# A Formal Version of the Third Way

I believe by using mereological sums, I avoid the charge of the quantifier shift fallacy.

D1: God is the x such there is not some y by which x receives the necessity it has, and x is a member of the essentially ordered causal series by which things receive their necessity .

P1. For all x, if it is possible that x does not exist, then there is a time at which x does not exist.

P2. If there is a time at which the mereological sum of everything does not exist, then there does not exist now the mereological sum of everything.

P3. If there exists now some x, then there exists now the mereological sum of everything.

P4. I exist now.

P5. If necessarily there exists the mereological sum of everything, then there is some x that necessarily exists, and x is a part of the mereological sum of everything.

P6. If there is some x that necessarily exists, then if for all x, x necessarily exists, then there is some y such that x receives the necessity it has from y, only if there is an essentially ordered causal series by which things receive their necessity and it does not regress finitely.

P7. For all z it is not the case that there is an x, such that both x is a member of the essentially ordered causal series by which things receive z and it is not the case that z regresses finitely.

P8. For all x, if x necessarily exists, then x is a member of the essentially ordered causal series by which things receive their necessity.

P9. For all x, if there is not some y by which x receives the necessity it has, and x is a member of the essentially ordered causal series by which things receive their necessity, then for all z, there is not some y by which z receives the necessity it has, and z is a member of the essentially ordered series by which things receive their necessity, and z is identical to x.

C1. God necessarily exists.

Note: D1 tells us that God does not receive his necessity from any other cause, but, being a part of the causal series by which things receive their necessity, is the cause of necessity in other things.

Let:

E!x ≝ x exists

E!_{t} ≝ x exists at time t

Fx ≝ x regresses finitely

Oxy ≝ x is a member of essentially ordered causal series y

Rxy ≝ x receives the necessity it has from y

σ<x,P> ≝ the mereological sum of all x that P.

σ<e,E!> ≝ (∀x)[E!x ⊃ (x ≤ e)] & (∀y)[(y ≤ e) ⊃ (∃z)(E!z & (y ⊗ z)]^{1}

e ≝ everything

g ≝ (ɿx)[~(∃y)Rxy & Oxl]

i ≝ I (the person who is me)

l ≝ the causal series by which things receive their necessity

n ≝ now

1. (∀x)[♢~E!x ⊃ (∃t)~E!_{t}x] (premise)

2. (∃t)~E!_{t}σ<e,E!> ⊃ ~E!_{n}σ<e,E!> (premise)

3. (∃x)E!_{n}x ⊃ E!_{n}σ<e,E!>(premise)

4. E!_{n}i (premise)

5. ☐E!σ<e,E!> ⊃ (∃x)[☐E!x &(x ≤ e)] (premise)

6. (∃x)☐E!x ⊃ {(∀x)[☐E!x ⊃ (∃y)Rxy] ⊃ (∃x)[Oxl & ~Fl]} (premise)

7. (∀z)~(∃x)[Oxz & ~Fz] (premise)

8. (∀x)[☐E!x ⊃ Oxl] (premise)

9. (∀x){[~(∃y)Rxy & (Oxl & Fl)] ⊃ (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = x)]} (premise)

10. ♢~E!σ<e,E!> (IP)

11. ♢~E!σ<e,E!> ⊃ (∃t)~E!_{t}σ<e,E!> (1 UI)

12. (∃t)~E!_{t}σ<e,E!> (10,11 MP)

13. ~E!_{n}σ<e,E!> (2,12 MP)

14. (∃x)E!_{n}x (4 EG)

15. E!_{n}σ<e,E!> (3,14 MP)

16. E!_{n}σ<e,E!> & ~E!_{n}σ<e,E!> (13,15 Conj)

17. ~♢~E!σ<e,E!> (10-16 IP)

18. ☐E!σ<e,E!> (17 ME)

19. (∃x)[☐E!x &(x ≤ e)] (5,18 MP)

20. ☐E!μ & (μ ≤ e) (19 EI)

21. ☐E!μ (20 Simp)

22. (∃x)☐E!x (21 EG)

23. (∀x)[☐E!x ⊃ (∃y)Rxy] ⊃ (∃x)[Oxl & ~Fl] (6,22 MP)

24. ~(∃x)(Oxl & ~Fl)] (7 UI)

25. ~(∀x)[☐E!x ⊃ (∃y)Rxy] (23,24 MT

26. (∃x)~[☐E!x ⊃ (∃y)Rxy] (25 QN)

27. (∃x)~[~☐E!x ∨ (∃y)Rxy] (26 Impl)

28. (∃x)[~~☐E!x & ~(∃y)Rxy] (27 DeM)

29. ~~☐E!ν & ~(∃y)Rνy (28 EI)

30. ☐E!ν & ~(∃y)Rνy (29 DN)

31. ☐E!ν (30 Simp)

32. ☐E!ν ⊃ Oνl (8 UI)

33. Oνl (31,32 MP)

34. ~(∃x)[Oxl & ~Fl] (7 UI)

35. (∀x)~[Oxl & ~Fl] (34 QN)

36. ~[Oνl & ~Fl] (35 UI)

37. ~Oνl ∨ ~~Fl (36 DeM)

38. ~~Oνl (33 DN)

39. ~~Fl (37,38 DS)

40. Fl (39 DN)

41. ~(∃y)Rνy (30 Simp)

42. Oνl & Fl (33,40 Conj)

43. ~(∃y)Rνy & (Oνl & Fl) (41,42 Conj)

44. [~(∃y)Rνy & (Oνl & Fl)] ⊃ (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = ν)] (9 UI)

45. (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = ν)] (43,44 MP)

46. ~(∃y)Rνy & Oνl (33,41 Conj)

47. [~(∃y)Rνy & Oνl] & (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = ν)] (45,46 Conj)

48. [~(∃y)Rνy & Oνl] & (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = ν)] & ☐E!ν (31,47 Conj)

49. (∃x){[~(∃y)Rxy & Oxl] & (∀z)[(~(∃y)Rzy & Ozl) ⊃ (z = x)] & ☐E!x} (48 EG)

50. ☐E!g (49 Theory of Descriptions)

QED

^{1}Formulation of definition for everything based influenced by Filip, H. (n.d.) “Mereology”. Online: https://user.phil-fak.uni-duesseldorf.de/~filip/Mereology.pdf

Posted on February 29, 2016, in Arguments for God and tagged Aquinas, cosmological argument, God, mereology, modal logic, The Third Way. Bookmark the permalink. 1 Comment.

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