A Cosmological Argument

The following Cosmological Argument is based on the arguments of Alexander Pruss and Richard M. Gale1, and Jerome Gellman2.

D1. God is the timeless, immaterial, personal, omnipotent, omniscient, omnibenevolent, free being who is worthy of worship, and is the explanation for why the big conjunctive contingent fact is true in the actual world.
P1. For all propositions p, if proposition p is contingent, then there is a possible world where something explains why the proposition is true.
P2. The big conjunctive contingent fact is contingent.
P3. The big conjunctive contingent fact is the conjunction of all actual contingent atomic facts true at the actual world.
P4. For all worlds w, and all things x, if x explains the big conjunctive contingent fact is true at w and the big conjunctive contingent fact is the conjunction of all actual contingent atomic facts true at the actual world, then world w is identical to the actual world.
P5. For all propositions p, and things x, if x explains why p is true in the actual world and p is contingent, then x exists necessarily.
P6. For all propositions p, and things x, if x explains why p is true in the actual world, and the proposition “Matter Exists” is an element of p, then x is immaterial.
P7. For all propositions p, and things x, if x explains why p is true in the actual world, and the proposition “Time exists” is an element of p, then x is eternal.
P8. For all propositions p, and things x, if x explains why p is true in the actual world, and the set of all atomic propositions describing natural laws that exists is an element of p, then x is not natural.
P9. For all propositions p, and things x, if x explains why p is true in the actual world, x is either natural or personal.
P10. For all propositions p, and things x, if x explains why p is true in the actual world and p is contingent and x exists necessarily, then x is free.
P11. For all things x, if x is free, then it is not the case that there is something z such that the actions of x entirely depend on z.
P12. For all propositions p1 and all things x1, if x1 explains why p1 is true in the actual world and there is a world w and a proposition p2 such that p2 is the conjunction of all actual contingent atomic facts at w, and there is some y that explains why p2 is true at world w, and x is not identical to y, then there is something z such that the actions of x entirely depends upon z.
P13. For all propositions p1 and things x, if x explains why p1 is true at the actual world, and it is not the case that there is a world w and a proposition p2 such that p2 is the conjunction of all actual contingent atomic facts at w, and there is some y that explains why p2 is true at world w, and x is not identical to y, then x is omnipotent.
P14. For all x, if x is omnipotent and x is personal, then x is all knowing.
P15. For all x, if x is omnipotent, x is all-knowing, and x is free, then x is omnibenevolent.
P16. For all x, if x is omnibenevolent, and x is personal, and x is omnipotent, then x is worthy of worship.
P17. For all x, if x is timeless, immaterial, personal, omnipotent, omniscient, omnibenevolent, free, worthy of worship, and explains why the big conjunctive contingent fact is true in the actual world, then if anything y is timeless, immaterial, personal, omnipotent, omniscient, omnibenevolent, free, worthy of worship, and explains why the big conjunctive contingent fact is true in the actual world, then x is identical to y.
P18. The proposition “Matter exists” is an element of the big conjunctive contingent fact.
P19. The proposition “Time exists” is an element of the big conjunctive contingent fact.
P20. The set of all atomic propositions that describe the natural laws that exist is an element of the big conjunctive contingent fact.
C. God necessarily exists.

Let

P ∈ Q ≝ proposition P is an element of proposition Q
E!x ≝ x exists
Fx ≝ x is free
Gx ≝ x is omnibenevolent
Kx ≝ x is omniscient
M̅x ≝ x is immaterial
Nx ≝ x is natural
Ox ≝ x is omnipotent
Sx ≝ x is personal
T̅x ≝ x is eternal
Wx ≝ x is worthy of worship
Bpw ≝ p is the conjunction of all actual contingent atomic facts is true at world w
Dxy ≝ the actions of x entirely depends upon y
Expw ≝ x explains that p is true at world w
a ≝ the actual world
β ≝ the big contingent conjunctive fact
g ≝ (ɿx){[(T̅x ∧ M̅x) ∧ (Sx ∧ Ox)] ∧ [(Kx ∧ Gx) ∧ (Fx ∧ Wx)] ∧ Exβa}
M ≝ Matter exists.
T ≝ Time exist.
ℕ ≝ the set of all atomic propositions describing natural laws that exist.

1. (∀p)[(◊p ∧ ◊~p) ⊃ (∃w)(∃x)Expw] (premise)
2. ◊β ∧ ◊~β (premise)
3. Bβa (premise)
4. (∀w)(∀x)[(Exβw ∧ Bβa) ⊃ (w = a)] (premise)
5. (∀p)(∀x){[Expa ∧ (◊p ∧ ◊~p)] ⊃ □E!x} (premise)
6. (∀p)(∀x){[Expa ∧ (M ∈ p)] ⊃ M̅x} (premise)
7. (∀p)(∀x){[Expa ∧ (T ∈ p)] ⊃ T̅x} (premise)
8. (∀p)(∀x){[Expa ∧ (ℕ ∈ p)] ⊃ ~Nx} (premise)
9. (∀p)(∀x)[Expa ⊃ (Nx ∨ Sx)] (premise)
10. (∀p)(∀x){{[Expa ∧ (◊p ∧ ◊~p)] ∧ □E!x} ⊃ Fx} (premise)
11. (∀x)(Fx ⊃ ~(∃z)Dxz) (premise)
12. (∀p1)(∀x){{Exp1a ∧ (∃w)(∃p2){Bp2w ∧ (∃y)[Eyp2w ∧ (x ≠ y)]} ⊃ (∃z)Dxz} (premise)
13. (∀p1)(∀x){{Exp1a ∧ ~(∃w)(∃p2){Bp2w ∧ (∃y)[Eyp2w ∧ (x ≠ y)]}} ⊃ Ox} (premise)
14. (∀x)[(Ox ∧ Sx) ⊃ Kx] (premise)
15. (∀x){[(Ox ∧ Kx) ∧ Fx] ⊃ Gx} (premise)
16. (∀x){[(Gx ∧ Sx) ∧ Ox] ⊃ Wx} (premise)
17. (∀x){{[(T̅x ∧ M̅x) ∧ (Sx ∧ Ox)] ∧ [(Kx ∧ Gx) ∧ (Fx ∧ Wx)] ∧ Exβa} ⊃ (∀y){[(T̅y ∧ M̅y) ∧ (Sy ∧ Oy)] ∧ [(Ky ∧ Gy) ∧ (Fy ∧ Wy)] ∧ Eyβa ⊃ (x = y)]} (premise)
18. m ∈ β (premise)
19. t ∈ β (premise)
20. ℕ ∈ β (premise)
21. (◊β ∧ ◊~β) ⊃ (∃w)(∃x)Exβw (1 UI)
22. (∃w)(∃x)Exβw (2,21 MP)
23. (∃x)Exβω(22 ΕΙ)
24. Eμβω (23 ΕΙ)
25. (∀x)[(Exβω ∧ Bβa) ⊃ (ω = a)] (4 UI)
26. (Eμβω ∧ Bβa) ⊃ (ω = a) (25 UI)
27. Eμβω ∧ Bβa (3,24 Conj)
28. (ω = a) (26,27 MP)
29. Eμβa (24,29 ID)
30. Eμβa ∧ (◊β ∧ ◊~β) (2,29 Conj)
31. (∀x){[Exβa ∧ (◊β ∧ ◊~β)] ⊃ □E!x} (5 UI)
32. [Eμβa ∧ (◊β ∧ ◊~β)] ⊃ □E!μ (31 UI)
33. □E!μ (30,32 MP)
34. Eμβa ∧ (M ∈ β) (18,29 Conj)
35. (∀x){[Exβa ∧ (M ∈ β)] ⊃ M̅x} (6 UI)
36. [Eμβa ∧ (M ∈ β)] ⊃ M̅μ (35 UI)
37. M̅μ (34,36 MP)
38. Eμβa ∧ (T ∈ β) (19,29 Conj)
39. (∀x){[Exβa ∧ (T ∈ β)] ⊃ T̅x} (7 UI)
40. [Eμβa ∧ (T ∈ β)] ⊃ T̅μ (39 UI)
41. T̅μ (38,40 MP)
42. Eμβa ∧ (ℕ ∈ β) (20,29 Conj)
43. (∀x){[Exβa ∧ (ℕ ∈ β)] ⊃ ~Nx} (8 UI)
44. [Eμβa ∧ (ℕ ∈ β)] ⊃ ~Nμ (43 UI)
45. ~Nμ (42,44 MP)
46. (∀x)[Exβa ⊃ (Nx ∨ Sx)] (9 UI)
47. Eμβa ⊃ (Nμ ∨ Sμ) (46 UI)
48. Nμ ∨ Sμ (29,47 MP)
49. Sμ (45,58 DS)
50.(∀x){{[Exβa ∧ (◊β ∧ ◊~β)] ∧ □E!x} ⊃ Fx} (10 UI)
51. {[Eμβa ∧ (◊β ∧ ◊~β)] ∧ □E!μ} ⊃ Fμ (50 UI)
52. [Eμβa ∧ (◊β ∧ ◊~β)] ∧ □E!μ (30,33 Conj)
53. Fμ (51,52 MP)
54. Fμ ⊃ ~(∃z)Dμz (11 UI)
55. ~(∃z)Dμz (53,54 MP)
56. (∀x){{Exβa ∧ (∃w)(∃p2){Bp2w ∧ (∃y)[Eyp2w ∧ (x ≠ y)]} ⊃ (∃z)Dxz} (12 UI)
57. {Eμβa ∧ (∃w)(∃p2){Bp2w ∧ (∃y)[Eyp2w ∧ (μ ≠ y)]} ⊃ (∃z)Dxz (56 UI)
58. ~{Eμβa ∧ (∃w)(∃p2){Bp2w ∧ (∃y)[Eyp2w ∧ (μ ≠ y)]} (55,57 MT)
59. ~Eμβa ∨ ~(∃w)(∃p2){Bp2w ∧ (∃y)[Eyp2w ∧ (μ ≠ y)]} (58 DeM)
60. ~~Eμβa (29 DN)
61. ~(∃w)(∃p2){Bp2w ∧ (∃y)[Eyp2w ∧ (μ ≠ y)]} (59,60 DS)
62. (∀x){{Exβa ∧ ~(∃w)(∃p2){Bp2w ∧ (∃y)[Eyp2w ∧ (y ≠ z)]} ⊃ Ox} (13 UI)
63. {Eμβa ∧ ~(∃w)(∃p2){Bp2w ∧ (∃y)[Eyp2w ∧ (μ ≠ y)]}} ⊃ Oμ (62 UI)
64. Eμβa ∧ ~(∃w)(∃p2){Bp2w ∧ (∃y)[Eyp2w ∧ (μ ≠ y)]} (29,61 Conj)
65. Oμ (63,64 MP)
66. (Oμ ∧ Sμ) ⊃ Kμ (14 UI)
67. Oμ ∧ Sμ (49,65 Conj)
68. Kμ (66,67 MP)
69. [(Oμ ∧ Kμ) ∧ Fμ] ⊃ Gμ (15 UI)
70. Oμ ∧ Kμ (65,68 Conj)
71. (Oμ ∧ Kμ) ∧ Fμ (53,70 Conj)
72. Gμ (69,71 MP)
73. [(Gμ ∧ Sμ) ∧ Oμ] ⊃ Wμ (16 UI)
74. Gμ ∧ Sμ (49,72 Conj)
75. (Gμ ∧ Sμ) ∧ Oμ (65,74 Conj)
76. Wμ (73,75 MP)
77. {[(T̅μ ∧ M̅μ) ∧ (Sμ ∧ Oμ)] ∧ [(Kμ ∧ Gμ) ∧ (Fμ ∧ Wμ)] ∧ Eμβa} ⊃ (∀y){[(T̅y ∧ M̅y) ∧ (Sy ∧ Oy)] ∧ [(Ky ∧ Gy) ∧ (Fy ∧ Wy)] ∧ Eyβa ⊃ (μ = y)} (17 UI)
78. T̅μ ∧ M̅μ (37,41 Conj)
79. Sμ ∧ Oμ (49,65 Conj)
80. (T̅μ ∧ M̅μ) ∧ (Sμ ∧ Oμ) (78,79 Conj)
81. Kμ ∧ Gμ (68,72 Conj)
82. Fμ ∧ Wμ (53,76 Conj)
83. (Kμ ∧ Gμ) ∧ (Fμ ∧ Wμ) (81,82 Conj)
84. [(T̅μ ∧ M̅μ) ∧ (Sμ ∧ Oμ)] ∧ [(Kμ ∧ Gμ) ∧ (Fμ ∧ Wμ)] (80,83 Conj)
85. [(T̅μ ∧ M̅μ) ∧ (Sμ ∧ Oμ)] ∧ [(Kμ ∧ Gμ) ∧ (Fμ ∧ Wμ)] ∧ Eμβa (29,84 Conj)
86. (∀y){[(T̅y ∧ M̅y) ∧ (Sy ∧ Oy)] ∧ [(Ky ∧ Gy) ∧ (Fy ∧ Wy)] ∧ Eyβa ⊃ (μ = y)} (77,85 MP)
87. {[(T̅μ ∧ M̅μ) ∧ (Sμ ∧ Oμ)] ∧ [(Kμ ∧ Gμ) ∧ (Fμ ∧ Wμ)] ∧ Eμβa} ∧ (∀y){[(T̅y ∧ M̅y) ∧ (Sy ∧ Oy)] ∧ [(Ky ∧ Gy) ∧ (Fy ∧ Wy)] ∧ Eyβa ⊃ (μ = y)} (85,86 Conj)
88. {[(T̅μ ∧ M̅μ) ∧ (Sμ ∧ Oμ)] ∧ [(Kμ ∧ Gμ) ∧ (Fμ ∧ Wμ)] ∧ Eμβa} ∧ (∀y){[(T̅y ∧ M̅y) ∧ (Sy ∧ Oy)] ∧ [(Ky ∧ Gy) ∧ (Fy ∧ Wy)] ∧ Eyβa ⊃ (μ = y)} ∧ □E!μ (33,87 Conj)
89. (∃x){{[(T̅x ∧ M̅x) ∧ (Sx ∧ Ox)] ∧ [(Kx ∧ Gx) ∧ (Fx ∧ Wx)] ∧ Exβa} ∧ (∀y){[(T̅y ∧ M̅y) ∧ (Sy ∧ Oy)] ∧ [(Ky ∧ Gy) ∧ (Fy ∧ Wy)] ∧ Eyβa ⊃ (x = y)} ∧ □E!x} (88 EG)
90. □E!g (89 theory of descriptions)

QED

Footnotes:

1 A.R. Pruss & R.M. Gale. (1999). “A New Cosmological Argument.” In Religious Studies. Vol. 35. 461-476

2 J. Gellman. (2000). “Prospects for a Sound Stage 3 of Cosmological Arguments.” In Religious Studies. Vol. 36 159-201

Posted on August 8, 2016, in Arguments for God and tagged , , , , , . Bookmark the permalink. Leave a comment.

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