Category Archives: Arguments for God

Combining Aquinas and the MOA

Here is a variation on my argument from Anselm to Plantinga:

P1) Possibly, there is an absolutely metaphysically simple being.
P2) Necessarily, that there is an absolutely metaphysically simple being implies that there is a maximally great being.
P3) If it’s possible that something is maximally great, then it’s possible that necessarily there is an omnipotent, omniscient, and omnibenevolent.
C) There is an omnipotent, omniscient, and omnibenevolent being.

Defense of P1: An absolutely metaphysically simple being, insofar as it is being, is attributed positively, cannot contain a part that negates its essential nature, which means it does not contain inconsistent properties or attributes.  Now, it has been objected, by none other than Plantinga, that the concept of a metaphysically simple being is incoherent, but as Vallicella (2019) points out, one need not adopt the metaphysical framework by which that incoherence is pressed.  Thus the metaphysical possibility of an absolutely metaphysical being will depend on the supposition of a “constituent”  metaphysical frame work.  Vallicella (2019) writes, the “constituent” metaphysicians “…did not think of individuals as related to their properties as to abstracta external to them, but as having properties as ontological constituents.”  This roughly tracks Aristotelian realism over Platonic realism, which I think is a decisively preferable metaphysical framework, given the third-man objection to Platonism.  With these considerations in mind, I think it is highly plausible to defend the metaphysical possibility of an absolutely simple being.

Defense of P2: Aquinas demonstrates that an absolutely metaphysically simple being is metaphysically necessary (since its has existence essentially, see [3]-[4]), omnipotent (since God is infinite, which is derived from His simplicity), omniscient (see, in particular, [3]), and the good of every good (see [3]) and the highest good (see [5]), so omnibenevolent.  Now one might object that a maximally great being has many divine attributes and is, therefore, not absolutely metaphysically simple, but Aquinas explains that the plurality of divine attributes is not opposed to divine simplicity.  Since the attributes of a maximally great being can be deduced from an absolutely simple being, we can conclude that the existence of an absolutely simple being necessarily implies an maximally great being (where maximal greatness is defined as a necessarily existing, omnipotent, omniscient, and morally perfect being).

Defense of P3: This implication follows from Plantinga’s stipulative definitions of maximal greatness, and maximal excellence, with a slight deviation from moral perfection to omnibenevolence, defined in Thomistic terms.  So this is an analytically true implication.

Let,

Mx ≝ x is maximally great
Ox ≝ x is omnipotent, omniscient, and omnibenevolent
Sx ≝ x is absolutely metaphysically simple
Theorem of K: ☐(p → q) → (♢p → ♢q)
Theorem of S5: ♢☐p → ☐p
Axiom M: ☐p → p

1. ♢(∃x)Sx (premise)
2. ☐[(∃x)Sx → (∃y)My](premise)
3. ♢(∃y)My → ♢☐(∃z)Oz (premise)
4. ☐[(∃x)Sx → (∃y)My]] → [♢(∃x)Sx → ♢(∃y)My] (Theorem of K)
5. ♢(∃x)Sx → ♢(∃y)My (2,4 MP)
6. ♢(∃y)My (1,5 MP)
7. ♢☐(∃z)Oz (3,6 MP)
8. ♢☐(∃z)Oz → ☐(∃z)Oz (Theorem of S5)
9. ☐(∃z)Oz (7,8 MP)
10. ☐(∃z)Oz → (∃z)Oz (Axiom M)
11. (∃z)Oz (9,10 MP)

QED

References:

Vallicella, William F., “Divine Simplicity”, The Stanford Encyclopedia of Philosophy (Spring 2019 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2019/entries/divine-simplicity/&gt;.

 

Anselm’s God to Plantinga’s God

Prefatory Remarks:

There is a slight difference in the way Anselm and Plantinga define God.  Anselm’s definition is that God is that than which none greater can be conceived.  Plantinga’s God is a maximally great being, i.e. a necessarily existing being that has omnipotence, omniscience, and morally perfection.  Anselm’s definition is negative, while Plantinga’s is positive.  Anselm’s definition fits with the apophatic tradition of a negative theology, i.e. God is not among those things of which a greater can be conceived.  It is because Anselm’s definition is negative that I contend that Thomas Aquinas is incorrect in his central critique of the ontological argument.  Anselm isn’t offering a positive account of God’s essential nature.  I agree with Aquinas that a positive account of God’s essential nature cannot be completely and univocally known to us, but I should also say that although Plantinga’s definition is positive, it is not claimed to be complete and it need not be interpreted as perfections of “power”, “knowledge”, and “goodness” as those terms are understood univocally.

There is still a strong relationship between the Anselmian definition of God and the Plantingan definition.  Namely, one can derive from the Anselmian definition various divine attributes like necessary existence, omnipotence, omniscience, and moral perfection, among other perfections.  So one can argue that if there is an Anselmian God, then there is a maximally great being, in the Plantingan sense.  Indeed, that impication necessarily holds, given that it analytically follows from the Anselmian definition.  As an aside, I would argue that the two definitions are not equivalent in that one cannot derive the Anselmian definition from the Plantingan definition.  So, the existence of a maximally great being would not necessarily imply the existence of Anselm’s God.

Another interesting aspect of Anselm’s definition is that, since it is negative, I think the case for its metaphysical possibility can be firmly established.  Now, I am not suggesting that Anselm makes a modal inference that the metaphysical possibility of God, as he defines it, entails his actual existence.  Still, it is often disputed that conceivability does not entail metaphysical possibility. However, in this particular case, the conceivability of the Anselmian God makes the following falsehood self-evident, viz. that it is somehow metaphysically necessary that for any object, there will always be something else one could conceive of which would be greater.

Given that Plantinga’s maximally great being is a necessarily existing omnipotent, omnicient, and morally perfect being, I think there may be a powerful way to combine the fact that we can understand the Anselmian God, and show the Anselmian God possible, and use that to demonstrate the existence of a being that is omnipotent, omniscient, and morally perfect.  In what follows, I exploit Anselm to vindicate Plantinga.

Informal Expression of the Argument:

P1) If I can understand the Anselmian definition of God, then it is not necessarily the case that, for any given thing, there will be something conceivably greater.
P2) If it is possible that there is something than which none greater can be conceived, then it is possible that there is an Anselmian God.
P3) The existence of the Anselmian God necessarily implies the existence of a maximally great being.
P4) I can understand the Anselmian definition of God.
P5) If it’s possible that something is maximally great, then it’s possible that there is a necessarily existing, omnipotent, omniscient, and morally perfect being.
C) There is an omnipotent, omniscient, and morally perfect being.

A Formal Expression of the Argument:

P1) If it is possible that the Anselmian God is in the understanding, then it is not necessary that, for all x, it is conceivable that there is y and y is greater than x.
P2) If it is possible that there is something, x, such that it is not conceivable that there is some y and y is greater than x, then it is possible that there is something, z, and z is the Anselmian God.
P3) Necessarily, if there is something that is the Anselmian God, then there is something that is maximally great.
P4) It is possible that the Anselmian God is in the understanding
P5) If it is possible that there is something that is maximally great, then it is possibly necessary that there is something that is omnipotent, omniscient, and morally perfect.
C) There is something that is omnipotent, omniscient, and morally perfect.

Formal Deductive Proof of the Argument:

Let,

Mx ≝ x is maximally great
Ox ≝ x is omnipotent, omniscient, and morally perfect
Ux ≝ x is in the understanding
Gxy ≝ x is greater than y
©… ≝ it is conceivable that…
g ≝ (ɿx)(~©(∃y)Gyx)
Theorem of K: ☐(p → q) → (♢p → ♢q)
Theorem of S5: ♢☐p → ☐p
Axiom M: ☐p → p

1. ♢Ug → ~☐(∀x)©(∃y)(Gyx) (premise)
2. ♢(∃x)~©(∃y)(Gyx) → ♢(∃z)(z = g) (premise)
3. ☐[(∃z)(z = g) → (∃x)Mx] (premise)
4. ♢Ug (premise)
5. ♢(∃x)Mx → ♢☐(∃y)Oy (premise)
6. ☐[(∃z)(z = g) → (∃x)Mx] → [♢(∃z)(z = g) → ♢(∃x)Mx] (Theorem of K)
7. ~☐(∀x)©(∃y)(Gyx) (1,4 MP)
8. ~~♢~(∀x)©(∃y)(Gyx)(7 ME)
9. ♢~(∀x)©(∃y)(Gyx)(8 DN)
10. ♢(∃x)~©(∃y)(Gyx)(9 QN)
11. ♢(∃z)(z = g) (2,10 MP)
12. ♢(∃z)(z = g) → ♢(∃x)Mx (3,6 MP)
13. ♢(∃x)Mx (11,12 MP)
14. ♢☐(∃y)Oy (5,13 MP)
15. ♢☐(∃y)Oy → ☐(∃y)Oy (Theorem of S5)
16. ☐(∃y)Oy (14,15 MP)
17. ☐(∃y)Oy → (∃y)Oy (Axiom M)
18. (∃y)Oy (16,17 MP)

QED

That an Omnipotent Individual Exists

Here is an argument that an omnipotent individual exists:

P1) All potentialities are things, or states of affairs, that can be realized by an actually existing individual or an actually existing mereological sum.

P2) All metaphysical possibilities are potentialities.

C1) All metaphysical possibilities are things, or states of affairs, that can be realized by an actually existing individual or an actually existing mereological sum (P1,P2 Modus Barbara).

P3) If all metaphysical possibilities are things, or states of affairs, that can be realized by an actually existing individual or an actually existing mereological sum, then some individual is an omnipotent being or some mereological sum is an omnipotent being.

C2) Some individual is an omnipotent being or some mereological sum is an omnipotent being. (C1,P3 Modus Ponens).

P4) No thing that is contingent is an omnipotent being.

P5) All mereological sums are things that are contingent.

C3) No mereological sum is an omnipotent being (P4,P5 Modus Celarent).

C4) It is not the case that some mereological sum is an omnipotent being (C3 Contradiction).

C5) Some individual is an omnipotent being (C2,C4 Disjunctive Syllogism).

C6) There is an individual that is an omnipotent being (C5 Semantic Equivalence).

QED

Defense of premises:

Support for P1: This is a statement of actualism, the metaphysical thesis that anything that is potentially real must be grounded in something that is actually real.  That is, potentials are the powers that actualities possess.

Support for P2: Here, I defend this implication as following from the definition of what a metaphysical possibility is, namely, a real potential that can be actualized.  That is, these are genuine possibilities, and not mere epistemic possibilities, and so are properly potentially real things, or states of affairs.

Support for P3:  The implication, here, is that there is either an individual or set of things that is the actuality by which all potentials can be realized. That is, if all potentials can be realized by something actual, then that actuality, be it individually or collectively, is omnipotent.  This is the definition of omnipotence.  Note that this premises is neutral on the question of whether the set of “all metaphysical possibilities” is finite or infinite.  However, to be omnipotent, it is sufficient that one has the power to actualize all of the metaphysical possibilities there are.  It need not be established that the set is infinite, though I suspect it is.  To be omnipotent, one must possess the ability to actualize all of the metaphysical possibilities that there are.

Support for P4: A thing that is contingent is not the source of its own existence, and therefore cannot be the actuality by which its own existence obtains.  The potential for a contingent thing to exist must exist in some other actuality beyond itself.

Support for P5: A mereological sum is a collection of things that, grouped together, compose some whole.  All collections of things are contingent on their parts, and the arrangement or structure by which those parts really compose a whole, just as a human is contingent upon the atoms which compose his body.

An Inductive Way of Thinking about the Modal Ontological Argument

P1. If philosophers of religion over the past 50+ years have successfully defended the coherence of the concept of a maximally great God, then probably a maximally great God is metaphysically possible.
P2. The metaphysical possibility of a maximally great God entails that a maximally great God exists.
P3. Philosophers of religion over the past 50+ years have successfully defended the coherence of the concept of a maximally great God.
C. Probably a maximally great God exists.

I think this argument also helps to distinguish between epistemic possibility (I think it is probable because of sustained intellectual scrutiny) and metaphysical possibility.

Also, I should note that by the coherence of the concept of a maximally great God, I mean more than mere consistency among the attributes, or even self-consistency of each attribute, but also the coherence of theism with other facts, necessary or contingent, e.g. evil or suffering.

 

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)
16. E!g (15 DN)

QED

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.

Ontological Argument Improved Again

Let,

Rx ≝ 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…

g ≝ (ɿx)(~©Gx ∧ ~©(∃y)Gyx)

1. (∀x)[(Ix ∧ ~Rx) ⊃ ©Rx] (premise)
2. (∀x)G[Rx,~Rx] (premise)
3. (∀x){[[~Rx ∧ G] ∧ ©Rx] ⊃ ©Gx}(premise)
4. Ig (premise)
5. ~Rg (IP)
6. Ig ∧ ~Rg (4,5 Conj)
7. (Ig ∧ ~Rg) ⊃ ©Rg (1 UI)
8. ©Rg (6,7 MP)
9. G[Rg,~Rg] (2 UI)
10. ~Rg ∧ G[Rg,~Rg] (5,9 Conj)
11. [~Rg ∧ G[Rg,~Rg]] ∧ ©Rg (8,10 Conj)
12. {[~Rg ∧ G[Rg,~Rg]] ∧ ©Rg} ⊃ ©Gg (3 UI)
13. ©Gg (11,12 MP)
14. (∃x){{[~©Gx ∧ ~©(∃y)Gyx] ∧ (∀z){[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = x)]}} ∧ ©Gx} (13 theory of descriptions)
15. {[~©Gμ ∧ ~©(∃y)Gyμ] ∧ (∀z){[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = μ)]}} ∧ ©Gμ (14 EI)
16. {(∀z){[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = μ)]} ∧ [~©Gμ ∧ ~©(∃y)Gyμ]} ∧ ©Gμ (15 Comm)
17. {(∀z){[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = μ)]} ∧ [~©(∃y)Gyμ ∧ ~©Gμ]} ∧ ©Gμ (16 Comm)
18. {(∀z){[[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = μ)]} ∧ ~©(∃y)Gyμ] ∧ ~©Gμ} ∧ ©Gμ (17 Assoc)
19. (∀z){[[~©Gz ∧ ~©(∃y)Gyz] ⊃ (z = μ)]} ∧ ~©(∃y)Gyμ] ∧ {~©Gμ ∧ ©Gμ} (18 Assoc)
20. ~©Gμ ∧ ©Gμ (19 Simp)
21. ~~Rg (5-20 IP)
22. Rg (21 DN)

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.

A Formal Version of the Third Way

I believe by using mereological sums, I avoid the charge of the quantifier shift fallacy.

D1: God is the x such there is not some y by which x receives the necessity it has, and x is a member of the essentially ordered causal series by which things receive their necessity .
P1. For all x, if it is possible that x does not exist, then there is a time at which x does not exist.
P2. If there is a time at which the mereological sum of everything does not exist, then there does not exist now the mereological sum of everything.
P3. If there exists now some x, then there exists now the mereological sum of everything.
P4. I exist now.
P5. If necessarily there exists the mereological sum of everything, then there is some x that necessarily exists, and x is a part of the mereological sum of everything.
P6. If there is some x that necessarily exists, then if for all x, x necessarily exists, then there is some y such that x receives the necessity it has from y, only if there is an essentially ordered causal series by which things receive their necessity and it does not regress finitely.
P7. For all z it is not the case that there is an x, such that both x is a member of the essentially ordered causal series by which things receive z and it is not the case that z regresses finitely.
P8. For all x, if x necessarily exists, then x is a member of the essentially ordered causal series by which things receive their necessity.
P9. For all x, if there is not some y by which  x receives the necessity it has, and x is a member of the essentially ordered causal series by which things receive their necessity, then for all z, there is not some y by which z receives the necessity it has, and z is a member of the essentially ordered series by which things receive their necessity, and z is identical to x.
C1. God necessarily exists.

Note: D1 tells us that God does not receive his necessity from any other cause, but, being a part of the causal series by which things receive their necessity, is the cause of necessity in other things.

Let:
E!x ≝ x exists
E!t ≝ x exists at time t
Fx ≝ x regresses finitely
Oxy ≝ x is a member of essentially ordered causal series y
Rxy ≝ x receives the necessity it has from y
σ<x,P> ≝ the mereological sum of all x that P.
σ<e,E!> ≝ (∀x)[E!x ⊃ (x ≤ e)] & (∀y)[(y ≤ e) ⊃ (∃z)(E!z & (y ⊗ z)]1
e ≝ everything
g ≝ (ɿx)[~(∃y)Rxy & Oxl]
i ≝ I (the person who is me)
l ≝ the causal series by which things receive their necessity
n ≝ now

1. (∀x)[♢~E!x ⊃ (∃t)~E!tx] (premise)
2. (∃t)~E!tσ<e,E!> ⊃ ~E!nσ<e,E!> (premise)
3. (∃x)E!nx ⊃ E!nσ<e,E!>(premise)
4. E!ni (premise)
5. ☐E!σ<e,E!> ⊃ (∃x)[☐E!x &(x ≤ e)] (premise)
6. (∃x)☐E!x ⊃ {(∀x)[☐E!x ⊃ (∃y)Rxy] ⊃ (∃x)[Oxl & ~Fl]} (premise)
7. (∀z)~(∃x)[Oxz & ~Fz] (premise)
8. (∀x)[☐E!x ⊃ Oxl] (premise)
9. (∀x){[~(∃y)Rxy & (Oxl & Fl)] ⊃ (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = x)]} (premise)
10. ♢~E!σ<e,E!> (IP)
11. ♢~E!σ<e,E!> ⊃ (∃t)~E!tσ<e,E!> (1 UI)
12. (∃t)~E!tσ<e,E!> (10,11 MP)
13. ~E!nσ<e,E!> (2,12 MP)
14. (∃x)E!nx (4 EG)
15. E!nσ<e,E!> (3,14 MP)
16. E!nσ<e,E!> & ~E!nσ<e,E!> (13,15 Conj)
17. ~♢~E!σ<e,E!> (10-16 IP)
18. ☐E!σ<e,E!> (17 ME)
19. (∃x)[☐E!x &(x ≤ e)] (5,18 MP)
20. ☐E!μ & (μ ≤ e) (19 EI)
21. ☐E!μ (20 Simp)
22. (∃x)☐E!x (21 EG)
23. (∀x)[☐E!x ⊃ (∃y)Rxy] ⊃ (∃x)[Oxl & ~Fl] (6,22 MP)
24. ~(∃x)(Oxl & ~Fl)] (7 UI)
25. ~(∀x)[☐E!x ⊃ (∃y)Rxy] (23,24 MT
26. (∃x)~[☐E!x ⊃ (∃y)Rxy] (25 QN)
27. (∃x)~[~☐E!x ∨ (∃y)Rxy] (26 Impl)
28. (∃x)[~~☐E!x & ~(∃y)Rxy] (27 DeM)
29. ~~☐E!ν & ~(∃y)Rνy (28 EI)
30. ☐E!ν & ~(∃y)Rνy (29 DN)
31. ☐E!ν (30 Simp)
32. ☐E!ν ⊃ Oνl (8 UI)
33. Oνl (31,32 MP)
34. ~(∃x)[Oxl & ~Fl] (7 UI)
35. (∀x)~[Oxl & ~Fl] (34 QN)
36. ~[Oνl & ~Fl] (35 UI)
37. ~Oνl ∨ ~~Fl (36 DeM)
38. ~~Oνl (33 DN)
39. ~~Fl (37,38 DS)
40. Fl (39 DN)
41. ~(∃y)Rνy (30 Simp)
42. Oνl & Fl (33,40 Conj)
43. ~(∃y)Rνy & (Oνl & Fl) (41,42 Conj)
44. [~(∃y)Rνy & (Oνl & Fl)] ⊃ (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = ν)] (9 UI)
45. (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = ν)] (43,44 MP)
46. ~(∃y)Rνy & Oνl (33,41 Conj)
47. [~(∃y)Rνy & Oνl] & (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = ν)] (45,46 Conj)
48. [~(∃y)Rνy & Oνl] & (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = ν)] & ☐E!ν (31,47 Conj)
49. (∃x){[~(∃y)Rxy & Oxl] & (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = x)] & ☐E!x} (48 EG)
50. ☐E!g (49 Theory of Descriptions)

QED

1Formulation of definition for everything based influenced by Filip, H. (n.d.) “Mereology”. Online: https://user.phil-fak.uni-duesseldorf.de/~filip/Mereology.pdf

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