# Author Archives: Daniel Vecchio

## 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)

## Self-Referential Unsound Modus Ponens

[Image Source Credit: TeX]

An argument is sound if and only if it is valid and the premises are true. If those conditions are met, the conclusion must be true.

Consider the following argument:

P1. If God does not exists, this argument is unsound.
P2. God does not exist.
C. Therefore, this argument is unsound.

The argument is valid (Modus Ponens), so it is sound if the premises are true. But, if both premises are true, the conclusion is would have to be true, and the argument would both be sound and unsound. So consistency demands that we deny the soundness of the argument. At lease one of the premises must be false.
Consider whether P1 is false. It is a material conditional, and so it is false when the antecedent is true (it is true that God does not exist) and when the consequent is false (it is false that this argument is unsound).[1] So P1 is false only if the argument is sound, which means that the falsity of P1 leads to a contradiction, since the soundness of the argument entails P1 is true. So, P1 cannot be false.

P2 is the only premise that can be false. So given that the argument must be unsound, we must conclude that it is false that God does not exist.

So this unsound modus ponens proves the contradictory of the minor premise, whatever it might be!

I am probably not the first to note this, but it is new to me.

[1]The truth-table for the Material Conditional is as follows:

p  q | p → q
1. T  T        T
2. T  F        F*
3. F  T        T
4. F  F        T
*The material conditional is only false on line 2.

## 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) (Astump. 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 Dilemma Theodicy

1. By definition, God is a maximally great being, i.e. an omnipotent, omniscience, morally perfect being in every possible world.
2. Any argument against God’s existence that depends on a premise of the form “If God were to exist, then we would expect there to be x” (hereafter, the “counterfactual” premise) must have a justification, either by way of a trivial entailment, given the incoherence of the concept of God, and so the impossibility of the existence of God, or by way of the defense of a substantive counterfactual implication, given a thoroughgoing conceptual analysis of the concept of God, and the sorts of states of affairs implied by God’s existence.
3. If the justification for the “counterfactual” premise is by way of a trivial entailment, given the incoherence of the concept of God, and so the impossibility of the existence of God, then the justification for the “counterfactual” premise begs the question of any argument against God’s existence that depends upon the “counterfactual” premise, which means the argument containing the “counterfactual” premise is informally fallacious.
4. If the justification for the “counterfactual” premise is by way of a defense of a substantive counterfactual implication, given a thoroughgoing conceptual analysis of the concept of God, and the sorts of states of affairs implied by God’s existence, then the justification depends upon the metaphysical possibility of God, and the sorts of states of affairs that obtain in the nearest possible worlds where God exists, which also serves as a justification for the possibility premise of the modal ontological argument, by which the existence of God can be directly demonstrated from His metaphysical possibility, based upon an axiom of S5.
5. But, a successful argument cannot be informally fallacious, nor can a successful argument depend on a justification that directly implies the contradictory of the its conclusion.
6. So, no argument against God’s existence that depends on the “counterfactual” premise is successful.

Escaping the horns would require a substantive justification of the counterfactual premise that does not imply any real metaphysical possibility of God.  Would such a justification be compelling enough for a theist, or neutral party to accept the truth of the counterfactual premise?

## Dr. Tuggy’s response

I posted a critique of Dr. Dale Tuggy’s Trilemma a couple of weeks ago.

He offered a charitable analysis, and when I have time, I hope to respond.  Check it out here.

Or you can listen through Youtube:

## 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.

## Tuggy’s Trilemma

Dale Tuggy offers the following trilemma over at his excellent Trinities blog/podcast:

1. Jesus died.

2. Jesus was fully divine.

3. No fully divine being has ever died.

Tuggy explains that one cannot hold to all three, so at least one must go.  But which one? As a unitarian, he thinks the Biblical data requires the affirmation of 1 and 3, and so rejects 2.

I am going to respond to this Trilemma by adopting a “Two Natures” view as expressed by the doctrine of the hypostatic union.  So, I believe the Second Person of the Holy Trinity is a Divine hypostasis that has two natures.  Those natures are not mixed or confused.

Proposition 1:  Did Jesus die?

I accept that Jesus Christ died.  This is affirmed throughout scripture.  1 Peter 3:18 tells us that he was “put to death in the flesh”, in Matthew 27:50 John 19:30 it is said that Jesus “gave up the ghost.”  The death of Christ is a mystery of the Catholic faith, repeated at every Mass in both thr litergy and in the Nicene Creed.

So, I am inclined to accept (1).  I will note, however, that the plain reading of scripture suggests that death involves the flesh and separation or loss of the soul or spirit.  So, I would understand death as the separation of the soul from the body.  Tuggy defines death as the loss of all or most living functions and does not limit life-functions to biological or natural life functions.  The question might then be raised if, on the two-natures view, an individual hypostasis is dead if the life-functions of one of his natures are still fully operational even if the life-functions of the other nature become severally restricted.  It seems to me that when orthodox Christians claim that Jesus died, they mean that the human substance that he assumed at the incarnation was destroyed by the separation of Christ’s human soul from his human body, but that he also has a divine nature in which he is consubstantial with the Father and Holy Spirit.  That divine substance is essentially immortal.

So, would Tuggy say that I deny Proposition 1?  I don’t know, but I think there is a literal sense in which Jesus died.

Proposition 2: Was Jesus fully divine?

Here, I think we need to tease out different ways of understanding “fully”.  In one sense, a thing can be fully of a nature if that is the only nature it has.  For example, I am fully human and this implies that I am not anything non-human.  In this sense, it could not be said that Jesus is fully divine.  Jesus is divine, but on the two natures view, we must reject the implication that he is not anything non-divine.  He is human, and a human nature, even if assumed by a divine person, does not become a divine nature (lest we confuse the natures).

There is a sense in which I would say Jesus is fully divine though.  I would say that something is fully some nature if it lacks nothing essentially had by things of that nature.  So, again, I am fully human in this sense too, since I do not lack any of the essential attributes of a human.  We might imagine some monster, like the Minotaur, who is half-man and half-bull.  Such a creature may have some of the essential attributes of a human, and some of the attributes of a bull, but really could not be said to be fully human or fully bull.  That is not Christ’s situation, however.  He is not a monstrosity, but has a complete human nature and a complete divine nature.  So according to his human nature, he has a human body, human organs, a human mind, a human will, and so forth.  According to his divine nature, as I said above, Christ is of the very same substance as the Father and the Holy Spirit, and so according to that divine nature, shares in the Divine Essence and lacks nothing essential to the True God.  In this sense Jesus is fully divine.  That is, he is a hypostasis that has a divine nature identical to the divine ousia.  Would Tuggy agree with me that I can affirm Proposition 2, in some sense?  I am not sure.

Proposition  3:  Can a fully divine being die?

Again, there is a sense in which I affirm 3 and a sense in which it could be said that I deny 3.  As Aristotle tells, “being” is said in many ways.  In fact, he thought the primary sense of “being” is “ousia” (see Meta IV.2).  Another sense of “being” could be some individual x, which is how I understand the function of “hypostasis” or “supposite” in these debates.  The Father, Son, and Holy Spirit are persons insofar as they are rational individuals.  We could say that the Father is a rational being.  In fact, the Father is essentially rational insofar as he cannot fail to have an intellect and will.  However, he is not essentially rational insofar as he is an individual x, but insofar as his substance is essentially rational.  Substances have essential attributes, and individual have essential attributes only with reference to their substances.  They do not have essential attributes qua hypostasis or because they are an individual x.

So, I would say that essential immortality belongs to the divine substance (ousia).  Divine Persons, or Divine Hypostases are essentially immortal only in reference to their substantial nature.  It makes no sense to say that a Divine Person is essentially immortal because of the essential nature of being a hypostasis.

Can  the Divine Ousia die?  No, it is essentially immortal.  Can a divine hypostasis die when referencing their divine nature?  No.  Can a divine hypostasis assume a moral nature and die with respect to that nature.  Yes, and Thomas Aquinas agrees that each of the divine hypostases could have assumed a moral nature (I mention this not to appeal to his authority, but as a marker to show that I am not far off the reservation of orthodoxy).

Conclusion: So, there is a sense in which I affirm all three propositions.  I really affirm that Jesus died a human death, which is the separation of the human soul from the human body in which most of the living functions of the human substance ceased.  I really affirm that Jesus is a fully divine hypostasis insofar as he has a nature that lacks none of the essential divine attributes.  I really affirm that the fully divine ousia is essentially immortal.  I think these are ways to affirm what orthodox Christians mean when they say such things, though they may not be what Tuggy means.  So he might say that I reject all three propositions, even if I think I affirm them after making the distinctions I have made.  But then we would just be quibbling, and I could grant that I reject one or more of the propositions as Tuggy defines them and still safely be in orthodoxy.  Nonetheless, I see no contradiction in accepting the three propositions given my qualifications.

## Ontological Argument Improved Again

Let,

E!x ≝ x exists in re
Ix ≝ x exists in intellectu
Gx ≝ x admits of more greatness
G<Px,~Px> ≝ x having P is greater than x not having P
Gxy ≝ x is greater than y
©… ≝ it is conceivable that…

1. (∀x)[(Ix ∧ ~E!x) ⊃ ©E!x] (premise)
2. (∀x)G<E!x, (~E!x ∧ Ix)> (premise)
4. Ig (premise)
5. ~E!g (IP)
6. Ig ∧ ~E!g (4,5 Conj)
7. (Ig ∧ ~E!g) ⊃ ©E!g (1 UI)
9. G<E!g, (~E!g ∧ Ig)> (2 UI)
10. ~E!g ∧ G<E!g, (~E!g ∧ Ig)> (8,9 Conj)
11. [~E!g ∧ G<E!g, (~E!g ∧ Ig)>] ∧ ©E!g (5,10 Conj)
12. [[~E!g ∧ G<E!g, (~E!g ∧ Ig)>] ∧ ©E!g] ⊃ ©Gg (3 UI)
21. E!g (5-20 IP)

## 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…

1. (∀x)[(Ix ∧ ~E!x) ⊃ ©(Ix ∧~E!x)] (premise)
4. Ig (premise)
5. ~E!g (IP)
6. Ig ∧ ~E!g (4,5 Conj)
7. (Ig ∧ ~E!g) ⊃ ©(Ig ∧~E!g) (1 UI)
10. ©(Ig ∧ E!g) (8,9 MP)
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

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)
27. (∀y)[~©Gy ⊃ (y = μ)] (26 Simp)
28. (~©Gν ∧ Lν) ∧ (∀y)[(~©Gy ∧ Ly) ⊃ (y = ν)] (25 Simp)
29. ~©Gν ∧ Lν (28 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.

## Absolute and Relative Identity

I have seen some argue that any relative identity claim can be reduce to an absolute identity claim in the following manner:

1) x and y are the same F  ≝ is an Fis an F, and x = y.

However, I don’t think this works.  Part of the motivation for relative identity is that there may be circumstances like:

2) x and y are the same F, but x and y are not the same G.

But (1) and (2) are not compatible, since we would have to affirm and deny absolute identity between x and y.  So the relative identity theorist should reject (1) given his commitment to (2).

Relative identity is not just absolute identity, plus the idea that x and y fall under the same sortal.  Moreover, this would be to suggest that relative identity is derivative, and absolute identity is the more primitive notion.  I would argue that is it the other way around.  So I would define absolute identity in terms of relative identity in the following manner:

4) x = y ≝ for any sortal, S, if x is an S or y is an S, then x and are the same S.

In other words, the absolute identity between x and y is derived from the fact that for any sortal which belongs to either x or y, it is the case that x and y count as the same S.  I say “either x is an S, or y is an S” as opposed to “both x is an S and y is an S” to avoid situations where x can be counted as an S and some y cannot, but they are the same S for any sortal underwhich both can be counted.  For there to be absolute identity, it must be the case that all sortals that belong to x must also belong to y.  I believe (4) captures this.

So to say x and y are absolutely identical is to say that for any sortal underwhich x or y can be counted, x and y are the same sortal.