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) &

9œx[ö(x) 6 ø(x)] 6 P(ø). Theorem 1. P(ö) 6 ›xö(x). Def G. G(x) : œö[P(ö) 6 ö(x)]. Axiom 3. P(G).

Axiom 4. P(ö) 6 9P(ö). Def Ess. ö Ess x : ö(x) & œø[ø(x) 6 9œy[ö(y) 6 ø(y)]]. Theorem 2. G(x) 6 G

Ess x. Def NE. NE(x) : œö[ö Ess x 6 9›xö(x)]. Axiom 5. P(NE). Theorem 3. 9›xG(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. œx[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. 9›xG(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, 9œâ(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.

1.3 To avoid its engendering a modal collapse while preserving its validity, Koons confines the range of second order quantifiers in Gödel’s argument to ‘intrinsic properties’, “properties that are qualitative and non-relational”

(Koons 2005, p. 9), and adds axioms for the property of properties of being intrinsic sufficient, he says, “to make his proof work” (ibid.).7

A simpler and more conservative solution designed by Sreæko Kovaè would restrict the axiom-schema

Properties to properties abstracted from formulas in which no subformulas are closed or contains free variables: this prevents Pickwickian extrinsic properties such as being self-identical and such that snow is white, i.e., Ðx[x = x &

P], from doing their nasties in deductions from the axioms and definitions used in Gödel’s Ontological Proof.8 He writes that restricting Properties in this way (that is, “[restricting] the ë-Intro rule” in this manner) is an “(easy) way

6Cf. (Sobel 2004, p. 137). For an illustration of ‘the absurd’, Thomas Nagel has offered ‘declaring one’s love to a recorded announcement’. (Nagel 1971.) I know what he means. I think I may love Melanie who helps me add money to my Pay-as-You- Go plan. I mean, “What a voice! And such patience!” For another illustration, there is Jack Benny’s, “I’m thinking,” said in response to hold-up guy’s, “Well?” There has been a long pause since the guy’s, “Your money or your life!”

7Koons writes: “There is nothing in Gödel’s argument that rules out this interpretation of his second-order variables.” (Koons 2005, P. 9). In fact, Gödel himself uses the ‘property’ of being non-self-identical, a property that nothing has, in his deduction of Theorem 1, though there is another way, limned by Koons, that does not. Koons’s way with this small problem is to stipulate that impossible properties count as intrinsic :). He continues: “To make his proof work...we need only the following properties of...intrinsic properties: IN1. If F is intrinsic, so is -F. INS2. The conjunction of a set of intrinsic properties is itself intrinsic. INS3. Everything has at least one intrinsic property in every world (satisfied if the property of being self-identical counts as intrinsic), and an impossible (such as being non-self-identical) counts as intrinsic. These are quite plausible assumptions. Furthermore, Gödel’s axioms make perfect sense under this new interpretation [of properties they are about – It is in this sense that they are sufficient “to make his proof work under this interpretation”: these axioms for intrinsicness play no rôles in deductions of its theorems. JHS]. Under this interpretation, Sobel’s modal collapse proof does not go through, since being such that snow is white is not a property (under the intended interpretation) [it is not among ‘properties that are qualitative and non-relational’].” (Ibid., bold emphasis added.) I suppose that being such that snow is white is not an ‘intrinsic’ property of mine, say, though it is less obvious that it is neither a qualitative nor a non-relational property of mine. More needs to be said, I think, for it to be quite clear that Koons’s metaphysical restriction on properties rules out the peculiar ‘properties’ that collapse modalities in Gödel’s system given his presumably unrestricted syntactical characterization of property-abstraction terms.

8A God-like being would have such a property for every truth P. A God-like being would have all and only positive properties: Definition of God-likeness, and Theorem 4 of Logic and Theism. Every positive property, since God-likeness is necessarily instantiated (Theorem 3), is itself necessarily instantiated. So, if such properties are ‘covered’ by Properties, it follows that, Theorem 9, if it is true that P, then it is true that it is necessary that P.

4 to block the unrestricted necessitations of sentences of the system [Theorem 9], since Sobel’s proof (the only one known)...is dependent...on a ë introduction that leaves a closed subformula within the scope of [the] ë operator”

(Kovaè 2003, p. 569).

1.4 Kovaè’s nice syntactical restriction, though it targets for exclusion not all predicates for ‘non-intrinsic’ properties (for relational or non-qualitative properties, the properties explicitly excluded by Koons’s metaphysical restriction),9 but only predicates for Pickwickian properties, would not have recommended itself to Gödel as a solution to the modal collapse of his 1970 proof, unless he had by then either changed his mind about kinds of properties to which he was committed in 1944, or he was prepared to move away from the idea of ‘essences of individuals’ as ‘total properties of individuals’ (compare to Leibniz’s ‘complete individual concepts’).10

In his contribution to the Schilpp volume for Bertrand Russell, Gödel indicates assumptions he considers available for a ‘collapsing argument’ that says that there is exactly one fact or true proposition, and exactly one falsehood (‘Gödel’s slingshot’[discussed in Section 14 of Chapter VIII, “Bertrand Russell’s Theory of Descriptions,” in (Sobel 2004b)]). One ‘assumption’ he makes is that

“every proposition ‘speaks about something,’ i.e., can be brought to the form ö(a)”

(Gödel 1944, p. 129.) Stephen Neale suggests, to illustrate a general device, that Gödel could say that the proposition expressed by ‘all men snore’ can be brought to the form of ‘Clinton is an x such that all men snore’.

(Neale 1995, p. 778.) One would expect Gödel to say that the proposition expressed by that sentence is about

Clinton, and that it ascribes a property to Clinton, perhaps the property of being self-identical while all man snore

(!): this would-be property could be named by ‘Ðx[x is self-identical & all men snore]’. Kovaè’s restriction on

Properties would make this property-abstraction ‘inert’ in derivations.

9That would be an advantage of Kovaè’s restriction for traditionalists who would like being the creator of the world to be part of Gödelian ‘God-likeness’. Another advantage is that it clearly rules out of Gödel’s system the ‘properties’ I use to prove that modalities collapse in it.

10Koons’s ‘emendation’ would, in Gödel’s view, include such re-thinking of ‘essence’, if, in Gödel’s view, the Koonsian intrinsic properties of God would not be all of God’s properties, as they would not be on any ordinary understanding of His properties. God would, on a common view, have the properties of creating (a relation) these (demonstrative pronoun) things around us, and of loving (another relation) these creations, which properties would presumably not fall into “[t]he class of intrinsic properties” which, Koons explains, is “the class of properties that are qualitative and non-relational: that pertain or fail to pertain to a thing because of its internal constitution.” (P. 9, bold emphasis added.)

5 1.5 At least in 1944, Gödel, if he had wanted a solution to the modal collapse that we see as plaguing his axiomatic proof, would have wanted a different one from Koons’s or Kovaè’s. “If he had wanted a solution that collapse?”

Well, yes, for there is strong evidence that Gödel thought his principles entailed this collapse, and was not bothered by this, that he indeed considered it a welcomed feature of the Leibnizian metaphysics he was developing for himself. (Cf., Adams 1995, pp. 100-1.)

1.5.1 To this evidence. On an undated page in a notebook begun it seems after April 1946 and before 1955 (op. cit., p.429) titled “Ontological Proof” he writes,

“If ö(x) e Nö(x) is assumed [as following from the essence of x], then it is easily provable that for every compatible

system of properties there is a thing, but that is the inferior way. Rather ö(x) e Nö(x) should follow first from the

existence of God.” (Gödel 1995, p. 435 – brackets original.)

Gödel seems to have thought that it follows from things’s essences being their ‘complete properties’, that if a thing has a property, then it is necessary that it has it, 9œöœx[ö(x) 6 9ö(x)]. If he thought that he was mistaken: all that follows in this direction from the identification of a thing’s essence with its ‘complete property’ is that if a thing has a property, then it is necessary that, if it exists, it has this property.

9œöœx(ö(x) 6 9[E!x 6 ö(x)]), follows from,

Def Ess. 9œöœx(öEss x : ö(x) & 9œø[ø(x) 6 9œy[ö(y) 6 ø(y)]),

given axioms for essences,

Essences (everything has at least one). 9œx›ö öEss x, and Essentiality of Essences. 9œöœx[öEss x 6 9[E!x 6 ö(x)]),

of which Gödel would approve. (A2 of the Appendix to the extended on-line version of this, for which please see note

4 above, contains a confirming derivation.)

1.5.2 Join that mistake which Gödel may have made with the thought “that every proposition ‘speaks about something,’ i.e., can be brought to the form ö(a)” (Gödel 1944, p. 129n: commented on above), expressed thus, that

9œP›x›ö9(P : ö(x)], and one has a short way to Theorem 9, 9œQ(Q 6 9Q),11 that does not traffic in any principles of the ontological proof of 1970 other than DefEss, and that makes no use of property-abstractions.

‘11 Q ’ is here a proposition-variable; propositions are here functions from possible worlds to truth values; UD, UI, and EG are valid for proposition generalizations. A3 of the Appendix contains a confirming derivation.

6 Furthermore, given Theorem 9, there is a short way, with Theorem 1 (that comes from the ‘good half’ of Axiom

1 and Axiom 2), and Axiom 3, to Theorem 3, 9›xGx. Neither the way to Theorem 9 of the previous paragraph, nor this way on to Theorem 3, uses any of Axiom 4, Axiom 5, Def NE, and Def G. (A4 of the Appendix to the extended on-line version of this, for which please see note 4, contains a confirming derivation.) Gödel would have said this is an ‘inferior way’. He would prove first the existence of God, and then somehow infer from it the collapsing principle [ö(x) 6 9ö(x)]. (Strange!)

We can be generous to Leibniz and not say that he made the mistake of thinking that from his doctrine of the complete individual concepts of things that if a thing has a property, then it is necessary that this thing has property, that is, that there is in every possible world a counterpart of this thing that has this property. We can say that he saw, and was bothered by, that this doctrine entails that individuals are specific to possible worlds and that when human beings act could not logically have acted otherwise. The main present point, however, is not that Gödel seems to have made a mistake of ‘the logic of essences’ that we should not say that Leibniz made, but that Gödel was comfortable, as Leibniz most certainly would not have been, with the universal necessity that that mistake would have entailed for him given his idea of a thing’s essence, and view that every proposition “can be brought to the form

ö(a)” (Gödel 1944, p. 129n).12

2. KOONS AGAINST AXIOM 5

2.1 “[T]here is a fatal flaw in Gödel’s argument....The flaw concerns axiom A5, the positivity of necessary existence....Gödel’s necessary existence [defined thus,

[Def NE] NE(x) : œö(öEss x 6 9›xö(x)), ] is provably equivalent to....the condition of being ‘contingency free’ [defined thus,

[Def CF] CF(x) : œö(ö(x) 6 9›xF(x)),]

....on the assumption, which Sobel rightly endorses, that everything necessarily has at least one essence.” (Pp. 12-3)

There is a derivation in Section A5 of the Appendix to the extended on-line version of this, for which please see note 4

above, of,

12Appendix A of Chapter VI, Sobel 2004, includes a note, p. 231f, on Leibniz and his doctrine of complete individual concepts.

7 Logical Equivalence13 of NE and CF. 9œx[NE(x) : CF(x)].

The derivation uses necessitations of universal closures of Def NE and Def CF, and axioms for essence

Essences. 9œx›ö öEss x,

and,

Logical Equivalence of Essences of a Thing. 9œöœøœx(öEss x & øEss x 6 9œx[ö(x) : ø(x)]),

of which Gödel would approve.

“So NE is positive if and only if CF is.” (p. 13)

The necessitation of that, 9 [P(NE) : P(CF)]

is an easy consequence of the Logical Equivalence NE and CF and

Axiom 2. 9œöœø[P(ö) & 9œx[ö(x) 6 ø(x)] 6 P(ø).

(A6 of the Appendix to the extended on-line version of this, for which please see note 4 above, contains a

confirming derivation.)

Now, edging closer to the problem Koons sees for Axiom 5, P(NE), we find that Koons writes: “whether CF...[is] positive depends on whether it is true that all positive properties are necessarily instantiated” (p. 13, emphasis added). Why is that? It is because (backing up in Koons’s text): “CF is the property of having only necessarily instantiated properties....CF is equivalent to the infinite conjunction of...the...complements of those properties that are not necessarily instantiated (i.e., that are possibly uninstantiated). CF is positive iff none of the properties that are possibly uninstantiated are themselves positive. If, instead, there is a positive property F that is not necessarily instantiated, then CF entails not having F, which would make CF a negative property (any property that entails not

13A discrepancy between my modal semantics and what we should say is the semantics of Gödel for this ‘ontological proof’ prevents me from saying with Koons that these properties are “necessarily coextensive” (Koons 2005, p. 13). I identify properties in an interpretation with functions that assign to worlds subsets of its universal domain, and make names rigid designators, while allowing a name’s designatum at a world not to exist in this world or be a member of its domain: whether or not the designatum of a name exists in a world of every property and its negation this designatum will have one of them at this world. Gödel should be said to identify properties with functions that assign to worlds subsets of their domains of existents. There is no evidence in the texts directly relevant to Gödel’s thoughts about ontological proofs concerning his policy for names in modal logic. We agree in the range we assign to individual quantifiers at a world: we make their range the domain of existents at this world. One consequence of this difference is that their properties are necessarily ‘existence-entailing’, while mine need not be. Under my semantics something can HAVE a property at a world, though this property IS NOT INSTANTIATED in this world: that ‘F(A)’ is true at a world does not entail that ‘›xFx’ is true at this world. Another consequence is that logically equivalent properties are for them, but not for me, necessarily co-extensive.

8 having some positive property must itself be negative).” (P. 13, bold emphasis added.) These boldly emphasized things are true, in Gödel’s system.

That “any property that entails not having some positive property must itself be negative,”

(P(ø) & 9œx[ö(x) 6 -(øx)]) 6 -P(ö),

follows rather obvious from Axiom 1 (the good half), œö[P(-ö) 6 -P(ö)], and Axiom 2, œöœø[P(ö) & 9œx[ö(x) 6

ø(x)] 6 P(ø)], and from nothing less in Gödel’s system. There is a derivation in A7 of the Appendix to the extended

on-line version of this, for which please see note 4 above, for the principle that “whether CF...[is] positive depends on

whether it is true that all positive properties are necessarily instantiated” (p. 13),

P(CF) 6 œö[P(ö) 6 9›xö(x)],

from the same two axioms of Gödel’s system. I think that no derivation of it from less of Gödel’s system as

explicitly developed in Scott’s notes is possible. Are these things true ‘outside’ Gödel’s system and its axioms

that partially define ‘positive’? Huh?!

And now to the problem Koons sees for Axiom 5, P(NE), given that “whether CF (and NE) are positive depends on whether it is true that all positive properties are necessarily instantiated.” The problem, given that, is that “[i]f some positive property is possibly uninstantiated, then CF and NE are clearly themselves negative. Thus, we have no reason to accept Axiom 5, unless we already believe that all the positive properties (including of course G) are necessarily instantiated. We have no reason to accept Axiom 5 unless we [already] know that God exists necessarily.” (P. 13.)

2.2 To respond on Gödel’s behalf, FIRST, that “all positive properties are necessarily instantiated” (p. 13) is not a consequence of Axiom 5 together with definitions and natural assumptions concerning Gödelian essences, and nothing substantial about, or partially definitive of, positive properties. That all positive properties are necessarily instantiated has only been implicitly shown by Koons to be a consequence of the conjunction of Axiom 5 with that of the good half of Axiom 1 and Axiom 2 (see the remark headed, “These things are true, in Gödel’s system,” in the previous section). However, Koons’s result is not without interest, since it means that Theorem 3, 9›xGx, can be reached with less axiomatic baggage than Gödel, as constructed by Scott, uses: Theorem 3 is evidently derivable from the conjunction of the good half of Axiom 1, Axiom 2, Axiom 3, and Axiom 5.

9 SECOND, even if Axiom 5 did, without other Gödelian assumptions about positive properties, ‘presuppose’ or entail that all positive properties are necessarily instantiated, it would not follow that no one could have a reason for accepting this axiom unless he already believed or knew that God exists necessarily. John Findlay, in an ‘ontological disproof of the existence’ of God, wrote:

“we are led irresistibly, by the demands inherent in religious reverence; to hold that an adequate object of our worship

must possess its various qualities in some necessary sense....we can’t help feeling that a worthy object of our worship

can never be a thing that merely happens to exist....his own non-existence must be wholly unthinkable” (Findlay 1955

[1948], pp. 51 and 52).

Axiom 5, that Gödelian necessary existence is a positive property, entails that a God-like being who possesses all and only positive properties, satisfies this necessary condition for worshipfulness of Findlay’s. It is condition for worshipfulness endorsed at least implicitly by all Anselmian perfect-being theists. If we understand by positiveness in the present context, being a property that either tends to make worshipful or is necessary to being worshipful,14 then Findlay had this Anselmian reason for counting necessary existence as positive: he had this reason when he did not ‘already believe or know that God exists necessarily,’ for he had it when he was engaged in a disproof of God’s existence: he had it when writing an essay titled, “Can God’s Existence Be Disproved?” in which essay he argues,

Yes, to its question.

And, THIRD AND LAST, though the good half of Axiom 1 is secure on any interpretation of ‘positive’, the same cannot be said of Axiom 2. It is a consequence of this axiom that, if there is at least one positive property, then every property that is necessarily instantiated by everything, such as the tautological properties of being either God-like or not God-like, and being either red or not red, is a positive. That strikes me as ‘seriously wrong’ on a

‘moral/aesthetic’ interpretation, and, more importantly, on a ‘spiritual/religious’ interpretation such as the one

14In a context improved for theological purposes, there would be axioms for ‘negative’ properties as well as ‘positive’ properties, that did not equate being negative (positive) with being not positive (negative), and ‘God-likeness’ would be defined as having every positive, and no negative, property. In this context, ‘positiveness’ (‘negativeness’) could be interpreted as tending to make ‘worshipful’ (‘unworshipful’). Under these interpretations, absences of properties necessary for being worshipful (or equivalently presences of properties sufficient for being unworshipfulness) would be limiting cases of negative properties.

10 recently floated.15 This negative opinion of mine is contrary to Koons’s attitude towards Axiom 2 (‘not to mention’ the bad half of Axiom 1): he writes that “there’s nothing seriously wrong with Axioms 1-4" (p. 14).

It is true that such ‘disjunctive properties’ that featured elementary properties would be positive on the logical/ ontological interpretation that Gödel floats according to which a property is positive if “[its] disjunctive normal in terms of elementary properties contains a member without negation” (Gödel 1995, p. 404). This, however, tells against the gloss that Gödel offers on “pure ‘attribution’ as opposed to...containing privation” (ibid., italics original), if as he implies, “being and goodness are [in his view] convertible” (Koons, p. 2).

3. A6 TO THE RESCUE FOR A ‘MINIMALIST GÖDELIAN PROOF’

3.1 Persuaded that Axiom 5 is, on pain of begging the question, not available to Gödel, Koons writes: “...without

Axiom 5, Gödel’s ontological proof is unsuccessful. There is, however, a simple repair that might do the job: replace Axiom 5 with Axiom 6:

A6. P(F) 6 P(9F)

If a property F is positive, then so is the property of being F in every possible world. [Proceeding with Axiom 6 in place instead of Axiom 5] since Godlikeness is positive, it follows that being Godlike in every possible world is also positive.16 Positive properties are always possibly instantiated, so being necessarily Godlike is possibly instantiated.

In S5, it follows that Godlikeness is necessarily, and thus also actually, instantiated.” (Pp.14-5)

There is a derivation in A8 of the Appendix to the extended on-line version of this, for which please see note 4, that spells

out this informal deduction of

9›xG(x),

with the aid an axiom regarding positive properties that is not made explicit by Gödel, or for him by Scott. The derivation is

from,

15Having some property can be ‘necessary’, in the sense intended, to being worshipful only if it is a property that not everything of which one can speak necessarily has. Being either red or not red is a property that everything of which one can speak has. It is a property that even impossible things such as, an opponent of Anselm’s argument might say, a being that which no greater being can be conceived. Intuitively, tautological properties should be neither ‘positive’ nor ‘negative’ in a context such as described in note 12 above in which being positive (negative) is not the same as being not negative (positive).

16Let being necessarily ö be being ö at every possible world. If ö is existence-entailing for possible things (things that are instantiated in some worlds), so that something has ö at a world only if it exists at this world, then being necessarily ö is equivalent to existing necessarily and being essentially ö, (9E!x & 9[E!x 6 ö(x)]).

11 Axiom 6. P(ö) e P[Ðx(9[ö(x)])],17

Axiom 3. P(G),

Theorem 1. 9[P(ö) e ›xö(x)] [which is derivable from Axiom 2, together with the ‘good half’ Axiom 1, or

instead of this, David Johnson has observed, the principle that not every property is positive (Johnson 1999,

p. 99), ¬œöP(ö), which is a corollary of the ‘good half’ of Axiom 1].

and the unstated until now axiom,

God-likeness Is for Possible Things Existence-Entailing, (G/Poss/EE). 9œx9(Gx 6 E!x),

of which Gödel would approve (for my semantics). Why not simply 9œx(Gx 6 E!x)? Because that can go without saying in

my semantics, since in it individual universal quantifiers range, at a world, over precisely the domain of this world’s

existents: for every property ö, 9œx(öx 6 E!x). In my semantics, properties of an interpretation are functions that assign to

worlds subsets of its ‘universal domain’ which need not be the union of its world domains. In this scheme things can have

properties at worlds in which they do not exist; indeed things that exist in no worlds – impossible things – can have

properties at worlds.18 In my semantics, (G/Poss/EE) says of every x that exists in any world – that is, of every possible

thing x – that at every world w, x has G at w only if x exists in x.19 ‘Not every property can make that claim’ (in my

semantics).

The derivation, which is in Section A8 of the Appendix to the extended on-line version of this, for which please see note 4 above, uses,

Properties. Every formula, 9œâ(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.

It could use a version of Properties restricted to formulas in which no subformulas are closed (Kôvac’s restriction).

17While Axiom 6 plays no rôle in Gödel’s Ontological Proof of February 10, 1970, or in Scott’s notes, in entries in a notebook on the project of ontological proof Gödel did more than merely endorse it with reference to that project: he wrote that “[t]hat the necessity of a positive property is positive is the essential presupposition for the ontological proof.” (Gödel 1995, p. 435, emphasis added; cited above in Section 1.4.)

18The case is otherwise for Gödel, given what we should say is his implicit semantics. If, as I suggest in note 11 above, he make properties functions that assign to worlds subsets of their domains of existents, and also (for a new observation) identify universal domains of interpretations with the union of the domains of existents of its worlds, then the possibility envisioned of something that does not exist at any world’s having G at some world cannot arise.

19More explicitly, ‘9œx9(Gx 6 E!x)’ says that, for every possible world, w, and for each things, x, that exists in w, at every possible world, w’, it is true at w’ that Gx, only it is true at w’ that x exists.

12 3.2 That is, in the resources it draws from Gödel’s system, a very economical Gödelian proof. As observed, it can get Theorem 1 from Gödel’s Axiom 2, and Johnson’s Principle that not every property is positive, ¬œöP(ö): a derivation is given in Section A9 of the Appendix. It does not need either half of Gödel’s Axiom 1, P(¬ö) : ¬P(ö).

Nor does it need either his Axiom 4, P(ö) 6 9P(ö), or his Axiom 5, P(NE). It does not presuppose his definitions – it does not presuppose any definitions – of ‘essence’ or of ‘necessary existence’. And, perhaps most remarkable, it makes the question of the nature of ‘Godlikeness’ open to detailing according to one’s idea of what would make a being most worshipful, subject only to the conditions on ‘positiveness’ of Johnson’s principle, and Axioms 2 and 6.

“But surely that ‘minimalist’ proof is too good to be true. Evidence of at least that can be found in the possibility of a

parallel demonstration that Devil-likeness, D, is necessarily instantiated from analogues of Johnson’s Principle and the

rest for analogues of the polar opposite, N, of ‘positivity’ whatever one’s idea of positivity is.” I think this is right,

and recommend for the reader’s consideration that the principal fly in the ointment of this ‘minimalist’ proof for Gods

and Devils is Gödel’s appealing-to-the-logically-minded Axiom 2, at which a finger was pointed towards the end of

Section 2.2 above.20

4. KOONS AGAINST A6

“Does A6 suffer from exactly the same flaw as A5? Unfortunately, the answer is Yes. If there is a positive property F

that is possibly uninstantiated, then A6 will fail in that case, since [(i)] in that case 9F or, more precisely, x89F(x),

(being F in every possible world), will be an impossible property, and so [(ii)] negative rather than positive. Thus,

A6 presupposes that every positive property (including the conjunction of all of them) is instantiated of necessity, but

this is just what the ontological argument was supposed to establish.” (P. 15, emphasis, bold and italics, added).

There is in Section A10 below of the Appendix a derivation of Koons’s [(i)], that if there is a positive property ö that is possibly uninstantiated, [P(ö) & ¬›xö(x)], then the property Ðx[9ö(x)] of being ‘necessarily ö’ (that is, of being ö at every possible world) is ‘an impossible property’ (that is, not possibly instantiated property),

œö([P(ö) & ¬›xö(x)] 6 ¬›yÐx[9ö(x)]y), from the unstated until now axiom,

Positive Properties are for Possible Things Existence-Entailing. 9œö(P(ö) 6 9œx9[ö(x) 6 E!x]),

20Suppose one’s idea of a ‘positive’ property is that of property that would tend to make a thing worthy of worship. Then to conjure with in the ‘minimalist’ system of this section, augmented by analogues of its principles for ‘N’ standing for the polar opposite of this ‘positiveness’, and ‘D’ standing for Devil-likeness, would be the disjunctive property (G w D) and the theorems thereof, P(G w D) and N(G w D).

13 of which Gödel would approve for my semantics. See the paragraph in the previous section on (G/Poss/EE).

However, it is plain that Koons’s [(ii)], according to which from the property Ðx[9ö(x)]’s being not possibly instantiated it follows that this property is not positive, depends on Theorem 1. So it has not been shown that Axiom

6 presupposes or entails, without aid of other principles for positiveness, that every positive property is instantiated of necessity, œö[P(ö) 6 9›xö(x)]. If it did, then it, Axiom 6, would all by itself entail Theorem 1, 9œö[P(ö) 6

›xö(x)]. The most that the argument stated establishes is that the conjunction of Axiom 6 and Theorem 1 entails that œö[P(ö) 6 9›xö(x)], and that thus the conjunction of Axiom 6, Theorem 1, and Axiom 3, P(G), entails the object of Gödel’s exercise, Theorem 3, 9›xGx.

5. Weighing A5 and A6

If, as seems, Axiom 6 needs less help from Gödel’s axioms and theorems than does Axiom 5 for a proof of Theorem

3, Axiom 6 may be felt to be closer to something like question-begging than Axiom 5.21 That could encourage an ontological reasoner to go back to Axiom 5, and the idea that Necessary Existence as defined by Gödel is a positive property in a moral/aesthetic sense, or better for the object of the exercise, in the spiritual sense of tending to make worshipful. Though we may expect an ontological reasoner to hold on to the substance of Axiom 6, whether or not he uses in his proof. Axiom 6 says, for example, that if being good is a positive property then being essentially good, or necessarily good, is if anything ‘more so’. The substance of Axiom 6 has been prominent in the thinking of ontological reasoners from the first one through Gödel.22 To call up again Findlay as my witness, he writes: “we are

21Axiom 6 works with Theorem 1 and Axiom 3 for a derivation of Theorem.3. Axiom 5 works with Theorem 1, Theorem 2, and Axiom 3 in an articulation of Scott’s sketch of Gödel’s deduction of Theorem 3 in (Sobel 2004) on page 150, and, it has been noted recently, works with Axioms 1, 2, and 3.

22A possible problem for Axiom 6 assuming a ‘moral/aesthetic’ interpretation of positiveness concerns presumptively positive moral properties such as veracity and generosity, and their ‘essential enhancements’. Could it contribute to a person’s that moral perfection that he was not merely perfectly good in acts and intentions, but essentially so, with the consequence that he could not be bad and lie or ‘under tip’, not if he wanted to, that is, if he had no choice but to be in his conduct perfectly good? Would his being essentially honest and kind be consistent with his being praiseworthy for his honesty and kindness? It would be for a person who was essentially good not only in actions but in intentions, sentiments, and so on, absolutely impossible that he should want to lie or cheat, let alone do these bad deeds. Peter Geach was scandalized by the suggestion that he found in Richard Price’s writing that “it must not be absolutely impossible for God to do something wicked....or [else] God isn’t free and isn’t therefore laudable for his goodness” (Geach 1973a, p. 16). There lurks for Axiom 6 substantially the same problem assuming a ‘spiritual/religious’ interpretation of positiveness according to which it is ‘worshipful, adorable, deserving-of-gratitude making or enabling’, the question being whether anything could contribute to a being’s worshipfulness, and so on that was for this being logically inescapable. For example, could a being be worshipful and deserving of ‘cosmic gratitude’ thanks in part to the goodness and bounty of its creation, if it is not logically possible that its creation should fail to be good and bountiful, that is, if, as a matter of logical necessity, it had no choice in this (continued...)

14 led on irresistibly, by the demands inherent in religious reverence, to hold that an adequate object of our worship must possess its various qualities in some necessary manner” (Findlay 1955, p. 53)23.

The substance of Axiom 6 should be a ‘keeper’ for every child of Anselm. With Axiom 5 returned to its proper place in Gödel’s scheme conservatively emended to avoid modal collapse, Axiom 6 should be securable as a theorem, a confirmatory theorem for anyone of an Anselmian mind.24 Such a person would have all that he sought for worship, he would have a necessarily existing being that had every ‘positive’ property essentially, “a being [who was] in every way inescapable” (Findlay 1955[1948], p. 54[182]: see note 3 above). As for me, I have no interest in worshipping anything, and if I did , would find in ‘a being who was in every way inescapable’ – that is, in a necessarily existent being who was essentially everything it was – the last thing I was looking for.

APPENDIX OF DERIVATIONS25

A1. Theorem 9. Q 6 9Q from Theorem 2. œx[G(x) 6 G Ess x], and Theorem 3. 9›xG(x), using

Properties. Every formula,

22(...continued) matter? Price would presumably say, “No, in that case the being would be neither laudable nor worshipful nor deserving of gratitude for the goodness and bounty of his creation.” But if not for that, then for what could a being be worshipful and so on?

23Which, Findlay would have been happy to clarify, is not to say that ‘we’ are led to an object whose acts are essential to it. ‘We’ (I do not include myself) are presumably led to an object that while ‘inescapable in its existence and qualities’ is both actively and passively contingently related to ‘us’. That is the problem for Findlay. ‘We’ are led the idea of an object, that though it does not obviously entail a priori a contradiction, he in 1957 (as in 1948) considers to be of a metaphysically or logically impossible object.

24The main idea for a derivation, that I have not written down for a well-emended system, is that a God-like being would not only have each positive property ö, but would necessarily have it, and would thus have the property of having it necessarily, Ðx[9ö(x)], and have this property necessarily. Since a God-like being would have only positive properties necessarily or essentially, it should be provable in some number of steps in a well-emended system that therefore, P(ö) 6 P(Ðx[9ö(x)]), which is what we have labelled ‘Axiom 6'. (I think I see rôles in such a derivation for additional ‘just right’ axioms for Gödelian essences (I stress Gödelian essences), viz.: Exclusive Essences – 9œöœxœy(öEss x & øEss y 6 x = y); and Essentiality of Essences – 9œöœx[öEss x 6 9(E!x 6 öEss x).]

25Coming derivations take place in the logical system of “Second order quantified modal logic for ontological arguments of Kurt Gödel and David Johnson” which an extension of that of “Quantification natural deduction, and quantified modal logic” which is an extension of that of “Sentential natural deduction, and sentential modal logic.” These files are linked to a web-page for Logic and Theism – URL: http://www.scar.utoronto.ca/%7Esobel/L_T ) which is linked to my home page.

15 9 œ â ( Ð á [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.

The derivation also uses,

Def Ess. œöœx(ö Ess x : ö(x) & œø[ø(x) 6 9œy[ö(y) 6 ø(y)]])

1. SHOW Q 6 9Q CD +)))))))))))))))))))))))))))))))))) 2. *Q assumption 3. *E!a & G(a) Theorem 3, N, FrEI(a) 4. *a = a & Q 3, S, Self-Identity, 2, Adj 5. *Ða[a = a & Q](a) 4, Properties, N, UI, BC, MP 6. *G Ess a Theorem 2, 3, S, FrUI(a), 3, S, MP 7. *G Ess a : G(a) & œø[ø(a) 6 9œy[G(y) 6 ø(y)]] Des Ess, UI(G), 3, S, FUI(a) 8. *9œy[G(y) 6 Ða[a = a & Q](y)] 7, BC, 6, MP, S, UI(Ða[a = a & Q]), 5, MP 9. *SHOW 9[›yG(y) 6 ›yÐa[a = a & Q](y)] ND *+)))))))))))))))))))))))) 10. **9œy[G(y) 6 Ða[a = a & Q](y)] 8, R 11. **›yG(y) 6 ›yÐa[a = a & Q](y) 10, N, Quantifier Distribution *.)))))))))))))))))))))))) 12. *SHOW 9Q ND *+))))))))))))))))))))))))) 13. **9[›yG(y) 6 ›y Ða[a = a & Q](y)] 9, R 14. **9›yG(y) 6 9›y Ða[a = a & Q](y) 13, Modal Distribution 15 **9›xG(x) Theorem 3 16 **9›y Ða[a = a & Q](y) 15, AV, 14, MP 17. **E!b & Ða[a = a & Q](b) 16, N, FrEI(b), 18. **9œx(Ða[a = a & Q](x) : a = a & Q) Properties 19. **Q 18, N, 17, S, FrUI(b), BC, 17, MP, S *.))))))))))))))))))))))))) .))))))))))))))))))))))))))))))))))

A2. 9 œ ö œ x(ö(x) 6 9[E!x 6 ö(x)]) from,

Essences. 9œx›ö ö Ess x, and the new axiom,

Essentiality of Essences. 9œNœx[N Ess x 6 9[E!x 6 N(x)]), of which Gödel would approve.

The derivation uses,

Def Ess. 9œNœx(N Ess x : N(x) & 9œR[R(x) 6 9œy[N(y) 6 R(y)]),

1. SHOW 9œöœx(ö(x) 6 9[E!x 6 ö(x)]) (ND, 2: [(8), (10), (12)]) +)))))))))))))))))))))))))))) 2. *SHOW œöœx(ö(x) 6 9[E!x 6 ö(x)]) (UD, 3: [‘ö’]) *+))))))))))))))))))))))))))))))))))))))))))) 3. **SHOW œx(ö(x) 6 9[E!x 6 ö(x)]) (FrUD, 4: [‘x’]) **+)))))))))))))))))))))))))))))))))))))))))) 4. ***SHOW E!x 6 (ö(x) 6 9[E!x 6 ö(x)]) (CD, 6) ***+))))))))))))))))))))))))))))))))))))))))) 5. ****E!x ****+))))))))))))))))))))))))))))))))))))))))

16 6. *****SHOW N(x) 6 9[E!x 6 N(x)] (CD, 13) *****+)))))))))))))))))))))))))))))))))))))))) 7. ******N(x) 8. ******9œx›N N Ess x Essences 9. ******P Ess x 8, N, 5, FrUI, EI 10. ******9œNœx(N Ess x : N(x) & 9œR[R(x) 6 9œy[N(y) 6 R(y)]) Def Ess 11. ******P Ess x : P(x) & 9œR[R(x) : 9œy[P(y) 6 R(y)] 10, N, UI(P), 5, FrUI 12. ******9œNœx[N Ess x 6 9[E!x 6 N(x)]) Essentiality of Essences 13. ******SHOW 9[E!x 6 N(x)] (ND, 14: (16), (17)]) ******+)))))))))))))))) 14. *******SHOW E!x 6 N(x) (CD, 19) *******+))))))))))))))) 15. ********E!x 16. ********9œy[P(y) 6 N(y)] 11, BC, 9, MP, S, N, UI(N), BC, 7, MP 17. ********9[E!x 6 P(x)] 12, N, UI(P), 9, MP 18. ********P(x) 6 N(x) 16, N, 15, FrUI 19. ********N(x) *******.))))))))))))) ******.)))))))))))))))) *****.)))))))))))))))))))))))))))))))))))))))))) ****.))))))))))))))))))))))))))))))))))))))))))) ***.)))))))))))))))))))))))))))))))))))))))))))) **.))))))))))))))))))))))))))))))))))))))))))))) *.)))))))))))))))))))))))))))))))))))))))))))))) .)))))))))))))))))))))))))))))))))))))))))))))))

A3. Theorem 9, 9œQ(Q 6 9Q) from the mistake

* 9œNœx[N(x) 6 9N(x)], and ** 9œP›x›N9(P : N(x)].

1. SHOW 9œQ(Q : 9Q) (ND, 2: [(5), (8)]) +)))))))))))))))))) 2. *SHOW œQ(Q : 9Q) (UD, 3: [‘Q’]) *+))))))))))))))))) 3. **SHOW Q 6 9Q (CD, 10) **+)))))))))))))))) 4. ***Qassumption CD 5. ***9œP›x›N9(P : N(x)] ** 6. ***E!a & ›N9(Q: N(a)] 5, N, UI(Q), FrEI 7. ***9[Q : R(a)] 6, S, EI 8. ***9œNœx[N(x) 6 9N(x)] * 9. ***R(a) : 9R(a) 8, N, UI(R), 6, S, FrUI(a) 10. ***SHOW 9Q (ND, 13: [(11), (12)]) ***+))))))))) 11. ****9R(a) 7, N, BC, 4, MP, 9, BC, MP 12. ****9[Q : R(a)] 7, R 13. ****Q 12, N, BC, 11, N, MP ***.))))))))) 14. **.)))))))))))))))) 15. **SHOW 9Q 6 Q (CD, 17) **+))))))))))) 16. ***9Q assumption CD 17. ***Q 16, N

17 **.))))))))))) 18. **Q : 9Q 3, 15, CB *.))))))))))))))))) .))))))))))))))))))

A4. Theorem 3, 9›xGx from, Theorem 1 and Theorem 9.

1. SHOW 9›xGx (ND, 2: [(4), (6), (10)]) +))))))))))))))))) 2. *SHOW ›xGx (ID, 6, 14) *+)))))))))))))))) 3. **¬›xGx assumption ID 4. **9P(G) Axiom 3 5. **9œö[P(ö) 6 ›xö(x)] Theorem 1 6. **›xGx 5, N, UI(G), 4, N, MP 7. **SHOW 9¬›xG(x) (ND, 8: [(10), (12)]) **+)))))))))))) 8. ***SHOW ¬›xG(x) (ID, 11, 13) ***+))))))))))) 9. ****›xGx assumption ID 10. ****9œQ(Q 6 9Q) Theorem 9 11. ****9›xGx 10, N, UI[›xGx], 9, MP 12. ****9¬›xGx 3, P, UP 13. ****¬9›xGx 12, N, MdlNeg ***.))))))))))) **.)))))))))))) 14. **¬›xGx 7, MdlNeg *.)))))))))))))))) .)))))))))))))))))

A5. Logical Equivalence of NE and CF. 9œx[NE(x) : CF(x)] from Essences. 9œx›ö öEss x and Logical Equivalence of Essences. 9œöœøœx(öEss x & øEss x 6 9œx[ö(x) : ø(x)])

The derivation uses,

Def NE. 9œx(NE(x) : œö[öEss x 6 9›xö(x)]), and Def CF. 9œx[CF(x) : œö[ö(x) 6 9›xö(x)].

1. SHOW 9œx[NE(x) : CF(x)] (ND, 6: [(2), (3), (4), (5), (26)]) +))))))))))))))))))))))))))))))))))))))))))))) 2. *9œx›ö öEss x Essences 3. *9œx(NE(x) : œö[öEss x 6 9›xö(x)]) Def NE 4. *9œx[CF(x) : œö[ö(x) 6 9›xö(x)] Def CF 5. *9œöœøœx(öEss x & øEss x 6 9œx[ö(x) : ø(x)]) Logical Equivalence of Essences 6. *SHOW œx[NE(x) : CF(x)] (FrUD, 7: [‘x’]) *+))))))))))))))))))))))))))))))))))))))))))))) 7. **SHOW E!x 6 [NE(x) : CF(x)] (CD, 9) **+)))))))))))))))))))))))))))))))))))))))))))) 8. ***E!x 9. ***SHOW NE(x) : CF(x) (DD, 29) ***+))))))))))))))))))))))))))))))))))))))))))) 10. ****SHOW NE(x) 6 CF(x) (CD, 19)

18 ****+)))))))))))))))))))))))))))))))))))))))))) 11. *****NE(x) 12. *****SHOW œö[ö(x) 6 9›xö(x)] (UD, 13: [(ö)]) *****+))))))))))))))))))))))))))))))))))))))))) 13. ******SHOW ö(x) 6 9›xö(x) (CD, 18) ******+)))))))))))))))))))))))))))))))))))))))) 14. *******ö(x) assumption for CD 15. *******œö[öEss x 6 9›xö(x)] 12, N, 4, FrUI 16. *******ø Ess x 11, N, 4, FrUI, EI(ø) 17. *******ö Ess x 5, N, UI, UI, 8, FrUI,, 16, 17, Adj, MP, N, 8, FrUI, BC, 16, MP 18. *******9›xö(x)] 15, UI, 17, MP ******.)))))))))))))))))))))))))))))))))))))))) *****.))))))))))))))))))))))))))))))))))))))))) 19. *****CF(x) 4, N, 8, FrUI, BC, 12, MP ****.)))))))))))))))))))))))))))))))))))))))))) 20. ****SHOW CF(x) 6 NE(x) (CD, 29) ****+)))))))))))))))))))))))))))))))))))))))))) 21. *****CF(x) assumption CD 22. *****SHOW œö[öEss x 6 9›xö(x)] (UD, 23: [‘ö’]) *****+))))))))))))))))))))))))))))))))))))))))) 23. ******SHOW öEss x 6 9›xö(x) (CD, 28) ******+)))))))))))))))))))))))))))))))))))))))) 24. *******ö Ess x 25. *******œö[ö(x) 6 9›xö(x)] 4, N, 8, FrUI, BC, 21, MP 26. *******9œöœx(öEss x : ö(x) & 9œø[ø(x) 6 9œy[ö(y) 6 ø(y)] Def Ess 27. *******ö(x) 26, N, UI, 8, FrUI, BC, 24, MP, S 28. *******9›xö(x) 25, UI, 27, MP ******.)))))))))))))))))))))))))))))))))))))))) *****.))))))))))))))))))))))))))))))))))))))))) ****.)))))))))))))))))))))))))))))))))))))))))) 29. ****NE(x) 3, N, 8, FrUI, BC, 22, MP ***.))))))))))))))))))))))))))))))))))))))))))) 30. ***NE(x) : CF(x) 10, 20, CB **.)))))))))))))))))))))))))))))))))))))))))))) *.))))))))))))))))))))))))))))))))))))))))))))) .))))))))))))))))))))))))))))))))))))))))))))))

A6. 9[P(NE) : P(CF)] from Logical Equivalence of NE and CF. 9œx[NE(x) : CF(x)] and Axiom 2. 9œöœø[P(ö) & 9œx[ö(x) 6 ø(x)] 6 P(ø).

1. SHOW 9[P(NE) : P(CF)] (ND, (2): [(12), (14)]) +))))))))))))))))))))))))) 2. *SHOW P(NE) : P(CF) (DD, 18) *+)))))))))))))))))))))))) 3. **SHOW P(NE) 6 P(CF) (CD, 16) **+))))))))))))))))))))))) 4. ***P(NE) assumption CD 5. ***SHOW 9œx[NE(x) 6 CF(x)] (ND, 6: [(12), (15)]) ***+)))))))))))))))))))))) 6. ****SHOW œx[NE(x) 6 CF(x)] (FrUD, 6: [‘x’]) ****+))))))))))))))))))))) 7. *****SHOW E!x 6 [NE(x) 6 CF(x)] (CD, 9) *****+)))))))))))))))))))) 8. ******E!x assumption CD 9. ******SHOW NE(x) 6 CF(x) (CD, 13)

19 10. ******+))))))))))))))) 11. *******NE(x) assumption CD 12. *******9œx[NE(x) : CF(x)] Logical Equivalence of NE and CF 13. *******CF(x) 12, N, 8, FrUI, BC, 11, MP ******.))))))))))))))) *****.)))))))))))))))))))) 14. *****9œöœø[P(ö) & 9œx[ö(x) 6 ø(x)] 6 P(ø)Axiom 2 15. *****P(NE) & 9œx[NE(x) 6 CF(x)] 6 P(CF) 14, UI(NE), UI, CF 16. *****P(CF) 4, 5, Adj, 15, MP ****.))))))))))))))))))))))))))))) ***.)))))))))))))))))))))))))))))) **.))))))))))))))))))))))))))))))) 17. **SHOW P(CF) 6 P(NE) **+))))))))))))))))))))))))))))))) ***Similar to lines 4 - 16 **.))))))))))))))))))))))))))))))) 18. **P(NE) : P(CF) 2, 17, CB *.)))))))))))))))))))))))))))))))) .)))))))))))))))))))))))))))))))))

A7. P(CF) 6 œö[P(ö) 6 9›xö(x)] from Axiom 1. (the good half). 9œ[P(¬ö) 6 ¬P(ö)], and Axiom 2. 9œöœø[P(ö) & 9œx[ö(x) 6 ø(x)] 6 P(ø).

The derivation uses,

Negated Properties (Complementary Properties). 9œö[(¬ö)x : ¬ö(x)], and Def CF. 9œx[CF(x) : œö[ö(x) 6 9›xö(x)].

1. SHOW P(CF) 6 œö[P(ö) 6 9›xö(x)] (CD, 3) +))))))))))))))))))))))))))) 2. *P(CF) assumption CD 3. *SHOW œö[P(ö) 6 9›xö(x)] (UD, 4:[]) *+))))))))))))))))))))))))))))))))))) 4. **SHOW P(ö) 6 9›xö(x) (CD, 6) **+)))))))))))))))))))))))))))))))))) 5. ***P(ö) assumption (CD 6. ***SHOW 9›xö(x) (ID, 32, 34) ***+))))))))))))))))))))))))))))))))) 7. ****¬9›xö(x) assumption ID 8. ****SHOW 9œx[CF(x) 6 (¬ö)x] (ND, 9: [(18), (24)]) ****+))))))))))))))))))))))))))))))) 9. *****SHOW œx[CF(x) 6 (¬ö)x] (FrUD, 10: [‘x’]) *****+)))))))))))))))))))))))))))))) 10. ******SHOW E!x 6 [CF(x) 6 (¬ö)x] (CD, 12) ******+))))))))))))))))))))))))))))) 11. *******E!x assumption CD 12. *******SHOW CF(x) 6 (¬ö)x (CD, 14) *******+)))))))))))))))))))))))))))) 13. ********CF(x) assumption CD ********+))))))))))))))))))))))))))) 14. *********SHOW (¬ö)x (ID, 23, 25) *********+)))))))))))))))))))))))))) 15. **********¬(¬ö)x assumption ID 16. **********SHOW ö(x)

20 **********+)))))))))))))) 17. ***********¬ö(x) assumption ID 18. ***********9œö[(¬ö)x : ¬ö(x)] Negated Properties (Complementary Properties) 19. ***********(¬ö)x 18, UI, BC, 17, MP 20. ***********¬(¬ö)x 15, R **********.)))))))))))))) 21. **********CF(x) 13 22. **********9œx[CF(x) : œö[ö(x) 6 9›xö(x)] Def CF 23. **********9›xö(x) 22, N, 11, FrUI, BC, 21, MP, UI, 16, MP 24. **********9¬›xö(x) 7, MdlNeg, UP 25. **********¬9›xö(x) 24, N, MdlNeg *********.)))))))))))))))))))))))))) ********.))))))))))))))))))))))))))) *******.)))))))))))))))))))))))))))) ******.))))))))))))))))))))))))))))) *****.)))))))))))))))))))))))))))))) ****.))))))))))))))))))))))))))))))) 26. ****9œöœø[P(ö) & 9œx[ö(x) 6 ø(x)] 6 P(ø)Axiom 2 27. ****P(CF) & 9œx[CF(x) 6 (¬ö)x] 6 P(¬ö) 26, N, UI(CF), UI(¬ö) 28. ****P(¬ö) 2, 8, Adj, 27, MP 29. ****9œö[P(¬ö) 6 ¬P(ö)] Axiom 1 (the good half) 30. ****¬P(ö) 29, N, UI, 28, MP 31. ****P(ö) 26, N ***.)))))))))))))))))))))))))))))))) **.))))))))))))))))))))))))))))))))) *.)))))))))))))))))))))))))))))))))) .)))))))))))))))))))))))))))))))))))

A8. 9›xGx from, Axiom 6. 9œö(P(ö) e P[Ðx(9[ö(x)])]),

Properties. Every formula, 9œâ(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.

Axiom 3. 9P(G)

Theorem 1. 9œö[P (ö) e ›xö(x)] and the axiom,

God-likeness is for Possible Things Existence-Entailing. 9œx9(Gx 6 E!x)

0. SHOW 9›xGx (ND, 1: [(2), (3), (14), (5), (29)]) +))))))))))))))))))))))))))))))))) 1. * SHOW ›xGx (DD, 40) *+)))))))))))))))))))))))))))))))) 2. **P(G) Axiom 3, N 3. **P(G) e P[Ðx(9[G(x)])] Axiom 6, N, UI 4. **P[Ðx(9[G(x)])] 2, 3, MP 5. **›y[Ðx(9[G(x)])]y Theorem 1, N, UI, AV 6. **SHOW ›y9[G(y)] (ID, 21, 22) **+))))))))))))))))))))))))))))))) 7. ***¬›y9[G(y)] 8. ***SHOW 9¬›y[Ðx(9[G(x)])]y (ND, 9: [(18)]) ***+))))))))))))))))))))))))))))))

21 9. ****SHOW ¬›y[Ðx(9[G(x)])]y (ID, 17, 18) ****+))))))))))))))))))))))))))))) 10*****›y[Ðx(9[G(x)])]y assumption ID 11*****E!a & [Ðx(9[G(x)])]a 12*****SHOW (z)[Ðx(9[G(x)])]z : 9[G(z)] (FrUI, 13: [‘z’]) *****+)))))))))))))))))))))))))))) 13******SHOW E!z e [Ðx(9[G(x)])]z : 9[G(z)] (CD, 14) ******+))))))))))))))))))))))))))) 14*******[Ðx(9[G(x)])]z : 9[G(z)] Properties, N ******.))))))))))))))))))))))))))) *****.)))))))))))))))))))))))))))) 15*****9[G(a)] 11, S, 12, FrUI, BC, 11, S, MP 16*****›y9[G(y)] 11, 15, FrEG 17*****›y9[G(y)] 15, P 18*****9¬›y9[G(y)] 7, MP 19*****¬›y9[G(y)] 18, MdlNeg 20****.))))))))))))))))))))))))))))) ***.)))))))))))))))))))))))))))))) 21***¬›y[Ðx(9[G(x)])]y 8, MdlNeg 22***›y[Ðx(9[G(x)])]y 5, R **.))))))))))))))))))))))))))))))) 23**SHOW 9›yG(y) (ID, 37, 38) **+))))))))))))))))))))))))) 24***¬9›yG(y) assumption 25***SHOW 9¬›y9[G(y)] (ND, 26: [(29), (35)]) ***+)))))))))))))))))))))))) 26****SHOW ¬›y9[G(y)] (ID, 30, 36) ****+))))))))))))))))))))))) 27*****›y9[G(y)] assumption ID 28*****E!b & 9[G(b)] 27, FrEG 29*****9œx9[G(x) 6 E!x] God-likeness is for Possible Things Existence-Entailing, N 30*****SHOW 9›yGy (ND, 34: [(31), (32)]) *****+))))))))))) 31******9[G(b) 6 E!b] 29, N, 28, S, FrUI(b) 32******9[G(b)] 28, S 33******E!b 31, N, 32, N 34******›yGy 33, 32, N, FrEG *****.))))))))))) 35*****9¬9›yG(y) 24, MdlNeg 36*****¬9›yG(y) 35, N ****.)))))))))))))))))))))) ***.))))))))))))))))) 37***¬›y9[G(y)] 25, MdlNeg 38***›y9[G(y)] 6, R **.))))))))))))))))))))))))) 39**9›yG(y) 23, UN 40**›yG(y) 40, N *.)))))))))))))))))))))))))))))))) .)))))))))))))))))))))))))))))))))

22 A9. 9(ö)[P(ö) e ›xö(x)] from Johnson’s Principle. 9¬(ö)P(ö) and Axiom 2. 9(ö)(ø)[(P(ö) & 9(x)[ö(x) e ø(x)]) e P(ø)] 1. SHOW 9(ö)[P(ö) e ›xö(x)] (ND, 2: [(7), (16)]) +)))))))))))))))))))))))))))))))))) 2. *SHOW (ö)[P(ö) e ›xö(x)] (UD, 3: [‘ö’]) *+))))))))))))))))))))))))))))))))) 3. **SHOW P(ö) e ›xö(x) (CD, 5) **+)))))))))))))))))))))))))))))))) 4. ***P(ö) assumption for CD 5. ***SHOW ›xö(x) (ID, 8, 18) ***+))))))))))))))))))))))))))))))) 6. ****¬›xö(x) assumption for ID 7. ****9¬(ö)P(ö) Johnson’s Principle 8. ****¬P(ø) 7, N, QN, EI(ø) [‘ø’ is a variable new to the derivation] 9. ****SHOW 9(x)[ö(x) eø(x)] (ND, 10: [(13)]) ****+)))))))))))))))))))))) 10. *****SHOW (x)[ö(x) e ø(x)] (FrUD, 11) *****+))))))))))))))))))))) 11. ******SHOW E!(x) e [ö(x) e ø(x)] (CD, 15) ******+)))))))))))))))))))) 12. *******E!(x) assumption for CD 13. *******9¬›xö(x) 6, MdlNeg 14. *******¬ö(x) 13, N, MdlNeg, 12, FrUI 15. *******ö(x) e ø(x) 14, ADD, DisjCond ******.)))))))))))))))))))) *****.))))))))))))))))))))) ****.)))))))))))))))))))))) 16. ****9(ö)(ø)[(P(ö) & 9(x)[ö(x) e ø(x)]) e P(ø)] Axiom 2 17. ****(P(ö) & 9(x)[ö(x) e ø(x)]) e P(ø) 16, UI, UI, 18. ****P(ø) 4, 9, Adj, 17, MP ***.))))))))))))))))))))))))))))))) **.)))))))))))))))))))))))))))))))) *.))))))))))))))))))))))))))))))))) .))))))))))))))))))))))))))))))))))

23 A10. œö([P(ö) & ¬›xö(x)] 6 ¬›yÐx[9ö(x)]y), from the axiom,

Positive Properties are for Possible Things Existentice Entailing. 9œö(P(ö) 69œx9[ö(x) 6 E!x]).

1. SHOW œö([P(ö) & ¬›xö(x)] 6 ¬›yÐx[9ö(x)]y) (UD, 2: [‘ö’]) +))))))))))))))))))))))))))))))))))))))) 2. *SHOW [P(ö) & ¬›xö(x)] 6 ¬›yÐx[9ö(x)]y (CD, 3) *+)))))))))))))))))))))))))))))))))))) 3. **[P(ö) & ¬›xö(x)] assumption CD 4. **9œx9[ö(x) 6 E!x]) Positive Properties are for Possible Things Existentice Entailing, N, UI, 3, S, MP 5. **SHOW ¬›yÐx[9ö(x)]y (ID, 6, 23) **+)))))))))))))))))))))))) 6. ***›yÐx[9ö(x)]y assumption ID 7. ***SHOW 9¬›yÐx[9ö(x)]y (ND, 8: [(14), (21)]) ***+))))))))))))))))))))))) 8. ****SHOW ¬›yÐx[9ö(x)]y (ID, 15, 22) ****+)))))))))))))))))))))) 9. *****›yÐx[9ö(x)]y assumption ID 10. *****E!a & Ðx[9ö(x)]a 8, FrEI 11. *****9œöœy[(Ðx[9ö(x)]y : 9ö(y)] Properties 12. *****Ðx[9ö(x)]a : 9ö(a) 11, N, UI, 10, S, FrUI 13. *****9ö(a) 12, CB, 10, S, MP 14. *****9œx9[ö(x) 6 E!x])] 4, R 15. *****SHOW 9›xö(x) (ND, 20: [(16), (17)]) *****+))))))))))) 16. ******9ö(a) 13, R 17. ******9[ö(a) 6 E!a])] 14, N, 10, S, FrUI 18. ******ö(a) 16, N 19. ******E!a 17, N, 18, MP 20. ******›xö(x) 18, 19, FrEG *****.))))))))))) 21. *****9¬›xö(x) 3, S, UP 22. *****¬9›xö(x) 21, N, MdlNeg ***.))*.)))))))))))))))))))))) **.)))))))))))))))))))))))) .))*.))))))))))))))))))))))))))))))))))))))) )

REFERENCES

Findlay, J. N. 1955, "Can God's Existence Be Disproved?" Mind 57, 1948, reprinted in New Essays in Philosophical Theology, eds. A. Flew and A. MacIntyre, New York: Macmillan.

Geach, P. T. 1973a, "Omnipotence," Philosophy 48 (2004, in Philosophy and Faith: A Philosophy of Religion Reader, edited by David Shatz, Boston: McGraw-Hill).

Gödel, Kurt 1944, "Russell's Mathematical Logic," in The Philosophy of Bertrand Russell, edited by P. A. Schilpp, Evanston: Northwestern University Press, 123-53.

Gödel, Kurt 1995, Collected Works, Volume III Unpublished Essays and Lectures, Oxford: Oxford University Press.

Koons, Robert C. 2005, “Sobel on Gödel’s Ontological Proof” (linked to http://www.scar.utoronto.ca/%7Esobel/OnL_T).

24 Kovaè, Sreæko 2003, “Some Weakened Gödelian Ontological Systems,” Journal of Philosophical Logic 32, 565- 88.

Johnson, David 1999, Hume, Holism, and Miracles, Ithaca: Cornell University Press.

Nagel, Thomas 1971, "The Absurd," Journal of Philosophy.

Neale, Stephen 1998, “Grammatical Form, Logical Form, and Incomplete Symbols,” Definite Descriptions: A Reader, edited by Gary Ostertag, Cambridge, Mass.: MIT Press.

Sobel, Jordan Howard 2004, Logic and Theism: Arguments For and Against Beliefs in God, Cambridge University Press.

Sobel, Jordan Howard 2004b, Words and Symbols, Proofs and Invalidations (http://www.utsc.utoronto.ca/%7Esobel/Wds_Symbls).

Sobel, Jordan Howard 2005, “Robert Koons on Gödel’s Ontological Proof” (linked to http://www.scar.utoronto.ca/%7Esobel/OnL_T).

25