirmation to the principle of Instantiation since every instance of C can be inferred from it together with the schema by which the notion of application was introduced.

2.3. Application and Cognition

One thing a theory of propositions is supposed to provide is an account of why, for instance, thinking that two is prime entails thinking about two, and why thinking that two is prime entails ascribing the property of being prime to two. The following two principles strike me as quite natural:

• Attitude: For $x$ to think about $y$ is for $x$ to entertain $App\text{(}f,y\text{)}$ for some $f$.

• Ascription: For $x$ to ascribe $f$ to something is for $x$ to believe $App\text{(}f,x\text{)}$ for some $x$.²³

To think about something is to entertain a proposition about that thing and to ascribe a property is to believe a proposition that predicates that property. These principles unify various norms on belief and ascription. Truth is a norm of belief just as instantantiation is a norm of ascription. One should believe $p$ only if $p$ is true and one should ascribe $f$ only if $f$ is instantiated. Given the proposed theory the latter norm follows from the former. For any $x$, one should believe $App\text{(}f,x\text{)}$ only if $App\text{(}f,x\text{)}$ is true. So by Instantiation one should believe $App\text{(}f,x\text{)}$ only if $x$ instantiates $f$. And then by the principle Ascription, it follows that one should ascribe $f$ only if $f$ is instantiated.²⁴

It’s worth pausing here for a moment to say something about my treatment of propositional attitudes. Hanks (2015) has recently objected to primitivist views of propositions on the grounds that primitivist about propositions inevitably leads to primitivist about propositional attitudes:

[I]f propositions are simple and unstructured, we cannot take this act of endorsement [judgment] to consist in a mental operation performed on the constituents of a proposition. Furthermore … we cannot say that to endorse a proposition is to accept it as true… . If to accept $p$ as true is to judge that $p$ is true then we’ve analyzed one judgment, judging that $p$, in terms of another, judging that $p$ is true. This leads to regress … it looks as though [the primitivist] is going to have to view judgment as a primitive attitude one can bear to a proposition. (Hanks 2015: 45)

I find this argument unconvincing. On the one hand, the idea that simplicity of relata has something to do with the simplicity of a relation seems to me misguided. For instance, suppose that one sees an electron. Then one bears the seeing relation to a simple item that lacks any internal structure. Does this mean that seeing must be simple, and unalyzable? Of course not. Seeing something simple is compatible with a wide variety of analyses of seeing in general. Thus the idea that we could argue for primitivism about the attitudes from primivitism about their relata is a bit suspect on its face.

Now Hanks does propose a couple of analyses and points out that they fail; one on general grounds and the other because of its putative requirement of the structured theory of propositions. But he fails to show that many of our leading accounts of propositional attitudes conflict with primitivism: functionalism, interpretationism, causal theories, optimal conditions accounts etc.²⁵

As far as I can see, all of these accounts are consistent with primitivism. Indeed since many of these accounts have been developed under the assumption the propositions are sets of possible worlds, some of them are actually more naturally combined with views on which propositions are sets of possible worlds, since they generally do not make clear in virtue of what attitudes could differ in fine grained content. Since the view that propositions are sets of possible worlds does not differ structurally from the view on which propositions are primitive and form a complete, atomic Boolean algebra, it is hard to see how there are going to be any in-principle problems with primitivism when it comes to the propositional attitudes.²⁶

2.4. Laws and Action

The theory offered thus far shows how to define various representational properties and relations from truth and application. One way to broaden the explanatory ambitions of the theory is to show that aboutness and predication as they arise in other domains can be accounted for in the theory of propositions. Suppose, for instance, that we regard the thesis that $\varphi$, the fact that $\varphi$, the law that $\varphi$ and the act that $\varphi$ as the proposition that $\varphi$ under different guises. If that’s correct we can immediately account for any aboutness or predication these entities exhibit in terms of the theory of application. I will look at two examples.

Suppose that the law that $\varphi$ is simply the proposition that $\varphi$. This follows from the plausible theory that propositions are the referents of ‘that’-clauses. Some people have maintained that laws of nature are purely general. Laws of nature do not mention any particular individuals. We can formulate this thesis more precisely in the present framework as follows: for any law $l$, there is no individual $x$ and property $f$ such that $l=App\text{(}f,x\text{)}$ (or perhaps some generalization of this idea). This provides a language-independent account of the generality that laws exhibit.

Another example comes from the theory of action. Suppose that you and I both pick up a pen. There is a sense in which we have done the same thing and a sense in which we have not done the same thing. What are these senses? Well suppose that actions are propositions—things we make true. I made the following proposition true: that I pick up the pen. And you made the following proposition true: that you pick up the pen. The sense in which we’ve both done the same thing is that we’ve both made propositions true that predicate the property of picking up the pen. Then sense in which we’ve done different things is that that the proposition you’ve made true is about you whereas the proposition that I’ve made true is about me. More generally, say that $p$ is the same action as $q$ if and only if $p$ is an action and $p=q$. Say that $p$ is the same type of action as $q$ if and only if $p$ and $q$ are both actions, and for any property $f,p$ predicates $f$ if and only if $q$ predicates $f$. Then in the above example we performed different actions, but nevertheless performed actions of the same type.

2.5. The Aboutness of Properties

A more radical extension of the theory attempts to explain the distinction between qualitative properties and haecceitistic properties in terms of propositional aboutness. Consider the property of being identical to John. We can recognize some sense in which this property is about John. At the very least it is more closely related to John than the property of being identical to the person wearing the blue shirt, even provided that John is wearing the blue shirt. Even if one were disinclined to accept that being identical to John is about John, the property nevertheless seems to stand to John in the same way that the proposition that John is identical to John stands to John. Now consider the proposition that $x$ is identical to John, for an arbitrary individual $x$. If we are not too fine-grained about the individuation of propositions, we can take this proposition to be an application of the following complex property to John: the property of being a $y$ such that $x$ is identical to $y$. If that is correct, then we can say that property of being identical to John is about John because its application to an arbitrary individual is about John. More generally, I propose the following theory of property aboutness:

• Property Aboutness: For a property $f$ to be about $x$ is for $App\text{(}f,y\text{)}$ to be about $x$, for all $y$.

With a notion of property aboutness in hand, we can say that a property is qualitative if it is not about any particular thing and haecceitistic otherwise. So for instance, being blue is qualitative because, plausibly, there is no $x$ such that for any individual $y$, the proposition that $y$ is blue is about $x$. Being identical to John is not qualitative because the proposition that $y$ is identical to John is about John, for any individual $y$.

Call the theory just outlined the minimal theory of application. The minimal theory of application provides analyses of many of the representational properties of propositions and agents in terms of application and so demonstrates some of the potential explanatory power a primitivist view that makes use of application can have. The theory is of course incomplete in many ways. A full theory would generalize application to $n$-ary relations and show that it can be consistently combined with one’s desired theory of propositional fineness of grain. I won’t do that here. Instead, in the next section, I will situate the minimal theory within the literature to get a better sense of how it compares with more recent attempts to account for the representational dimension of propositions. In the final section I will evaluate this theory against these other theories and argue that it is to be preferred on broadly abductive grounds.

3.1. The Problem of the Unity of the Proposition

As is often acknowledged, there is no single clear problem that is “the problem of the unity of the proposition,” but rather a family of related problems. I want to first suggest that at least one of the problems that has gone under this heading can be formulated in terms of application. The basic idea is that once we acknowledge that propositions are applications of properties to individuals, we certainly want some account of this operation that explains the distinctive traits of its outputs. In particular we want an account of application that explains the representational features of propositions.

We might conceive of this problem by analogy with the general composition problem.²⁷ Pace composition as identity theory, I am not merely my parts. I am one thing whereas my parts are many. But I am the result of applying some operation to my parts: I am the fusion of my parts. The operation of fusion takes some things—my parts—and delivers one thing, me. The general composition problem is essentially that of providing an illuminating account of fusion. Hence the problem of the unity of the proposition, as I am conceiving of it, stands to application as the general composition problem stands to fusion.

We can be a bit more precise about the analogy. Peter van Inwagen calls the general composition question the question “What is it for some xx to compose $y$?” The general composition problem is then simply the problem of coming up with an answer to the general composition question. Call the general application question the question “What is it for $p$ to be the application of $f$ to $x$?” The general application problem is then the problem of coming up with an answer to the general application question. The theory I proposed is that application is primitive and so no answer to this question can be given that invokes more basic notions. But just like we can answer the question “What is it to be a set?” by providing some postulates on being a set, so too we can answer the general application question, I would argue, by providing postulates on application.

Recall that the special composition question, as opposed to the general one, is the question, “What are the necessary and sufficient conditions for some things to compose something?”²⁸ To answer this question, one must find some relation $r$ such that such that xx compose something if and only if $r$ holds of xx (or rather find some informative description of $r\text{)}$. Just as we can draw an analogy between the general composition question and the general application question, we can draw an analogy between the special composition question and the special application question. Say that a property f applies to an individual $x$ if and only if $App\text{(}f,x\text{)}$ exists. Then we can ask for the necessary and sufficient conditions for a property $f$ to apply to an object $x$. More precisely, the special application question is the question “What are the necessary and sufficient conditions for a property $f$ to apply to an individual $x$?” The special application problem is that of finding some relation $r$ such that $f$ applies to $x$ if and only if $r$ holds of $f$ and $x$ (or rather finding some informative description of $r\text{)}$.

Oftentimes when the problem of the unity is posed, it is posed as if it were the problem of finding an answer to the special application question.²⁹ For instance Hanks says in describing the problem:

Since the proposition [that Clinton is eloquent] is one thing, and the constituents [Clinton and eloquence] are two things, there must be something about the proposition that joins [Clinton] and [eloquence] together into a single thing. The constituents must bear a relation [my emphasis] to one another that unifies them into a proposition. (Hanks 2015: 43)

He then goes on to introduce the problem of unity as that of finding this relation that “unifies” $f$ and $x$ into a proposition. Similarly, Jeff King says

Presumably the constituents of a proposition are related somehow in that proposition, with the relation imposing structure on them. I’ll put this by saying the relation holds the constituents together. Answering [the unity question] requires saying which relations hold the constituents of propositions together. (King 2009: 259)

Although neither King nor Hanks makes explicit use of application in formulating these questions, the questions formulated seem closer to the special application question as opposed to the general application question. The metaphor of “holding together” suggests that we are looking for a relation that holds of the constituents of a proposition if and only if they form that proposition. And this is just the special application problem.

Despite these appearances, I will argue that the theories that Hanks and King both give, and many others for that matter, do not provide answers to the special application question. Rather, they provide answers to the general application question (suggesting that despite appearances, this is the question they mean to be asking in the first place). And this is as it should be, since, I will also argue, the special application question is the wrong question to ask.³⁰

3.2. Why the Special Application Question Is the Wrong Question

To see that the special application question is the wrong question to ask, it is helpful to reflect for a minute on what Peter van Inwagen says concerning the relation between the special and general composition questions:

What singular terms might be appropriate substituends for ‘$y$’ in the ‘the $x$s compose $y$’, given that Contact [xx compose something if and only if xx are in contact] is the correct answer to the Special Composition Question? There is no way of answering this question, for neither Contact nor any other answer to the Special Composition Question tells us anything about the identity, or even the qualitative properties, of any composite object. Moreover, no answer to the Special Question Composition will tell us what composition is. (Van Inwagen 1990: 38)

Van Inwagen is not putting forward any kind of controversial theory of the relation between the two questions in this passage but is making a straightforward logical point. To say that xx compose something if and only if they are in contact is consistent with any hypothesis concerning the kind of thing they compose. For instance, it is consistent to say that these two blocks compose something if and only if they are in contact, and what they compose is the entirety of the earth; it is consistent to say that two blocks compose something if and only if they are in contact and what they compose is the number π; it is consistent to say that two blocks compose something if and only if they are in contact and what they compose is the block on the left. The answer one gives to the special composition question on its own tells you absolutely nothing about the entity they compose. All it tells you is when (in a modally robust sense of ‘when’) some things compose.³¹

This point applies equally to the special application question. Suppose for definiteness that one thought that the application of being prime to two existed if and only if there is a state of affairs having the property of being prime as its universal component and the number two as its singular component. In brief: the application of being prime to two exists if and only if two and being prime are in contact in a state of affairs. This theory is logically consistent with any hypothesis whatsoever concerning what kind of thing the application is. For instance, it is consistent to say that the application of being prime to two exists if and only if two and being prime are in contact in some state of affairs and the application is identical to me; it is consistent to say that the application exists if and only if they are in contact in a state of affairs and the application is identical to the property of being prime. The answer one gives to the special application question on its own tells you absolutely nothing about what the application is. All it provides is the modal profile of the application.

This point can be easily overlooked but it is significant. Recall that one thing we want out of a response to the problem of unity is an account of the distinctive representational behavior of propositions. But since any answer to the special application question is logically consistent with any hypothesis whatsoever concerning the qualitative properties of propositions, any answer to the special application question is logically consistent with any hypothesis concerning the representational properties of propositions. Those who have been attempting to account for the representational properties of propositions merely by providing an account of what unifies the constituents of a proposition have been attempting the impossible. Thus, insofar as we want an account of certain qualitative properties of propositions, we should not be attempting to answer the special application question.

Now one might respond that really what we want is an answer to both the special application question and the general application question. A general theory of propositions will tell us what it is for $p$ to be the application of $f$ to $x$ and also tell us when the application of $f$ to $x$ exists. This more general theory will hope to account for the representational properties in terms of its analysis of application rather than the modal profile it assigns to applications. But I think even this more nuanced approach embodies a mistake. I’ll make this argument again by drawing another analogy to the case of composition.

Suppose that one started out accepting universalism about composition, according to which for any xs necessarily the xs compose something. What would one say to the special composition question? I’m inclined to think one should dismiss the question as having no answer at all. There is no relation that makes it the case that some things compose because all some things need to do to compose is exist. This is in fact the way Peter van Inwagen introduces universalism:

It is impossible for one to bring it about, [according to the universalist], that something is such that the xs compose it, because, necessarily …, something is such that the xs compose it… . One can’t bring it about that the $x$s compose something because they already do; they do so “automatically.” (Van Inwagen 1990: 72)

But we are in a similar situation with respect to propositions. Most authors grant that it is metaphysically necessary that whenever the property of being blue exists and the cup exists, the proposition that the cup is blue exists. In my preferred terminology, necessarily whenever $f$ is a property and $x$ is an individual, $App\text{(}f,x\text{)}$ exists. There is nothing that one can do to bring it about that the application of a property to an object exists; it exists “automatically.” Hence just as the universalist has no need to answer the special composition question, no theorist has any need to answer the special application question, since all theorists are universalists about application.

The search for some relation that “unifies” being prime and two into the proposition that two is prime is thus confused twice over. Since they form a proposition automatically, there is no such relation to be found. Moreover, since what we really want to know is why the proposition has certain qualitative features, the search for such a relation turns out to be completely irrelevant to what we really care about.

3.3. Reductive Answers to the General Application Question

This leaves us with something of a puzzle. Both King and Hanks and many others have formulated problems that on their face appear to be the special application question. They then go on to provide what they say are answers to this question and also claim of these answers that they account for the representational features of propositions. What is going on? When one looks more closely at proposed answers to the special application question, one quickly sees that they are not, after all, answers to this question, but are rather answers to the general application question. It is common in the literature for someone to highlight some relation $r$ and call it the unifying relation. But it is never the case that according to their theories the application of $f$ to $x$ exists if and only if $f$ bears $r$ to $x$; rather, the proposed theory says that there is some other operation $O$ such that the application of $f$ to $x$ is identical to $O\text{(}r,f,x\text{)}$. That is, the theories invariably are just analyses of application in terms of a further relation and a further operation. I’ll give three examples of this.

Consider first Jeff Speaks’s (2014a) recent theory of propositions. According to Speaks, the proposition that two is prime is the property of being such that two instantiates being prime. The application of $f$ to $x$ can exist without $x$ instantiating $f$. Instantiation does not unify the constituents of a proposition. Rather the theory is that there is some three-place operation, the property of being such that $x$ bears $r$ to $y$, whose application to two, instantiation, and being prime, delivers the proposition that two is prime. And this is just an answer to the general application question:

• Speaks: $App\text{(}f,x\text{)}=$ the property of being such that $x$ instantiates $f$. ³²

Peter Hanks and Scott Soames’s views can be similarly formulated.³³ According to Soames (roughly) the proposition that two is prime is the act of ascribing primehood to two. Since the proposition that two is prime can exist even if no one actually ascribes being prime to two, ascription is not the relation that unifies the constituents of a proposition. Rather there is some operation, the act of ascribing $f$ to $x$, whose application to being prime and two, delivers the proposition that two is prime. And this is just an answer to the general application question:

• Soames: $App\text{(}f,x\text{)}=$ the act of ascribing $f$ to $x$.

Finally consider the position put forward by King (2014). According to King, the proposition that two is prime is the fact that two bears a certain relation $r$ to being prime. This relation $r$ is quite complex and is defined by quantifying over linguistic items and their meanings; the details needn’t concern us here. So on King’s view, the proposition that two is prime is the result of applying some three place operation to $r$, being prime, and two. It is the operation denoted by ‘the fact that $x$ bears $y$ to $z$’. So King provides an answer to the general application question:

• King: $App\text{(}f,x\text{)}=$ the fact that $x$ bears $r$ to $f$. ³⁴

Appearances to the contrary, many recent theories of propositions are thus better construed as answers to the general application question as opposed to the special application question. And once these theories are presented this way, it becomes clear that they are in competition with my own primitivist account. For instance, each of Speaks, Soames and King takes as primitive one or more notions that the minimal theory of application provides analyses of. Speaks makes use of instantiation; Soames makes use ascriptions and actions; and King makes use of facts. In this respect, the minimal theory of application recommends itself on the basis of its unifying power. The phenomena of aboutness and predication as they arise in the theory of properties, beliefs, actions and facts, on this view, are unified by the notion of application.³⁵ This strikes me as a point in its favor. In the next section I will further develop this argument for my theory.