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What does Kant mean by "Existence is not a predicate"?
How does that invalidate the ontological arguments? and how can he show that it's not a predicate?
By predicate, I think he means a "property" of the entity, for example, the predicate of being tall. This is the meaning that I'm aware of and which is the meaning we use in mathematical logic.
Mathematical logic, and the associated notion of the existential quantifier, were invented only after Kant's time. Kant used other, more traditional concepts.
The ontological proof (or at least the version that Kant criticized) is related to the idea that God exists by necessity, that existence is an essential property of God. When Kant asserted that "existence is not a real predicate", what he meant was that existence cannot be an essential property of anything (that it was an inherently accidental property), and therefore cannot be an essential property of God.
Kant meant that existence was similar to, say, location. Joe can be today at New York and tomorrow at Washington. Joe's location would change, but Joe himself would not change. If Joe ceases to exist, in a way he himself would change.. but arguably his concept would not. That's why Kant went on to argue in terms of concepts and judgements, rather than in terms of entities and properties. He argued that predicating existence of a thing does not "enlarge" or "increase" that thing's concept. Relatedly, he argued that the judgement "x exists" is always synthetic rather than analytic (that is tautological, true by virtue of mere meaning).
Kant's proof, that existence is inherently accidental, is roughly as follows: Suppose that the existence of some A enlarges A. In that case, A and (A + existence) were different concepts. And then the proposition "A exists" would be necessarily false. Because if A exists, then it is actually (A + existence) which exists, and (A + existence) is, we assumed, different from A.
A hundred real dollars contain no more than a hundred possible dollars. For, as the latter indicate the conception, and the former the object, on the supposition that the content of the former was greater than that of the latter, my conception would not be an expression of the whole object, and would consequently be an inadequate conception of it. But in reckoning my wealth there may be said to be more in a hundred real dollars than in a hundred possible dollars—that is, in the mere conception of them. For the real object—the dollars—is not analytically contained in my conception, but forms a synthetical addition to my conception (which is merely a determination of my mental state), although this objective reality—this existence—apart from my conceptions, does not in the least degree increase the aforesaid hundred dollars. (Critique of Pure Reason "Of the Impossibility of an Ontological Proof of the Existence of God")
By predicate, I think he means a "property" of the entity, for example, the predicate of being tall. This is the meaning that I'm aware of and which is the meaning we use in mathematical logic.
Exactly; in mathematical logic "existence" is a quantifier acting on a predicate; we read:
∃xPx
as: "there is an object having property P".
The existence of such an object is a fact that we have to ascertain through an empirical verification or a proof.
Existence is not part of the "concept" P itself.
But there are other possibilities; see Alexius Meinong and Nonexistent Objects, as well as logics with an existence predicate in addition to the existential quantifier.
(I will leave aside mathematicianly quibbling over whether a predicate is a property, and just stick with the original language.)
If existence is a predicate, it should be possible to clearly identify which things do not satisfy it. But it offers no such clarity.
Do unicorns exist? Well, sort of, they exist as potential objects and do not exist as instantiated objects. So those two variations of existence might be predicated of things, as in Meinongianism. But the naive concept of existence as a whole is not clear enough to use as a predicate.
The very idea that everything that exists must do so in some given way undercuts all arguments that try to handle 'exists' as an unmodified predicate. In particular, the idea that 'existence' is a single state of which there is a perfect variant requires us to treat existence itself as a single state, which it just isn't.
The varieties of existence do not form that kind of hierarchy: Which is more ideal, the kind of existence a perfect circle has, or the kind of existence the pizza in front of me has? I would rather have the pizza not have the kind of existence the circle has, as I would wish for it to remain attainable. And I am fine with the idea of the circle not having the kind of existence the pizza has, as otherwise we have crazy "actual infinity" paradoxes in the philosophy of mathematics. So neither of these two ways of existing is definitely better.
Not having a common root predicate, the varieties of existence overlap indiscriminately and do not all fall under a common realization. There is no perfect form of existence, and it is pointless to try to order them and choose the one that God should have.
Kant believes that the descriptor of "existence" doesn't actually change the concept of the idea in itself, just relates it to the world. E.g. 100 imaginary dollars and 100 real dollars both have the same features just one is material and one not.
MODERN STATEMENT
According to Kant, existence is not a real predicate, that is, 'a predicate which is added to the concept of a subject and enlarges it';1 and modern philosophical analysis would seem to support Kant's view. One argument to show that existence is not a predicate is the following. In order to predicate something of X, it must be presupposed that X exists. So, if 'exists' is a predicate, then, for example, 'Tame tigers exist' will be tauto- logous and 'No tame tigers exist' will be self-contradictory, but, since neither of these is the case, 'exists' cannot be a predicate. (Vera Peetz, 'Is Existence a Predicate?', Philosophy, Vol. 57, No. 221 (Jul., 1982), 395-401 : 395.)
BACK TO THE KANTIAN ORIGINAL
Or take this passage from the Critique of Pure Reason :
By whatever and by however many predicates we may think a thing - even if we completely determine it - we do not make the least addition to the thing when we further declare that this thing is [exists : GT]. Otherwise, it would not be exactly the same thing that exists, but something more than we had thought in the concept; and we could not, therefore, say that the exact object of my concept exists.(B628). (Forgie, 567.)
THE KANTIAN ORIGINAL ORIGINAL
Kant's arguments that existence is not a first-level predicate antedate the first Critique. (Hereafter when the word 'predicate' is used without qualification it will mean "first-level" predicate.) To my knowledge the earliest such arguments appear in the following passage from the 1763 essay, "Der einzig mogliche Beweisgrund zu einer Demonstration des Daseins Cottes" (hereafter "Beweisgrund").
'Existence is not a predicate or determination of anything whatsoever. ... Take any subject you like, for example Julius Caesar. Combine in it all its conceivable predicates (not excepting those of time and place). You will then see that, with all these determinations, it may or may not exist. The being which gave existence to the world and to this hero was able to recognize all these predicates - not a single one excluded - and could still regard him as a merely possible thing which, save for His decree, did not exist. Who can deny that millions of things that really do not exist are, with all the predicates they would contain if they existed, merely possible; that in the conception which the highest being has of them, not one of these predicates is lacking, although existence is not among them. For He knows them only as possible things. Therefore, it cannot occur that if they exist they contain one more predicate; for in the possibility of a thing according to its complete determination, no predicate whatsoever can be missing. And if it had so pleased God to create another series of things, another world, then it would have existed with all the determinations, and nothing more, which He discerned in it, although it is only merely possible.' ( Quoted in J. William Forgie, 'Kant and the Question "Is Existence a Predicate?"', Canadian Journal of Philosophy, Vol. 5, No. 4 (Dec., 1975), pp. 563-582 : 563-4.)
For something to satisfy non-existence means the thing must exist, such that it can satisfy the predicate of non-existence.
