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]:

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

M ≝ Only finitely many things have existed to date

P ≝ Something presently exists

N ≝ 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. Bμ (9,10 MP)
12. Bμ ⊃ 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. (Tν ⊃ ~Cν) & (~Cν ⊃ Tν) (22 Equiv)
24. (~Cν ⊃ 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>.