Individuals: an essay in revisionary metaphysics

Abstract

We naturally think of the material world as being populated by a large number of individuals. These are things, such as my laptop and the particles that compose it, that we describe as being propertied and related in various ways when we describe the material world around us. In this paper I argue that, fundamentally speaking at least, there are no such things as material individuals. I then propose and defend an individual-less view of the material world I call “generalism”.

Appendix on language G

First I will define the syntax of language G and give a more complete and precise explanation of what the term-functors mean. Then I will define a notion of logical implication on G and a relation of equivalence between G and PL. Third, I will outline the proof that the relation of equivalence satisfies Sufficiency, Modesty, and Preservation, as required. Finally, I will respond to an objection to the way these tasks are carried out.

1.1 The syntax of G

The vocabulary of G consists of a countable set of atomic terms, each associated with an integer n ≥  0, including a term ${\user2{I}}^{\bf 2}$ associated with the number 2. If a term of G is associated with the integer n, we call it an n-place term. The vocabulary of G also includes six term-functors ${\boldsymbol{\&},\,\boldsymbol{\sim,}}\, {\user2{c},\,\user2{p}},\, {\boldsymbol{\imath}},\,\hbox{and}\,{\boldsymbol{\sigma}}$. The set of terms are then defined inductively as follows:

• 1. All atomic n-place terms are n-place terms.

• 2. If ${\user2{P}}^{\user2{n}} \hbox{ and } {\user2{Q}}^{\user2{m}}$ are n-place and m-place terms respectively, then

  • (a) ${\boldsymbol{\sim}} {\user2{P}}^{\user2{n}},\, {\boldsymbol{\imath}}{\user2{P}}^{\user2{n}},\,\hbox{and}\,{\boldsymbol{\sigma}}{\user2{P}}^{\user2{n}}$ are n-place terms,

  • (b) ${\user2{cP}}^{\user2{n}}$ is an (n − 1)-place term unless n = 0, in which case it is a 0-place term.

  • (c) ${\user2{pP}}^{\user2{n}}$ is an (n  +  1)-place term, and

  • (d) ${\bf (}{\user2{P}}^{\user2{n}}\,\boldsymbol{\&}\,{\user2{Q}}^{\user2{m}}{\bf )}$ is a max(n, m)-place term.

• 3. That’s all; nothing else is a term.

Finally, the vocabulary of G includes a single predicate, ${\user2{x\,obtains}}.$ A sentence of G is an expression of the form ${\user2{P}}^{\bf 0}\,{\user2{obtains}},\,\hbox{ where }\,{\user2{P}}^{\bf 0}$ is a 0-place term.⁴⁰

1.2 The term-functors

In the text I gave a partial explanation of what the term-functors mean by saying things like ‘if ${\user2{L}}^{\bf 2}$ is the 2-place property of loving, which we ordinarily understand as holding between individuals x and y if and only if x loves y, then ${\bf (}{\user2{L}}^{\bf 2}\,{\boldsymbol{\&}\,\boldsymbol{\sim}\boldsymbol{\sigma}}{\user2{L}}^{\bf 2}{\bf )}$ is the 2-place property of loving unrequitedly, which we ordinarily understand to hold between individuals x and y if and only if x loves y and y does not love x.’

Here I shall give a more precise and general explanation of what the term-functors mean in this form. But there are two points worth keeping in mind. First, I am explaining what the term-functors mean in terms that we ordinarily understand, namely in terms of individuals. But of course this is just an explanatory convenience, and we should keep in mind that the properties are ultimately to be understood independently of a domain of individuals. For a discussion of the legitimacy of this explanatory technique, see the end of this appendix. The second thing to keep in mind is that in the above quotation from the text the ‘if and only if’ is not a material bi-conditional—if it were, it would not allow me to pick out the intended properties. I shall not discuss exactly what bi-conditional it is, but I take it that we understand the quotation above well enough to be getting on with.

Now for the explanation of the term functors. Let ${\user2{P}}^{\user2{n}}$ be the n-place property we ordinarily understand as holding of individuals x ₁…x _n if and only if ϕ(x ₁…x _n ). And let ${\user2{Q}}^{\user2{m}}$ be the m-place property we ordinarily understand as holding of individuals y ₁…y _m if and only if ψ(y ₁…y _m ). Then

• 1. ${\boldsymbol{\sim}}{\user2{P}}^{\user2{n}}$ is the n-place property we ordinarily understand as holding of x₁…x _n if and only if it is not the case that ϕ(x₁…x _n );

• 2. ${\bf (}{\user2{P}}^{\user2{n}}\,{\boldsymbol{\&}}\,{\user2{Q}}^{\user2{m}}{\bf )}$ is the max(n, m)-place property that we ordinarily understand as holding of x₁…x _k if and only if ϕ(x₁…x _n ) and ψ(x₁…x _m ), where k = max(n, m);

• 3. ${\boldsymbol{\sigma}} {\user2{P}}^{\user2{n}}$ is the n-place property we ordinarily understand as holding of x₁…x _n if and only if ϕ(x _n x₁x₂…x_n-1);

• 4. ${\boldsymbol{\imath}} {\user2{P}}^{\user2{n}}$ is the n-place property we ordinarily understand as holding of x₁…x _n if and only if ϕ(x₂x₁x₃…x _n );

• 5. If n ≥ 1, then ${\user2{cP}}^{\user2{n}}$ is the (n − 1)-place property that we ordinarily understand as holding of x ₂…x _n if and only if there is something x ₁ such that ϕ(x₁…x _n ); otherwise ${\user2{cP}}^{\user2{n}}$ is the 0-place property ${\user2{P}}^{\user2{n}};$

• 6. ${\user2{pP}}^{\user2{n}}$ is the (n  +  1)-place property that we ordinarily understand as holding of x₁…x_n+1 if and only if ϕ(x₂…x _n x_n+1).

Finally, ${\user2{I}}^{\bf 2}$ is identity, the 2-place property we ordinarily understand as holding between x and y if and only if x is identical to y.

1.3 Predicate functor logic

Our task now is to define a relation of logical consequence on G and a relation of equivalence between G and PL in such a way that we can prove Sufficiency, Modesty, and Preservation. We shall reduce this task to one already accomplished by defining a closely related language of “predicate functor logic” I will call Q.⁴¹ There is already a well-defined relation of consequence on Q and a relation of equivalence between Q and PL for which analogues of these three theorems hold, and we will use this fact to induce corresponding relations on G and prove our theorems.

Syntactically speaking, the language Q is just the same as G with the exception that (i) it does not contain G’s single predicate ${\user2{x\,obtains}},$ and (ii) its expressions are written in normal font rather than the bold font of G. The terms in G are, when written in the normal font of Q, to be understood as predicates instead. More precisely, the vocabulary of Q consists of a countable set of atomic n-place predicates, including a 2-place predicate I ², and six predicate functors ∼, &, c, p, $\imath\,\hbox{and}\,\sigma.$ An expression of the form P ⁿ is an atomic n-place predicate of Q if and only if ${\user2{P}}^{\user2{n}}$ is an atomic n-place term of G; an expression of the form ∼P ⁿ is an n-place predicate of Q if and only if ${\boldsymbol{\sim}}{\user2{P}}^{\user2{n}}$ is an n-place term of G; and so on.

We can define a notion of logical implication on Q model-theoretically. Without loss of generality, we can assume that the predicates of Q are the predicates of PL. So we can give a semantics for Q using the standard models of PL (i.e. predicate logic with identity but without constants).⁴² These models are pairs of the form M = (D, v), where D is a non-empty set and v is a function from n-place predicates (other than I²) to sets of n-tuples of D. Given a predicate P ⁿ of Q, a model M = (D,v) and a sequence d = (d ₁, d ₂…) of elements of D, we say that d satisfies P ⁿ in M, written d⊧ _M P ⁿ, if and only if

• 1. If P ⁿ is atomic then $(d_1,\ldots, d_n) \in v(P^n)$,

• 2. If P ⁿ = I ² then d ₁ = d ₂,

• 3. If P ⁿ = ∼ Q ⁿ then it is not the case that $d \models_M Q^n$,

• 4. If $P^{n}=(Q^m\,\&\,R^r)$ then $d\models_M Q^m$ and $d\models_M R^r$,

• 5. If P ⁿ = c Q ⁿ⁺¹ then there is an $x \in D$ such that ($x, d_1, d_2, \ldots)\models_M Q^{n+1}$,

• 6. If P ⁿ = p Q ⁿ⁻¹ then ($d_2, d_3, \ldots)\models_M Q^{n-1}$,

• 7. If $P^{n}=\imath Q^n$ then ($d_2, d_1, d_3, d_4, \ldots) \models_M Q^n$, and finally

• 8. If $P^n={\sigma}Q^n$ then ($d_n, d_1, d_2,\ldots, d_{n-1}, d_{n+1}, \ldots)\models_M Q^n$.

We can then say that P ⁿ is true on M, written $\models_M P^{n}$, if and only if $d \models_M P^{n}$ for all sequences d = (d ₁, d ₂,…) of elements of D.

With this semantics in place, it is now easy to define a relation of logical implication on Q and a relation of equivalence between Q and PL. First, the relation of logical implication can be defined as follows: a predicate P ⁿ logically implies a predicate Q^m, written P ⁿ⊧ Q ^m, if and only if for any model M, if ⊧ _M P ⁿ then ⊧ _M Q ^m.

Now for the relation of equivalence between Q and PL. Remember, the models M = (D, v) are models of PL too. So let us assume that the notion of an n-ary formula $\phi(x_1\ldots \,x_n)$ of PL being true on a model M, written $\models_M \phi(x_1\ldots\, x_n)$, is defined in the normal way.⁴³ Then we can say that an n-ary formula $\phi(x_1\ldots\, x_n)$ of PL and a predicate P ⁿ of Q are equivalent if and only if for every model M, $\models_M \phi(x_1 \ldots\, x_n)$ if and only if $\models_M P^{n}$.

For our purposes the important results are these:

• Q-Sufficiency Every n-ary formula of PL is equivalent to some n-place predicate of Q.

• Q-Modesty Every n-place predicate of Q is equivalent to some n-ary formula of PL.⁴⁴

With these results in hand, our task is now easy.

1.4 Implication, equivalence, and our three theorems

First, we use Q to define a relation of consequence on G and a relation of equivalence between G and PL. Then we prove our three theorems.

The relation of logical implication on Q naturally induces a relation $\preceq$ between terms of G: ${\user2{P}}^{\user2{n}}\preceq {\user2{Q}}^{\user2{m}}$ if and only if $P^{n}\models Q^m$. In turn, this induces a relation of logical implication on sentences of G: ${\user2{P}}^{\bf 0}\,{\user2{obtains}}$ logically implies ${\user2{Q}}^{\bf 0}\, {\user2{obtains}},$ written ${\user2{P}}^{\bf 0}\, {\user2{obtains}} \models {\user2{Q}}^{\bf 0}\, {\user2{obtains}},$ if and only if ${\user2{P}}^{\bf 0} \preceq {\user2{Q}}^{\bf 0};$ that is, if and only if $P^{0}\models Q^{0}.$ (The sign ‘$\models$’ is being used for logical implication in both languages, but I take it that no confusion will result.)

The relation of equivalence between Q and PL also induces a relation of equivalence between G and PL: a sentence ${\user2{P}}^{\bf 0}\, {\user2{obtains}}$ of G is equivalent to a sentence p of PL if and only if P ⁰ is equivalent to p.

Preservation is now easy to prove. Suppose a sentence ${\user2{P}}^{\bf 0} \,{\user2{obtains}}$ in G is equivalent to a sentence p in PL, and suppose a sentence ${\user2{Q}}^{\bf 0} \,{\user2{obtains}}$ in G is equivalent to q in PL. We must show that ${\user2{P}}^{\bf 0}\,{\user2{obtains}} \models {\user2{Q}}^{\bf 0}\,{\user2{obtains}}$ if and only if $p \models q.$ Now, ${\user2{P}}^{\bf 0}\,{\user2{obtains}} \models {\user2{Q}}^{\bf 0}\,{\user2{obtains}}$ if and only if $P^0 \models Q^{0}$ (by definition of implication in G). But P⁰ is equivalent to p and Q⁰ is equivalent to q (by our hypothesis and the definition of equivalence between G and PL), and therefore $P^{0} \models Q^{0}$ if and only if $p \models q$ (by model-theoretic reasoning). So ${\user2{P}}^{\bf 0}\,{\user2{obtains}} \models {\user2{Q}}^{\bf 0}\, {\user2{obtains}}$ if and only if $p\models q,$ as required.

Finally, Sufficiency and Modesty follow straight from Q-Sufficiency and Q-Modesty along with the above definitions.

1.5 Using a domain of individuals

Earlier in the appendix and in the text I explained what the term-functors of G mean by appealing to our ordinary understanding of properties being instantiated by a domain of individuals. But the generalist claims that individuals are not constituents of the fundamental facts of the world. So was that method of explanation legitimate?

Similarly, I just used the semantics for Q to define a relation of implication on G and a relation of equivalence between G and PL. But the semantics for Q used ordinary models of PL of the form M = (D, v), where D is a domain of individuals. Was that legitimate?

I think it was. The key is to distinguish between metaphysical and conceptual priority. Metaphysically, the generalist claims that the fundamental facts of the world are those expressed by G. But conceptually, I have explained the view to you in terms that you are familiar with, namely in terms of individuals. Once you are competent with G you can then, if you so wish, become a generalist by giving up your understanding of G in terms of a domain of individuals and simply taking the term-functors and predicate of G as undefined primitives.

There is no space to defend the cogency of the strategy here in full. I shall only plead innocence by association, for there are many other cases in which we adopt it. The case of velocity is, unsurprisingly, a good example. Our ordinary concept of the relative velocity between two material things is defined as the difference between their absolute velocities. Once you are competent with the concept, you can then (if you so wish) dispense with absolute velocity and either think of relative velocity as a primitive concept or define it in other terms. And this is precisely what we do when we adopt a Galilean conception of space-time.