Russell on Existence in TPLA III: The Argument from Transferability

In the last post we explained what Russell's views on existence are and how they entail a "higher-order" theory of existence according to which existence is not a feature of individuals but of some "higher-order" things, viz. propositional functions. Since this is, initially, a very counter-intuitive proposal ("Socrates exists" is meaningless on Russell's view!), Russell ought to have some arguments to defend his view. This is going to be a long series of posts, so we'll discuss several of Russell's arguments from The Philosophy of Logical Atomism, but in the next two posts we'll discuss one of his arguments in particular: What I will call the "Transferability Argument." But before that I'll briefly mention Russell's motivation for having a theory of existence like his in the first place.

In the first place, the whole notion of existence comes up in connection with what we might call “negative existential” statements. A negative existential statement is a statement saying that something does not exist: For instance, that Socrates does not exist, or that dogs do not exist. These present an initial puzzle. On the one hand, if they are true, then it seems “Socrates” and “dogs” do not refer to anything, and so it’s not clear what could make the sentences true. On the other hand, they seem to be saying that something, “Socrates” or “dogs,” has the feature of “not existing.”

Now, this doesn’t immediately support Russell’s view on existence, but it does give one impetus to develop some sort of view that would address the question of negative existentials. It is interesting to see how Russell’s view deals with the problem. In the first place, ‘Socrates does not exist’ is simply meaningless according to Russell, since it doesn’t make sense to attribute existence to an individual, and so neither does it make sense to deny existence of an individual. On the other hand, since existence is a property of propositional functions, “dogs do not exist,” is easy to deal with: it is the same as saying ‘x is a dog’ is impossible. This involves no shady references to non-existent dogs or anything of that sort. One need only say that ‘x is a dog’ is never true.

With that said, it is not enough to point out that Russell’s view gives an answer to this question. Russell’s view is still prima facie implausible, and there might also be other positions available. Hence, Russell needs to give some direct arguments specifically for his view and arguments against alternatives. We will discuss just one of the arguments that Russell gives, which I call “the Transferability Argument.” The argument is quite subtle in fact, and it is rather complicated. But I think it is worth thinking through because it incorporates several interesting assumptions from logic and the philosophy of language.

Before delving into it, I want to define what we will call a ‘transferable predicate’. Russell does not use this terminology himself, but he uses the concept, and his argument is easier to state with this terminology. Now, a predicate F is transferable in my sense just in case (i) F can  be meaningfully applied to some kind G, and (ii) for any kind G that F applies to, ‘G’s are F’ is true only if every individual x that is a G is also F. For instance, the predicate ‘green’ is transferable: It applies to a generic kind term like ‘men’, since we can say ‘men are green’, and ‘men are green’ is true only if each man is himself green. The predicate ‘green’ “transfers” to the individual men. The predicate ‘numerous’ on the other hand is non-transferable: While we can say ‘men are numerous’, it does not imply any particular man is himself numerous. Indeed, this last statement is meaningless.

With that said, Russell’s Argument from Transferability can be reconstructed as follows:

• (1’) ‘Unicorns exist’ is false, but meaningful. [Premise]
• (2’) If there is an individual sense of ‘exists’, then ‘exists’ is transferable. [Premise]
• (3’) If ‘exists’ is transferable, then ‘Unicorns exist’ implies ‘a exists’, for some proper name ‘a’ of some particular unicorn. [Premise]
• (4’) So, if there is an individual sense of ‘exists’, then ‘Unicorns exist’ implies ‘a exists’, for some proper name ‘a’ of some particular unicorn. [By 2’ and 3’]
• (5’) If ‘a’ is a proper name then ‘a is F’ is meaningful only if ‘a’ refers. [Premise]
• (6’) So ‘a exists’ is meaningful only if ‘a’ refers. [5’, Universal Instantiation]
• (7’) Suppose there is an individual sense of ‘exists’. [Supposition for Reductio]
• (8’) Then ‘Unicorns exist’ implies ‘a exists’ for some proper name ‘a’ of some particular unicorn. [By 4’ and 7’]
• (9’) If ‘unicorns exist’ is false, then ‘a’ does not refer. [Premise]
• (10’) So ‘a’ does not refer. [By 1’ and 9’]
• (11’) So ‘a exists’ is meaningless. [By 10’ and 6’]
• (12’) No meaningful statement can imply a meaningless statement. [Premise]
• (13’) So, ‘unicorns exist’ is meaningless. [By 8’, 11’, and 12’]

But this contradicts our assumption in (1’). Hence, we must reject our assumption in 7’:

• (14’) There is no individual sense of ‘exists’. [By 7’ – 13’ and Reductio ad Absurdum]

This is an extremely interesting and rich argument. It is the best reconstruction I can give of Russell’s argument. The argument is clearly valid. It has a total of six premises: 1’, 2’, 3’ 5’, 9’, and 12’. I think it is useful to isolate the premises so that we can see precisely the principles at work here:

• (1’) ‘Unicorns exist’ is false, but meaningful.
• (2’) If there is an individual sense of ‘exists’, then ‘exists’ is transferable.
• (3’) If ‘exists’ is transferable, then ‘Unicorns exist’ implies ‘a exists’, for some proper name ‘a’ of some particular unicorn.
• (5’) If ‘a’ is a proper name then ‘a is F’ is meaningful only if ‘a’ refers.
• (9’) If ‘unicorns exist’ is false, then ‘a’ does not refer.
• (12’) No meaningful statement can imply a meaningless statement.

The first premise is uncontroversial I assume. The third premise seems to follow from the definition of ‘transferable’.  12’ also seems straightforward: If, by hypothesis, p is meaningful, then it does not imply anything meaningless, and this is just what 12’ says. That leaves 2’, 5’, and 9’ as the crucial premises.

I take it that the motivation behind 2’ is that if ‘existence’ is just another predicate of individuals, like ‘green’, say, then it should be transferable in precisely the way they are. After all, how could it be that ‘frogs are green’ is true but that ‘green’ does not transfer to all of the individual frogs? But if ‘exists’ is just like ‘green’ then it should behave in the same way.

5’ seems to be motivated by the fact that we are supposing ‘a’ to be a proper name in Russell’s sense. Recall that, according to Russell, a logically proper name is a word whose meaning just is a particular object; in other words, the proper name ‘a’ is meaningful only if, and because, ‘a’ refers. So if ‘a’ is meaningless, then the whole sentence ‘a is F’ will be meaningless too.

Finally, 9’ is motivated by the fact that if ‘unicorns exist’ is false then there simply aren’t any unicorns for ‘a’ to refer to, and so ‘a’ cannot have a reference.

On the face of it, there is some reasonableness about all of these premises. However, I think there are worries for all of them. In the next post I will try to raise some of those worries.

Russell on Existence in TPLA II: Russell's Second-Order View of Existence

In my last post I introduced the topic of Russell on existence. Now I'll deliver. Let’s see what Russell thinks.

First, it is helpful to understand some of Russell’s technical vocabulary. In particular, for our purposes, we should consider his notion of a name, of a definite description, of a proposition, and of a propositional function.

For Russell, a logically proper name (or, for short, just a name) is a word whose meaning is a particular, i.e., an individual object or entity. For instance, intuitively, the name “Socrates” directly denotes the particular object, Socrates. Or the name “Paco” directly denotes my Chihuahua, Paco. Now, this is simplifying a little bit, since Russell has a whole theory of what a particular is and which words actually are proper names, but this isn’t really essential to his account of existence. One could hold to views about existence that are basically the same as Russell’s even if one modified his account of particulars and the extent of the proper names.

What is important, however, is that proper names be contrasted with definite descriptions. A definite description is some phrase that is meant to describe a particular, unique individual. For instance “the Chihuahua that I have had since 6th grade” is a definite description. (As it turns out, it does successfully denote something: my dog Paco.) “The dragon flying above my head” is a definite description too, though to my knowledge it is one that does not refer to anything.

Note: Names are not definite descriptions and definite descriptions are not names. The meaning of a name is just the object it refers to; the meaning of a definite description includes all of the predicates mentioned in the description (for instance, in the last example, "dragon," "flying", and "above my head" are all part of the meaning of the description).

It is important to bring up this contrast between definite descriptions and proper names because Russell gives a separate account of existence statements for each. What we are interested in when talking about “individual existence” statements is existence statements whose subject term is a proper name. This is the type of existence statement Russell will say is meaningless.

A proposition for Russell is, in essence, something that can be asserted, or something that can be true or false. For instance, that it is raining is a proposition, or that Paco is black is a proposition. Once again, this is simplifying a bit, but the particular details of Russell’s views on propositions are not essential here.

Finally, there is the notion of a propositional function. Russell says that a propositional function is “any expression containing an undetermined constituent, or several undetermined constituents, and becoming a proposition as soon as the undetermined constituents are determined.” Examples include ‘x is a man’ or ‘n is a number’ or ‘(x+y)(x-y)=x2-y2’. So, if we were to “fill in the blanks” so to speak we would have a full proposition. For instance, replacing ‘x’ with ‘Paco’ gives the proposition that Paco is a man. Replacing 'n' with '2' gives the proposition that 2 is a number.

Russell's propositional functions can be necessary, possible, or impossible. Russell defines this as follows. A propositional function is:

• Necessary, when it is always true;
• Possible, when it is sometimes true;
• Impossible, when it is never true.

Russell technically says that we have to take at least one of these locutions – “always true,” “sometimes true,” etc. – as undefined. But intuitively, “always true” means that every instance of the propositional function is true. For example, ‘x is x’ is a propositional function that is “necessary’ in Russell’s sense, since it is “always true,” whereas ‘x is a man’ is a propositional function that is possible but not necessary. These locutions are clearly not meant in a temporal sense.

(Interesting side-note: Obviously Russell's definition of "possible" and the like is not at all the definition we would immediately think of when we hear these words. What's interesting is that it's not clear whether he even meant to capture what we do with possible worlds semantics. He makes it explicit that he thinks previous thinking about modality is confused and problematic in some way, but it's not clear whether his discussion of modality is trying to capture some sort of traditional modal phenomenon as opposed to just making stipulations, nor whether his attitude toward traditional notions of modality is one of revision or rejection. Another interesting question: Is there any way to modernize Russell here? Is he on to anything at all? Anyway, enough of this digression...)

That brings us finally to Russell’s theory of existence. Russell’s official view is that “existence is a predicate of a propositional function.” In particular, if F is a type or kind of entity, then to say that F’s exist is just a shorthand way of saying that the propositional function ‘x is F’ is possible:

• (EXIST): F’s exist iff ‘x is F’ is possible (in Russell’s sense above).

For instance, dogs exist iff ‘x is a dog’ is possible. Or men exist iff ‘x is a man’ is possible.

This makes Russell's view a "second-order" or "second-level" view of existence. If we think of individual objects or entities as the "first level" and we think of things that apply to individuals -- propositional functions -- as the second level, then existence is a property of things at the second level, since it is a property of propositional functions. Hence Russell's view has been variously described as a "second-order", "higher-order", or "higher-level" view of existence.

So, according to Russell, “It is of propositional functions that you can assert or deny existence.” On the other hand, to say of a particular thing in the world that it is exists or not is “strictly nonsense.” After all, it doesn’t make sense to say of a particular object a that it is “possible” or “sometimes true.” Hence, individual existence statements are meaningless.

This is of course rather shocking on the face of it. We seem to make true individual existence claims all the time. But on Russell’s view, “John exists” isn’t simply false. It isn’t even a loose way of speaking. It’s simply nonsense. Moreover, the seemingly indubitable inference from “I think” to “I exist” is not invalid on this view; it isn’t even an argument, since arguments have to have propositions as their conclusions, and “I exist” isn’t even a comprehensible thought. What one might have thought incorrigible turns out to be unintelligible.

Nonetheless, as repugnant to common sense as this might seem at first, common sense is not infallible. And to be fair, we have only laid out Russell’s views and have not presented his arguments for them. In the next post I'll consider some of the reasons why, exactly, Russell might have come to this conclusion.

[Part III is here!]

Russell on Existence in TPLA I: Why Care?

In the next few posts I'm going to talk about Bertrand Russell's views on existence as one finds them in his The Philosophy of Logical Atomism (TPLA for short). (Note: I have already written the posts, so I will actually deliver!)

In TPLA Bertrand Russell offers a brief but intriguing account of the notion of existence. Russell holds forcefully to the view that existence cannot be said to apply to individual objects – at least, not without descending into nonsense. According to Russell we cannot meaningfully say this or that particular thing exists; instead, only types or kinds of things can be said to exist.

I will first try to make clear what, exactly, Russell’s views on the matter of existence are, at least insofar as he talks about it in TPLA, and I will clarify his technical terminology along the way. I will then attempt to lay out what are, so far as I can tell, Russell’s arguments for his views, as well as some of the problems concerning existence that motivate him to have a view in the first place. After questioning the soundness of Russell’s arguments I will lay out an alternative view that deals with some of the problems of existence he has identified. This alternative view of existence is more similar to that held by the majority of people before him, including the medieval Scholastics. Ironically, it turns out that this view is actually similar to some of what Russell says about propositions and propositional functions (as well as his own earlier view before TPLA).

Before I begin, however, I’d like to explain why I think it is worthwhile to think about Russell’s views on existence and to bother to critique them. After all, Russell gave the Lectures for TPLA nearly a hundred years ago, and hardly anyone would agree with the precise details of his account of existence (let alone his general metaphysic).

It is probably true that Russell’s precise views are not generally accepted, and arguably parts of his technical “machinery” are archaic. Nevertheless, Russell’s spirit lives on. Russell’s claim that existence is not a predicate of individuals and that existence should ultimately be defined in a “higher-order” way, in terms of quantification, has been widely accepted by many prominent philosophers.  Indeed, the slogan that “existence is what is expressed by ‘existential quantification’” is standard orthodoxy nowadays. (See, for instance, Frege’s  Foundations of Arithmetic sec. 53, Quine’s “On What There Is,” C.J.F. Williams’ What is Existence?, and Peter van Inwagen’s “Being, Existence and Ontological Commitment,” just for a few examples.)

So, aside from the fact that Russell was a great thinker, and the general guideline that it is worthwhile to interact with great thinkers, we should think about his views on existence because views like his are held in one form or another even today.

[Part II is here.]

Quantifier Variance and the Semantics of Quantifiers

In my previous post I explained the basic idea behind quantifier variance. Now I want to criticize it. In particular, I said I want to point out some problems with the quantifier variantist's simultaneously affirming the following two statements:

(i) the different quantifiers behave the same logically; and

(ii) the different quantifiers have different meanings.

Let's do a little basic semantics. Let's define the truth function τ[ψ]^U,g relative to models U and g for the cases of quantified formulas ψ as follows. The following definitions are true for all models M, all variable assignments s, all variables x, and all formulas φ. If a formula is not assigned to T it is assigned to F:


τ : {<ψ,U,g>|ψ is a formula, U a model, g a var. assign.} → {T,F}

• (τ-∀): τ[∀xφ]^M,s = T ⇔ for all variable assignments s′, if for all variables v, s(v) ≠ s′(v) ⇒ v = x, then τ[φ]^M,s′ = T 
• (τ-∃): τ[∃xφ]^M,s = T ⇔ for some variable assignment s′, for all variables v, s(v) ≠ s′(v) ⇒ v = x, and τ[φ]^M,s′ = T 

Basics of Quantifier Variance

When I say that there are tables is it unambiguous what I'm saying? Quantifier variantists say no. Or at least they would say that in certain contexts it is not. In particular, the sentence is ambiguous when we are engaging in metaphysical debate about the existence of the table, as in the following case.

Consider the debate between what I will call compositionalism and anti-compositionalism. Compositionalism is the thesis that there are composite material objects, while anti-compositionalism is the thesis that there are not. Take the case of a world with just a table and its parts, and suppose we are considering a form of compositionalism which says there are tables. Assume further that there are exactly n atoms which, according to this form of compositionalism, are proper parts of the table. Note that we are using a philosophical definition of 'atom', according to which an atom is a material object which has no proper parts. Anti-compositionalism says there is no table; there are just the n atoms. 

In essence, compositionalism says (A) there are n+1 distinct things (viz. the n atoms, plus the table), while anti-compositionalism says (B) there are n things and there are no more than n things. Note that (A) and (B) can be adequately translated into a quantified language which only contains variables, quantifiers, sentential connectives, and the identity sign with the usual interpretation. For example, (A) would be translated as follows:

∃x₁∃x₂...∃x_n((x₁≠x₂ ∧ ... ∧ x₁≠x_n+1) ∧ (x₂≠x₃ ∧ ... ∧ x₂≠x_n+1) ∧ ... ∧ (x_n≠x_n+1))

William Lane Craig on God and Analogy

William Lane Craig, as with most contemporary analytic philosophers of religion, objects to the classical Thomistic idea that God cannot be said to be a being in the sense that we can. I aim to argue that at least some of his claims here are unreasonable.

Craig objects: "One of the aspects of Thomas Aquinas’ thought that I find most disturbing is his claim that we can speak of God only in analogical terms. Without univocity of meaning, we are left with agnosticism about the nature of God, able to say only what God is not, not what He is."

This does not follow in the slightest; in fact, I wonder if Craig actually understands what analogy is, since it is one of the main points of Aquinas's theory of analogy to avoid this problem. Aquinas sought to show, in contrast to Maimonides, that though we can't predicate attributes of God in the same sense as us we can still speak meaningfully and make positive predications about him. Craig fails utterly to show how from the semantic analogy of the term 'being' we are left only with negative theology.

Next, Craig says: "When in discussions with atheists I affirm, 'God exists' and they reply, 'God does not exist,' we may need to be sure that we mean the same thing by 'God,' but there is no equivocation on the meaning of 'exists.'"

I guess what Craig is trying to say here is something like this: If the word 'exists' is analogical, then when I affirm God exists and when atheists affirm God does not exist, we are both equivocating past each other. But this is a genuine ontological dispute and so there is no equivocation; hence, 'exists' is not analogical. The problem is that the main premise is simply not true; if analogy is true, then we affirm that God exists in one sense, and the atheist simply denies that God exists in any sense, including the one I am predicating of God.

In the next paragraph, Craig says this: "The problem you pose brings us to the heart of my current work on divine aseity. What makes God more than just one being among many is precisely His aseity: God alone is self-existent; everything else exists contingently. Only God exists of Himself (a se); everything else exists through another (ab alio). That makes God the source of being for everything apart from Himself."

Now, I really like this, and I agree with this completely. The only problem is that Craig himself doesn't; for if what Craig says is literally true then divine simplicity is true, from which it is a small step to the doctrine that talk about God is analogous. Here's why:

(1) Whatever is non-identical to God is created by God. [conceded by Craig]
(2) If God has metaphysical proper parts ('parts' hereafter), then at least one of these parts is not created by God. [prem]
(3) Either God has parts or he doesn't. [LEM]
(4) Suppose he does have parts. [assp]
Then
(5) All of God's parts are created by God. [by 1]
But
(6) One of God's parts is not created by God [by 2,4]
This is a contradiction. Hence, we must reject our assumption. Hence:
(7) God has no parts.

So by 'metaphysical proper parts' here I mean things like ontological constituents, such as a property-instance (or trope or accident or whatever). (1) is just Craig's own thoughts on the matter, and (2) is true because clearly God doesn't create his essential properties; he depends on those for his existence, since if they didn't exist then neither would he. The rest follows by the meanings of the terms and the rules of logic.

Craig says that he considers God to be a substance, presumably in the same manner we are: "Not a physical substance, of course, but a spiritual substance like a mind."

However, the case is even more clear if Craig thinks God's mind and will are distinct; for if he does, granting Craig's doctrine of aseity, then from (1) it follows God's will must be created by God. But it is absurd to suppose God creates his own will; after all, he must have a will to do that! So, either Craig's doctrine of aseity is false (which I agree with Craig it isn't) or God is not distinct from his will (which I think is right, but is really only intelligible given divine simplicity).

Craig thinks getting rid of Platonism will solve the problems concerning God's aseity; but it doesn't, since even if there are no abstract properties in us there are clearly ontological constituents (my brownness, my height, my shape, etc.). Even taking 'parts' in this sense, I think the above argument shows that if he wants to hold on to the strong doctrine of aseity set out in the quote above he needs to get rid of the idea that God has any parts at all. And if God has no parts in the metaphysical sense then it can be shown speech about God is analogical; for in our case, to say I am good is to say the quality of goodness inheres in me as an accident (or is exemplified as a property, or inheres as a trope or whatever). But since God has no parts in any of these senses, to say God is good cannot be to say this about him. And the same with any of the divine attributes. Thus our terms must be said analogically of God.

[Edit: Craig's reply here. My reply here.]

Is 'Existence' Univocal Because 'All' Is Univocal?

In this post, Bill Vallicella presents another argument by Peter van Inwagen for the univocity of existence and questions it (he posted and refuted the first argument here). I think Vallicella has a point. Still, I might grant that the Quinean like Van Inwagen can translate a singular existential statement so as to have the same form as a general existential statement and argue the conclusion still does not follow. Quine, in his famous paper "On What There Is," proposes that we treat for instance the relation '__ = Pegasus' as a single-place predicate '=Pegasus'; one can call this 'pegasizing' or formally 'P'. Then 'Pegasus exists' will just be 'something pegasizes', which will just be translated to, '(Ex)(Px)'. Even ignoring the problematic aspects here I would pose a different objection. I would accuse Van Inwagen's argument of being a 'non sequitur'. Vallicella states the argument thus:

(1) 'Every' is univocal.

(2) 'Exist(s)' and 'every' are interdefinable: 'Fs exist' is equivalent to 'It is not the case that everything is not an F.'

Therefore

(3) 'Exist(s)' is univocal.

Clearly, as is, this argument is not valid. To make it valid we need some further premise. I'm not sure what sort of plausible premise Van Inwagen is using to get to his conclusion, but maybe it is something like

(2.5) If two terms are interdefinable then each of the terms' uses share the same sense relation.

(Just a clarificatory point: Sense relations are things like 'univocity' or 'equivocity', and Van Inwagen thinks that all the uses of 'exists' are univocal.) How are we to understand 'interdefinable' here? Surely not as meaning that for each 'exists' statement there is a semantically identical 'every' statement, i.e. one with the exact same meaning, for that would be utterly question-begging. We must construe it then as something like 'for each 'exists' statement there is a logically equivalent 'every' statement'. The problem is that (2.5) is not obviously true on this interpretation. I'll explain.

I think we can admit that 'some' and 'every' are univocal, that these two are interdefinable in the sense that logically equivalent statements can be expressed in terms of each, but still say that 'some' doesn't fully capture the meaning of 'exists', and thus neither does 'every'. Of course, every 'some' statement is logically equivalent to another 'there exists' statement, but that does not imply they are semantically identical.

On the idea that 'exists' is analogical, the natural language quantifier 'there exists' has many senses, but all beings can be said to exist in one of those senses; thus the range of this quantifier includes all beings (regardless as to which sense of 'being' can be said of them). And since there are no non-existent beings, the range of the quantifier 'some' is over all beings. So the two quantifiers range over the same domain of discourse; and since for any 'some' statement there is a logically equivalent 'there exists' statement, it follows that we can translate logically equivalent statements involving either of them with the same symbol in predicate-logic, '(Ex)'. This is also why they are each logically equivalent to at least one 'all' statement. But it simply doesn't follow that they all share the same sense relation (univocal, equivocal, etc.). It is true that our 'some' quantifier ranges over only and all beings, but it ranges over them regardless as to which of the many analogous senses of 'being' can be said of them. So it's consistent with both 'some' and 'all' being univocal that 'being' or 'exists' are not.