Anselm without Defining God

One of the most common objections that I hear to my ontological arguments is that they use definitions that beg the question.  However, I am careful to note four things that I think block the charge that I “define God into existence”: 1) I require that stipulated definitions be defended as coherent, 2) I specify that I am setting my arguments within the context of free logic, 3) my definitions cannot directly entail, or be semantically equivalent to the conclusion, and 4) I must provided at least one premise that is justified independently from any definition of God, or from the conclusion.

However, I think the concern over the definition is overblown, and we could just derive the conclusion that there is at most one being such that it is not conceivable that there is something greater.  Call it “God” or “Banana Smoothie”.  It really doesn’t matter.  A term like God is emotionally loaded anyways, so maybe there is some rhetorical strategy in abandoning the word “God” altogether.

Here is the argument:

P1) For all x, if x is that than which none greater can be conceived, and there is some other z, which is that than which none greater can be conceived, and x and z are not the same, then  it is conceivable that there is something that can be combined with x as a mereological sum to make a composite whole of it and x as proper parts.

P2) For all x, and all z1, if it is conceivable that there is some mereological sum, which is the whole composed of x and z1 as proper parts, then conceivably there is some thing greater than x (namely the whole, out of which x is a proper part)

P3) All things that are not fictional beings are things that exist in reality.

P4) All fictional beings are things of which a greater can be conceived.

P5) There is something than which none greater can be conceived and either it is fictional or it is not fictional.

C) There is exactly one being in reality such that it is not conceivable that there is something greater.

Defense of Premises:

We have to at least define some predicates, so let,

Fx ≝ x is a fictional being
Rx ≝ x exists in reality
Gxy ≝ x is greater than y
∑xyz ≝ x is the mereological sum of  the proper parts, y and z
©… ≝ it is conceivable that…

1. (∀x){~©(∃y)Gyx ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = x)]} ⊃ ©(∃z1)(∃y)∑yxz1} (premise)
2. (∀x)(∀z1){©(∃y)∑yxz1] ⊃ ©(∃y)Gyx}(premise)
3. (∀x)(~Fx ⊃ Rx) (premise)
4. (∀x)(Fx ⊃ ©(∃y)Gyx) (premise)
5. (∃x)[~©(∃y)Gyx ∧ (Fx ∨ ~Fx)] (premise)
6. (∃x){~©(∃y)Gyx ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = x)]} (IP)
7. ~©(∃y)Gyμ ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = μ)] (6 EI)
8. {~©(∃y)Gyμ ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = μ)]} ⊃ ©(∃z1)(∃y)∑yμz1 (1 UI)
9. ©(∃z1)(∃y)∑yμz1 (7,8 MP)
10. ©(∃y)∑yμν (9 EI)
11. (∀z1)[©(∃y)∑yμz1 ⊃ ©(∃y)Gyμ (2 UI)
12. ©(∃y)∑yμν ⊃ ©(∃y)Gyμ (11 UI)
13. ©(∃y)Gyμ (10,12 MP)
14. ~©(∃y)Gyμ (7 Simp)
15. ©(∃y)Gyμ ∧ ~©(∃y)Gyμ (13,14 Conj)
16. ~(∃x){~©(∃y)Gyx ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = x)]} (6-15 IP)
17. (∀x)~{~©(∃y)Gyx ∧ (∃z)[~©(∃y)Gyz ∧ ~(z = x)]} (16 QN)
18. (∀x){~~©(∃y)Gyx ∨ ~(∃z)[~©(∃y)Gyz ∧ ~(z = x)]} (17 DeM)
19. (∀x){©(∃y)Gyx ∨ ~(∃z)[~©(∃y)Gyz ∧ ~(z = x)]} (18 DN)
20. (∀x){©(∃y)Gyx ∨ ~(∃z)~[©(∃y)Gyz ∨ (z = x)]} (19 DeM)
21. (∀x){©(∃y)Gyx ∨ (∀z)[©(∃y)Gyz ∨ (z = x)]} (20 QE)
22. (∀x){©(∃y)Gyx ∨ (∀z)[~~©(∃y)Gyz ∨ (z = x)]} (21 DN)
23. (∀x){©(∃y)Gyx ∨ (∀z)[~©(∃y)Gyz ⊃ (z = x)]} (22 Impl)
24. (∀x)(~©(∃y)Gyx ⊃ Fx) (IP)
25. ~©(∃y)Gyμ ∧ (Fμ ∨ ~Fμ) (5 EI)
26. ~©(∃y)Gyμ ⊃ Fμ (24 UI)
27. Fμ ⊃ ©(∃y)Gyμ (4 UI)
28. ~©(∃y)Gyμ ⊃ ©(∃y)Gyμ (26,27 HS)
29. ~©(∃y)Gyμ (25 Simp)
30. ©(∃y)Gyμ (28,29 MP)
31. ©(∃y)Gyμ ∧ ~©(∃y)Gyμ (29,30 Conj)
32. ~(∀x)(~©(∃y)Gyx ⊃ Fx)(24-31 IP)
33. (∃x)~(~©(∃y)Gyx ⊃ Fx)(32, QN)
34. (∃x)~(~~©(∃y)Gyx ∨ Fx) (33 Impl)
35. (∃x)~(©(∃y)Gyx ∨ Fx)(34 DN)
36. (∃x)(~©(∃y)Gyx ∧ ~Fx) (35 DeM)
37. ~©(∃y)Gyμ ∧ ~Fμ (36 EI)
38. ~Fμ ⊃ Rμ (3 UI)
39. ~Fμ (37 Simp)
40. Rμ (38,39 MP)
41. ~©(∃y)Gyμ (37 Simp)
42. ©(∃y)Gyμ ∨ (∀z)[~©(∃y)Gyz ⊃ (z = μ)] (23 UI)
43. (∀z)[~©(∃y)Gyz ⊃ (z = μ)] (41,42 DS)
44. ~©(∃y)Gyμ ∧ (∀z)[~©(∃y)Gyz ⊃ (z = μ)] (41,43 Conj)
45. ~©(∃y)Gyμ ∧ (∀z)[~©(∃y)Gyz ⊃ (z = μ)] ∧ Rμ (40,44 Conj)
46. (∃x){~©(∃y)Gyx ∧ (∀z)[~©(∃y)Gyz ⊃ (z = x)] ∧ Rx} (45 EG)

But now 46 say that there is exactly one being than which none greater exists and it exists in reality.

QED

Suppose we add to our lexicon:

g ≝ (ɿx)~©(∃y)Gyx

Then we could easily reach:

47. Rg (46 theory of descriptions given the def of “g”)

And this is precisely the conclusion I reach here.  So using a definite description at the outset saves space, but requires additional defenses for the premise that are made explicit here.

Posted on February 20, 2020, in Uncategorized. Bookmark the permalink. Leave a comment.

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