Deflating truth (4): trivial Platonism

There is only one concept more important in (traditional) metaphysics than the concept of truth: the concept of being, existence, or reality. If we can interpret Aristotle as being the first deflationist philosopher about truth (when he asserts in his Metaphysics, IX, that “it is true to say of what is that is, and of what it is not that it is not”), we can consider Kant as the first deflationist about existence, when in his Critique of Pure Reason, and in particular in his criticism of St. Anselm’s ‘ontological argument’ for the existence of god, he denies that ‘existence’ has to be interpreted as a predicate, in a similar way as we saw in the previous entries with respect to the notion of truth. Kant’s idea was that we do not attribute a special property to a thing when we say that this thing exists. This view was developed in a clearer and more systematic way with the development of formal logic by Frege, Russell and others: as probably all of you know, in first order logic, i.e., in the modern logic of predicates (like “…is green”, o “…is the father of…”), the linguistic elements in charge of making existential claims are not predicates but quantifiers. When we say “there is a green bug on the table”, modern logic reconstructs this proposition as something like “there is an x such that x is green, x is a bug, and x is on the table”, or formally:

Ǝx(Gx & Bx & Oxt)

where t is the proper name of the table, and O is the binary predicate “…is on…”. The grammatical distinction between the predicates G, B and O, on the one hand, and the existential quantifier Ǝ, on the other hand, is just the modern expression of this (Kantian) idea that ‘being is not a property’. But, if it is not a property, what it is? Well it actually is a logical operator (remember that “…is true” was also a kind of prosentence-forming operator), i.e., it is structurally more similar to logical connectives like &, ◊, and ⌐, than to real predicates. In particular, it is closely analogous to disjunction (v), for, if the list of entities of our ‘universe of discourse’ is finite and we have a name (a, b, c…) for every one of the objects we are talking about, a proposition like

ƎxGx

would be logically equivalent to the following disjunction

Ga v Gb v Gc…

(by the way, this is the reason why the notation used for the quantifier Ǝ in some logical or mathematical texts is V). Stated differently, the relation between Ǝ and v would be completely analogous to the relation between the mathematical symbols ∑ and + ¹.

So, what do we assert about something when we say of it that it exists? The American philosopher Willard Quine² expressed it with a famous slogan: “to be is to be the value of a variable bound by an existential quantifier”, i.e., to be is to be the x in a proposition like ƎxGx, where the predicate G refers to a property x has, and Ǝ simply says that we do not specify which x is the one we are talking about, only that for some x, Gx is true. If a is an entity such that Ga is true, since ƎxGx logically follows from Ga (as Ga v Gb also follows from it), then ƎxGx is also true (as Ga v Gb also is) ³

This idea can be used to offer a deflationist view about one of the classical problems of ontology: whether abstract entities like numbers or literary characters exist or not. Deflationism would just recommend us to consider what propositions we accept (i.e., what propositions we take to be true), and apply to them the logical rule we have just employed in the previous paragraph (the one allowing us to derive ƎxGx from Ga). The list of consequences so derived will be an indirect list of the things we accept as existing, no more, no less. Hence, do mathematical objects exist? Well, if we admit that 13 is a number prime (i.e., if we admit that the proposition “13 is a number primer” is true), so it follows that there is a prime number, so prime numbers exist. If we admit that no real number solves the equation x² = -1, it follows that the expression “a real number which is the square root of -1” does not refer to anything that exists. So, to accept the existence of mathematical entities merely consists in applying the derivation ƎxGx-from-Ga to the mathematical theorems we admit as true (of course, many mathematical theorems directly have the form ƎxGx).

What about ‘fictious’ beings, like Batman? Well, in this case we do not accept that Batman exists, because all the propositions from which we could derive its existence (those narrating his stories, say) are taken by us as literally false. Actually, to say that something is a fictitious entity must be understood precisely in the sense that we don’t think it exists, though there are some false stories talking about that entity. Hence, numbers exist, but fictitious characters do not, what means that numbers are not fictions, they are real, as Plato claimed 25 centuries ago.

If this sounds to you as if we had brought metaphysics by the back door, take into account that we are defending a deflationist meaning of “existence”: to say that primes exist is just a trivial logical consequence from accepting that 13 is a prime. Existence is not a property, so we are not attributing any property at all to prime numbers when we say of them that they exist. In particular, we are not attributing them causal properties. The only properties we can know number 13 has are the properties that the mathematical theorems (accepted as true) say it has, and these will obviously be mathematical properties. Our deflationist Platonism is trivial in the sense that it does not force us to accept the most delicate part of Plato’s metaphysics: his opinion that numbers (or “ideas”, or “forms”) play a causal role in the structure and properties of the world. The causes of a physical event are (for us) other physical events; that these events can be described with the help of mathematical concepts is not a reason to think that mathematical “events” have causal influence on physical facts. Hence, prime numbers exist in the exactly the same sense as protons or kangaroos exist (i.e., in the sense that some affirmative propositions talking about them are true), but they have not the same properties as animals or elementary particles: they are not subject to physical forces, nor bear new animals, etc., etc.

Stated differently, deflationism recommends to consider that “existential” problems (whether something really exists or not) are not philosophical problems, but scientific problems. I.e., whether some mathematical entities exist or not, is something that mathematicians try to proof with their theorems, as physicist are the ones in charge of finding out whether certain particles exist or not. What properties do these existing entities have, is also a problem for the relevant scientists. So, contrarily to what has been the tradition, the deflationist position is that the philosopher has relatively little to say about these questions.