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A Slingshot from S4 to S5 establishing the Modal Ontological Argument?

…Or why the “strong” atheologian, i.e. the atheologian who holds that there is no omniscient, omnipotent, and omnibenevolent being, must say that ♢Θ semantically entails ☐Θ in S4.

Θ is the proposition that necessarily there is an omniscient, omnipotent, and omnibenevolent being.  

That is:

Kx ≝ x is omniscient
Px ≝ x is omnipotent
Bx ≝ x is omnibenevolent
Θ ≝ ☐(∃x)[(Kx ∧ Px) ∧ Bx]

Consider the following:

1. It is false that ♢Θ semantically entails ☐Θ in S4.

If that is true, then:

2. There is a world in which the valuation of ♢Θ at that world in S4 is true, and the valuation of ☐Θ at that world in S4 is false.

But this is just to say…

3. ♢♢Θ

That is, there is a world in which it is true that ♢Θ.  Moreover, it is an axiom of S4 that ♢♢p → ♢p, and so:

4. ♢Θ

But given our definition for “Θ”, we can say:

5. ♢☐(∃x)[(Kx ∧ Px) ∧ Bx]

Since S5 is just an extension of S4, if something is possible in S4 it is also possible in S5.  Given that ♢☐p → ☐p is an axiom in S5:

6. ☐(∃x)[(Kx ∧ Px) ∧ Bx]

And since ☐p → p in S5 (axiom M/T), we can conclude:

7. (∃x)[(Kx ∧ Px) ∧ Bx]

Hence, the committed “strong” atheologian must say that ♢Θ semantically entails ☐Θ in S4.  Moreover, since S4 is strongly complete, the atheologian is committed to: 

♢Θ ⊢S4 ☐Θ

I’d like to see that deduction. 

[Update]: One objection that I have encountered is that the move from 5 to 6 seems to switch frameworks from S4 to S5, and so the argument is invalid. The argument does not presume S4 as the framework, but rather attempts to exploit an intuition about what is semantically entailed about ♢Θ in S4. In other words, if you grant that such entailment doesn’t hold in S4, I think it follows that you are committed to ♢♢Θ in S4 and S5, which of course is just to say that you are committed to ♢Θ in S5. So from the framework of S5, and its related axioms, you would have to be committed to Θ.

In an attempt to more clearly show how I am not applying axioms of S5 in S4, here is a more formal representation of the argument. Add to our key, the following:

T ≝ true
F ≝ false
V(ω)M(P) = … the valuation at ω in model M of proposition p equals…

1. (∀p)(∀q)~[p ⊨S4 q] → (∃ω){[V(ω)S4(p) = T] ∧ [V(ω)S4(q) = F] (premise)
2. (∃ω){[V(ω)S4(p) = T] ∧ [V(ω)S4(q) = F] → ⊨S4♢p} (premise)
3. (∀p){⊨S4♢p → (∃ω){[V(ω)S5(p) = T]} (premise)
4. (∀p)(∃ω){[V(ω)S5(p) = T] → ⊨S5♢p} (premise)
5. (∀p)[⊨S5♢♢☐p → ⊢S5☐p] (premise)
6. ~[♢Θ ⊨S4 ☐Θ] (premise)
7. (∀q)~[♢Θ ⊨S4 q] → (∃ω){[V(ω)S4(♢Θ) = T] ∧ [V(ω)S4(q) = F] (1 UI)
8. ~[♢Θ ⊨S4 ☐Θ] → (∃ω){[V(ω)S4(♢Θ) = T] ∧ [V(ω)S4(☐Θ) = F] (7 UI)
9. (∃ω){[V(ω)S4(♢Θ) = T] ∧ [V(ω)S4(☐Θ) = F] (6,8 MP)
10. [V(w)S4(♢Θ) = T] ∧ [V(w)S4(☐Θ) = F (9 EI)
11. [V(w)S4(♢Θ) = T] (10 Simp)
12. (∃ω)S4(♢Θ) = T] (11 EG)
13. (∃ω){[V(ω)S4(♢Θ) = T] → ⊨S4♢♢Θ (2 UI)
14. ⊨S4♢♢Θ (12,13 MP)
15. ⊨S4♢♢Θ → (∃ω){[V(ω)S5(♢Θ) = T] (3 UI)
16.(∃ω){[V(ω)S5(♢♢Θ) = T] → ⊨S5♢♢Θ (4 UI)
17. ⊨S4♢♢Θ → ⊨S5♢♢Θ (15,16 HS)
18. ⊨S5♢♢Θ (14,17 MP)
19. ⊨S5♢♢☐(∃x)[(Kx ∧ Px) ∧ Bx] (18 Def “Θ”)
20. ⊨S5♢♢☐(∃x)[(Kx ∧ Px) ∧ Bx] → ⊢S5☐(∃x)[(Kx ∧ Px) ∧ Bx] ∧ Bx] (5 UI)
21. ⊢S5☐(∃x)[(Kx ∧ Px) ∧ Bx] ∧ Bx](19,20 MP)

The Dilemma Once More

P1. If it is possible that necessarily there is an omniscient, omnipotent, omnibenevolent being, necessarily there is an omniscient, omnipotent, omnibenevolent being. (From axiom 5 of S5)[1]

P2. Either the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being” entails the proposition “there is gratuitous evil and suffering” or it is not the case the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being” entails the proposition “there is gratuitous evil and suffering”. (From the Law of the Excluded Middle)[2]

P3. For all propositions p if there is some proposition q such that it is not the case that p entails q, then possibly p. (Contraposition of the Principle of Explosion)[3][4]

C1. If it is not the case the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being” entails the proposition “there is gratuitous evil and suffering”, it is possible that necessarily there is an omniscient, omnipotent, omnibenevolent being. [From P3][5]

C2. If it is not the case the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being” entails the proposition “there is gratuitous evil and suffering”, necessarily there is an omniscient, omnipotent, omnibenevolent being. [From P1 and C1, Hypothetical Syllogism][6]

P4. If the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being” entails the proposition “there is gratuitous evil and suffering”, gratuitous evil and suffering is not counter-evidence to the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being”.[7]

C3. Either necessarily there is an omniscient, omnipotent, omnibenevolent being, or gratuitous evil and suffering is not counter-evidence to the proposition “necessarily there is an omniscient, omnipotent, omnibenevolent being.” (From P2,C2,P4 Constructive Dilemma)[8][9]

[1] The axiom in S5 can be found here: https://en.m.wikipedia.org/wiki/S5_(modal_logic). So, given the axiom 5 of S5: ♢p → ☐♢p

Here is the proof for P1:

Let

Kx ≝ x is omniscient
Px ≝ x is omnipotent
Bx ≝ x is omnibenevolent

1 ~ ☐(∃x)[(Kx ∧ Px) ∧ Bx] (Assump. CP)
2 ~ ☐~~(∃x)[(Kx ∧ Px) ∧ Bx] (1 DN)
3 ♢~(∃x)[(Kx ∧ Px) ∧ Bx] (2 ME)
4 ☐♢~(∃x)[(Kx ∧ Px) ∧ Bx] (3 Axiom 5)
5 ☐~~♢~(∃x)[(Kx ∧ Px) ∧ Bx] (4 DN)
6 ☐~☐(∃x)[(Kx ∧ Px) ∧ Bx] (5 ME)
7 ~☐(∃x)[(Kx ∧ Px) ∧ Bx] → ☐~☐(∃x)[(Kx ∧ Px) ∧ Bx] (CP 1-6)
8 ~☐~☐(∃x)[(Kx ∧ Px) ∧ Bx] → ~~☐(∃x)[(Kx ∧ Px) ∧ Bx] (7 Contra)
9 ~☐~☐(∃x)[(Kx ∧ Px) ∧ Bx] → ☐(∃x)[(Kx ∧ Px) ∧ Bx] (8 DN)
10 ♢☐(∃x)[(Kx ∧ Px) ∧ Bx] → ☐(∃x)[(Kx ∧ Px) ∧ Bx] (9 ME)

[2] The Law of the Excluded Middle can be found here: https://en.m.wikipedia.org/wiki/Law_of_excluded_middle

[3] Contraposition can be found here: https://en.m.wikipedia.org/wiki/Contraposition

[4] The Principle of Explosion can be found here: https://en.m.wikipedia.org/wiki/Principle_of_explosion

Here is the proof that P3 is the contrapositive of the Principle of Explosion, which we will state as follows: (∀p)[~♢p → (∀q)(p ⊨ q)], for all propositions p, if p is impossible, then for all propositions q1, p entails q.

1 (∀p)[~♢p → (∀q)(p ⊨ q)] (Principle of Explosion)
2 ~♢φ → (∀q)(φ ⊨ q) (1 UI)
3 ~(∀q)(φ ⊨ q) → ~~♢φ (2 Contra)
4 (∃q)~(φ ⊨ q) → ~~♢φ (3 QN)
5 (∃q)~(φ ⊨ q) → ♢φ (4 DN)
6 (∀p)(∃q)~(p ⊨ q) → ♢p] (5 UG)

[5] Here is the proof that C1 follows from P3:

Let

G ≝ ☐(∃x)[(Kx ∧ Px) ∧ Bx]
E ≝ ‘there is gratuitous evil and suffering’

1 (∀p)(∃q)~(p ⊨ q) → ♢p] (P3)
2 ~(G ⊨ E) (Assump. CP)
3 (∃q)~(G ⊨ q) → ♢G (1 UI)
4 (∃q)~(G ⊨ q) (2 EG)
5 ♢G (3,4 MP)
6 ~(G ⊨ E) → ♢G (205 CP)
7 ~(G ⊨ E) → ♢☐(∃x)[(Kx ∧ Px) ∧ Bx] (6 def. of ‘G’)

Thus Line 7 (C1) follows from Line 1 (P3), QED.

[6] Hypothetical Syllogism can be found here: https://en.m.wikipedia.org/wiki/Hypothetical_syllogism

[7] This premise is defended on given a Bayesian interpretation of counter-evidence:
(∀p)(∀q){[P(p|q)<P(p)] ⊃ Cqp} (read as: for all proposition p and q, if the probability of q given p is less than the probability of q unconditioned, then q is counter-evidence for p).

If we assume G ⊨ E, then by Logical Consequence P(E|G) = 1, but if E is counter-evidence to G, then it must be the case that P(G|E) < P(G). But both of these statements about probabilities cannot be true.

According to Bayes’ Theorem:

P(E|G) = [P(E)/P(G)] x P(G|E)

So given P(E|G) = 1

We can infer:

P(G)/P(G|E) = P(E)

But given 0 ≤ P(E) ≤ 1, it is not possible for P(G)/P(G|E) = P(E) and P(G|E) < P(G), as whenever the denominator is less than the numerator, the result is greater than 1.

Hence, we must reject the assumption that [P(E|G) = 1] ∧ [P(G|E) < P(G)]. This provides us with the following defense of P4:

1 ~{[P(E|G) = 1] ∧ [P(G|E) < P(G)]} (Result from the proof by contradiction above)
2 ~[P(E|G) = 1] ∨ ~[P(G|E) < P(G)] (1 DeM)
3 [P(E|G) = 1] → ~[P(G|E) < P(G)] (2 Impl)
4 [G ⊨ E] → [P(E|G) = 1] (by Logical Consequence)
5 [G ⊨ E] → ~[P(G|E) < P(G)] (3,4 HS)

And line 5 is just what is meant by P4.

[8] Constructive Dilemma can be found here: https://en.m.wikipedia.org/wiki/Constructive_dilemma

[9] The proof of the entire argument is as follows:

1 ♢☐(∃x)[(Kx ∧ Px) ∧ Bx] → ☐(∃x)[(Kx ∧ Px) ∧ Bx] (Premise)
2 (G ⊨ E) ∨ ~(G ⊨ E) (Premise)
3 (∀p)(∃q)~(p ⊨ q) → ♢p] (Premise)
4 [G ⊨ E] → ~[P(G|E) < P(G)] (Premise)
5 ~(G ⊨ E) (Assump CP)
6 (∃q)~(G ⊨ q) → ♢G (3 UI)
7 (∃q)~(G ⊨ q) (5 EG)
8 ♢G (6,7 MP)
9 ~(G ⊨ E) → ♢G (5-8 CP)
10 ~(G ⊨ E) → ♢☐(∃x)[(Kx ∧ Px) ∧ Bx] (9 definition of ‘G’)
11 ~(G ⊨ E) → ☐(∃x)[(Kx ∧ Px) ∧ Bx] (1,10 HS)
12 ☐(∃x)[(Kx ∧ Px) ∧ Bx] ∨ ~[P(G|E) < P(G)] (2,4,11 CD)

The Cartesian Ontological Argument

D1. God is the x such that for all attributes Y, if Y is a perfection, Y belongs to x.
P1. Necessarily existing is a perfection
P2. For all x, if it is not the case that x exists, possibly it is not the case that x exists.
C. God exists.

Let,

E!x ≝ x exists
P(Y)≝ Y is a perfection

g ≝ (ɿx)(∀Y)(P(Y)⊃ Yx)

1. P(☐E!) (premise)
2. (∀x)[~E!x ⊃ ♢~E!x] (premise)
3. ~E!g (IP)
4. (∃x){[(∀Y)(P(Y) ⊃ Yx) ∧ (∀y)[(∀Y)(P(Y) ⊃ Yy) ⊃ (y = x)]] ∧ ~E!x} (3 theory of descriptions)
5. [(∀Y)(P(Y)⊃ Yμ) ∧ (∀y)[(∀Y)(P(Y)⊃ Yy) ⊃ (y = μ)]] ∧ ~E!μ (4 EI)
6. ~E!μ ⊃ ♢~E!μ (2 UI)
7. ~E!μ (5 Simp)
8. ♢~E!μ (6,7 MP)
9. ~☐E!μ (8 MN)
10. (∀Y)(P(Y) ⊃ Yμ) ∧ (∀y)[(∀Y)(P(Y) ⊃ Yy) ⊃ (y = μ)] (5 Simp)
11. (∀Y)(P(Y) ⊃ Yμ) (10 Simp)
12. P(☐E!) ⊃ ☐E!μ (11 UI)
13. ☐E!μ (1,12 MP)
14. ☐E!μ ∧ ~☐E!μ (9,13 Conj)
15. E!g (3-14 IP)

In the definition, I am just setting down what I take g to mean… all I mean by g is that it is the something that, for any attribute, if that attribute is a perfection, then it has that perfect attribute. So God is the being that has all perfections (as I define God).  You might say that there is no definite description of a perfect being, i.e. there could be multiple perfect beings.  However, I would argue that there cannot be two omnipotent beings, since a simple reductio would rule out this possibility.  That is, if there are two omnipotent beings, then any power the one has would be limited by whether or not the other being wills to bring about a contradictory state of affairs.  Since they cannot both bring about contradictory states of affairs, they cannot both be omnipotent.  So there cannot be two beings that have all perfections, given that omnipotence is a perfection that implies uniqueness.  Descartes says that his idea of a supremely perfect being is clear and distinct, which in turn justifies is appeal to the definition (even incoherent stipulated definitions can be rejected).  Leibniz famously demanded a more rigorous proof that the definition is coherent, and sought to prove all perfections cohere.  I think it is a mistake to then interpret Leibniz’s ontological argument in terms of using God’s possibility to infer his necessary existence via S5 in modal logic.  Rather, I think he is doing what Descartes is doing, namely trying to show that the definition of God is self-consistent.

Leibniz’s proof for the self-consistency of the concept of a supremely perfect being is through an analysis of a perfection, which he says is simple, positive, and unlimited.  If any two perfections are inconsistent, one of them would have to be negative, or contain a part that is negative.  But a perfection cannot, by definition, be negative, or contain parts.  So any two perfections can cohere.  Leibniz reasons that if this is so, then all perfections cohere, and so a being that has all perfections is coherent.

P1 say necessary existence is a perfection because a perfection is any attribute that is of a simple kind that is positively complete. Omniscience is a perfection of because it is an attribute of the simple kind (knowledge) that is positively complete. Whatever has omniscience lacks nothing with respect to knowledge. So we recognize omniscience as a kind of perfection regarding knowledge. So necessary existence is an attribute regarding the simple kind “modes of existence” that is positively complete. Whatever exists necessarily exists in all possible situations, so it does not lack positive existence given any other state of affairs.

P2 is axiomatically true given that if something is necessarily true (system M of modal logic), then it is true. Assume P2 is false: ~(~E!x ⊃ ♢~E!x), this is logically equivalent to saying ~E!x ∧ ☐E!x (x does not exist and necessarily x exists). Given system M, ☐E!x implies E!x, so P2 cannot be false. In order to object to P2, you would have to say that some necessary truths are not actually true, which I think is a rather absurd position to take.

A Possible Interpretation of Proslogion 2

One of my struggles in trying to understand Proslogion 2 is how Anselm gets to the actual existence of God rather than what he arrives at in Proslogion 3, namely the inconceivability of God’s non-existence.  I’ve also struggled with the notion of using a two-place predicate like “greater than”, since Anselm tells us that if God exists in the mind alone, a greater could be conceived, i.e. to think of God as existing in reality.  Here, we are saying that we could conceive of one and the same concept in greater ways rather than conducting a comparison of the God concept to other items in the world.  The following interpretation approximates what Anselm seems to be arguing, and I would say that it is a sound argument for God’s existence.

D1. God is defined as that which cannot be conceived to admit of more greatness.
P1. For all x, if x exists in intelletu and not in re, then it can be conceived that x exists in intellectu and not in re.
P2. For all x, if it can be conceived that x exists in intellectu and not in re, then it can be conceived that x exists in intellectu and in re.
P3. For all x, if it can be conceived that x exists in intellectu and not in re and it can be conceived that x exists in intellectu and in re, then it is conceivable that x admits of more greatness.
P4. God exists in intellectu.
C. Therefore, God exists in re.

Let,

E!x ≝ x exists in re
Ix ≝ x exists in intellectu
Gx ≝ x admits of more greatness
©… ≝ it is conceivable that…

g ≝ (ɿx)~©Gx

1. (∀x)[(Ix ∧ ~E!x) ⊃ ©(Ix ∧~E!x)] (premise)
2. (∀x)[©(Ix ∧ ~E!x) ⊃ ©(Ix ∧ E!x)] (premise)
3. (∀x){[©(Ix ∧ ~E!x) ∧ ©(Ix ∧ E!x)] ⊃ ©Gx} (premise)
4. Ig (premise)
5. ~E!g (IP)
6. Ig ∧ ~E!g (4,5 Conj)
7. (Ig ∧ ~E!g) ⊃ ©(Ig ∧~E!g) (1 UI)
8. ©(Ig ∧~E!g) (6,7 MP)
9. ©(Ig ∧ ~E!g) ⊃ ©(Ig ∧ E!g) (2 UI)
10. ©(Ig ∧ E!g) (8,9 MP)
11. ©(Ig ∧~E!g) ∧ ©(Ig ∧ E!g) (8,10 Conj)
12. ©(Ig ∧ ~E!g) ∧ ©(Ig ∧ E!g)] ⊃ ©Gg (3 UI)
13. ©Gg (11,12 MP)
14. (∃x){{~©Gx ∧ (∀y)[~©Gy ⊃ (y = x)]} ∧ ©Gx} (13 theory of descriptions)
15. {~©Gμ ∧ (∀y)[~©Gy ⊃ (y = μ)]} ∧ ©Gμ (14 EI)
16. {(∀y)[~©Gy ⊃ (y = μ)] ∧ ~©Gμ} ∧ ©Gμ (15 Comm)
17. (∀y)[~©Gy ⊃ (y = μ)] ∧ {~©Gμ ∧ ©Gμ} (16 Assoc)
18. ~©Gμ ∧ ©Gμ (17 Simp)
19. E!g (5-18 IP)

QED

[Edit: My friend, Matt, thinks my argument may be susceptible to parody.  Here is my response]

Generally, I think parodies fail because such supposed objects, like islands of which none greater can be conceived, do not really exist in the intellect for the very same reason round squares are not abstract objects in the mind.  The phrase is nonesense, and so does not pick out any object of the understanding.

Islands just are the sorts of things that admit of degrees of greatness, so are other objects used in parody. For example, islands are present in a specified location that is surrounded by water, but it is unclear how big an island should be when considering its greatness.  It certainly cannot be omnipresent and be an island.  How many trees, island beauties, or sandy beaches ought there to be on the island which cannot be conceivably greater?  

My argument can motivate this response by proving that the greatest conceivable island is not an object that exists in the intellect.  This is because specifying that there is an island than which none greater can be conceived leads to the conclusion that God is an island, and that seems like a good reductio of the idea such a concept can be conceived.

So, if we grant the parody, I could prove that island can be predicated of God, or a being than which a greater cannot be conceived. But since islands are essentially contingent and admit of degrees of greatness, island cannot be a predicate of God, who is the being than which none greater can be conceived. So, we must reject the assumption that a greatest conceivable island exists in intellectu and we can base it on the somewhat reasonable premise that God is not an island. I would argue as follows:

Let,

Lx ≝ x is an island

i ≝ (ɿx)(~©Gx ∧ Lx)

20. ~Lg (premise)
21. (∃x){{~©Gx ∧ (∀y)[~©Gy ⊃ (y = x)]} ∧ E!x} (19 theory of descriptions)
22. Ii (IP)
23. (∃x){{(~©Gx ∧ Lx) ∧ (∀y)[(~©Gy ∧ Ly) ⊃ (y = x)]} ∧ Ix} (22 theory of descriptions)
24. {~©Gμ ∧ (∀y)[~©Gy ⊃ (y = μ)]} ∧ E!μ (21 EI)
25. {(~©Gν ∧ Lν) ∧ (∀y)[(~©Gy ∧ Ly) ⊃ (y = ν)]} ∧ Iν (23 ΕΙ)
26. ~©Gμ ∧ (∀y)[~©Gy ⊃ (y = μ)] (24 Simp)
27. (∀y)[~©Gy ⊃ (y = μ)] (26 Simp)
28. (~©Gν ∧ Lν) ∧ (∀y)[(~©Gy ∧ Ly) ⊃ (y = ν)] (25 Simp)
29. ~©Gν ∧ Lν (28 Simp)
30. ~©Gν (29 Simp)
31. ~©Gν ⊃ (ν = μ) (27 UI)
32. ν = μ (30,31 MP)
33. ~©Gμ ∧ Lμ (29,32 ID)
34. (~©Gμ ∧ Lμ) ∧ (∀y)[~©Gy ⊃ (y = μ)] (27,33 Conj)
35. ~©Gμ ∧ {Lμ ∧ (∀y)[~©Gy ⊃ (y = μ)]} (34 Assoc)
36. ~©Gμ ∧ {(∀y)[~©Gy ⊃ (y = μ)] ∧ Lμ} (35 Comm)
37. {~©Gμ ∧ {(∀y)[~©Gy ⊃ (y = μ)]} ∧ Lμ (36 Assoc)
38. (∃x){{~©Gx ∧ {(∀y)[~©Gy ⊃ (y = x)]} ∧ Lx} (37 EG)
39. Lg (38 theory of descriptions)
40. ~Lg ∧ Lg
41. ~Ii (22-40 IP)

So as long as you can provide the premise that God is not an island, not a pizza, etc. the proof works to show that such objects really are not in the intellect.

A Remix of Anselm’s Conceptual Ontological Argument

st-20anselm20weninger

D1. God is defined as the x such that there is not something, y, where y is conceivably greater than x.
P1. For all x, if x is conceivable, then there is something, y, such that either y is identical to x and y exists or there is something, z, such that z is identical to x, z does not exist, and y is conceivably greater than z.
P2. There is some x such that x is conceivable and it is not the case that there is some y such that y is conceivably greater than x.
P3. For all x and y, either x is conceivably greater than y or y is conceivably greater than x, or if it is not the case that either x is conceivably greater than y or that y is conceivably greater than x, there is some z such that z is the mereological sum of x and y, and either z is conceivably greater than x or z is conceivably greater than y.
C. God exists.1

E!x ≝ x exists
Cx ≝ x is conceivable
Gxy ≝ x is conceivably greater than y
σ<x,y> ≝ the mereological sum of x and y
g ≝ (ɿx)~(∃y)Gyx

1. (∀x){Cx ⊃ (∃y){[(y = x) ∧ E!y] ∨ (∃z)[(z = x) ∧ (~E!z ∧ Gyz)]}} (premise)
2. (∃x)(Cx ∧ ~(∃y)Gyx) (premise)
3. (∀x)(∀y){[Gxy ∨ Gyx] ∨ {~(Gxy ∨ Gyx) ⊃ (∃z)[(z = σ<x,y>) ∧ (Gzx ∨ Gzy)]}} (premise)
4. Cμ ∧ ~(∃y)Gyμ (2 EI)
5. ~(∃y)Gyμ (4 Simp)
6. (∃z)[~(∃z1)Gz1z ∧ ~(z = μ)] (IP)
7. ~(∃z1)Gz1ν ∧ ~(ν = μ) (6 EI)
8. (∀y){[Gνy ∨ Gyν] ∨ {~(Gνy ∨ Gyν) ⊃ (∃z)[(z = σ<ν,y>) ∧ (Gzν ∨ Gzy)]}} (3 UI)
9. [Gνμ ∨ Gμν] ∨ {~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)]} (8 UI)
10. (∀y)~Gyμ (5 QN)
11. ~Gνμ (10 UI)
12. ~(∃z1)Gz1ν (7 Simp)
13. (∀z1)~Gz1ν (12 QN)
14. ~Gμν (13 UI)
15. Gνμ ∨ [Gμν ∨ {~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)]}] (9 Assoc)
16. Gμν ∨ {~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)]} (11,15 DS)
17. ~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)] (14,16 DS)
18. ~Gνμ ∧ ~Gμν (11,14 Conj)
19. ~(Gνμ ∨ Gμν) (18 DeM)
20. (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)] (17,19 MP)
21. (ζ = σ<ν,μ>) ∧ (Gζν ∨ Gζμ) (20 EI)
22. Gζν ∨ Gζμ (21 Simp)
23. ~Gζμ (10 UI)
24. Gζν (22,23 DS)
25. ~Gζν (13 UI)
26. Gζν ∧ ~Gζν (24,25 Conj)
24. ~(∃z)[~(∃z1)Gz1z ∧ ~(z = μ)] (6-23 IP)
25. (∀z)~[~(∃z1)Gz1z ∧ ~(z = μ)] (24 QN)
26. (∀z)[~~(∃z1)Gz1z ∨ ~~(z = μ)] (25 DeM)
27. (∀z)[~(∃z1)Gz1z ⊃ ~~(z = μ)] (26 Impl)
28. (∀z)[~(∃z1)Gz1z ⊃ (z = μ)] (27 DN)
29. {Cμ ∧ ~(∃y)Gyμ} ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = μ)] (4,28 Conj)
30. Cμ ∧ {~(∃y)Gyμ ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = μ)]} (29 Assoc)
31. {~(∃y)Gyμ ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = μ)]} ∧ Cμ (30 Comm)
32. (∃x){~(∃y)Gyx ∧ (∀z)[~(∃z1)Gz1z ⊃ (z =x)]} ∧ Cx} (31 EG)
33. Cg (32 theory of descriptions)
34. Cg ⊃ (∃y){[(y = g) ∧ E!y] ∨ (∃z)[(z = g) ∧ (~E!z ∧ Gyz)]} (1 UI)
35. (∃y){[(y = g) ∧ E!y] ∨ (∃z)[(z = g) ∧ (~E!z ∧ Gyz)]} (33,34 MP)
36. [(ξ = g) ∧ E!ξ] ∨ (∃z)[(z = g) ∧ (~E!z ∧ Gξz)] (35 EI)
37. (∃z)[(z = g) ∧ (~E!z ∧ Gξz)] (IP)
38. (ν = g) ∧ (~E!ν ∧ Gξν) (37 EI)
39. ~E!ν ∧ Gξν (38 Simp)
40. Gξν (39 Simp)
41. (ν = g) (38 Simp)
42. Gξg (40,41 ID)
43. (∃x){~(∃y)Gyx ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = x)]} ∧ Gξx} (42 theory of descriptions)
44. {~(∃y)Gyζ ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = ζ)]} ∧ Gξζ (43 EI)
45. ~(∃y)Gyζ ∧ (∀z)[~(∃z1)Gz1z ⊃ (z = ζ)](44 Simp)
46. ~(∃y)Gyζ (45 Simp)
47. (∀y)~Gyζ (46 QN)
48. ~Gξζ (47 UI)
49. Gξζ (44 Simp)
50. Gξζ ∧ ~Gξζ (48,49 Conj)
51. ~(∃z)[(z = g) ∧ (~E!z ∧ Gξz)] (37-50 IP)
52. (ξ = g) ∧ E!ξ (36,51 DS)
53. (ξ = g) (52 Simp)
54. E!ξ (52 Simp)
55. E!g (53,54 ID)

QED

1 Some aspects of this argument are influenced by Oppenheimer & Zalta (1991), i.e. the existential quantifier carries no existential import and is analogous to Anselm’s existence in intellectu whereas E! is a predicate that indicates existence in re. One weakness of Oppenheimer & Zalta’s argument is that it depends on a non-logical axiom regarding Gxy such that it is connected. In other words, either Gxy or Gyx or (x = y). This requires all individuals to stand in a greater than relationship. It is plausible, though, that two non-identical individuals could share equal greatness. I am able to derive the uniqueness of the being than which none greater can be conceived by appealing to the notion that the merelogical composite of two equally great individuals is at least greater than one of its proper parts, which I take to be a modest premise. The interesting thing about my formulation is the first premise, which distinguishes in intellectu from in re existence, and captures Anselm’s claim that a greater could be conceived than a being that exists in the understanding alone without begging the question that this greater thing actually exists—it is merely conceptually greater.  See P.E Oppenheimer & E.N. Zalta. (1991). “On the Logic of the Ontological Argument.” In Philosophical Perspectives. Vol. 5. 509-529.

An Argument based on Maydole’s Interpretation of Proslogion 2

Robert Maydole uses definite descriptions and Russell’s theory of descriptions to explicate Anselm’s first ontological argument in Proslogion 2.  I like the idea of using definite descriptions in the argument, and broadly agree with Maydole that Anselm intends to treat “that than which none greater can be conceived” as a definite description.  I do have some issues with Maydole’s formulation, however.  1) I think of Anselm’s argument as a reductio, but that isn’t how Maydole formulates it, 2) there are extra premises in Maydole’s formulation that are ultimately unnecessary, in my opinion, e.g. his seventh premise below 3) there is a typological error’s in Maydole’s argument, which is a minor quibble, but this seems to be a common problem with Maydole’s arguments in the Blackwell Companion to Natural Theology. It doesn’t appear that the editors proofed his arguments very well, to be honest. This is not to say that Maydole’s arguments are not ingenuiously formulated.

Maydole’s argument is formulated as follows:

Ux ≝ x is understood
Sy ≝ the concept of y exists-in-the-understanding
Ex ≝ x exists-in-reality
Gxy ≝ x is greater than y
Fxy ≝ x refers to y
Dx ≝ x is a definite description
d ≝ the definite description “(ɿx) ~©(∃y)Gyx”
g ≝ (ɿx)~©(∃y)Gyx
P(Y) ≝ Y is a great-making property
©… ≝ it is conceivable that…

Here then is our logical reconstruction of Anselm’s ontological argument:

A1 The defi nite description “that than which it is not conceivable for something to be greater” is understood. (Premise)

A2 “That than which it is not conceivable for something to be greater” refers to that than which it is not conceivable for something to be greater. (Premise)

A3 The concept of whatever a defi nite description that is understood refers to has existence-in-the-understanding. (Premise)

A4 It is conceivable that something is greater than anything that lacks a great-making property that it conceivably has. (Premise)

A5 Existence-in-reality is a great making property. (Premise)

A6 Anything the concept of which has existence-in-the-understanding conceivably has existence-in-reality. (Premise)

A7 It is not conceivable that something is greater than that than which it is not conceivable for something to be greater. (Premise)

Therefore,

A8 That than which it is not conceivable for something to be greater exists-in-reality.

The following deduction proves that this argument is valid:

Deduction

1. Dd & Ud pr
2. Fdg pr
3. (x)(y)((Dx & Fxy & Ux) ⊃ Sy) pr
4. (x1)(Y)[(P(Y) & ~Yx1 & ©Yx1) ⊃ ©(∃x2)Gx2x1] pr
5. P(E) pr
6. (x)(Sx ⊃ ©Ex) pr
7. ~©(∃y)Gyg pr
8. Fdg & ~©(∃y)Gyg 2, 7 Conj
9. (∃x)[~©(∃y)Gyx & (z)(~©(∃y)Gyx ⊃ z=x) & (Fdx & ~©(∃y)Gyx)] 8, theory of descriptions1
10. ~©(∃y)Gyν & (z)(~©(∃y)Gyz ⊃ z=ν) & (Fdν & ~©(∃y)Gyν) 9, EI
11. ~©(∃y)Gyν 10, Simp
12. Fdν 10, Simp
13. (P(E) & ~Eν & ©Eν) ⊃ ©(∃x2)Gx2ν 4 UI
14. (Dd & Fdν & Ud) ⊃ Sν 3 UI
15. (Dd & Fdν & Ud) 1, 12, Simp, Conj
16. Sν 14, 15 MP
17. Sν ⊃ ©Eν 6, UI
18. ©Eν 16, 17 MP
19. ~(P(E) & ~Eν & ©Eν) 13, 11 MT
20. ~((P(E) & ©Eν) & ~Eν) 19 Com, Assoc
21. ~(P(E) & ©Eν) ∨ ~~Eν) 20, DeM
22. P(E) & ©Eν 5, 18 Conj
23. Eν 21, 22, DS, DN
24. ~©(∃y)Gyν & (z)(~©(∃y)Gyx) ⊃ z=ν) 10 Simp
25. ~©(∃y)Gyν & (z)(~©(∃y)Gyx) ⊃ z=ν) & Eν 23, 24 Conj
26. (∃x)[~©(∃y)Gyx & (z)(~©(∃y)Gyx) ⊃ z=x) & Ex] 25 EG
27. Eg 26, theory of descriptions
(Maydole 2012, 555-557).

My version is adapted from Maydole and runs this way:

P1. Possibly, God, the x such that there is not some y such that y conceivably has greater capacities, exists in the understanding.

P2. For all x, if possibly x exists in the understanding, it is conceivable that x exists in reality.

P3. For all x, if it is not the case that x exists in reality, and x can exist in the understanding such that it is conceivable that x exists in reality, then there is some y such that y is the proposition “x exists in reality” and there is some z such that y refers to z, z can exist in the understanding and z conceivably has greater capacities than x.

C1. The x such that there is not some y such that y conceivably has greater capacities than x, i.e. God, exists in reality.

The formal deduction is as follows, let:

Cx ≝ it is conceivable that x exists in reality
Ix ≝ x exists in intellectu
Rx ≝ x exists in re
Fxy ≝ x refers to y
Gxy ≝ x conceivably has greater capacities than y
g ≝ (ɿx)~(∃y)Gyx

1. ♢Ig (premise)
2. (∀x)[♢Ix ⊃ Cx] (premise)
3. (∀x){[~Rx & (♢Ix &Cx)] ⊃ (∃y)[(y = ⌜Rx⌝) & (∃z)((Fyz &♢Iz) & Gzx)]} (premise)
4.♢Ig ⊃ Cg(2 UI)
5.♢Ig ⊃ (♢Ig & Cg) (4 Exp)
6.♢Ig & Cg (1,5 MP)
7. ~Rg (IP)
8. ~Rg & (♢Ig & Cg) (6,7 Conj)
9. [~Rg & (♢Ig & Cg)] ⊃ (∃y)[(y = ⌜Rg⌝) & (∃z)((Fyz & ♢Iz) & Gzg)](3 UI)
10. (∃y)[(y = ⌜Rg⌝) & (∃z)((Fyz & ♢Iz) &Gzg)] (8,9 MP)
11. (μ = ⌜Rg⌝) & (∃z)((Fμz & ♢Iz) & Gzg) (10 EI)
12. (Fμν &♢Iν) & Gνg (11 EI)
13. Gνg (12 Simp)
14. (∃y)Gyg (13 EG)
15. (∃x){[~(∃y)Gyx & (∀z)(~(∃y)Gyz ⊃ (z = x))] & (∃y)Gyx} (14 theory of descriptions)
16. [~(∃y)Gyμ & (∀z)(~(∃y)Gyz ⊃ (z =μ))] & (∃y)Gyμ (15 EI)
17. ~(∃y)Gyμ & (∀z)(~(∃y)Gyz ⊃ (z =μ)) (16 Simp)
18. ~(∃y)Gyμ (17 Simp)
19. (∃y)Gyμ (16 Simp)
20. (∃y)Gyμ & ~(∃y)Gyμ (18,19 Conj)
21. ~~Rg (7-20 IP)
22. Rg (21 DN)

QED

1This line has an error and should be: (∃x)[~©(∃y)Gyx & (z)(~©(∃y)Gyz ⊃ z=x) & (Fdx & ~©(∃y)Gyx)

Reference:
Maydole R. 2012. “The Ontological Argument”. In The Blackwell Companion to Natural Theology. Ed. W.L. Craig & J.P. Moreland. Malden, MA: Blackwell Publishing, pp. 555-557.

A Modest Formulation of the Ontological Argument

In this post, I have formulated Anselm’s argument for the necessary existence of a being than which none greater can be conceived.  However, I have noted that the argument depends upon a two-place “greater than” predicate that functions with something like the Neo-Platonic “Great Chain of Being” in mind.  Some thing, x, is conceived to be greater than y in the sense that x is understood to have more capacities or has an essence that can be actualized to a greater degree. For example, a plant is understood to contingently exists, grows, takes in nutrients, and reproduces. An animal is understood to be greater in the sense that it too contingently exists, grows, takes in nutrients, and reproduces, but it also has capacities like sentience, and can self-move, etc. So the greater something is, the more powers/more capacities it is understood to have. If God exists, then God would be that being which none more powerful could be conceived, which is just to say “none greater”. I find the metaphysics where a two-place “conceivably greater than” predicate can be objectively exemplified to be extremely plausible. There is an objective sense in which I have greater capacities and abilities than a flea.

The argument is as follows:

D1. Some x is an Anselmian God if and only if x is conceivable, it is not the case that there is something that is conceivably greater than x, and x necessarily exists.

P1. There is some x conceivable such that there is nothing conceivably greater than x.

P2. For all x, if the possibility of failing to conceive of x implies the possibility that x doesn’t exist, x is mentally dependent (premise).

P3. For all x, if x is mentally dependent, there is some y such that y is conceivably greater than x (premise).

P4. If there is some x such that necessarily there is some z and z is identical to x, and x is an Anselmian God, then necessarily there exists an Anselmian God.

Therefore,

C1. Necessarily, there is an Anselmian God.

That is the argument in ordinary language. To show that it is a formally valid syllogism, I offer the following formal deduction:

Let,

Cx ≝ x is conceived
Mx ≝ x is mentally dependent
Gxy ≝ x is conceived to be greater than y
Θx ≝ (∃x){[♢Cx & ~(∃y)♢Gyx]& ☐(∃z)(z=x)} (Def Θx)

1. (∃x)[♢Cx & ~(∃y)♢Gyx] (premise)
2. (∀x){[♢~Cx ⊃ ♢~(∃z)(z=x)] ⊃ Mx} (premise)
3. (∀x)[Mx ⊃ (∃y)♢Gyx] (premise)
4. (∃x)[☐(∃z)(z=x)& Θx] ⊃ ☐(∃x)Θx (premise)
5. (∀x){[♢Cx & ~(∃y)♢Gyx] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (IP)
6. ♢Cμ & ~(∃y)♢Gyμ (1 EI)
7. [♢~Cμ ⊃ ♢~(∃z)(z=μ)] ⊃ Mμ (2 UI)
8. Mμ ⊃ (∃y)(♢Gyμ) (3 UI)
9. [♢~Cμ ⊃ ♢~(∃z)(z=μ)] ⊃ (♢Gyμ)(7,8 HS)
10. ♢Cμ & ~(∃y)♢Gyμ] ⊃ [♢~Cμ ⊃ ♢~(∃z)(z=μ)] (5 UI)
11. ♢~Cμ ⊃ ♢~(∃z)(z=μ) (6,10 MP)
12. (∃y)♢Gyμ (7,9 MP)
13. ♢Gνμ (12 EI)
14. ~(∃y)♢Gyμ (6 Simp)
15. (∀y)~(♢Gyμ) (14 QN)
16. ~♢Gνμ (15 UI)
17. ♢Gνμ & ~♢Gνμ (13,16 Conj)
18. ~(∀x){[♢Cx & ~(∃y)♢Gyx] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (5-17 IP)
19. (∃x)~{[♢Cx & ~(∃y)♢Gyx] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (18 QN)
20. (∃x) ~{~[♢Cx & ~(∃y)♢Gyx] ∨ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (19 Impl)
21. (∃x){~~[♢Cx & ~(∃y)♢Gyx] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (20 DeM)
22. (∃x){[♢Cx & ~(∃y)♢Gyx] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (21 DN)
23. (∃x){[♢Cx & ~(∃y)♢Gyx] & ~[~♢~Cx ∨ ♢~(∃z)(z=x)]} (22 Impl)
24. (∃x){[♢Cx & ~(∃y)♢Gyx] & ~[☐Cx ∨ ♢~(∃z)(z=x)]} (23 ME)
25. (∃x){[♢Cx & ~(∃y)♢Gyx] & [~☐Cx & ~♢~(∃z)(z=x)]} (24 DeM)
26. (∃x){[♢Cx & ~(∃y)♢Gyx] & [~☐Cx & ☐(∃z)(z=x)]} (25 ME)
27. [♢Cμ & ~(∃y)♢Gyμ] & [~☐Cμ & ☐(∃z)(z=μ)] (26 EI)
28. ~☐Cμ & ☐(∃z)(z=μ) (27 Simp)
29. ☐(∃z)(z=μ) (28 Simp)
30. [♢Cμ & ~(∃y)♢Gyμ] (27 Simp)
31. [♢Cμ & ~(∃y)♢Gyμ] & ☐(∃z)(z=μ) (29,30 Conj)
32. Θμ (31 Def “Θx”)
33. ☐(∃z)(z=μ) & Θμ (29,32 Conj)
34 (∃x)[☐(∃z)(z=x) & Θx] (33 EG)
35. ☐(∃x)Θx (4,34 MP)

QED

Indeed, I find the above argument very persuasive. However, there may be some who are resistant to the notion that the two-place “conceivably greater-than” predicate can actually and objectively be exemplified. For such a person, I propose a more modest version of the argument. The more modest version is that, since C1, i.e. “☐(∃x)Θx”, is provable given P1-P4,one can argue that if P1-P4 are jointly possible, C1 is possible, and so an Anselmian God necessarily exists. This follows given S5 in modal logic, which says that ◊☐P entails ☐P. The argument can be formally proved as follows:

Let, also:

P1 ≝ (∃x)[♢Cx & ~(∃y)♢Gyx]
P2 ≝ (∀x){[♢~Cx ⊃ ♢~(∃z)(z=x)] ⊃ Mx}
P3 ≝ (∀x)[Mx ⊃ (∃y)♢Gyx]
P4 ≝ (∃x)[☐(∃z)(z=x) & Θx] ⊃ ☐(∃x)Θx
C1 ≝ ☐(∃x)Θx

36. ◊[(P1 & P2) & (P3 & P4)] (premise)
37. [(P1 & P2) & (P3 & P4)] ⊢ C1 (premise; proved by 1-35)
38. [◊[(P1 & P2) & (P3 & P4)]& {[(P1 & P2) & (P3 & P4)]⊢ C1}] ⊃ ◊C1 (premise)
39. ◊[(P1 & P2) & (P3 & P4)] & {[(P1 & P2) & (P3 & P4)] ⊢ C1} (36,37 Conj)
40. ◊C1 (38,39 MP)
41. ◊☐(∃x)Θx (40 Def “C1”)
42. ☐(∃x)Θx (41 by “S5”)

QED (again)

Since (37) is established, and (38) merely argues that if premises are jointly possible, and those premises prove some conclusion, then the conclusion is possible, (38) is relatively uncontroversial.  So, if one objects that P1-P4 are not actually true, but admits that they are at least broadly logically, or metaphysically compossible, then one ought to agree that, necessarily, an Anselmian God exists.

The Modesty of Maydole’s Temporal Contingency Argument

In a recent discussion that I had, my interlocutor claimed that “contingency” was an outdated scholastic concept. Really it is just a modal property. Sometimes it is called “two-way” possibility, i.e. x is contingent iff possibly and possibly not x. Temporal contingency the possibility of existing at some point in time and not existing at some point in time. We experience temporal contingency all the time. Anyways, I promised to explain how contingency is still relevant today in the philosophy of religion. In fact, I think it is relevant in one of the most powerful arguments for God’s existence. I can’t really imagine a good reason to deny any of the premises, and it is of course logically valid. So I am compelled to conclude that it is a sound argument for the existence of a supreme being, which I call “God”.

In a sense, The argument originates with Thomas Aquinas’s third way, but is developed by Robert Maydole, who fuses it with a modal ontological argument to devise an ingenious new argument.

Maydole defines a supreme being as follows:

D1. A supreme being is such that it is not possible that there exists anything greater than it and it is not possible that it is not greater than anything else that is non-identical to it.

He then proves the following, which we will call T1:

T1. If possibly a supreme being exists, then a supreme being exists.

Maydole does this by making use of a few theorems, like Barcan Formula, and other theorems in modal logic (I will reproduce the argument below, for those who are interested, see the conditional proof on lines 4-19 for the exact proof). Then Maydole constructs an argument for the possibility of a supreme being. He lists the following premises (but don’t attack them straight off, something interesting happens):

P1. Something presently exists.
P2. Only a finitely many things have existed to date.
P3. Every temporally contingent being begins to exist at some time and ceases to exist at some time.
P4. Everything that begins to exist at some time and ceases to exist at some time exists for a finite period of time.
P5. If everything exists only for a finite period of time, and there have been only a finitely many things to date, then there was a time when nothing existed.
P6. If there was a time when nothing existed, then nothing presently exists.
P7. A being is temporally necessary if and only if it is not temporally contingent.
P8. Everything has a sufficient reason for its existence.
P9. Anything that has a sufficient reason for its existence also has a sufficient reason for its existence that is a sufficient reason for its own existence.
P10. No temporally contingent being is a sufficient reason for its own existence.
P11. Every temporally necessary being that is a sufficient reason for its own existence is a being without limitations.
P12. A being without any limitations is necessarily greater than any other being.
P13. It is not possible for anything to be greater than itself.
P14. It is necessarily the case that “greater than” is asymmetric.

From P1-P14 one can prove C1:

C1. A supreme being exists.

The proof from P1-P14 to C1 is a bit long, and I believe Maydole even made a few typographical mistakes along the way. Here is my adaptation of this part of the argument, if you are interested.

Next consider what was said, before, that if it is possible that a supreme being exists, then a supreme being exists, i.e. T1. Maydole’s argument is surprisingly modest. What he does is argue that POSSIBLY (P1-P14) is true. Since C1 is provable from (P1-P14), we can say POSSIBLY C1 is true, which is to say that possibly a supreme being exists. Given T1 and the possibility that a supreme being exists, we can conclude that a supreme being exists (which is rightly called God)!

Now, the argument is very strong, because it is plausible that P1-P14 are actually true. However, Maydole only requires that the premises be possibly true rather than actually true, which is to say that they are not logically or metaphysically incoherent, or that they are true in some metaphysically possible world (as contemporary modal logicians would say). The deduction is valid, and it is very hard for me to think any of the premises are false. So I am compelled to think that this is, indeed, a sound argument for God’s existence.
So the proof looks something like this:

Let

Gxy ≝ x is greater than y
Sx ≝ (~◊(∃y)Gyx & ~◊(∃y)(x≠y & ~Gxy))

1. ◊(P1-P14) (premise)
2. (P1-P14) ⊢ C1 (premise that C1 is provable from P1-P14)
3. {◊(P1-P14) & [(P1-P14) ⊢ C1]} ⊃ ◊C1 (premise)
4. ◊(∃x)Sx (Assump CP)
5. ◊(∃x)Sx ⊃ (∃x)◊Sx (BF theorem)
6. (∃x)◊Sx (4,5 MP)
7. ◊Su (6 EI)
8. ◊(~◊(∃y)Gyu & ~◊(∃y)(u≠y & ~Guy)) (7, df “Sx”)
9. ◊(~◊(∃y)Gyu & ~◊(∃y)(u≠y & ~Guy)) ⊃ (◊~◊(∃y)Gyu & ◊~◊(∃y)(u≠y & ~Guy)) (theorem)
10. ◊~◊(∃y)Gyu & ◊~◊(∃y)(u≠y & ~Guy) (8,9 MP)
11. ◊~◊(∃y)Gyu (10 Simp)
12. ◊~◊(∃y)(u≠y & ~Guy) (10 Simp)
13. ◊~◊(∃y)Gyu ⊃ ~◊(∃y)Gyu (theorem, by “S5”)
14. ◊~◊(∃y)(u≠y & ~Guy) ⊃ ~◊(∃y)(u≠y & ~Guy) (theorem, by “S5”)
15. ~◊(∃y)Gyu (11,13 MP)
16. ~◊(∃y)(u≠y & ~Guy) (12,14 MP)
17. ~◊(∃y)Gyu & ~◊(∃y)(u≠y & ~Guy) (15,16 Conj)
18. Su (17, df “Sx”)
19. (∃x)Sx (18 EG)
20. ◊(∃x)Sx ⊃ (∃x)Sx (4-19 CP, which proves T1)
21. {◊(P1-P14) & [(P1-P14) ⊢ C1] (1,2 Conj)
22. ◊C1 (3,22 MP)
23. ◊(∃x)Sx (22, def “C1”)
24. (∃x)Sx (20,23 MP)

QED

To me, it is P11 that needs more explanation. It certainly seems right that a temporally necessary being who is the sufficient reason for its own existence has the sort of existence that is not limited by time nor by the existence of any other thing. But to say that the existence of x is not limited by time nor any thing seems a bit different from saying thag such a being is essentially without limitations. I believe the idea is that if there is no time nor state of affairs in which such a being would cease to exist or lack a reason for existing, then it is not limited by anything at all, and must be greater than every other thing.

Another person noted that P5 did not make sense to him because time is something that exists, so there could never be a time when nothing exists. Maydole, however, is quantifying over things in a way that is distinct from moments (in his “Modal Third Way” you see a more careful distinction between moments and things). With the right qualifications, and stipulations, this worry can be alleviated, e.g. one might say “no concrete things” or “no subsitent things” rather than “nothing”.

Reference:
Maydole, R. 2012. “The Ontological Argument”. In The Blackwell Companion to Natural Theology. Ed. W.L. Craig & J.P. Moreland. Malden, MA: Blackwell Publishing, pp. 580-586.

Some Proposed Corrections to Maydole’s Temporal Contingency Argument

Robert Maydole presents an interesting argument for a supreme being, called the temporal contingency argument.  The argument is a long deduction, and so is seen as somewhat difficult to comprehend. The version that I am critiquing appears in the Blackwell Companion to Natural Theology and appears as follows (with highlighted lines that I believe are problematic)[1]:

Maydole-TCA3

These errors are not fatal to the argument, however.  Here is a quick workaround that I think preserves the spirit of Maydole’s deduction (using nested conditional proofs and the identity rule, for example).  I’ve simplified some of the lexicon, but if pretty much follows Maydole’s definitions.  A revised deduction is as follows:

Bx ≝ x begins to exist at some time and ceases to exist at some time
Tx ≝ x is temporally necessary
Cx ≝ x is temporally-contingent
Fx ≝ x exists for a finite period of time
≝ Only finitely many things have existed to date
≝ Something presently exists
≝ There was a time when nothing existed
Sxy ≝ x is a sufficient reason for the existence of y
Wx ≝ x is without any limitations
Gxy ≝ x is greater than y
Sx ≝ (~◊(∃y)Gyx & ~◊(∃y)(x≠y & ~Gxy))
 
Deduction

1. P (premise)
2. M (premise)
3. (
∀x)(Cx ⊃ Bx) (premise)
4. (∀x)(Bx ⊃ Fx) (premise)
5. ((∀x)Fx & M) ⊃ N (premise)
6. N ⊃ ~P (premise)
7. (
∀x)(Tx ≡ ~Cx) (premise)
8. (∀x)Cx (IP)
9. Cμ ⊃ Bμ (3 UI)
10. Cμ (8 UI)
11. B
μ (9,10 MP)
12. B
μ ⊃ Fμ (4 UI)
13. Fμ (11,12 MP)
14. (∀x)Fx (13 UG)
15. (∀x)Fx & M (2,14 Conj)
16. N (5,15 MP)
17 ~P (6,16 MP)
18. P & ~P (1,17 Conj)
19. ~(
∀x)Cx (8–18 IP)
20. (∃x)~Cx (19 QN)
21. ~Cν (20 EI)
22. Tν ≡ ~Cν (7 UI)
23. (T
ν ⊃ ~Cν) & (~Cν ⊃ Tν) (22 Equiv)
24. (~C
ν ⊃ Tν) (23 Simp)
25. Tν (21,24 MP)
26. (∃x)Tx (25 EG)
27. (
∀x)(∃y)Syx (premise)
28. (∀x)[(∃y)Syx ⊃ (∃z)(Szx & Szz)] (premise)
29. (∀x)(∀y)[(Tx & Syx) ⊃ ~Cy] (premise)
30. (∀y)[(Ty & Syy) ⊃ Wy] (premise)
31. (∀y)[Wy ⊃ ☐(∀z)(z≠y ⊃ Gyz)] (premise)
32. ~◊(∃y)Gyy (premise)
33. 
☐(∀x)(∀y)(Gxy ⊃ ~Gyx) (premise)
34. (∃y)Syν (27 UI)
35. (∃y)Syν ⊃ (∃z)(Szν & Szz) (28 UI)
36. (∃z)(Szν & Szz) (34,35 MP)
37. Suν & Suu (36 EI)
38. (∀y)[(Tν & Syν) ⊃ ~Cy] (29 UI)
39. (Tν & Suν) ⊃ ~Cu (38 UI)
40. Suν (37 Simp)
41. Tν & Suν (25,40 Conj)
42. ~Cu (39,41 MP)
43. Tu ≡ ~Cu
 (7 UI)
44. (Tu ⊃ ~Cu) & (~Cu ⊃ Tu) (43 Equiv)
45. ~Cu ⊃ Tu (44 Simp)
46. Tu (42,45 MP)
47. Suu (37 Simp)
48. Tu & Suu (46,47 Conj)
49. (Tu & Suu) ⊃ Wu (30 UI)
50. Wu ⊃ 
☐(∀z)(z≠u ⊃ Guz) (31 UI)
51. Wu (48,49 MP)
52. ☐(∀z)(z≠u ⊃ Guz) (50,51 MP)
53. ☐(∀z)(~z≠u ∨ Guz) (52 Impl)
54. ☐(∀z)(~z≠u ∨ ~~Guz) (53 DN)
55. ☐(∀z)~(z≠u & ~Guz) (54 DeM)
56. ☐~(∃z)(z≠u & ~Guz) (55 QN)
57. ~◊(∃z)(z≠u & ~Guz) (56 MN)
58. 
☐~(∃y)Gyy (32 MN)
59. ☐(∀y)~Gyy (58 QN)
60. (∀y)~Gyy (CP)
61. μ=ν (CP)
62. ~Gμμ (60 UI)
63. ~Gμν (61,62 IR)
64. μ=ν ⊃ ~Gμν (61-63 CP)
65. (∀y)~Gyy ⊃ (μ=ν ⊃ ~Gμν) (60-64 CP)
66. ☐[(∀y)~Gyy ⊃ (μ=ν ⊃ ~Gμν)] (65 NI)
67. ☐(μ=ν ⊃ ~Gμν) (59,66 MMP)
68. ☐(∀x)(∀y)(Gxy ⊃ ~Gyx) & ☐(∀z)(z≠ν ⊃ Gνz) (33,52 Conj)
69. [☐(∀x)(∀y)(Gxy ⊃ ~Gyx) & ☐(∀z)(z≠ν ⊃ Gνz)] ⊃ ☐[(∀x)(∀y)(Gxy ⊃ ~Gyx) & (∀z)(z≠ν ⊃ Gνz)] (theorem)
70. ☐[(∀x)(∀y)(Gxy ⊃ ~Gyx) & (∀z)(z≠ν ⊃ Gνz)] (68,69 MP)
71. {[(∀x)(∀y)(Gxy ⊃ ~Gyx) & (∀z)(z≠ν ⊃ Gvz)] ⊃ (μ≠ν ⊃ ~Gμν)} (theorem)
72. ☐(μ≠ν ⊃ ~Gμν) (70,71 MMP)
73. [☐(μ=ν ⊃ ~Gμν) & ☐(μ≠ν ⊃ ~Gμν)] ⊃ ☐[(μ=ν ∨ μ≠ν) ⊃ (~Gμν ∨ ~Gμν)] (theorem)
74. ☐(μ=ν ⊃ ~Gμν) & ☐(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 (theorem)
82. ☐(∀z)~Gzν (80,81 MP)
83. ☐~(∃z)Gzν (82 QN)
84. ~◊(∃z)Gzν (83 MN)
85. ~◊(∃z)Gzν & ~◊(∃z)(z≠ν & ~Gνz) (57,84 Conj)
86. Sν (85 def “S”)
87. (∃x)Sx (86 EG)

[1]R. Maydole. 2012. “The Ontological Argument”. The Blackwell Companion to Natural Theology. Ed. W.L. Craig & J.P. Moreland. Malden, MA: Blackwell Publishing. Document image retrieved from <http://commonsenseatheism.com/wp-content/uploads/2009/05/irrefutable.png>.

An Ontological Argument from Actuality

Here is a refinement on my ontological argument from actuality:

1. Something is an Anselmian God if and only if it is conceivable, nothing can be conceived of which is more actual, and it necessarily exists (definition Θ).

2. There is something conceivable such that nothing can be conceived of which is more actual (premise).

3. For all x, if the possibility of failing to conceive of x implies the possibility that x doesn’t exist, x is mentally dependent (premise).

4. For all x, if x is mentally dependent, there is something conceivable that is more actual than x (premise).

Therefore,

5. An Anselmian God exists.

I start out with a definition of an Anselmian God, which is a stipulation, but is rooted in the idea that a Being of Pure Actuality is arguably perfect and possesses a good number of divine attributes.  

As I noted in a previous post, the traditional argument uses a “greater than” relation, which some find suspect.  “Greatness” would have been understood by Anselm as something that can be evaluated objectively on a scale, as in the Neo-Platonic notion of the Great Chain of Being.  To the contemporary ear, “greatness” seems subjective and vague.  I think “actual” in the Thomistic-Aristotelian sense is a fair approximation of greatness, but we can have a better sense of what “actual” means.  Thomas is able to derive the divine attributes from a being of Pure Actuality, so “most actual” is plausibly a divinely loaded superlative.  Moreover, it seems to me that the act-potency distinction is not something the contemporary ear would take to be dependent on subjective opinions.  So, I think (2) is fairly impeccable.  

I think (3) is a bit clunky, but it basically means that if something is merely a concept, then it is mentally dependent.  So, in the case of God, if God is merely a concept in the mind, then the possibility that God could fail to be conceived by all minds that exist implies that God, as a mere concept, could fail to exist, and so depends upon minds to continue to exist.  Put another, if God is merely a concept, then there was no God in the Jurassic period, as William Lane Craig once suggested to John Dominic Crossan.

Finally, (4) says that if something is mentally dependent, then something is conceivable that is more actual than it.  Some people think, for instance, that moral values are mind-dependent.  So, for instance, the actuality of the value of human life, VHL, depends on there being an actual community of minds that actually conceive of human life as valuable.  Were such a community to cease to exist, the VHL would only potentially exist, even if humans existed.  If the VHL were an objective fact grounded in human nature, then the actuality of the VHL would obtain whenever humans actual exist.  There is a certain assymetry that suggests that grounding the VHL in human nature is to view VHL as more actual than grounding VHL in the subjective opinions of a community of minds.  For the VHL to be actual in one case, there need only be actual humans exemplifying human nature, where as in the latter, there needs to be actual humans exemplifying human nature and an actual community of minds that actually is of the opinion that human life is valuable.  For, without the humans, a community of minds that endorses the VHL would really just be saying that VHL potentially exists and would be actual upon the occassion of human life.  We could say, then, that x is more actual than y iff the existence of x depends upon the actualization of fewer potentials than y depends upon.  VHL grounded in the actuality of human nature depends upon the actualization of fewer potentials than VHL grounded in subjective opinions about humans. So (4) just tells us that for any x that depends upon the mental for its actuality, it is conceivable that there is something that is more actual (and less dependent) than x, e.g. to conceive that x can actually exist independent of mentally conceiving of x.

Let

Cx – x is conceived
Mx – x is mentally dependent
Axy – x is more actual than y
Θx- x is an Anselmian God, 

that is: 

1. (∀x){Θx ≝ ([♢Cx & ~(∃y)(Ayx & ♢Cy)] & ☐(∃z)(z=x))} (Def Θ)
2. (∃x)[♢Cx & ~(∃y)(Ayx & ♢Cy)] (premise)
3. (∀x){[♢~Cx ⊃ ♢~(∃z)(z=x)] ⊃ Mx} (premise)
4. (∀x){Mx ⊃ [(∃y)(Ayx & ♢Cy)]} (premise)
5. (∀x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (IP)
6. ♢Cu & ~(∃y)(Ayu & ♢Cy) (2 EI)
7. [♢~Cu ⊃ ♢~(∃z)(z=u)] ⊃ Mu (3 UI)
8. Mu ⊃ [(∃y)(Ayu & ♢Cy)] (4 UI)
9. [♢~Cu ⊃ ♢~(∃z)(z=u)] ⊃ [(∃y)(Ayu & ♢Cy)] (7,8 HS)
10. [♢Cu & ~(∃y)(Ayu & ♢Cy)] ⊃ [♢~Cu ⊃ ♢~(∃z)(z=u)] (5 UI)
11. ♢~Cu ⊃ ♢~(∃z)(z=u) (6,10 MP)
12. (∃y)(Ayu & ♢Cy) (9,11 MP)
13. Avu & ♢Cv (12 EI)
14. ~(∃y)(Ayu & ♢Cy) (6 Simp)
15. (∀y)~(Ayu & ♢Cy) (14 QN)
16. ~(Avu & ♢Cv) (15 UI)
17. (Avu & ♢Cv) & ~(Avu & ♢Cv) (13,16 Conj)
18. ~(∀x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (5-17 IP)
19. (∃x)~{[♢Cx & ~(∃y)(Ayx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (18 QN)
20. (∃x) ~{~[♢Cx & ~(∃y)(Ayx & ♢Cy)] ∨ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (19 Impl)
21. (∃x){~~[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (20 DeM)
22. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (21 DN)
23. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[~♢~Cx ∨ ♢~(∃z)(z=x)]} (22 Impl)
24. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[☐Cx ∨ ♢~(∃z)(z=x)]} (23 ME)
25. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & [~☐Cx & ~♢~(∃z)(z=x)]} (24 DeM)
26. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & [~☐Cx & ☐(∃z)(z=x)]} (25 ME)
27. [♢Cu & ~(∃y)(Ayu & ♢Cy)] & [~☐Cu & ☐(∃z)(z=u)] (26 EI)
28. ~☐Cu & ☐(∃z)(z=u) (27 Simp)
29. ☐(∃z)(z=u) (28 Simp)
30. [♢Cu & ~(∃y)(Ayu & ♢Cy)] (27 Simp)
31. [♢Cu & ~(∃y)(Ayu & ♢Cy)] & ☐(∃z)(z=u) (29,30 Conj)
32. Θu (1,31 “Def Θ”)
33. (∃x)Θx (32 EG)

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