3, 12 MP
3, 12 MP
14 ◊(∃x)(Rx & Ex) ⊃ ◊(∃x)(Rx & Ex)
theorem
15 ◊(∃x)(Rx & Ex)
13, 14 MP
16 ◊(∃x)(Rx & Ex) ⊃ (∃x)(Rx & Ex)
theorem
17 (∃x)(Rx & Ex)
15, 16 MP
2b. On the Truth of the Descartes–Leibniz Premises
D1 is analytically true. D2 and D4 are synthetic a priori metaphysical truths. D2 is true, according to Descartes, because the concept of being supremely perfect includes the having of all nonexistential perfections, and the property of necessarily existing if at all is a nonexistential perfection. But why is the conditional property of existing necessarily if at all a perfection (great-making)? Descartes says in “Meditation V” that existence (ˆy[Ey]) is a perfection (1952, p. 94).17 If it is, then it should also be true that the categorical property of existing necessarily (ˆy[Ey]) is a perfection. But property ˆy[Ey] entails the conditional property of existing necessarily if at all, ˆy[Ey ⊃ Ey]. If we then accept that perfections entail only perfections, as I think we should, it follows that property ˆy[Ey ⊃ Ey] is a nonexistential perfection. Schematically,
1 P(ˆy[Ey])
2 P(ˆy[Ey]) ⊃ P(ˆy[Ey])
3
(x)(ˆy[Ey]x ⊃ ˆy[Ey ⊃ Ey]x)
16. The word “theorem” in these annotated proofs and deductions refers to theorems of the logic Q2S5. See Appendix 1.
17. The proposition that existence is a perfection is consistent with not including the property of existence in the concept or essence of a supremely perfect being, per the Principle of Existential NonInclusion.
4 (Y)(Z)(P(Y)
&
(x)(Yx ⊃ Zx)) ⊃ P(Z))
P(ˆy[Ey ⊃ Ey])
Or, as John Findlay has argued, supremely perfect beings are beings that by nature are worthy of worship, and beings that by nature are worthy of worship cannot exist contingently – because to be worthy of worship is to be absolutely perfect in every possible respect (1998, pp. 95–6). So supremely perfect beings must exist necessarily if at all.
Findlay also notes that a supremely perfect being cannot “possess its various excellences in some adventitious or contingent manner”(1998, p. 95). But the property of being supremely perfect is surely one of the excellences (great-makers). Therefore, if a being is supremely perfect, it must be supremely perfect necessarily. So D4 is true.
Descartes was clearly aware of the need to show that it is possible for a supremely perfect being (God) to exist (D3), even though he does not explicitly include it among the premises of the ontological argument he formulates in “Meditation V . ” Father Mersenne pointed it out to him in “The Second Set of Objections” that Descartes had attached to his fi rst publication of the Meditations, and he responded to Mersenne in his appended “Reply,” thus: But though we conceive of God only inadequately . . . this does not prevent its being certain that His nature is possible, or not contradictory; nor does it prevent our affi rming truly that we have examined it with suffi cient precision in order to know that necessary existence apper-tains to this same Divine nature. For all contradictoriness or impossibility is constituted by our thought . . . it cannot reside in anything external to the mind, because by the very fact that it is outside the mind it is clear that it is not contradictory, but is possible. Moreover, contradictoriness in our concepts arises merely from their obscurity and confusion. Hence it suffi ces us to understand clearly and distinctly those few things that we perceive about God . . . to note that among the other constituents of this idea . . . necessary existence is found . . . [and] . . . to maintain that it contains no contradiction. (Descartes 1952, p. 127)
In short, God is possible, according to Descartes, because our concept of God is clear and distinct. Or formally,
1 I (Descartes) have a clear and distinct idea of a supremely perfect being (God).
2 Whatever someone has a clear and distinct idea of possibly exists.
3 Therefore, a supremely perfect being possibly exists.
This is a valid argument for D3. But even if we grant the somewhat controversial assumption that clarity and distinctness are suffi cient for logical possibility, the fi rst premise might well be challenged as being all too subjective. Consider the comments of Leibniz: The reasoning of Descartes concerning the existence of the most perfect being assumed that the most perfect being can be known, or is possible.
For this being assumed . . . it immediately follows that that being exists. But the question is asked whether it is within our power to conceive such a being . . . and [whether it] can be clearly known without contradiction. For the opponents will say that such a notion of the most perfect being . . . is a chimera. Nor is it suffi cient for Descartes to appeal to experience and to allege that he perceives the same in such a manner in himself clearly and distinctly, for this is to break off, not complete the demonstration, unless he shows the method through which others also can attain the same experience . . . [otherwise] we wish to convince them by our authority alone. (1964, pp. 38–9)
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A RG U M E N T 569
Leibniz agrees with Descartes’ ontological argument except for the subjective and authoritative argument for D3. Instead, Leibniz sketches an argument for God’s possibility that is objective and a priori (1964, p. 38). Here is a valid reconstruction: L1 All perfections are compatible.
L2 Every essential property of a supremely perfect being (God) is a perfection.
L3 If something’s essential properties are perfections and all perfections are compatible, then its essential properties are compatible.
L4 If the essential properties of something are compatible, then it is possible that it exists.
Therefore,
D3 It is possible that a supremely perfect being (God) exists.
L2, L3, and L4 are self-evident. L1 is not. So Leibniz constructs an argument for L1 based on his defi nition of a perfection as a “simple quality which is positive and absolute, or expresses whatever it expresses without any limits” (1964, p. 37), and his belief that true propositions that express incompatibility are necessary truths which must be “either demonstrable or known per se” (1964, p. 38). A valid and arguably sound logical reconstruction of his argument is this:
S1 If any two perfections are compatible, then all perfections are compatible.
S2 If any two perfections are incompatible, then they are necessarily incompatible.
S3 If any two perfections are necessarily incompatible, then it is either self-evident that they are incompatible or it can be demonstrated that they are incompatible.
(Because necessary truths are a priori, and a priori truths are either self-evident or demonstrable.)
S4 It is not self-evident that any two perfections are incompatible.
S5 If it can be demonstrated that any two perfections are incompatible, then either one is the negation of the other or some part of the one is incompatible with the other.
S6 If one perfection is the negation of the other, then one of them is not positive.
S7 Perfections are simple, positive qualities.
S8 If some part a perfection is incompatible with another, then one of them is not simple.
Therefore,
L1 All perfections are compatible.
2c. Critiques of the Descartes–Leibniz Ontological Argument
There are three fairly well-known critiques of the Descartes–Leibniz ontological argument.
The fi rst is Kant’s claim that Descartes and Leibniz illicitly include existence as a property in the concept or essence of a supremely perfect being. According to Kant, existence is not a property at all. Even our logical reconstruction of the argument fails to avoid this critique
by merely including the putative conditional property of necessarily existing if at all in the concept or essence of a supremely perfect being, for if existence is not a property, then it is not a property to exist necessarily if at all. Moreover, we explicitly assume that existence is a perfection in our Cartesian justifi cation of D2.
Kant’s main worry, if I read him correctly, is less about whether existence is a property per se, and more about whether it makes sense to include existence in the concept or essence of something. “ ‘ Being’ is obviously not a real predicate; that is, it is not a concept of something which could be added to the concept of a thing” (1933, p. 504). Kant argues for this on the grounds that nothing new is said about anything that is said to exist.
By whatever and by however many predicates we may think a thing . . . we do not make the least addition to the thing when we further declare that it is. Otherwise, it would not be exactly the same thing that exists, but something more than we had thought in the concept . . . (1933, p. 505)
Kant is half right and half wrong. He is right that existence is not a property in the usual sense of being includable in the concept of a thing. His explanation is that existence is not a property at all. A better explanation would be that we beg the question of the thing’s very existence if we include existence in its concept or essence. True, we do not add to the concept of a thing when we say that it exists. So existential propositions are indeed synthetic. But, contrary to Kant, I think that we do predicate something new of a thing when we say that it exists. Kant seems to acknowledge as much when he says, “My fi nancial position is affected . . . very differently by a hundred real thalers than it is by the mere concept of them”
(1933, p. 505). Real money, because it exists, adds fi nancial value to things that exist. Merely possible money, because it does not exist, does not add fi nancial value to things that exist.
It would seem, then, that Kant could consistently hold that existence is not only a property but also that it is a perfection, so long as it is not included in a thing’s concept.18
The second critique is that the argument begs the question. But there is no evidence that it does because neither the Cartesian nor Leibnizian reasons for asserting the premises of the argument include the assertion that the conclusion is true. Perhaps the argument would beg the question were existence really included (by defi nition) in the concept of a supremely perfect being. However, our reconstruction of the argument does not make that assumption.
The third critique is that the argument is easily parodied. In the “First Set of Objections”
to the Meditations, Caterus attempts to parody Descartes’ ontological argument by saying that the same kind of argument could be used to prove the existence (in-reality) of an existent Lion, a being whose essence includes both being a lion and existing.
This complex existent Lion includes both lion and the mode existence . . . essentially . . . But now, has not God from all eternity had a clear and distinct knowledge of this complex? . . .
Yet . . . the distinct cognition of it which God possesses . . . does not constrain either part of the complex to exist, unless you assume that the complex does exist . . . Therefore . . . even though you have a distinct knowledge of the highest being, and granted that a being of supreme 18. Kant can still object to ontological arguments by arguing that a supremely perfect being is not an object of possible experience, and the synthetic proposition “a supremely perfect being exists” is neither a posteriori justifi able nor an a priori condition for the possibility of experience, the only two avenues of real knowledge for him.
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A RG U M E N T 571
perfection includes existence in the concept of its essence, yet it does not follow that its existence is anything actual. . . (1952, p. 107)
Descartes parries this parody in “Reply to First Objections” by saying that:
. . . even though other things are conceived only as existing, yet it does not thence follow that they do exist, but only that they may exist, because we do not conceive that there is any necessity for actual existence being conjoined with their other properties; but, because we understand that actual existence is necessarily and at all times linked to God’s other attributes, it follows that God actually exists. (1952, p. 113)
What I think Descartes is struggling to say here is that an existent Lion is a contingent being, one that exists only if it possibly exists and possibly does not exist. This is because an existent Lion is still a lion, and lions are, by nature, contingent. All that relevantly follows from saying that contingency is contained in the concept or essence of an existent Lion is that anything that is an existent Lion possibly exists and possibly does not exist, if it exists.
So if we assume that existent Lions possibly exist, we can only relevantly infer that it is possible that they possibly exist, and possible that they possibly do not exist. A supremely perfect being, by contrast, is a necessary being; one that exists noncontingently by nature.
And, as we saw earlier, what follows from saying that the property of being a necessary being is contained in the concept or essence of a supremely perfect being is that a supremely perfect being exists necessarily if it exists at all. If we assume that a supremely perfect being possibly exists, we can validly infer that it does in fact exist. It appears, then, that Caterus’
putative parody fails to refute Descartes’ ontological argument.
Suppose, however, that we defi ne a W* as a nonsupreme necessary being. We then get the parody “D1, K2, K3, K4/ KC” of Descartes’ ontological argument, where K2, K3, K4, and KC are the same as D2, D3, D4, and DC, respectively, except for “W*” in place of
“supremely perfect.” Call this parody “the necessary being parody.” Unlike Caterus’ invalid putative parody of Descartes’ argument, the necessary being parody is valid. But is it refuting?
Since the conclusion of the necessary being parody is obviously absurd, at least one of its premises must be false. So if each premise of the necessary being parody is at least as justifi able as the corresponding premise of Descartes’ ontological argument, then the necessary being parody refutes it.
But it is not at all clear that each premise of the necessary being parody is at least as justifi able as the corresponding premise of Descartes’ argument. Consider K3. We gave two arguments earlier for why D3 is true: Descartes’ clear and distinct ideas argument, and Leibniz’s compatibility of perfections argument. Might similar arguments be mustered in support of K3? I doubt it. An analogue of the Cartesian argument will not work because, I would guess, no one really has a clear and distinct idea of a W*. And a Leibnizian analogue will not work, because it must be false that every essential property of a W* (a nonsupreme being) is a perfection: otherwise, W* beings would be supremely perfect.
It is also doubtful that K4 could be justifi ed in the same way that we justifi ed D4 because it is not at all clear that W* is a perfection. And even if we were disposed to say that K4 is true because it is analytically or conceptually true that every of kind of thing is a thing of that kind essentially, then some other premise of the necessary being parody would have to be false. (It would also thereby make D4 analytically or conceptually true.) But both D1
and K2 are analytically true. So K3 would then defi nitely have to be false. Yet D3 is arguably true. Hence, the necessary being parody is nonrefuting.
3. Ontological Arguments of the Twentieth Century
Even though we made use of modal logic in proving the validity of our logical reconstruction of the ontological argument of Descartes and Leibniz, philosophers do not explicitly use modal reasoning in ontological arguments until the twentieth century. In this section, I shall present and very briefl y discuss logical reconstructions of three explicitly modal arguments that are modeled on Anselm’s ontological argument but do not assume
that existence-in-reality is great-making. In the next section, I shall give an exposition and analysis of the lesser-known modal ontological argument of Gödel, which looks to be fashioned, in part, after Leibniz’s proof of the compatibility of all perfections. And in the last two sections I shall present and briefl y discuss two of my own ontological arguments: the modal perfection argument (MPA), and the temporal-contingency argument (TCA).
Norman Malcolm thought that Anselm had actually presented a convincing modal
argument in Proslogion III. Here is a logical reconstruction of his 1960 rendition of that argument:
C1 It is possible that the greatest conceivable being exists.
C2 The greatest conceivable being is unlimited.
C3 Everything that is unlimited is so if and only if it does not depend on anything else for its existence or nonexistence and it neither just happens to exist nor just happens not to exist.
C4 Everything that does not depend on anything else for its existence or nonexistence is such if and only if no other being causes it to begin to exist and no other being causes it to cease to exist.
C5 Anything that begins to exist is caused to begin to exist by some other being, or it just happens to begin to exist.
C6 Anything that ceases to exist is caused to cease to exist by some other being, or it just happens to cease to exist.
C7 Anything that neither begins nor ceases to exist exists necessarily if it exists at all, and fails to exist necessarily if it exists at all.
Therefore,
C8 The greatest conceivable being exists.
Charles Hartshorne, who was also inspired by Anselm’s Proslogion III, formulated a very elegant modal argument “1962, pp. 47–57”. Here is a reconstruction:
H1 It is possible that a perfect being exists.
H2 Necessarily, if a perfect being exists, then a perfect being necessarily exists. (Anselm’s Principle)
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A RG U M E N T 573
Therefore,
H3 A perfect being exists.
Then in 1974, Alvin Plantinga developed what he calls “A Victorious Modal Version” of the Anselmian argument that he embeds in the extensional language of possible worlds. A reconstruction is this:
P1 The property of being maximally great is exemplifi ed in some possible world.
P2 The property of being maximally great is equivalent, by defi nition, to the property of being maximally excellent in every possible world.
P3 The property of being maximally excellent entails the properties of omniscience, omnipotence, and moral perfection.
P4 A universal property is one that is exemplifi ed in every possible world or none.
P5 Any property that is equivalent to some property that holds in every possible world is a universal property.
Therefore,
P6 There exists a being that is essentially omniscient, omnipotent, and morally perfect (God).
These three modal arguments are valid.19 But are they sound? A pretty good case might be made for C2–C6, H2, and P2–P5. But C7 is clearly questionable, especially if the modalities are construed logically, because an eternal being that does not exist in all possible words certainly is possible. And what should we think about the fi rst premise of each of these arguments, which effectively says in each case that it is possible for God to exist? None of the authors of these respective arguments is particularly sanguine about proving this. Hartshorne merely suggests that we might “employ one or more of the other theistic proofs . . . to demonstrate that perfection must at least be conceivable” (1962, p. 52). Plantinga treats the possibility premise as a philosophical hypothesis, which he says it is rational to accept because otherwise “we should fi nd ourselves with a pretty slim and pretty dull philosophy” (1974, p. 221). (Hardly the highest standard for what counts as rational!) And Malcolm says that he does “not know how to demonstrate the concept of God . . . is not self-contradictory” (1967, p. 318). Yet he assumes that it is not self-contradictory because it has “a place in the thinking and lives of human beings” (1967, p. 318).
One consequence of not attempting to prove that it is possible for God to exist is that the arguments of Malcolm, Hartshorne, and Plantinga do not beg the question. On the other hand, they thereby become particularly vulnerable to being refuted by parodies. For example, one might easily validly argue contra Hartshorne that Anselm’s Principle, and the premise that it is possible that a supremely perfect being does not exist, jointly entail that a supremely perfect being does not exist. If we were merely to postulate that, possibly, a supremely perfect being exists, then we could also rightfully postulate that, possibly, a 19. See Appendix 2 for the deductions that prove that these are valid arguments.
supremely perfect does not exist. But then the parody would refute Hartshorne’s argument because we should not be able to rightfully claim that the premises of the parody are less justifi able than those of Hartshorne’s argument.
4a. Gödel’s Ontological Argument
Gödel’s “Ontological Proof ” consisted of two very cryptic and highly technical handwritten pages dated February 10, 1970, which he subsequently shared with Dana Scott. It was fi rst published as an appendix to Sobel’s “Gödel’s Ontological Proof ” and then posthumously in the Collected Works of Gödel. Yet Gödel’s ontological “proof ” is still not widely known in the philosophical and theological communities, with only a dozen or so discussions of it in print, mostly by logicians, and is quite technical.
Gödel develops his ontological argument as a formal axiomatic theory with a theorem that says that there exists a so-called God-like being, where a being is God-like just in case it has every positive property. Although the second-order predicate “positive” is left undefi ned, Gödel suggests that it should be understood in either a moral-aesthetic sense (independent of the accidental structure of the world) or in the sense of pure attribution (as opposed to privation).20 He cautions, however, not to interpret “positive” in the moral-aesthetic sense to mean the same thing as “good” (in the ordinary utilitarian sense) because
“good” (in the ordinary utilitarian sense) means “greatest advantage + smallest disadvantage [which] is negative” (1995b, p. 435). Rather, he says that “positive” could be interpreted as “perfective,” meaning “purely good” and implying nothing negative (1995b, p. 435). It is thus tempting to view the property of being positive in the moral-aesthetic sense as coextensive with the property of being a Platonic form. In other words, each Platonic form is positive, and each positive property is a Platonic form.
But “positive” in the sense of pure attribution rings more Leibnizian. In a footnote to his “Ontological Proof ” Gödel says that a property (or proposition) that is expressed in
“disjunctive normal form in terms of elementary properties [that] contains a member without negation” illustrates pure attribution (1995a, p. 404). And in his “Text” he says,
“the positive properties are precisely those that can be formed out of the elementary ones through application of the operations &, V, ⊃ ” (1995b, p. 437). Adams interprets these cryptic remarks in Leibnizian fashion and suggests that “the purely positive properties will be those that involve no negation at all in their construction from elementary properties (provided the disjunction operation here too is inclusive)” (1995, p. 398).
Gödel’s theory does not assume or presuppose that either existence or existence-in-reality is a property, and his logic has full existential import: the job of saying that something exists is performed by existential quantifi cation. It has one primitive, three defi ned notions, fi ve axioms, and three important theorems.21
Primitive
Property Y is positive.
P ( ) =
1 Y
df Property Y is positive
20. Koons argues that the two conceptions of positivity “coincide perfectly” (2005, p. 3).
21. The formulation presented here is Scott’s. But see Gödel’s “Ontological Proof ” (1995a, p. 403) or Sobel’s
“Gödel’s Ontological Proof ” (1987, pp. 256–7) for Gödel’s own cryptic formulation.
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Defi nitions
Df 1 A being has the property of being God-like (G1) if and only if it has every positive property.
G
= ( )( ( ) ⊃ )
1x
df
Y P1 Y
Yx
Df 2 A property is an essence (E1) of something if and only if it has the property, and the property entails each of its properties.
E (
) =
( )( ⊃ (
)( ⊃ ))
1 Y, x
df Yx & Z
Zx
y Yz
Zy
Df 3 Something has the property of being a necessary being (N2) if and only if every essence it has is necessarily instantiated.
N
= ( )( ( , ) ⊃ (
∃ ) )
2 x
df
Y E1 Y x
z Yz
Axioms
Ax 1 A property is positive if and only if its negation is not positive.
(Z)(P
∼
ˆ ∼
1 (Z ) ≡
P1 (y[ Zy]))
Ax 2 Positive properties entail only positive properties.
(Y)(Z)(P
1 (Y ) &
(x)(Yx ⊃ Zx)) ⊃ P1(Z))
Ax 3 God-likeness is positive.
P (
)
1 G1
Ax 4 Positive properties are necessarily positive.
(Y)(P
1 (Y ) ⊃
P1 (Y))
Ax 5 The property of being a necessary being is a positive.
P (
)
1 N2
Theorems
Tm 1 It is possible that something is God-like.
◊(∃y)G1y
Tm 2 God-likeness is an essence of whatever is God-like.
(x)(G1x ⊃ E1(G1, x))
Tm 3 Something is God-like .
(∃y)G1y
Proof-sketch of Tm 1.22 Assume that it is not possible for something God-like to exist. Then God-likeness is an impossible property. Since impossible properties entail all properties, God-likeness entails the negation of God-likeness. Now God-likeness is positive by Ax 3.
So the negation of God-likeness must be positive by Ax 2. But the negation of God-likeness cannot be positive by Ax 1. Therefore, by reductio ad absurdum, it must be possible that something is God-like.
22. I construct formal proofs of these three theorems in Appendix 2.
Proof sketch of Tm 2. Suppose that anything x is God-like, and that x has property Θ.
Then Θ is positive: otherwise the negation of Θ would be positive by Ax 1; and if x had the negation of Θ, it would not have Θ. But then, Θ necessarily is positive by Ax 4. Now every positive property is possessed by anything that is God-like. So necessarily every property that is necessarily positive must be possessed by anything that is God-like. Since Θ is necessarily positive, it follows that necessarily Θ must be possessed by x. In other words, God-likeness entails Θ. Therefore, God-likeness is an essence of anything that is God-like.
Proof-sketch of Tm 3. Assume that something x is God-like. Since the property of being a necessary being is positive, and things that are God-like have all positive properties, x must be a necessary being. But things are necessary beings only if, for each of their essences, there is something that necessarily has that essence. Since God-likeness is an essence of x, it follows that something necessarily is God-like. Therefore, if it is possible that something is God-like, then it is possible that it is necessary that something is God-like. Now it is possible that something is God-like by Tm 1. So it is possible that it is necessary that something is God-like. But whatever is possibly necessary is necessary. And whatever is necessary is actually the case. Therefore, something is God-like.
4b. On Whether Gödel’s Argument is Sound
Gödel’s axioms imply that something is God-like.23 While the argument is valid, Sobel shows that Gödel’s axioms also imply the absurdity that every true proposition is necessarily true – the so-called modal collapse argument (1987, p. 253).24 A modifi cation of that argument shows that Gödel’s argument cannot be sound:
Assume that Gödel’s argument is sound, and that it proves the existence of the God-like being g1. By Tm 2 the property of being God-like is an essence of g1. By the proof of Tm 3, Godlikeness is necessarily instantiated. Let p be any contingent truth, and let Q be the property that a thing has if and only if p. It follows from the Principle of Abstraction that g1 has Q if and only if p. This and the truth of p imply that g1 has Q. Since the property of being God-like is an essence of g1, the property of being God-like entails Q. But then property Q is necessarily instantiated, because the property of being God-like is necessarily instantiated. From Abstraction and Necessity Introduction, we have it that Q is necessarily instantiated if and only if it is necessarily the case that p. Therefore, p is necessarily true, which contradicts our assumption that p is contingent. So Gödel’s argument cannot be sound.
At least one of Gödel’s axioms must be false. Sobel objects to Ax 2, Ax 3, and Ax 5. His argument against Ax 2 is this:
23. Gödel’s fi ve axioms are the premises of his ontological argument, and Tm 3 is the conclusion.
24. Sobel also shows that Gödel’s axioms imply that if everything has an essence, (x)(∃Y)E1(Y, x)), then everything is a necessary being, (x)N2x. But that result does not prove that Gödel’s argument is unsound. It makes perfect sense to think that free or contingent things do not have essences in Gödel’s sense of essence. Ironically, Sobel notes that things with no essence are necessary beings: (x)(~(∃Y)(E1(Y, x) ⊃ ((Y)(E1(Y, x) ⊃ (∃z)Yz)). That is true, vacuously; and there is no warrant thereby for claiming that everything is a necessary being, or that some essence of a free being is necessarily instantiated (1987, p. 252).
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Gödel’s generous interpretation of properties is at least awkward for an axiological interpretation of Axiom 2, since, according to it, if there is a positive property, then every necessarily universal property such as being self-identical, and being either red or not red, is a positive property. (2004, p. 120)
There are two quick challenges to this argument. First, Gödel’s argument for a God-like being is valid even if tautological or necessarily universal properties are positive. Second, if Ax 2 were replaced by a modifi cation which says that nontautological properties which are entailed by a positive property are positive, and Ax 3 were replaced by a modifi cation which says that God-likeness is nontautological and positive, then the resulting ontological argument would still be valid, but its premises would not imply that tautological properties are positive.
Hàjek argues that there are diffi culties with either Ax 2 or Ax 3. Let Devil-likeness (D1) be the property of having all properties that are not positive. Now God-likeness entails the property of being either God-like or Devil-like. So by Ax 2 and Ax 3, the property of being either God-like or Devil-like (G1 ∨ D1) must be positive. Hájek believes that that result is
“counterintuitive.” Maybe it is and maybe it is not; but even if it is counterintuitive, that does not mean that either Ax 2 or Ax 3 is false.
Sobel says that the diffi culty is worse than counterintuitive. “There is prima facie no more reason for saying that that [a] disjunctive property is positive than there is for saying that it is not positive; it is entailed by a property that is positive according to Axiom 3 . . . and it is entailed by a property that is ‘equally negative’ ” (2004, p. 122). True enough, it is reasonable to assume that D1 is negative (not positive), and D1 surely entails (G1 ∨ D1).
But Sobel is surely wrong to presuppose that these two things imply that (G1 ∨ D1) is negative. Properties that are negative entail properties that are positive, but not vice versa.
The property of being morally evil, for example, entails the property of having some intelligence.
Ax 5 is the axiom that Sobel objects to the most. He maintains that nothing worthy of worship can exist necessarily. And he argues that if the property of being a necessary being is positive, and if any being whose essence is to be worthy of worship can be assumed have to have all positive properties, which is plausible, then any being worthy of worship would have to have the property of being a necessary being and would, therefore, exist necessarily if all.25 So if we abandon Ax 5, we not only block the proof of Tm 3, as well as the modal collapse argument for why Gödel’s argument is not sound, we also mollify Sobel’s angst about objects of worship being necessary beings. I shall return later to the issue of Ax 5.
25. Sobel holds that Gödel’s god (or the demonstrable god of any ontological argument) cannot be the God of Theism (1987, pp. 254–5). When fully stated in standard logical form, his argument appears to be this: 1 Gödel’s god is a necessary being.
2 Every necessary being is an abstract entity.
3 No abstract entity is worthy of being worshipped.
4 The God of Theism is worthy of being worshipped.
5 Therefore, Gödel’s god is not the God of Theism
Although valid, the second premise of this argument is arguably false. I see no good reason for not thinking that something with causal powers could exist in every possible world. But it is hard to imagine an abstract entity with causal powers. So some necessary beings might not be abstract entities.
What about Ax 1? Anderson splits Ax 1 into Ax 1a and Ax 1b:
Ax 1a If a property is positive, then its negation is not positive
(Z)(P
∼
ˆ ∼
1 (Z ) ⊃
P1 (y[ Zy]))
Ax 1b If the negation of a property is not positive, then the property is positive.
(Z)(∼P ˆ
1 (∼ y [∼ Z] ⊃ P1 (Z)))
He then persuasively argues that Ax 1a is true, and he correctly notes that Ax 1b is false.26
Many properties and their negations appear not to be positive, such as the property of being red and the property of not being red. But if we abandon Ax 1 in favor of Ax 1a, and make no other changes, we also block the proof of Tm 3. Interestingly, Ax 1b is also used in our version of the Sobel modal collapse argument, because an instance of Ax 1b occurs in line 3 of the proof of Tm 2.
Anderson, Hazen, Koons, and Hàjek have all formulated different emendations
of Gödel’s theory that successfully dodge Sobel’s modal collapse refutation of Gödel’s ontological argument. Since space here will not permit a detailed analysis, comparison, and evaluation of all four emendations, I shall provide just a summary of the key ideas of Anderson’s emendation. I shall then briefl y discuss a class of ingenious parodies designed by Graham Oppy that he says infl ict Anderson’s theory, and quite possibly the others, too.
Anderson’s emendation has three new defi nitions, Df 1a, Df 2a, and Df 3a. Its axioms are Ax 1a, Ax 2, Ax 3a, Ax 4, and Ax 5a.
Df 1a A being has the property of being Goda-like (Ga1) if and only if its essential properties are all and only those properties that are positive.27
Df 2a A property Y is an essencea (Ea1) of being x if and only if, for every property Z, x has Z essentially if and only if Y entails Z.
Df 3a Something has the property of being a necessarya being (Na2) if and only if every essencea it has is necessarily instantiated.
Ax 3a Goda-likeness is positive.
Ax 5a The property of being a necessarya being is positive.
Anderson’s axioms imply that something is Goda-like (Anderson’s ontological argument).
But Sobel’s modal collapse argument is not valid in Anderson’s theory.28 One very interesting feature of Anderson’s theory, unlike Gödel’s, is that it allows for the possibility that Goda-like beings have some nonpositive properties contingently. Another is that the concept of essencea closely resembles the common philosophical concept of essence, whereas an essence for Gödel is best understood as a complete characterization.
Oppy’s theory (1996) is just like Anderson’s except for Df 1* in place of Df 1a, Ax 3* in place of Ax 3a, and Ax 5* in place of Ax 5a.
26. Anderson deduces Ax 1a from plausible principles about intrinsic preferability (1990, p. 295).
27. Although Anderson includes a “*” in the names of his key terms and axioms to distinguish them from those of Gödel, we instead include an “a” in the names of Anderson’s key terms and axioms in order to distinguish them from names used by Oppy, who also includes a “*” in the names of still other Gödel-like terms and axioms.
28. Our modifi cation of Sobel’s argument shows why. The property Q that a thing has if and only if p (where p is some contingent truth) cannot be an essential property of a Goda-like being, and Goda-likeness does not entail Q.
O N TO LO G I C A L
A RG U M E N T 579
Df 1* A being has the property of being God*-like (G*1) if and only if its essential properties are those and only those which are positive, except for Φ1, . . . , Φn.
Ax 3* God*-likeness is positive.
Ax 5* Necessary existence is a positive property, and distinct from Φ1, . . . , Φn.
His axioms imply the absurdity that there are almost as many God*-like beings as there are positive properties (Oppy’s ontological parody).29
Oppy misinterprets his ontological parody as showing that at least one of Anderson’s axioms must be false, and he only conjectures that an “obvious candidate is [Ax 3a]” (2000, p. 2).30 It is also certain that not all of Oppy’s axioms can be true because the conclusion of his valid parody is absurd. Oppy’s parody will, however, substantively refute Anderson’s ontological argument if and only if Ax 3* and Ax 5* are at least as justifi able as Ax 3a and Ax 5a, respectively, assuming that Ax 1a, Ax 2, and Ax 4 are at least modestly justifi able (more so than not); and Oppy’s parody will vacuously refute Anderson’s ontological argument if either Ax 1a, Ax 2, or Ax 4 is not even modestly justifi able.
We could show that Oppy’s parody does not substantively refute Anderson’s ontological argument if we could assume that the property of not having Φi essentially is not positive, given that Φi is a positive property excluded by defi nition from being an essential property of anything that is God*-like. For we could then prove that Ax 3* is false with the following Gettings (1999)-style argument:
1 The property of being God*-like entails the property of not having Φi essentially.
(Derived from the defi niens of “God*-like.”)31
2 If the property of being God*-like is positive, and the property of being God*-like entails the property of not having Φi essentially, then the property of not having Φi essentially is positive. (Ax 2)
3 The property of not having Φi essentially is not positive.
The property of being God*-like is not positive.
It would be diffi cult to know, however, whether Oppy’s parody substantively refutes Anderson’s argument without knowing more about what a positive property is, and whether it is true that the property of not having Φi essentially is not positive.32 In the next section, 29. Oppy constructs a slightly different parody of a slight modifi cation of Anderson’s argument that includes the new premise that if a property Z is positive then the property ˆy[Zy] is positive (2007, pp. 16–7).
30. Instead of arguing directly for the falsity of Ax 3a, Oppy merely says, “atheists and agnostics may (perhaps should) say that the property of being [Goda-like] is positive only if it is exemplifi ed,” and then notes that if this new proposition were to replace Ax 3a of Anderson’s argument, then the resulting argument would beg the question (2000, p. 2). This is a red herring, and it fails to show that Ax 3a itself is false.
31. Symbolically: (x)[(Y)((P1(Y) & Y ≠ Φi) ≡ Yx) ⊃ ~Φix]. Use a conditional proof and utilize the fact that
“Φi = Φi” is a logical truth.
32. If the mere presence of the word “not” in a description of a property were a guarantor of nonpositivity, then the property of not having property Φi essentially would be nonpositive. But the presence of “not” does not guarantee nonpositivity. Otherwise, the property of not having the negation of a positive property would be nonpositive, which it is not.
We might attempt to prove that the property of not having Φi essentially is not positive with another Gettings (1999)-style argument:
a. P1(Φi)
given
b. (Y)(P1(Y) ⊃ P1(ˆy[Yy)) new
axiom
c. P1(Φi) ⊃ ~P1(ˆy[~Φiy]) Ax
1a
~P1(ˆy[~Φiy])
The problem is that the “new axiom” is no more obvious than the conclusion.
I shall replace the predicate “is a positive property” with the predicate “is a perfection,” and I shall replace “is Goda-like” with “is supreme.” I shall then argue that the analogues of Ax 1a, Ax 2, and Ax 3a are true, but that the analogue of Ax 3* is false. Those three analogues will constitute the premises of the MPA. The analogues of Ax 4 and Ax 5a will not be needed.
5. The Modal Perfection Argument
The Modal Perfection Argument (MPA) is an ontological argument that is rooted in the ontological arguments of Anselm, Descartes–Leibniz, and Gödel.33 Think of a perfection (P2) as a property that it is necessarily better to have than not; and defi ne the property of being supreme (S1) as the property that a thing has if and only if it is impossible for something to be greater and impossible for there to be something else than which it is not greater: S1x =df (~◊(∃y)Gyx & ~◊(∃y)(x≠y & ~Gxy)). The conclusion of MPA is that exactly one supreme being exists,34 and the premises are the following:
M1 A property is a perfection only if its negation is not a perfection.
M2 Perfections
entail
only perfections.
M3 The property of being supreme is a perfection.
We can show that MPA is valid by fi rst showing that M1, M2, and M3 jointly imply that it is possible that a supreme being exists. The proof is the same as our proof of Gödel’s Tm 1 in Appendix 2, save for “P2” in place of “P1” and “S1” in place of “G1.” (The annotation also substitutes M1, M2, and M3 for A1, A2, and A3, respectively, and drops “Equiv” and
“Simp” from line 10.) We can then prove that the possibility of a supreme being implies the existence of a supreme being as follows:
Deduction
1 ◊(∃x)S1x pr
2 ◊(∃x)S1x ⊃ (∃x)◊S1x theorem35
3 (∃x)◊S1x
1, 2 MP
4 ◊S ν
1 3,
EI
5 ◊(~◊(∃y)Gyν & ~◊(∃y)(ν≠y & ~Gνy)) 4,
df
“S1”
6 ◊(~◊(∃y)Gyν & ~◊(∃y)(ν≠y & ~Gνy)) ⊃
theorem
(◊~◊(∃y)Gyν & ◊~◊(∃y)(ν≠y & ~Gνy))
33. I fi rst formulated a version of MPA in November 2001, and I presented it at The Second Annual Saint Anselm Conference held in April 2002 at Saint Anselm College, Manchester, NH. Philo published an improved version of MPA by Maydole (2003), replies by Oppy (2004) and Metcalf (2005), and my counterreplies (2005a and 2005b).
34. MPA does not assume or presuppose that existence is a property, and its quantifi ers have existential import.
35. “◊(∃x)S1x ⊃ (∃x)◊S1x” is an instance of the controversial Barcan Formula (BF). Plantinga argues against BF (1974, pp. 59–60). I refute his argument (Maydole 1980, pp. 140–2). And I argue for BF (Maydole 2003, pp. 303–7).
O N TO LO G I C A L
A RG U M E N T 581
7 (◊~◊(∃y)Gyν & ◊~◊(∃y)(ν≠y & ~Gνy))
4, 5 MP
8 ◊~◊(∃y)Gyν 7
Simp
9 ◊~◊(∃y)(ν≠y & ~Gνy) 7
Com,
Simp
10 ◊~◊(∃y)Gyν ⊃ ~◊(∃y)Gyν theorem
11 ◊~◊(∃y)(ν≠y & ~Gνy) ⊃ ~◊(∃y)(ν≠y & ~Gνy) theorem
12 ~◊(∃y)Gyν
8, 10 MP
13 ~◊(∃y)(ν≠y & ~Gνy)
9, 11 MP
14 ~◊(∃y)Gyν & ~◊(∃y)(ν≠y & ~Gνy)
12, 13 Conj
15 S ν
1 14,
df
“S1”
16 (∃x)S1x 15
EG
It is also logically true that, at most, one thing is supreme. (Maydole 2003, p. 302). Therefore, exactly one supreme being exists.
I show that M1, M2, and M3 are true, and that MPA is resistant to sundry salient parodies (Maydole 2003). Here is a small snapshot of how I argue there for the premises: M1 is true because it is better to have a property than not only if it is not better to not have that property than not; M2 is true because it is always better to have that which is a necessary condition for whatever it is better to have than not; and M3 is true because it is reasonable to assume that a thing is supreme if and only if it is necessarily greater than everything else solely by virtue of having some set of perfections, making the extension of the property of being supreme identical with the intersection of the extensions of those perfections. Premises of a different argument for the truth of M3 might be as follows:
M31 For every Z, all of the nontautological essential properties entailed by Z are perfections if and only if the property of being a Z is a perfection.
M32 Every nontautological essential property entailed by the property of being supreme is a perfection.
Therefore,
M3 The property of being supreme is a perfection.
This argument is clearly valid, and the premises are plausible. M31 is arguably true because (1) it is necessarily better to have a property if and only if the property endows whatever has it with nontautological properties that are necessarily better to have than not, and (2) for any properties Y and Z, if Z endows something with Y, then Z entails Y, and (3) perfections are properties that are better to have than not. M32 is arguably true because (4) all the nontautological essential properties entailed by the essence of a supreme being are perfections, and (5) anything entailed by the essence of a thing of kind Z is entailed by the property of being a Z.
There is no evidence to indicate that MPA begs the question. And we can easily show that an Oppy-style parody based on the idea of being almost supreme does not refute MPA, where something is almost supreme just in case it is impossible for anything to be almost greater: from M31, and the fact that it is not the case that every nontautological essential property entailed by the property of being almost supreme is a perfection, it follows that the property of being almost supreme is not a perfection.
6. The Temporal-Contingency Argument
We have shown that the fi rst three premises of MPA imply that it is possible that a supreme (greatest possible) being exists. But any valid argument for the existence of a supreme being that has copossible premises would show the same thing, even if one or more of its premises happens to be false. Consider the Third Way of St Thomas Aquinas.
The third way is taken from possibility and necessity and runs thus. We fi nd in nature things that are possible to be and not possible to be, since they are found to be generated and corrupted. But it is impossible for these always to exist, for that which can not-be at some time is not. Therefore, if everything can not-be, then at one time there was nothing in existence.
Now if this were true then even now there would be nothing in existence, because that which does not exist begins to exist only through something already existing. Therefore if at one time nothing was in existence, it would have been impossible for anything to have begun to exist; and thus now nothing would be in existence – which is absurd. Therefore, not all beings are merely possible, but there must exist something the existence of which is necessary. But every necessary thing has its necessity caused by another, or not. Now it is impossible to go on to infi nity in necessary things which have their necessity caused by another, as has already been proved in regard to effi cient causes. Therefore, we cannot but admit the existence of some being having of itself its own necessity, and not receiving it from another, but rather causing in others their necessity. This all men speak of as God. (1998, pp. 4–5)
Aquinas’ Third Way is invalid per se because the proposition that everything fails to exist at some time does not entail the proposition that there is a time when everything fails to exist.
Temporally contingent things might be eternal in the actual world yet fail to exist at some time in some other possible world. But the Third Way is fertile, and it can easily be transformed into a valid argument for the existence of a supreme being that has copossible premises.
Think of something as generated just in case there is a time when it exists and an earlier time when it does not; and as corrupted just in case there is a time when it exists and a later time when it does not. Defi ne something as temporally necessary if and only if it is necessarily not generated and necessarily not corrupted. And defi ne something as temporally contingent if and only if it is possibly generated or possibly corrupted. Then our modifi ed Third Way is this.
T1
Something presently exists.
T2 Only
fi nitely many things have existed to date.
T3
Every temporally contingent being begins to exist at some time and ceases to exist at some time.
T4
Everything that begins to exist at some time and ceases to exist at some time exists for a fi nite period of time.
T5
If everything exists for only a fi nite period of time, and there have been only fi nitely many beings to date, then there was a time when nothing existed.
T6
If there was a time when nothing existed, then nothing presently exists.
T7
A being is temporally necessary if and only if it is not temporally contingent.
T8
Everything has a suffi cient reason for its existence.
T9
Anything that has a suffi cient reason for its existence also has a suffi cient reason for its existence that is a suffi cient reason for its own existence.
O N TO LO G I C A L
A RG U M E N T 583
T10 No temporally contingent being is a suffi cient reason for the existence of a temporally necessary being.
T11 Every temporally necessary being that is a suffi cient reason for its own existence is a being without any limitations.
T12 A being without any limitations is necessarily greater than any other being.
T13 It is not possible for anything to be greater than itself.
T14 It is necessarily the case that “greater than” is asymmetric.
Therefore,
T15 There exists a supreme being.
The Temporal-Contingency Argument (TCA) is the argument “◊(T1 & T2 & . . . & T14)
/ (∃x)S1x.” The later deduction for “(T1, T2, . . . , T14) / (∃x)S1x” plus the deduction (in the previous section) for “◊(∃x)S1x / (∃x)S1x” together prove that TCA is valid.36 First, let us add to our lexicon, thus
B2x =df x begins to exist at some time and ceases to exist at some time
T2x =df x is temporally necessary
C2x =df x is temporally-contingent
F2x =df x exists for a fi nite period of time
M2 =df Only fi nitely many things have existed to date
P3 =df Something presently exists
N2 =df There was a time when nothing existed
S2xy =df x is a suffi cient reason for y for the existence of y
W2x =df x is without any limitations
Deduction37
1 P3 pr
1
2 M2 pr 2
3 (x)(C2x ⊃ B2x) pr
3
4 (x)(B2x ⊃ F2x) pr
4
5 ((x)F2x & M2 ) ⊃ N2 pr
5
6 N2 ⊃ ~P3 pr
6
7 (x)(T2x ≡ ~C2x) pr
7
8 (x)C2x AIP
9 C μ
μ
2
⊃ B2 3
UI
10 C μ
2
8
UI
11 B μ
2
9, 11 MP
36. The modifi ed Third Way that I present here is a variation on a modifi cation of Aquinas’ Third Way that I discuss in “A Modal Model . . .” (1980, pp. 139–40). It is also different from the central argument (MTW) of my
“The Modal Third Way” (2000, pp. 1–28). MTW is sound, but the supreme being it proves is not defi ned as a greatest possible being; and it is not necessarily the God of Anselm, unless we postulate that such a supreme being is a greatest possible being. By contrast, the supreme being of TCA is a greatest possible being by defi nition.
37. This deduction departs slightly from the norm of beginning with all premises.
12 B2u ⊃ F2u 4
UI
13 F2u 11,12
MP
14 (x)F2x 13
UG
15 (x)F2x & M2
2, 14 Conj
16 N2
5, 15 MP
17 ~P3
6, 16 MP
18 P3 & ~P3
1, 17 Conj
19 ~(x)C2x 8–19
IP
20 (∃x)~C2x 19
QN
21 ~C ν
2 1 20
EI
22 T ν
ν
2 1 ≡ ~C2 1 7
UI
23 (T ν
ν
ν
ν
2 1 ⊃ ~C2 1) & (~C2 1 ⊃ T2 1) 22
Equiv
24 (~C ν
ν
2 1 ⊃ T2 1)
23 Com, Simp
25 T ν
2 1
21, 24 MP
26 (∃x)T2x 25
EG
27 (x)(∃y)S2yx pr
8
28 (x)[(∃y)S2yx ⊃ (∃z)(S2zx & S2zz)] pr
9
29 (x)(y)[(T2x & S2yx) ⊃ ~C2y] pr
10
30 (y)[(T2y & S2yy) ⊃ W2y] pr
11
31 (y)[W2y ⊃ (z)(z≠y ⊃ Gyz)]
pr 12
32 ~◊(∃y)Gyy pr
13
33 (x)(y)(Gxy ⊃ ~Gyx)
pr 14
34 (∃y)S2yν1 27,
UI
35 (∃y)S2yν1 ⊃ (∃z)(S2zν1 & S2zz) 28,
UI
36 (∃y)(S2zν1 & S2zz)
34, 35 MP
37 S νν
νν
2
1 & S2
36,
EI
38 (T ν
νν
ν
2 1 & S2
1) ⊃ ~C2
29, UI twice
39 S νν
2
1 37
Simp
40 T ν
νν
2 1 & S2
1
25, 39 Conj
41 ~C ν
2
38, 40 MP
42 T ν
ν
2 ≡ ~C2 7,
UI
43 (T ν
ν
ν
ν
2 ⊃ ~C2 ) & (~C2 ⊃ T2 ) 42
Equiv
44 ~C ν
ν
2 ⊃ T2
43 Com, Simp
45 T ν
2
44, 41 MP
46 S νν
2
37 Com, Simp
47 T ν
νν
2 & S2
45, 46 Conj
48 (T ν
νν
ν
2 & S2
) ⊃ W2 30
UI
49 W ν
2 ⊃ (z)(z≠ν ⊃ Gνz) 31
UI
50 (z)(z≠ν ⊃ Gνz)
47, 48, 49 MP
51 (z)(~z≠ν ∨ Gνz) 50
Impl
52 (z)(~z≠ν ∨ ~~Gνz) 51
DN
53 (z)~(z≠ν & ~Gνz) 52
DeM
54 ~(∃z)(z≠ν & ~Gνz) 53
QN
55 ~◊(∃z)(z≠ν & ~Gνz) 54
ME
56 ~(∃y)Gyy 32
ME
57 (y)~Gyy 56
QN
58 (y)~Gyy
ACP
O N TO LO G I C A L
A RG U M E N T 585
59 ~Gμμ 58
UI
60 ~Gμμ ∨ ν≠μ 59
Add
61 ν≠μ ∨ ~Gμμ 60
Com
62 ν=μ ⊃ ~Gμμ 61
Impl
63 μ=ν ⊃ ν=μ theorem
64 μ=ν ⊃ ~Gμν
62, 63 HS
65 (y)~Gyy
⊃ (μ=ν ⊃ ~Gμν) 58–64
CP
66 [(y)~Gyy ⊃ (μ=ν ⊃ ~Gμν)] 65
(theorem)
NI
67 (μ=ν ⊃ ~Guν)
57, 66 MMP
68 (x)(y)(Gxy ⊃ ~Gyx) & (z)(z≠ν ⊃ Gνz)
33, 50 Conj
69
[(x)(y)(Gxy ⊃ ~Gyx) & (z)(z≠ν ⊃ Gνz)] ⊃
theorem
[(x)(y)(Gxy ⊃ ~Gyx) & (z)(z≠ν ⊃ Gνz)]
70 [(x)(y)(Gxy ⊃ ~Gyx) & (z)(z≠ν ⊃ Gνz)]
68, 69 MP
71
{[(x)(y)(Gxy ⊃ ~Gyx) & (z)(z≠ν ⊃ Gvz)] ⊃
theorem
(μ≠ν ⊃ ~Gμν)}
72 (μ≠ν ⊃ ~Gμν)
70, 71 MMP
73 [(μ=ν ⊃ ~Gμν) & (μ≠ν ⊃ ~Gμν)] ⊃
theorem
[(μ=ν ∨ μ≠ν) ⊃ (~Gμν ∨ ~Gμν)]
74 (μ=ν ⊃ ~Guν) & (μ≠ν ⊃ ~Gμν)
67, 72 Conj
75 [(μ=ν ∨ μ≠ν) ⊃ (~Gμν ∨ ~Gμν)]
73, 74 MP
76 (μ=ν ∨ μ≠ν) theorem
77 (~Gμν ∨ ~Gμν)
75, 76 MMP
78 (~Gμν ∨ ~Gμν) ⊃ ~Gμν theorem
79 ~Gμν
77, 78 MP
80 (z)~Gzν 79,
UG
81 (z)~Gzν ⊃ (z)~Gzν theorem38
82 (z)~Gzν
80, 81 MP
83 ~(∃z)Gzν 82,
QN
84 ~◊(∃z)Gzν 83,
ME
85 ~◊(∃z)Gzν & ~◊(∃z)(z≠ν & ~Gνz)
84, 55 Conj
86 S ν
1 85,
def
“S1”
87 (∃x)S1x 86
EG
The sole premise of TCA is true if and only if there is some possible world where the premises T1, T2 . . . and T14 are true. Now I know of no reason to believe that there could not be a possible world ω where the propositions T1, T2, T3, T6, T8, T9, T10, and T11
express logically contingent facts about ω. Propositions T4, T5, T7, T13, and T14 appear to be self-evident analytic truths which are true in every possible world, including ω. Only T12 requires special justifi cation:
Assume x is a being without limitations in ω. Then x possesses every great making property in ω. In particular, x possesses the property in ω of not being limited in world ω1 by anything.
38. Line 81 is an instance of another version of the BF. The two versions are equivalent, by the rules, to Trans, DN, ME, and QN. It is also interesting to note, however, that the validity of our modifi ed Third Way can be proven in the logic QS4, which is just like QS5, save for the rule ┌◊p / ◊◊p┐ in place of ┌◊p / ◊p┐. The BF is not a theorem of QS4.
In other words, if x is a being without any limitations in ω, then x possesses every great making property in ω. But the property of not being limited in ω1 is a great making property of ω.
So it is true in ω that it is true in ω1 that x is unlimited. But for any statement p, if it is true in world α that p is true in world β, then p is true in world β. Hence, x is unlimited in world ω1. Now if x is unlimited in ω1, then in ω1 x is greater than any other being in ω1; otherwise x would be limited by not possessing a great making property possessed by something else.
Hence it is true in ω1 that x is greater than every other being. Since ω1 is an arbitrarily selected possible world, it follows that it is true in every possible world that x is greater than every other being. Consequently, it is necessarily the case that x is greater than every other being. So T12 is true in ω. (Maydole 1980, p. 140)
TCA is a quasi-ontological argument that is arguably sound. There is also no evidence to indicate that it begs the question. And it seems that it would be particularly resistant to being parodied, given its dependence on sundry logically contingent facts about a possible world, and the historical absence of any parodies against Third Way arguments.
7. Conclusion
Ontological arguments are captivating. They convince some people but not others. Our purpose here was not to convince but simply to show that some ontological arguments are sound, do not beg the question, and are insulated from extant parodies. Yet good logic does convince sometimes. Other times, something else is needed.
References
Adams, R. M. (1994) Leibniz: Determinist, Theist, Idealist. New York: Oxford University Press.
Adams, R. M. (1995) Introductory note to *1970. In S. Feferman (ed.), Kurt Gödel, Collected Works, vol lll, 388–402. New York: Oxford University Press.
Anderson, C. A. (1990) Some emendations of Gödel’s ontological proof. Faith and Philosophy 7, 291–303.
Anselm (1962) Basic Writing, 2nd edn. Trans. S. N. Deane. La Salle, IL: Open Court.
Anselm (1967) Proslogion ( Chapters II–IV). Trans. A. C. McGill. In J. H. Hick and A. C. McGill (eds.), The Many-Faced Argument, 3–8. New York: Macmillan.
Anselm (1998) Proslogion. Trans. M. J. Charlesworth. In B. Davies and G. Evans (eds.), Anselm of Canterbury: The Major Works, 1–34. New York: Oxford University Press.
Thomas Aquinas (1998) Summa Theologica, First Part, Q. II, Art. 3. Trans. A. C. Pegis. Reprinted in Philosophy of Religion, 3rd edn. Ed. L. P. Pojman. Belmont, CA: Wadsworth.
Copleston, F. C. (1961) Medieval Philosophy. New York: Harper Torchbooks.
Descartes, R. (1952) Meditations on First Philosophy and Objections Against the Meditations and Replies (including “arguments demonstrating the existence of God . . . drawn up in geometrical fashion”). Trans. E. S. Haldane and G. R. T. Ross. In R. M. Hutchins (ed.), The Great Books, vol.
31, 69–293. Chicago: Encyclopedia Britannica.
Findlay, J. N. (1998) Can God’s existence be disproved? In L. P. Pojman (ed.), Philosophy of Religion, 3rd edn, 93–7. Belmont, CA: Wadsworth.
Gettings, M. (1999) Gödel’s ontological argument: a reply to Oppy. Analysis 59: 4, 309–13.
Girle, R. (2003) Possible Worlds. Montreal: McGill-Queens University Press.
O N TO LO G I C A L
A RG U M E N T 587
Gödel, K. (1995a) Ontological proof (*1970). In S. Feferman (ed.), Kurt Gödel, Collected Works, vol.
III, 403–4. New York: Oxford University Press.
Gödel, K. (1995b) Texts relating to the ontological proof. In S. Feferman (ed.), Kurt Gödel, Collected Works, vol. III, 429–37. New York: Oxford University Press.
Gustason, W. and Ulrich, D. E. (1989) Elementary Symbolic Logic. Prospect Heights, IL: Waveland Press.
Hàjek, P. (2002) A new small emendation of Gödel’s ontological proof. Studia Logica 71, 149–64.
Hartshorne, C. (1962) The Logic of Perfection and Other Essays in Neoclassical Metaphysics. La Salle, IL: Open Court.
Hazen, A. P. (1998) On Gödel’s ontological proof. Australian Journal of Philosophy 76, 361–77.
Kant, I. (1933) Critique of Pure Reason. Trans. N. K. Smith. New York: St. Martin’s Press.
Konyndyk, K. (1986) Introductory Modal Logic. Notre Dame, IN: University of Notre Dame Press.
Koons, R. C. (2005) Sobel on Gödel’s Ontological Proof, http://www.scar.utoronto.ca/~sobel/OnL_T/
(accessed October 12, 2007).
Kripke, S. (1959) A Completeness theorem in modal logic. The Journal of Symbolic Logic 24: 1, 1–14.
Leibniz, G. W. (1964) The Existence of God. Ed. J. Hick. New York: Macmillan. Reprinted from New Essays Concerning Human Understanding by G. W. Leibniz (1896). Trans. A. G Langley. New York: Macmillan, Appendix X.
Malcolm, N. (1967) Anselm’s ontological arguments. In J. H. Hick and A. C. McGill (eds.), The Many-Faced Argument, 301–20. New York: Macmillan. Reprinted from The Philosophical Review LXIX: 1 (1960).
Maydole, R. (1980) A modal model for proving the existence of God. American Philosophical Quarterly 17, 135–42.
Maydole, R. (2000) “The Modal Third Way.” International Journal for Philosophy of Religion 47: 1–28.
Maydole, R. (2003) The modal perfection argument for the existence of God. Philo 6, 299–313.
Maydole, R. (2005a) On Oppy’s objections to the modal perfection argument. Philo 8: 2, 123–130.
Maydole, R. (2005b) On Metcalf ’s objections to the modal perfection argument. Philo 8: 2, 134–6.
McGill, A. C (1967) Recent discussions of Anselm’s argument. In J. H. Hick and A. C. McGill (eds.), The Many-Faced Argument, 33–110. New York: Macmillan.
Metcalf, T. (2005) Entailment and ontological arguments: reply to Maydole, Philo 8: 2, 131–3.
Oppy, G. (1995) Ontological Arguments and Belief in God. Cambridge: Cambridge University Press.
Oppy, G. (1996) Gödelian ontological arguments. Analysis 56, 226–30.
Oppy, G. (2000) Response to Gettings. Analysis 60: 4, 363–7.
Oppy, G. (2004) Maydole’s 2QS5 argument. Philo 77: 2, 203–11.
Oppy, G. (2007) Ontological Arguments. Stanford Encyclopedia of Philosophy, http://plato.stanford.
edu/entries/ontological-arguments/ (accessed July 29, 2007).
Plantinga, A. (1974) The Nature of Necessity. Oxford: Oxford University Press.
Rowe W. L. (2001) Philosophy of Religion, 3rd edn. Belmont, CA: Wadsworth.
Sobel, J. H. (1987) Gödel’s ontological proof. In J. J. Thomson (ed.), On Being and Saying: Essays for Richard Cartwright, 241–61. Cambridge, MA: MIT Press.
Sobel, J. H. (2004) Logic and Theism. Cambridge: Cambridge University Press.
Appendix 1. Logic Matters
The strongest logic used in this chapter is a standard natural deduction system of second-order quantifi cational modal logic with identity (2QS5). It is equivalent to a standard second-order extension of Kripke’s 1959 system of fi rst-order modal logic. A weaker subsystem of 2QS5 is frequently used. The language of 2QS5 includes fi rst- (lowercase) and second-order (uppercase) variables, constants and pseudo-names, property abstracts, and the standard array of quantifi ers, connectives, punctuation marks, and so on.
The nonmodal propositional and quantifi cational inference rules of 2QS5 are from Gustason and Ulrich: Conjunction (Conj), Addition (Add), Simplifi cation (Simp), Disjunctive Syllogism (DS), Excluded Middle Introduction (E-M I), Modus Ponens (MP), Modus Tollens (MT), Hypothetical Syllogism (HS), Constructive Dilemma (CD), Com-mutation (Com), Distribution (Dist), Association (Assoc) Double Negation (DN), DeM-organ (DeM), Transposition (Trans), Exportation (Exp), Equivalence (Equiv), Existential Instantiation (EI), Existential Generalization (EG), Universal Instantiation (UI), Universal Generalization (UG), Quantifi er Negation (QN), Identity Introduction (II), Identity Elimination (IE), Conditional Proof (CP) and Indirect Proof (IP).39
The fi ve Modal Inference Rules of 2QS5 are as follows:
For every substitution instance of ┌p┐ and ┌q┐
NE (necessity elimination)
p / p
MMP (modal modus ponens)
(p ⊃ q), p / q
NI (necessity introduction)
If ┌p┐ is a theorem
then
┌p┐ is a theorem
ME (modal equivalence)
┌◊p┐ for ┌~~p┐ and
┌p┐ for ┌~◊~p┐
PN (possibility necessity)
◊p / ◊p
2QS5 also includes the Principle of Abstraction (Abs) as an axiom schema: (x)(ˆy[Ψy]x
≡ Ψx), where ┌ˆy[Ψy]┐ denotes the property of being a Ψ.40
Appendix 2. Formal Proofs of Some Modal Arguments
The validity of Malcolm’s ontological argument
Let
U1x =df x is unlimited
D1x =df x depends on something else for its existence or nonexistence
Hx =df x happens to exist
39. The quantifi cation rules apply to both fi rst-order and second-order variables, and no pseudo-names are allowed in the last line of a deduction.
40. Theorems and theorem schemata of Q2S5 are sometimes used in the proofs and deductions in this chapter.
It would make this chapter too long to include proofs of them. They are, however, fairly straightforward, and should be fairly common in the literature.
O N TO LO G I C A L
A RG U M E N T 589
Jx =df x happens not to exist
B1x =df x is caused to begin to exist by some other being
C1x =df x is caused to cease to exist by some other being
M1x =df x begins to exist
N1x =df x ceases to exist
Deduction
1 ◊Eg pr41
2 U1g pr
3 (x)(U1x ≡ (~D1x & ~Hx & ~Jx))
pr
4 (x)(~D1x ≡ (~B1x & ~C1x)) pr
5 (x)(M1x ⊃ (B1x ∨ Hx))
pr
6 (x)(N1x ⊃ (C1x ∨ Jx))
pr
7 (x)((~M1x & ~N1x) ⊃ ((Ex ⊃ Ex) & (~Ex ⊃ ~Ex))) pr
8 U1g ≡ (~D1g & ~Hg & ~Jg)
3, UI
9 (~D1g & ~Hg & ~Jg)
2, 8 Equiv, Simp, MP
10 ~D1g ≡ (~B1g & ~C1g) 4
UI
11 ~D1g 9
Simp
12 (~B1g & ~C1g)
10, 11 Equiv, Simp, MP
13 ~Hg
9
Assoc,
Simp
14 ~B1g 12
Simp
15 ~(B1g ∨ Hg)
13, 14 Conj, DeM
16 ~Jg
9
Assoc,
Simp
17 ~C1g 12
Simp
18 ~(C1g ∨ Jg)
16, 17 Conj, DeM
19 M1g ⊃ (B1g ∨ Hg)
5 UI
20 ~M1g
15, 19 MT
21 N1g ⊃ (C1g ∨ Jg)
6 UI
22 ~N1g
18, 21 MT
23 (~M1g & ~N1g) ⊃ ((Eg ⊃ Eg) & (~Eg ⊃ ~Eg)) 7
UI
24 (~M1g & ~N1g)
20, 22 Conj
25 ((Eg
⊃ Eg) & (~Eg ⊃ ~Eg))
23, 24 MP
26 (~Eg
⊃ ~Eg) 25
Simp
27 ~~Eg ⊃ Eg
26 Trans
28 ◊Eg ⊃ Eg
27 ME
29 Eg
1, 28 MP
The validity of Hartshorne’s ontological argument
Let
q =df There is a perfect being
Deduction
41. “Eg” could be replaced by “(∃x)x = g.” The existential quantifi er would then have to have existential import, as it does for both Hartshorne and Plantinga.
1 (q ⊃ q) pr
2 ◊q
pr
3 (q ⊃ q) ⊃ (◊q ⊃ ◊q) theorem
4 (◊q ⊃ ◊q)
1, 3 MP
5 ◊q
2, 4 MP
6 ◊q ⊃ q theorem
7 q 5,6
MP
8 q
7,
NE
The validity of Plantinga’s ontological argument
Let
Ax =df x is maximally great
Bx =df x is maximally excellent
W(Y) =df Y is a universal property
Ox =df x is omniscient, omnipotent, and morally perfect
Deduction
1 ◊(∃x)Ax pr
2 (x)(Ax ≡ Bx) pr
3 (x)(Bx ⊃ Ox)
pr
4 (Y)[W(Y) ≡ ((∃x)Yx ∨ (~(∃x)Yx)] pr
5 (Y)[(∃Z)(x)(Yx ≡ Zx) ⊃ W(Y)]
pr
6 (∃Z)(x)(Ax ≡ Zx) 2,
EG
7 [(∃Z)(x)(Ax ≡ Zx) ⊃ W(A)]
5, UI
8 W(A) ≡ ((∃x)Ax ∨ (~(∃x)Ax) 4,
UI
9 W(A)
6, 7 MP
10 W(A)
⊃ ((∃x)Ax ∨ (~(∃x)Ax)
8, Equiv, Simp
11 (∃x)Ax (~(∃x)Ax)
9, 10 MP
12 ~◊~~(∃x)Ax ∨ ((∃x)Ax)
11, Com, ME
13 ◊(∃x)Ax ⊃ (∃x)Ax DN,
Impl
14 (∃x)Ax
1, 13 MP
15 (x)(Ax ≡ Bx) ⊃ ((∃x)Ax ⊃ (∃x)Bx) theorem
16 (∃x)Bx
14, 15 MP (twice)
17 (x)(Bx ⊃ Ox) ⊃ ((∃x)Bx ⊃ (∃x)Ox) theorem
18 (∃x)Ox
16, 17 MP (twice)
19 (∃x)Ox 18,
NE
Complete proofs of Gödel’s ontological theorems
Proof of Tm 1
1 ~◊(∃y)G1y AIP
2 ~◊(∃y)G1y ⊃ (x)(G1x ⊃ ~G1x) theorem
O N TO LO G I C A L
A RG U M E N T 591
3 (x)(G1x ⊃ ~G1x)
1, 2 MP
4 (x)(ˆy[~G1y]x ≡ ~G1x) Abs,
NI42
5
((x)(G1x ⊃ ~G1x) & (x)(ˆy[~G1y]x ≡ ~G1x)) ⊃
theorem
(x)(G1x ⊃ ˆy[~G1y]x)
6 (x)(G1x ⊃ ˆy[~G1y]x)
3, 4, 5, Conj, MP
7 P1(G1) Ax
343
8 (P
ˆ
1(G1) & (x)(G1x ⊃ ˆ
y[~G1y]x)) ⊃ P1y[~G1y])
Ax 2, UI
9 P ˆ
1y[~G1y]
6, 7, 8, Conj, MP
10 P1(G1) ⊃ ~P1(ˆy[~G1y])
Ax 1, UI, Equiv, Simp
11 ~P1(ˆy[~G1y])
9, 10 MP
12 ◊(∃y)G1y 1–11
IP
Proof of Tm 2
1 G μ
1
& Θμ ACP
2 ~P1(Θ) AIP
3 ~P1(ˆy[~Θy]) ⊃ P1(Θ)
Ax 1, UI, Equiv, Simp
4 P1(ˆy[~Θy])
2, 3, DN, MT
5 P1(ˆy[~Θy]) ⊃ ˆy[~Θy]μ
1, Simp, df “G1”, UI
6 ˆy[~Θy]μ
4, 5 MP
7 ˆy[~Θy]μ ≡ ~Θμ Abs,
UI
8 ~Θμ
6, 7 Equiv, Simp, MP
9 Θμ & ~Θμ
1, 8 Simp, Conj
10 P1(Θ) 2–7
IP
11 (x)(G1x ⊃ (Y)(P1(Y) ⊃ Yx))
theorem, df “G1”
12
(x)(G1x ⊃ (Y)(P1(Y) ⊃ Yx)) ⊃ (x)(Y)(P1(Y) ⊃
theorem
(x)(G1x ⊃ Yx))
13 (x)(Y)(P1(Y) ⊃ (x)(G1x ⊃ Yx))
11, 12 MP
14 P1(Θ) ⊃ (x)(G1x ⊃ Θx) 13
UI
15 P1(Θ) ⊃ P1(Θ)
Ax 4, UI
16 P1(Θ)
10, 15 MP
17 (x)(G1x ⊃ Θx)
14, 16 MP
18 (G μ
1
& Θμ) ⊃ (x)(G1x ⊃ Θx)) 1–17
CP
19 (x)(Z)((G1x & Zx) ⊃ (x)(G1x ⊃ Zx))
18, UG
20
(x)(Z)((G1x & Zx) ⊃ (x)(G1x ⊃ Zx)) ⊃
theorem
(x)(G1x ⊃ (G1x & (Z)(Zx ⊃ (x)(G1x ⊃ Zx))))
21 (x)(G1x ⊃ (G1x & (Z)(Zx ⊃ (x)(G1x ⊃ Zx))))
19, 20, MP
22 (x)(G1x ⊃ E1(G1, x))
21, df “E1”
42. Scott and Sobel omit necessitated Abstraction in their proofs of Tm 1. (Anderson acknowledges that it is implicit.) Their proofs assume that it is necessarily true that self-difference is identical to the negation of self-identity. But howsoever obvious that identity might be, its proof requires necessitated Abstraction: (x)(yˆ[y y
≠ ]x ≡ x
∼ =x)
(x)(yˆ[ y
∼ =y]x ≡ x
∼ =x)
∴ ˆy [y y
≠ ] = ˆy[ y
∼ =y]
43. See section (4a) for a formal expression of Gödel’s axioms.
Proof of Tm 3
1 G μ
1
ACP
2 P1(N2) Ax
5
3 P
μ
1(N2) ⊃ N2
1, UI, df “G1”
4 N μ
2
2, 3 MP
5 E1(G1, μ) ⊃ (∃z)G1z
4, UI, df “N2”
6 G μ
μ
μ
1
⊃ (G1 & (Z)(Zx ⊃ (x)(G1 ⊃ Zμ)))
Tm 2, df “E1”, UI
7
(G μ
μ
μ
1
⊃ (G1 & (Z)(Zx ⊃ (x)(G1 ⊃ Zμ)))) ⊃
theorem
(G μ
μ
1
& (Z)(Zx ⊃ (x)(G1 ⊃ Zμ)))
8 E1(G1, μ)
6, 7 MP, df “E1”
9 (∃z)G1z
5, 8 MP
10 G μ
1
⊃ (∃z)G1z 1–9
CP
11 (x)(G1x ⊃ (∃z)G1z)
10 UG, NI
12 (x)(G1x ⊃ (∃z)G1z) ⊃ (◊(∃y)G1y ⊃ ◊(∃z)G1z) theorem
13 ◊(∃y)G1y ⊃ ◊(∃z)G1z
11, 12 MP
14 ◊(∃z)G1z
13 Tm 1, MP
15 ◊(∃z)G1z ⊃ (∃z)G1z theorem
16 (∃z)G1z
14, 15 MP
17 (∃z)G1z 16
NE
The Argument from Miracles:
A Cumulative Case for
the Resurrection of Jesus
of Nazareth
TIMOTHY MCGREW AND LYDIA MCGREW