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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)

The Ontological Argument From Transcendence 2.0

I’ve presented my own version of Anselm’s ontological argument here and I’ve also argued for an ontological argument using “more transcendent” rather than “greater” here. Combining the two, and refining the argument, I got this:

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

2. There is something conceivable such that nothing can be conceived of which is more transcendent (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 transcendent than x (premise). Therefore,

5. An Anselmian God exists.

Let

Cx – x is conceived
Mx – x is mentally dependent
Txy – x is more transcendent than y
Θx- x is an Anselmian God, that is: (∀x){Θx ≝ ([♢Cx & ~(∃y)(Tyx & ♢Cy)] & ☐(∃z)(z=x))} (Def Θ)

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

[Update 11/9/204] I’ve noticed that some did not understand why if possibility that failing to conceive x implied that x possibly didn’t exist, then a greater could be conceived than x.  I’ve tried to make this more explicit by explaining this in terms of mental dependence.  Here, a concept is not an abstract object, but an object in the mind.

A Voltairean Argument for God’s Existence

This post is a variation on what I have called an indispensability argument.  My original formulation can be found here, and I have made some further comments here. In this post, I thought I would do a take on the argument using Voltaire’s famous dictum “Si Dieu n’existait pas, il faudrait l’inventer” as an explicit premise (Epistle to the author of the book, The Three Impostors, 1768).  As an aside, it is commonly supposed that, since Voltaire was critical of organized religion, he was an atheist.  Voltaire was a deist. In fact, in the poem where he says that if God does not exist, it would be necessary to invent him, Voltaire doesn’t merely refer to God as some generic super-being, but as the “supreme essence.”  So it seems that he has something like a perfect being, or the God of the philosophers, in mind, at least in this poem.  I find the following argument cogent, and I think historical reflection makes the premises plausible. Thoughts are always appreciated, of course, though I anticipate some objections below.

  1. If God does not actually exist, it is necessary to invent the concept of God. [Voltairean Premise]
  2. For all x, if it is necessary to invent the concept of x, the concept of x is logically coherent. [Premise]
  3. If the concept of God is logically coherent, God actually exists. [By S5 and the logical possibility of God as a perfect being is necessarily existent and essentially perfect]
  4. For all x, x does not actually exist, or x actually exists. [Law of the Excluded Middle]

Therefore, God actually exists. Proof:

  1. Either God actually exists, or it is necessary to invent the concept of God. [From 1 Material Implication]
  2. If it is necessary to invent the concept of God, the concept of God is logically coherent. [From 2 Universal Instantiation]
  3. If it is necessary to invent the concept of God, God actually exists. [From 3, 6 Hypothetical Syllogism]
  4. God does not actually exist, or God actually exists. [From 4 Universal Instantiation]
  5. If God actually exists, God actually exists. [From 8 Material Implication]
  6. Either God actually exist, or God actually exists. [From 5, 7, and 9 Constructive Dilemma]
  7. God actually exists. [From 10 Tautology]

In this argument, I want to grant the Voltairean Premise, though I suspect most atheists would attack it with a Laplacean counter that “I have no need for that hypothesis.” Indeed, Laplace did not have a need to invoke God to explain the motion of the planets, but I don’t think that was Voltaire’s point. Rather, he was talking about the need of the concept of God for social cohesion. But, I think the concept of God plays a larger role than merely grounding natural law for a social contract, or putting the fear of hellfire into the hearts of the criminally minded and depraved. There is a necessity of God in many aspects of philosophical speculation. It is out of the concept of God that various philosophical concepts found further development, such as the notions of free will, personhood, simplicity, and aseity. The concept of God has helped thinkers clarify concepts surrounding Being, substance, essence, the relationship between eternality and time, etc. I suspect that the concept of God, a perfect being, was necessary in the intellectual development of our civilization. Whether one thinks that God is currently necessary to ground human rights and dignity, natural law, it happened that way historically. So it is important to note that the concept of this God, the God of the philosophers, is one that is both maximally great and fecund.

If the concept of God, a perfect being, is incoherent then such a history would be surprising, since incoherent concepts are not really all that necessary for anything. I take a concept to be incoherent if the sense of the concept is implicitly contradictory. Such a concept would not be any more necessary for deriving other philosophically interesting concepts than any other incoherent concept. For, impossibilities trivially imply anything and everything. Nonetheless, real work and reflection has gone into inferring the attributes and implications of the God of the philosophers. It is true that theologians and philosophers have come to contrary conclusions from the concept of God, but the steps by which they reach those contrary conclusions are comprehensible and not merely based on an explicit use of the principle of explosion or by being arbitrary. Often times the dispute is based on one philosopher taking an attribute or the concept of perfection to have a different sense than what another philosopher thinks. They genuinely disagree. So, it is not the case that they are simply picking out contradictions—contradictions that they would bashfully agree are there within the concept of God all along— and reaching contrary conclusions. They are actually disagreeing on basic definitions of terms that they think are implied by perfection. That being said, there is some consensus that has grown around perfect being theology. For instance, God’s power does not imply the ability to do the logically impossible, and God cannot make free-agents always do what is morally right. There are still genuine disputes over the details of a maximally great being, or a perfect being, but few would dispute that such a being would be necessary, omnipotent, omniscient, and morally perfect. Some dispute whether God’s omniscience is propositional. For instance, if God is absolutely simple, there can be no composition in God’s knowledge, and so it cannot be based on the composition of subject terms and predicates in the mind of God. But these sorts of disputes are not willy-nilly where anything goes. The disputes are rigorous, and based on careful definitions. So, it seems to me that while the concept of God is not settled upon by all philosophers, there are definite rules around how to do natural theology, and limits upon what the God concept entails.

One might also argue that, though the concept of God, the God of the philosophers, is incoherent, various aspects of the concept are coherent, and it is those aspects that have been fecund in the history of philosophy.  This doesn’t seem to be the case, however.  Rather, it is typically the confluence of various divine attributes that have generated so many ideas.  Perhaps even more to the point, if classical theism and absolute divine simplicity are granted, then these philosophers are not really considering a confluence of God’s attributes, but one essence that reveals itself to us an a variety of ways.  That thought itself has produced some of the most penetrating theology in Judaism, Christianity, and Islam.  The entire Summa Theologiae is built upon this foundation, brick by syllogistic brick.  Perhaps that brick was all straw, but the Summa itself has produced entire schools of philosophy and theology. I think the concept of God was a historical necessity and philosophical necessity, one born out of a reflection on the divine, the perfect, and the infinite. That this concept is both necessary and incoherent would be surprising since, as I have said, any incoherent concept could do the job of generating random inferences.  I don’t think that is what the concept of God has been doing in our history.  I don’t think it is there as an incoherency from which surprising and profound thoughts emerge.  The concept of God gives us traction in a way that a round-square does not.  Round-squares or squared-circles might in some sense be “meaningful” concepts, but they are not necessary to invent.  In fact, it is not entirely clear that they are concepts, at least in the sense that they can be conceived in the mind.  It seems more the case that one is conceiving of roundness and squareness and noting that they cannot be predicated of the same Euclidean plane figure.  They are more an oddity, a conceptual contradiction.  Their use is merely as a stand-in for any obvious instance of an impossibility.

As for premises (3) and (4), they are relatively uncontroversial rules of logic, and I will not go into defending them here. I know some people lament S5, but the issue is not whether the axiom is true, but rather, whether we genuinely know whether something is logically possible rather than, say, merely epistemically possible. I think my defense of (2) makes it clear that I am not merely saying that the God concept is conceivable, but that it contains no incoherency in it. So, I think that if Voltaire’s dictum is right, and the necessity of a concept implies its coherence, we have good reason to think God actually does exist.

An Argument from Transcendence

In a previous post, I attempted a version of the ontological argument that makes use of a comparative relation other than “greater than.” In the argument, I used “more actual.” Of course, I meant “actual” in a sort of Aristotelian-Thomistic sense, and so it depends upon understanding that particular set of metaphysical jargon. It occurs to me that one might make use of other comparatives to similar effect. This post can be considered a “Part 2,” as I attempt a similar argument with “more transcendent” as the comparative (though I have made a few modifications to the original version that I think strengthen it). Of course, there will still be some metaphysical unpacking to do. As I have said, it is impossible to avoid metaphysics when considering arguments for God’s existence. First, we must consider what it means to “transcend” and why it might be appropriate to define God this way. For, if an ontological argument is to be successful, the definition must at least implicitly contain the traditional divine attributes.  So we must consider if “transcendence” entails those other attributes.  I think it does. That which transcends goes beyond some limit, whether it is the physical laws, space, time, or any of the fundamental categories of existence. God is often defined as transcendent, but not in the sense that He is completely detached from us and in no way relates—transcendence and immanence seem to be related attributes of omnipresence, but from different perspectives.. For, in a sense, to say that God lacks immanence is to say that God cannot transcend those limits back into our finite existence. So I don’t see transcendence and immanence as true opposites, but as two perspectives by which one can refer to the omnipresence of God. The concept of transcendence can also help us understand other divine attributes. God is said to be omniscient, transcending anything that would limit knowledge. Likewise God is omnipotent, transcending, for instance, the physical laws that limit the amount of power finite creatures possess. God is also morally perfect, transcending anything that might limit His ability to perfectly express his being good and loving towards others.  So a if we conceive of God as that than which none more transcendent can be conceived, it seems that we will arrive at a being that exists infinitely perfect and a se (self-existent and not limited to depending on other things to exist). Such a being would be omnipotent, omniscient and morally perfect (so personal), and transcending time and space.  Now one might consider whether it is conceivable that God transcends existence itself.  Some theological traditions flirt with this idea, but I strongly suspect that the idea that something transcends existence is logically incoherent.  That is, it would be something not limited by being something.  I have no idea what that would mean, but if it means that God is something or someone who doesn’t exist, then it is incoherent.  It may just be that “transcending existence” is just meaningless. Anselm defined God as that than which none greater can be conceived, and I think we can understand “greatness” in terms of “transcendence.” As I mentioned in a previous blog, “greatness” can be difficult for some people to grasp. Is it supposed to be defined subjectively? For Anselm, “greatness” was conceived in terms of the Great Chain of Being, and so was an objective evaluation of existing things. But today, most people think of “greatness” as something that is in the eye of the beholder–and opinion with no factual basis whatsoever. When I have presented Anselm’s argument in the classroom, a few of my students inevitably ask, “Why must anyone agree that it is greater to exist in reality than in the mind alone?” Responding with neo-platonic metaphysics might help the student to understand what Anselm was thinking, but it makes the argument seem irrelevant, antiquated, and weak. Transcendence, though, is more clearly an objective feature. One may simply observe whether some being goes beyond a certain limit. Now one might say that it makes no sense to say that there is a limit if something transcends the limit. But we speak of things moving beyond limits all the time. Voyager I has recently transcended the limits of our solar system. When you drive too fast, you transcend the legal limit at which you can drive (though you might not use such a grandiose term as “transcend” when the cop pulls you over for speeding). So the existence of something surpassing a limit is not inconsistent with there being limits. The physical laws of nature are physical limits on the way natural objects can behave. If naturalism is true, there is nothing that transcends those limits. Whether something transcends those limits is an objective question, not a question dependent upon one’s opinions, desires, or tastes. So perhaps we can modify Anselm’s definition and say that God is that than which none more transcendent can be conceived. Just as Anselm initially invites us to consider, we may ask ourselves if such a God could merely exist in the mind, as some idea, imaginary thought, or mental construct. Surely ideas, imaginary thoughts, and mental constructs are limited by the mind in which they inhabit. They are limited not only by the mental capacity of the mind conceiving of them, but also insofar as their existence depends upon and so are limited by the very existence of minds. An argument from transcendence would look like this:

  1. God is that than which none more transcendent can be conceived. [Definition]
  2. If God exist only as an idea in the mind, something can be conceived more transcendent than God. [Premise]
  3. If it is not the case that something exists only as an idea in the mind, then it exists as an extra-mental reality. [Premise]
  4. If something can be conceived more transcendent than God, something can be conceived more transcendent than God. [Tautology]
  5. If something can be conceived more transcendent than God, something can be conceived more transcendent than that which none more transcendent can be conceived. [From 1, 4 Definition]
  6. If something can be conceived more transcendent than that than which none more transcendent can be conceived, then that than which none more transcendent can be conceived is not that than which none more transcendent can be conceived. [Premise]
  7. It is not the case that that than which none more transcendent can be conceived is not that than which none more transcendent can be conceived. [Law of Non-Contradiction]
  8. It is not the case that something can be conceived more transcendent than that which none more transcendent can be conceived. [From 6, 7 Modus Tollens]
  9. It is not the case that something can be conceived more transcendent than God. [From 5, 8 Modus Tollens]
  10. It is not the case that God exists only as an idea in the mind. [From 2, 9 Modus Tollens]

Therefore,

  1. God exists as an extra-mental reality. [From 3, 10 Modus Ponens]

Is this argument subject to parody? I think not, and much for the same reason I did not think an ontological argument using the comparative “more actual” is susceptible to parody. Consider, for instance, Gaunilo’s perfect island. It seems incoherent to define an island as “that island than which none more transcendent can be conceived.” The very nature of an island is such that it does not transcend the limits of water on all of its sides. Otherwise, it would cease to be an island! “Very well,” you might think, “let’s say that it does not transcend watery limitations, but it does transcend all other limits.” Well, which ones must such an island transcend? Must it transcend all limits as to the amount of sand it has on its beach? Would it have an actual infinity of sand on the beach? Perhaps there is some physical restriction you would want to place on the amount of sand, otherwise the gravitation would be so great that it would be more like a super-massive black hole than an island resort. Ah, but it transcends all physical limitations, and so it would not be bound to obey gravity or other physical laws. But now it is sounding less and less like an island, which seems to at least be bound by physical laws to do with water and land. Would it be limited such that it could not be conscious? Would it be limited in power? If you grant that it would, so that it could remain island-like, then it becomes more and more ad hoc that you should insist that one of the ways in which the island than which none more transcendent is transcendent is insofar as it must exist beyond the mental. If you insist that it would be conscious, even all knowing, and all powerful to boot, then it sounds less and less like you are really talking about an island, and more and more like you are really talking about God. Perhaps you are really just saying that God could choose to manifest himself as a physical island. And perhaps this is true. But then the island isn’t so much a parody as it is a reiteration of the actual proof, while insisting that we consider God in this odd manifested form. So, if we strictly hold to the concept of “island” it is not clear that the concept of an “island than which none more transcendent can be conceived” is coherent, or physically possible. If we ditch the idea that it really is an island, as islands are traditionally conceived, the parody crumbles apart and just becomes a reiteration of the proof. There are, of course, other objections to ontological arguments. Perhaps you could mention your objections to the argument in the comments below.

None More Actual

1579 drawing of the Great Chain of Being from Didacus Valades, Rhetorica Christiana (Wikipedia, Great chain of being)

Whenever I discuss the ontological argument with my atheistic friends, I find that they always get hung up on the same word, “greater”. They want to infuse it with moral or aesthetic meaning, and so suspect that it is subjectively defined. They don’t think there is any objective way to determine that one thing is ontologically greater than another (a flea is no greater than a child and the fact that you would swat one and not the other is just based on speciesist opinions). Indeed, to fully explain what Anselm meant by the definition, we would have to develop the neo-platonic notion of the Great Chain of Being, which is far more central to the argument than most contemporary philosophers of religion realize. Nonetheless, that requires some metaphysical assumptions from which many atheists will shy away. I want to sidestep that whole discussion by using something other than “greater.” My proposal is to run the ontological argument on a “more actual” relation. I think you can still derive the traditional divine attributes from this term, but it doesn’t suffer from seeming subjective (what is more actual is an objective question).  Nonetheless, understanding what is meant by “actual” will require some metaphysics.  When discussing proofs for God, metaphysics is inescapable.

What do I mean by “more actual”? I am appealing to the distinction between act and potency in the Aristotelian-Thomistic sense of the word. For Thomas, God is the only being that is purely actual. This is because God’s essence is His existence. God is “I am”. The distinction between act and potency is an important one in the history of philosophy. It is that distinction, which allowed Aristotle to provide a response to the Eleatics, who denied change. The Eleatics argued that change was impossible because it would have to involve being arising from non-being. Since nothing comes from nothing, change cannot arise from non-being.  Instead, Aristotle said that change occurs when a potential is actualized. So, a seed can become a plant because it is potentially a plant. And it undergoes that change when it is acted upon by actual things like water, soil, heat, etc.  We see change happen all around us, and it is rooted in the nature of things.  For instance, I am potentially bald, a potential that I am slowly actualizing with every lost hair follicle.  So, while act and potency are metaphysical concepts, they are fairly close to our commonsense.  The log is potentially fire, smoke, and ash.  The log is actually hard and damp.

An ontological argument that exploits the notion of actuality is a bit odd and perhaps shocking for my Thomistic friends. It is commonly thought that Thomas Aquinas did not accept the soundness of such arguments, a point that I am not going to discuss here. Nonetheless, I think the premises of such an argument could be defended. The argument would run like this:

1. God is that than which none more actual can be conceived (definition).
2. If God exists only in the mind, something more actual than God can be conceived (premise).
3. If something more actual than God can be conceived, something more actual than God can be conceived (tautology).
4. If something more actual than God can be conceived, something more actual than ‘that which none more actual can be conceived’ can be conceived (from 1 and 3).
5. Nothing more actual than ‘that which none more actual can be conceived’ can be conceived (premise).
6. Therefore, it is not the case that God exists only in the mind (from 2,4,5).
7. If it isn’t the case that something exists only in the mind, then it exists in reality (premise).
8. Therefore, God exists in reality (from 6 and 7).

Now, there are a few premises and a definition. The definition, I think, is fair. Aquinas takes great pains to show that whatever is pure actuality has the divine attributes. So a being than which none more actual can be conceived would be purely actual, and so simple, a se, necessary, immutable, eternal, omnipotent, omniscient, and good.

Furthermore, I think (2) is defensible. Generally that which exists merely as a conception is less actual, in some way, than its counterpart in reality. You can’t be cut by the thought of a knife.  Also, (5) seems plausible. For if something more actual than ‘that than which none more actual can be conceived’, a contradiction arises. Lastly, all that is meant in (7) is that if something doesn’t just exist in the mind, that means it exists independently of our minds, which is to say that it exists in reality.  I suspect someone might say that it is a false dichotomy to insist that if something doesn’t just exist in the mind, then it must exist in reality, but I can’t think of any alternative.  And if an alternative could be found, I am sure the argument could be adjusted in the relevant ways.

One last note is to consider whether this argument is susceptible to parody.  I think it is less susceptible.  Consider Gaunilo’s island.  Could we define an island than which none more actual can be conceived?  Well, every island is a composite of act and potency by nature.  So no island can be maximally or purely actual.  One can admit that islands that exist in reality are more actual than islands that exist in the mind, but this does not mean that ‘an island than which none more actual can be conceived’ would necessarily exist, since there is no such thing.  There are, at best, islands that are more actual than other islands, but that doesn’t lead to parody.

A Mystico-Ontological Argument

I was considering the idea of evidence in the last post. This argument occurred to me.  Criticisms, as always, are welcome (Also, this is my 100th post!!):

A Mystico-Ontological Argument

1. If the probability of a hypothesis is greater on a given piece of evidence than the probability of the hypothesis alone, and no fact makes the evidence impossible, then the probability of the hypothesis, given the evidence, is greater than zero.
2. If the probability of a hypothesis given the evidence is greater than zero, then possibly the hypothesis is true.
3. There is a hypothesis that necessarily there exists an all-perfect being.
4. There is evidence from the testimony from those who have had a mystical experience of an all-perfect being.
5. The probability of the hypothesis that there necessarily exists an all perfect being is greater on the testimonial evidence from mystical experience than the probability of the hypothesis alone.
6. No fact makes the testimonial evidence of the mystical experience of an all-perfect being impossible.
7. Therefore, an all-perfect being exists.

Deduction1
Let:
P(h|e) – the probability of hypothesis h given evidence e
P(h) – the unconditioned probability of h
Πx – x is all-perfect
Tx – x is testimony that one has mystical experienced an all-perfect being

1. (∀h)(∀e){[(P(h|e) > P(h)) & ~(∃φ)(P(e|φ) = 0)] → (P(h|e) > 0)} (premise)
2. (∀h)(∀e)[(P(h|e) > 0) → ◊h] (premise)
3. (∃h)(∃e){[(h = ☐(∃x)Πx) & (e = (∃y)Ty)] & [(P(e|h) > P(h)) & ~(∃φ)(P(e|φ) = 0)]} (premise)
4. (∃e){[(h = ☐(∃x)Πx) & (e = (∃y)Ty)] & [(P(e|h) > P(h)) & ~(∃φ)(P(e|φ) = 0)]} (3 EI)
5. [(h = ☐(∃x)Πx) & (e = (∃y)Ty)] & [(P(e|h) > P(h)) & ~(∃φ)(P(e|φ) = 0)] (4 EI)
6. [(P(e|h) > P(h)) & ~(∃φ)(P(e|φ) = 0)] (5 Simp)
7. (∀e){[(P(h|e) > P(h)) & ~(∃φ)(P(e|φ) = 0)] → (P(h|e) > 0)} (1 UI)
8. [(P(h|e) > P(h)) & ~(∃φ)(P(e|φ) = 0)] → (P(h|e) > 0) (7 UI)
9 (P(h|e) > 0) (6,8 MP)
10. (∀e)[(P(h|e) > 0) → ◊h] (2 UI)
11. (P(h|e) > 0) → ◊h (10 UI)
12. ◊h (9,11 MP)
13. (h = ☐(∃x)Πx) & (e = (∃y)Ty) (5 Simp)
14. h = ☐(∃x)Πx  (13 Simp)
15. ◊☐(∃x)Πx (12,14 ID)
16. ☐(∃x)Πx (15 axiom S5)
17. (∃x)Πx (16 NE)

Support for the premises:

Premise 1: This premise tells us that when there is evidence that raises the probability of a hypothesis even slightly, then the probability of the hypothesis on the evidence cannot equal zero. To counter the charge that the evidence is only apparent, I’ve added the condition that there should not be any fact that makes the evidence, itself, impossible.

Premise 2: If the probability for some hypothesis is greater than zero it has to be possible. Put another way, if a hypothesis is impossible, the probability for the hypothesis is not greater than zero. And that is just what it means to be impossible.

Premise 3: This tells us that there is a hypothesis on the table, namely that necessarily there exists a maximally great being. That is the God hypothesis, which is the central dispute in this debate. To deny that there is even a God hypothesis is absurd, for it is to deny the debate altogether.

Premise 4: This tells us there is evidence for that hypothesis in the form of testimony from religious experience.  Indeed, Mark Webb (2011, Religious Experience) confirms, “Some subjects of religious experiences report… [experiences of] an infinitely perfect, personal creator.”  Here are some examples of such testimony: i) There is the mystical writing of St. John of the Cross, “O gentle touch, and most gentle, for you touch me with your most simple and pure essence, which being infinite is infinitely gentle, therefore it is that this touch is so subtle, so loving, so deep, and so delicious that it savors of eternal life” (St. John of the Cross,The Living Flame of Love, Stanza II, emphasis mine). ii) There is the mystical writings of Pseudo-Dionysius the Aeropagite, “Through these, Its incomprehensible Presence is manifested upon those heights of Its Holy Places; that then It breaks forth, even from that which is seen and that which sees, and plunges the mystic into the Darkness of Unknowing, whence all perfection of understanding is excluded, and he is enwrapped in that which is altogether intangible, wholly absorbed in it that is beyond all, and in none else (whether himself or another); and through the inactivity of all his reasoning powers is united by his highest faculty to it that is wholly unknowable; thus by knowing nothing he knows That which is beyond his knowledge” (Mystical Theology, Ch. 1). Pseudo-Dionysius goes on to say, “…we can neither affirm nor deny it, inasmuch as the all-perfect and unique Cause of all things transcends all affirmation, and the simple pre-eminence of Its absolute nature is outside of every negation- free from every limitation and beyond them all” (Mystical Theology, Ch. 5, emphasis mine). iii) Augustine reports in the Confessions a mystical experience of God that he shared with his mother, Monica, in Ostia, “ Our colloquy led us to the point where the pleasures of the body’s senses, however intense and in however brilliant a material light enjoyed, seemed unworthy not merely of comparison but even of remembrance beside the joy of that life, and we lifted ourselves in longing yet more ardent toward That Which Is, and step by step traversed all bodily creatures and heaven itself, whence sun and moon and stars shed their light upon the earth. Higher still we mounted by inward thought and wondering discourse on your works, and we arrived at the summit of our own minds; and this too we transcended, to touch that land of never-failing plenty where you pasture Israel for ever with the food of truth. Life there is the Wisdom through whom all these things are made, and all others that have been or ever will be; but Wisdom herself is not made: she is as she always has been and will be forever. Rather should we say that in her there is no “has been” or “will be,” but only being, for she is eternal, but past and future do not belong to eternity. And as we talked and panted for it, we just touched the edge of it by the utmost leap of our hearts; then, sighing and unsatisfied, we left the first-fruits of our spirit captive there, and returned to the noise of articulate speech, where a word has beginning and end. How different from your Word, our Lord, who abides in himself, and grows not old, but renews all things” (Confessions IX, 24 emphasis mine). iv) Even the logical positivist and well-known atheist, A.J. Ayer, is reported to have had an religious experience of some sort, “I was confronted by a red light…Aware that this light was responsible for the government of the universe. Among its ministers were two creatures who had been put in charge of space…” (P. Foges 2010).  These mystical experiences of an infinite, all-perfect, self-abiding, eternal being is testimonial evidence that there is a being that has all-perfections, including omnipotence, omniscience, moral perfection, and necessary existence. That is precisely what our hypothesis is.

Premise 5: Testimonial evidence of the truth of some hypothesis raises the probability of that hypothesis higher than the hypothesis possesses intrinsically. This is testimonial evidence that some mystics have had experiences of a perfect personal God.

Premise 6: While one may be skeptical of such mystical experiences, or attempt to explain it away as a deception, neurological illusion, some other psychological delusion, or mere poetry none of these facts make the testimony that these people actual experiences a personal all-perfect being impossible. That is, attempts to explain this evidence away does not establish that there is zero probability that it is evidence at all. Nor is there any fact that makes a personal all-perfect being intrinsically possible, given that the plausibility that apparent inconsistencies in the divine attributes can be resolved, and arguments like Robert Maydole’s Modal Perfection argument or Alexander Pruss’s Gödelian Argument, which argues that positive perfections are compossible.

A Couple of Anticipated Objections:

1. One might attack (2) by saying that there is a shift between subjective probability and logical possibility, and that this is tantamount to shifting between conceivability and possibility. Then again, if the hypothesis is itself impossible, that should be established by the atheologian given the positive arguments for the coherence of h (found in 3D above). But given those positive arguments, combined with the testimonial evidence of mystics, we might say that we have some good positive reasons to think that the P(h|e) is higher than zero, and so possible. Perhaps this argument does shift between conceivability and logical possibility in (2). What it might tell us something interesting about our intuitions, namely, that if we have a sense that mystical testimony in anyway increases the probability that the hypothesis is true, then we should believe it. But if we have the sense that no amount of mystical evidence raises the probability that there is an all-perfect being, then that would be consistent with the impossibility of such a being.

2. One might argue that the testimonial evidence offered in (3) is not an encounter of an all-perfect, or maximally great being. But for this argument to be a defeater, they would need definitive proof that it was not, since even if it is the slightest bit probable that they did have such a genuine encounter, the probability of h is raised slightly, and we can conclude that h is possible.

1For the purposes of this argument, I’ve condensed premises 3-6 in natural language into the third premise of the formal deduction.