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To Comments Made by Robert Koons Jordan Howard Sobel University Of ON GÖDEL’S ONTOLOGICAL PROOF: To Comments Made by Robert Koons Jordan Howard Sobel University of Toronto I have claimed that “the best and smallest change that would obviate [both the problem of modal collapse engendered by principles of Kurt Gödel’s axiomatic proof, and philosophic problems with the worshipfulness of necessary beings is to] stop counting necessary existence as a positive property that a ‘God-like’ being properly so termed would have, give up on the idea of ontological arguments, and concede that no worshipful being could be [logically or metaphysically necessary]” (Sobel 2004, p. 135). Robert Koons argues that a better response to the problem of ‘modal collapse’ is to restrict the domain of properties in Gödel’s axiomatic proof in a manner that does not detract from his axioms for ‘positive’ properties, nor compromise lines of the necessary instantiation of ‘God- likeness’. He cautions, however, that this is not to say that this simple change yields a flawless proof for the necessary instantiation of ‘God-likeness’, since it leaves the serious problem that Anthony Anderson and I have overlooked that “we have no reason to accept Axiom 5 [that ‘necessary existence’ is a positive property] unless we already believe that all positive properties (including [Gödelian God-likeness]) are necessarily instantiated” (Koons 2005). Koons makes the interesting observation that replacing Axiom 5 with Axiom 6, the Anselmian principle that a property is positive only if the property of having it essentially or necessarily is positive, would preserve the validity of Gödel’s argument. However, he adds, this would not improve the argument, since Axiom 6 runs into the same problem: it too “presupposes that every positive property...is instantiated of necessity, [and so, amongst other things]...just what [Gödel’s] ontological argument was supposed to establish” (Koons). I offer, in Section I, elaboration of the business of ‘the collapse’, including Gödel’s personal relation to it, and then, in Sections II through IV, responses to the problem Koons has with Axiom 5 in the proof, comments on his alternative Gödelian proof that replaces Axiom 5 with Axiom 6 in an alternative Gödelian proof, and responses to the problem he has with Axiom 6 in this proof. 1. ON THE MODAL COLLAPSE IN THE SYSTEM OF GÖDEL’S ONTOLOGICAL PROOF “Photocopies of three handwritten pages titled ‘Godel’s Ontological Proof”...began to circulate in the early 1980's. The handwriting is Dana Scott’s; the ideas are Kurt Gödel’s. They agree with ideas conveyed in two pages of notes in Gödel's own hand dated 10 February 1970....” (Sobel 2004, p. 115.) Here, for ready reference, are Axioms, Definitions, and Theorems in the notation of Dana Scott’s notes, in order of their appearances in these notes. Axiom 1. P(¬ö) : ¬P(ö): ‘good half’, P(¬ö) 6 ¬P(ö); ‘bad half’, ¬P(ö) 6 P(¬ö).1 Axiom 2. P(ö) & 9x[ö(x) 6 ø(x)] 6 P(ø). Theorem 1. P(ö) 6 xö(x). Def G. G(x) : ö[P(ö) 6 ö(x)]. Axiom 3. P(G). Axiom 4. P(ö) 6 9P(ö). Def Ess. ö Ess x : ö(x) & ø[ø(x) 6 9y[ö(y) 6 ø(y)]]. Theorem 2. G(x) 6 G Ess x. Def NE. NE(x) : ö[ö Ess x 6 9xö(x)]. Axiom 5. P(NE). Theorem 3. 9xG(x). Each principle is intended as short for the necessitation of a universal closure of it. Primitives ‘P’, ‘Ess’, and ‘NE’ are for positiveness, a property of properties – abbreviation, P: ÷ is a positive property; Godlikeness, a property of individuals – abbreviation (s), G: a is Godlike (and Godlikeness); being an essence of, a specially defined relation of a property to an individual – abbreviation, Ess: ÷ is an essence of a; and necessary existence, a specially defined property of individuals – abbreviation(s), NE: a has the property of necessary existence (and Necessary Existence).2 1.1 Given the generous interpretation of properties in evidence in the system of Gödel’s Ontological Proof of 1970, and elsewhere in his work (particularly, in “Russell’s Mathematical Logic,” more of which below), according to which “anything is counted as a property which can be defined by ‘abstraction on a formula’ [in which no more than 1‘Bad’ according to (Anderson 1990, p. 291) on a ‘moral/aesthetic’ interpretation: for example, it is plausible that neither being not-red-all-over, nor being not-not-red-all over (that is, being red-all-over) is positive in this sense, though the ‘bad’ half of Axiom 1 says that one or the other of these properties in positive. The interested reader can find a problem for the ‘good’ half of Axiom 1 and 2 (which entail Theorem 1), DefEss, Axiom 3, the ‘bad’ half of Axiom 1, and any interpretation of ‘positive’ such that neither being not-red-all-over, nor being not-yellow-all-over is positive in the sense of this interpretation. A related problem can be found for any interpretation of ‘positive’ such that in its sense tautological properties (for example, being either red or not red) are not positive: it is a consequence of the corollary of Axiom 1 (whole) that there is at least one property that is positive, and Axiom 2, that tautological properties are all positive. 2‘G’ occurs in Gödel’s principles in both predicate or adjectival positions, and term of nominal positions. Similarly for ‘NE’. 2 one variable is free] ” (Anderson 1990, p. 292),3 it is a further theorem of the system, derived in (Sobel 2004), that every truth is a necessary truth. Theorem 9. (Q 6 9Q). This theorem is rigorously derived in Section A1 of the Appendix to an extended on-line version of this4 from, Theorem 2. x[G(x) 6 G Ess x], [derivable from the ‘bad half’ of Axiom 1 and Axiom 4, using Def G, and Def Ess] and Theorem 3. 9xG(x), [derivable from Theorem 1 (which is derivable from ‘good half’ of Axiom 1 and Axiom 2), Theorem 2, and Axiom 5, using Def G, and Def NE] using the principle for property-abstractions, Properties. Every formula, 9â(Ðá[F](â) : F' ) wherein á is an individual variable, â a term, F a formula, and F' is a formula that comes from F by proper substitution of â for á. Theorem 9 entails a ‘collapse of modalities’. It entails that propositions can be divided into two kinds: ones that are possible, true, and necessary, and ones that are impossible, false, and not necessary. 1.2 My reaction in Logic and Theism at once to this logical problem of modal collapse, and to a certain philosophical problem, is to delete Axiom 5, though I mention in a footnote that a “solution that is specific [to the logical problem] would consist in confining the essence of a thing to its 'intrinsic' properties” (Sobel 2004, p. 561n20.). I see Axiom 5 as bearing primary responsibility only for the philosophical problem, which is that of the possibility of a properly-termed -‘God-like’ being’s existing of necessity in the way of numbers, Platonic Forms, and propositions do, for a properly-termed-‘God-like’ being would be ‘properly worshipful’, and a necessary condition for that is, surely, being reachable by words and gestures of worship.5 It would be, for the reason suggested, absurd 3 What is more clearly in evidence in Gödel’s writings is the idea that for any formula öá in which exactly one variable á is +, free, Ðá[öáâ] names a property, the property that is had at a world w by something named by â if and only if the sentence ö that comes from ö by proper substitution of â for á is true at w. The greater liberality of ‘my’ property-conception is not exploited in my derivation of Theorem 9. 4This version is linked to “On Logic and Theism” – URL: http://www.scar.utoronto.ca/~sobel/OnL_T – which is linked to my home page. 5Cf.:“‘The religious frame of mind...desires the Divine...both to have an inescapable character ...and also the character of 'making a real difference' ....if God is to satisfy religious claims and needs, he must be a being in every way inescapable, One whose existence and whose possession of certain excellences we cannot possibly conceive away....It was indeed an ill day for Anselm when he hit upon his famous proof. For on that day he not only laid bare something that is of the essence of an adequate religious object, but also something that entails its necessary non-existence.’ (Findlay 1955[1948], p. 54[182].)” (Sobel 2004, p. 136.) I am comfortable speaking for myself of necessary conditions for being properly worshipful, but not, with Findlay and others with conditions that would tend to make an object properly worshipful, since being properly worshipful would be being an object that was objectively worthy of worship, a being that objectively ought to be worshipped by all, and, for Mackiean reasons I do not believe in “the possibility of a being who would be objectively worthy of worship” (Sobel 2004, p. 25, cf., p. 404). 3 to worship a number or a merely Ideal Person,6 and similarly, one may philosophically fear, for any necessary being. It has been said that ‘in logic one needs a robust sense of reality’ (Bertrand Russell). I don’t know about that. But I do think that in philosophy, especially in the philosophy of religion, it is good to have a robust sense of the absurd, even if, for broadly Pascalian reasons, this is not necessarily so in life.
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