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.

Posted on November 27, 2015, in Arguments for God and tagged , , . Bookmark the permalink. Leave a comment.

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