Existence

In his famous critique of the ontological argument, Kant said that existence is not a property. Frege and Russell gave a very influential response to Kant (though not framed as such): the existence of an object is not a property of the existent object, but it is a second-order property of an abstract object. Thus, the existence of Biden is the second-order property of being instantiated possessed not by Biden himself but by the abstract object Bidenness.

But now consider this very plausible principle:

1. The existence of an object is explanatorily prior to all the (other) properties of that object.

The parenthetical “other” is included to make (1) acceptable both to Frege-Russell and to “pre-Kantians” who think existence is a property of the existent object.

But combining (1) with the Frege-Russell account leads to an explanatory priority regress:

• Biden’s maleness is posterior to Biden’s existence.

• Biden’s existence is Bidenness’s being instantiated.

• Bidenness’s being instantiated is posterior to Bidenness’s existence.

• Bidenness’s existence is Bidennessness’s being instantiated.

• Bidennessness’s being instantiated is posterior to Bidennessness’s existence.

• Bidennessness’s existence is Bidennessnessness’s being instantiated.

• …

How can we arrest this regress? A natural move is to restrict the Frege-Russell view of existence to contingent entities. Thus, Biden’s existence is the instantiation of Bidenness, but Bidenness is a necessary entity, and its existence is not the instantiation of some further entity. Indeed, perhaps, the pre-Kantian view holds of Bidenness: Bidenness’s existence could just be a property of Bidenness.

Note that if the pre-Kantian view holds of necessary beings, then Kant’s critique of the ontological argument falls apart, since God is a necessary being.

But let’s think through the pre-Kantian view a little bit. Suppose that x is an object and e is its existence, and suppose e is a property of x. But how can x possess e without already existing prior to having e? (I.e., surely, (1) is true with the parenthetical “other” removed.) There seems to be one possible move here: perhaps x = e. That would be a view on which some objects are identical with their own existence—a view very much like St. Thomas’s, who held that God, and God alone, was identical with his own existence.

So, interestingly, thinking the Frege-Russell view through leads fairly naturally to a view like Thomas’s.

I am attracted to this variant of the Thomistic view:

2. Uncaused objects are identical with their existence.

3. The existence of a caused object is its being caused.

The worry that an object cannot possess a property without “already” existing does not appear pressing when the property in question is being caused.

Moreover, we might even more speculatively add:

4. An object’s being caused is its cause’s causing of it.

On this view, a contingent thing’s existence, like on the Frege-Russell view, is a property of something other than the thing: it is a property of the cause (perhaps an extrinsic property of the cause, when the cause is God, so as not to violate divine simplicity).

Ways of being and quantifying

Pluralists about ways of being say that there are multiple ways to be (e.g., substance and accident, divine being and finite being, the ten categories, or maybe even some indefinitely extendible list) and there is no such thing as being apart from being according to one of the ways of being. Each way of being comes with its own quantifiers, and there is no overarching quantifier.

A part of the theory is that everything that exists exists in a way of being. But it seems we cannot state this in the theory, because the "everything" seems to be a quantifier transcending the quantifiers over the particular ways of being. (Merricks, for instance, makes this criticism.)

I think there is a simple solution. The pluralist can concede that there are overarching unrestricted quantifiers ∀ and ∃, but they are not fundamental. They are, instead, defined in terms of more fundamental way-of-being-restricted quantifiers in the system:

1. ∀xF(x) if and only if ∀_BWoBb∀_bxF(x)

2. ∃xF(x) if and only if ∃_BWoBb∃_bxF(x).

The idea here is that for each way of being b, there are ∀_b and ∃_b quantifiers. But, the pluralist can say, one of the ways of being is being a way of being (BWoB). So, to use Merricks’ example, to say that there are no unicorns at all, one can just say that no way of being b is such that a unicorn b-exists.

Note that being a way of a being is itself a way of being, and hence BWoB itself BWoB-exists.

The claim that everything that exists exists in a way of being can now be put as follows:

3. ∀x(x = x → ∃_BWoBb∃_by(x = y)).

Of course, (3) will be a theorem of the appropriate ways-of-being logic if we expand out "∀x" in accordance with (1). So (3) may seem trivial. But the objection of triviality seems exactly parallel to worrying that it is trivial on the JTB+ account of knowledge that if you know something, you believe it. Whether we have triviality depends on whether the account of generic existence or knowledge, respectively, is stipulative or meant to be a genuine account of a pre-theoretic notion. And nothing constrains the pluralist to making (1) and (2) be merely stipulative.

Suppose, however, your motivations for pluralism are theological: you don’t want to say that God and humans exist in the same way. You might then have the following further theological thought: Let G be a fundamental way of being that God is in. Then by transcendence, G has to be a category that is special to God, having only God in it. Moreover, by simplicity, G has to be God. Thus, the only way of being that God can be in is God. But this means there cannot be a fundamental category of ways of being that includes divine and non-divine ways of being.

However, note that even apart from theological considerations, the BWoB-quantifiers need not be fundamental. For instance, perhaps, among the ways of being there might be being an abstract object, and one could hold that ways of being are abstract objects. If so, then ∀_BWoBbG(b) could be defined as ∀_BAb(WoB(b)→G(b)), where BA is being abstract and WoB(x) says that x is a way of being.

Coming back to the theological considerations, one could suppose there is a fundamental category of being a finite way of being (BFWoB) and a fundamental category of being a divine way of being (BDWoB). By simplicity, BDWoB=God. And then we could define:

4. ∀_BWoBbF(b) if and only if ∀_BDWoBbF(b) and ∀_BFWoBbF(b).

5. ∃_BWoBbF(b) if and only if ∃_BDWoBbF(b) or ∃_BFWoBbF(b).

Note that we can rewrite ∀_BDWoBbF(b) and ∃_BDWoBbF(b) as just F(God).

Intensions

The intension of a referring expression e in a language is a partial function I_e that assigns to a world w the referent I_e(w) of e there, when there is a referent of e in w. Thus, the intension of "the tallest woman" is a partial function that assigns to w the tallest woman in w.

The intension of a unary predicate P is a partial function I_P that assigns to a world w the extension I_P(w) of P at w, i.e., the set of all satisfiers of P there.

Intensions are meant to capture the semantic features of terms, with respect to intensional semantics. Now let e be the referring expression:

• The set of even integers.

Let E be the predicate

• is an even integer.

Then the intension of e assigns to w the set of all even integers, for each w. And the intension of E assigns to w the set of all even integers, too. So I_e=I_E. But e and E are plainly not semantically equivalent, even within intensional semantics. So intensions are insufficient for characterizing the semantic features of expressions, even with respect to intensional semantics.

A longshot: Perhaps something like this led Frege to his weird "The concept horse is not a concept" claim.

Hintikka's criticism of the Fregean view of quantifiers

On the Fregean view of quantifiers, quantifiers express properties of properties. Thus, ∀ expresses a property U of Universality and ∃ expresses a property I of instantiatedness. So, ∀xFx says that Fness has universality, while ∃xFx says that Fness has instantiatedness.

One of Hintikka's criticisms is that it is hard to make sense of nested quantifiers. Consider for instance

1. ∀x∃yF(x,y).

Properties correspond to formulae open in one variable. But in the inner expression ∃yFxy the quantifier is applied to F(x,y) which is open in two variables.

But the Fregean can say this about ∀x∃yFxy. For any fixed value of x, there is a unary predicate λyFxy such that (λyFxy)(y) just in case Fxy. The λ functor takes a variable and any expression possibly containing that variable and returns a predicate. Thus, λy(y=2y) is the predicate that says of something that it is equal to twice itself.

Now, for any predicate Q, there is a property of Qness. So, for any x, there is a property of (λyFxy)ness. In other words, there is a function f from objects to properties, such that f(x) is a property that is had by y just in case F(x,y). We can write f(x)=(λyFxy)ness.

Now, we can replace the inner quantification by its Fregean rendering:

2. (λyFxy)ness has I.

But (2) defines a predicate that is being applied to x, a predicate we can refer to as λx[(λyFxy)ness has I]. This predicate in turn expresses a property: (λx[(λyFxy)ness has I])ness. And then the outer ∀x quantifier in (1) says that this property has universality. Thus our final Fregean rendering of (1) is:

3. [λx[(λyFxy)ness has I]]ness has U.

We can now ask which proposition formation rules were used in the above construction. These seem to be it:

4. If R is an n-ary relation and 1≤k≤n, then for any x there is an (n−1)-ary relation R_k,x which we might call the <k,x>-contraction of R such that x₁,...,x_k−1,x,x_k+1,...,x_n stand in R if and only if X₁,...,x_k−1,x_k+1,...,x_n stand in R_k,x.
5. If p is a function from objects to propositions, then there is a property p* which we might call the propertification of p such that x has p* iff p(x) is true.
6. There are the properties I and U of instantiation and universality, respectively.

We can think of propertification and contraction as related in an inverse fashion. Given an n-ary relation, contraction can be used to define a function from objects to (n−1)-ary relations, and propertification takes a function from objects to 0-ary relations and defines a 1-ary relation from it (this could be generalized to an operation that takes a function from objects to (n−1)-ary relations and defines an n-ary relation from it).

Observe that if P is a property, i.e., a unary relation, then the contraction P_1,x is a proposition (propositions are 0-ary relations), equivalent to the proposition that says of x that it has P.

With these two rules and the relation R that is expressed by the predicate F, start by defining the function f(x) that maps an object x to the property R_1,x, and then define the function g(x) that maps an object x to the proposition I_1,f(x). Thus, g(x) says that x stands in R to something. Now, we can form the propertification g* of the function g, and to get (1) we just say that g* has U. Thus the proposition that is expressed by (1) will be U_1,g*.

One worry about proposition formation rules is that we might fear that if we allow too many, we will be able to form a liar-type sentence. A somewhat arbitrary restriction in the above is that we only get to form a propertification for functions of first-order objects.

Another worry I have is that I made use of the concept of a function, and I'd like to do without that.