The Nachlass of A.N. Prior
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Department of Information Studies - University of Copenhagen
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The Ontological Argument (Fragment 1)

By Arthur N. Prior on NA/NA/1965

This text has been transcribed by Martin Prior and David Jakobsen.

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1 The Ontological Argument (Fragment 1)by A.N. Prior 1. One version of the ontological argument can be very simply represented as follows: (1) Real existence is a perfection (2) Whatever has all perfections has real existence (3) Some thinkable has all perfections (4) Some thinkable has all perfections and has real existence (5) Something that has all perfections has real existence. Put this way, the conclusion certainly does follow by ordinary predicate calculus rules from the premises[Page 2] (1) and (2). Symbolising the object “real existence” by e, the one-place predicate “is a perfection” and “is a thinkable” by P and T, and the two-place-predicate “has” by , the whole comes out as (1) Pe (2) x (y(Py Hxy) Hxe); from (1) (3) x (Tx & y(Py Hxy)) (4) x (Tx & y (Py Hxy) & Hxe); from (2) and (3) (5) x (y(Py Hxy) & Hxe), from (4).
This text has been edited by David Jakobsen and Peter Øhrstrøm. The original MS is kept in the Prior collection at Bodleian Library, Oxford, Box 6. The original page numbers have been put in {...}. All underlinings in the text are Prior’s. Crossed out: (1) and (2). Using the system in E. J. Lemmon’s Beginning Logic, we could symbolize the object “real existence” by e, the predicates “is a perfection” and “is a thinkable” by P and T, and the two-place predicate “has” by H, and carry out the deduction thus: 1 (1) Pe A 2 (2) x(Tx & y(Py Hxy)) A [The last ‘)’ has been added. (Ed.)] 3 (3) Ta & y(Py Hay) A 3 (4) y(Py Hay) 3, & E 3 (5) Pe Hae 4, UE 1,3 (6) Hae 1,5, MPP 1,3 (7) y(Py Hay) & Hae 4,6 & I
2 This is exactly the form of the undoubtedly valid argument (1)Everest is a person in this room. (2) Whoever hits all persons in this room hits Everest (3) Some tramp hits all persons in this room. (4) Some tramp hits all persons in this room and hits Everest (5) Someone who hits all persons in this room hits Everest. The argument, naturally, also works if in place of “thinkable” (cogitabile) we put “possible” or
“possible being” (possibile), giving the argument
[page 3] a Leibnizian instead of an Aristotelian twist. 2. Still, we have presented the argument, so far, a little crudely. We have not merely treated existence as a predicate, but gone further and treated it as a subject. However, very little modification will alter this and leave the argument as valid as before. We re-state it thus: (1) Whatever is perfect really exists (2) Some cogitabile (or possibile) is perfect.
(3) Some cogitabile (or possibile) is perfect and really exists.
(4) Some perfect being really exists. (i.e. Something that really exists is perfect). Or in symbols, using P for “is perfect” and for “really exists”, (1) x (Px Rx) (2) x (Tx & Px) (3) x (Tx & Px & Rx) (4) x (Px & Rx) [page 4] Here (1) is justified, implicitly, by reality (R) being taken to be a component of perfection (P), i.e. Px is equated with Dx & Rx, where D is the conjunction of the remaining components of perfection, so that we could re-state the arguments as (1) x (Dx & Rx Rx)
3 (2) x (Tx & Dx & Rx) (3) x (Dx & Rx). Premise (1), being logically true, can now be dropped, so that we just have (1) x (Tx & Dx & Rx) (2) x (Dx & Rx), i.e. “Some cogitabile(or possibile) has all the other components of perfection and is real, therefore
something has all the other components of perfection and is real.” Put this way, the first premise doesn’t look very plausible, especially if we reflect that D can stand for anything at all, so that we could equally use this form of argument in the following way: (1) Some cogitabile(or possibile is a real dragon.
(2) Some dragon is real, or “Something real is a dragon”. This is just like (1) Some tramp is a blind sailor (2) Some sailor is blind. - valid enough; but we have to be sure of (1). 3. The stock objection to the premise “Some cogitabile (or possibile) has all the other components
of perfection and really exists”, and to the whole [page 5] argument, is that “existence isn’t a predicate”, so that our (1) and (2) aren’t properly representable as x (Tx & Dx & Rx) and x(Dx & Rx). We represent “Lions exist”, for example, simply as x Lx,“Something is a lion”, where the only predicate required is L (“is a lion”), and the notion of existence is carried by the quantifier. What bearing has this on the ontological argument? Well, it suggests this rewording: (1) Something is a cogitabile(or a possibile) and is perfect (or is a dragon)
(2) Something is a dragon, And this symbolic reformulation: (1) x (Tx & Dx) (2) x Dx
4 This is still valid, and the premiss still seems to have something wrong with it; but we cannot now say that what is wrong with it is that it treats existence as a predicate, because it doesn’t. 4. What is now wrong with the premiss, it might be suggested, is not that it treats “is real” as a predicate, but that it treats “is a cogitabile (or “is a possible”) as one. How, then, ought this to be
treated? One possible answer is that [Page 6] it ought to be treated in the same way as, according to the other theory, existence ought to be; that is, it ought to be thrown into the quantifier. This suggests a new way of looking at the so-called existential quantifier, according to which what it “carries with it” is not existence but imaginability or possibility. And it is not at all difficult to make out a case for such a change. The use of quantifiers, it may be said, always presupposes some “universe of discourse”, some range of objects from which it is understood that the values of our bound variables are to be drawn, and this range can be as wide or as narrow as we please. If we let it be undertakers, for example, xFx will mean “Some undertaker is F”; if we let it be real objects, xFx will mean “Some real object is F” (as on the ordinary view); and if we let it be cogitabilia(or possibilia), xFx will mean “Some cogitabile
(or possibile) is F”. So the correct way of rendering “Some cogitabileis perfect”, or “is a dragon”,
is not to write x (Tx & Rx), or x (Tx &Dx), and read this as “Some real thing is a cogitabile and
is perfect” (or “is a dragon”), but just to write xPx, or xDx, and understand that the x here does not mean “For [Page 7] some real x” but “For some imaginable x” or “For some possible x”. 5. Does this move, however, block the ontological argument? The first thing to notice about it is that it brings back existence as a predicate, in this way: Let us begin with the use of x in which the universe of discourse is confined to undertakers. Using our symbols this way, we have no way of saying “Something is an undertaker” or “Something is not an undertaker” or “Nothing is an undertaker”, since we do not have the predicate “is an undertaker” – being an undertaker is carried by the quantifier. But if we enlarge the universe of discourse to real objects, we can then introduce the predicate U for “is an undertaker” and say that some real object is an undertaker (xUx), or that some real object is not an undertaker (x –Ux) or that none are (–x Ux). And if we used another symbol say , for the reinterpreted quantifier we could define the original undertaker-quantifier in terms of this new one and the predicate U, i.e. we define the original xFx as x(Ux & Fx). The same procedure may be adopted when we [Page 8] start with x for “For some real object” and then widen the universe of discourse to cogitabiliaor
possibiliawith x for “For some object in this wider universe”, Within the new system we can
5 define xFx “Some real object is F”, as x(Rx & Fx), “Some cogitabile(or possibile) is real and is
F”. This now gives us a new way of stating the ontological argument, namely thus: (1) Every cogitabile that has all perfections really exists (since that is a perfection)
(2) Some cogitabile has all perfections
(3) Some cogitabile has all perfections and really exists
(4) Some real has all perfections. Or in symbols (using x for the universal quantifier with our widened universe of discourse), thus: (1) x (Px Rx) (2) x Px (3) x (Px & Rx) (4) x Px (by 3 and the definition of ) And if (1) is asserted on the ground that reality is a conjunctive component of perfection, we have (1) x (Dx & Rx Rx) (2) x (Dx & Rx) (3) x Dx, where (1) is superfluous. And this can be used for dragons too – from “Some imaginable thing is a [page 9] real dragon” we infer “Some real-thing is a dragon”. 6. It is still open to us, of course, to deny the premiss “Some imaginable-thing is a real dragon”,or “Some imaginable-thing is a real thing-than-which-no-greater-can-be-conceived”. But Anselm does not merely assert this premiss; he has an argument in its support. We can, he says, imagine (or conceive) a perfect being, and to imagine this property is to imagine it as real, so there is at least an imaginable-being that is perfect and therefore real. We can also (as Gaunilo rather than Anselm would say) imagine a real-dragon, from which we could conclude by the same argument that there is at least an imaginable-being that is a real-dragon. The general principle involved in this argument appears to be that if it is imaginable that there is an F, then there is an imaginable-object that is F. If we write IP for “It is imaginable that P”, we may symbolize this principle as follows: (A) IxFx xFx Letting our Fx be Dx & Rx, this immediately yields
6 (B) Ix (Dx & Rx) x(Dx & Rx) In itself, this is not obviously awkward. If, however, we combine it with our previous assumption that xFx, “There really is an F” is equivalent to “For some imaginable-, is an and is real.” [page 10] (treating the real as a simple sub-class of the imaginable - reals as those cogitabilia which have the
privilege of Reality), (B) immediately gives us (C) Ix (Dx & Rx) xDx, And indeed, since the same equation of xDx with x(Dx & Rx) gives us xDx = x(Dx & Rx) = x(Dx & Rx & Rx) = x(Dx & Rx), we have (D) I xDx xDx, i.e. whatever imaginably exists really does so. 7. To avoid this disaster, we must either deny the equation of xFx with x(Fx & Rx), or deny the principle (A), that IxFx implies xFx. The current tendency is probably to deny that xFx = x(Fx & Rx), i.e. to deny that the real is a sub-class of the imaginable (this is part of the denial that existence is a predicate); and I do not want to defend this equation. I do, however, want to attack the principle (A), as there seem to me to be powerful arguments against it even apart from its possible use in validating the ontological argument. [End of page 10] •  University of Copenhagen •  Aalborg University •