# Monthly Archives: October 2016

## An Argument Against Subjective Morality

P1. If morality is relative to the subject, then morality is a domain that is a matter of personal opinion.

P2. All domains that are matters of personal opinion, are domains where facts and evidence cannot determine correct belief.

P3. All domains where facts and evidence cannot determine correct belief are domains that lack propositions for which it is worth dying before giving assent under coercion.

P4. Morality is a domain with propositions for which it is worth dying before giving assent under coercion.

C. Morality is not relative to the subject.

P1 and P2 are not too objectionable. That is just what we mean when we say that morality is subjective. I think that if you are going to object to the argument, you will object to P3 or P4.

## A Remix of Anselm’s Conceptual Ontological Argument

D1. God is defined as the x such that there is not something, y, where y is conceivably greater than x.

P1. For all x, if x is conceivable, then there is something, y, such that either y is identical to x and y exists or there is something, z, such that z is identical to x, z does not exist, and y is conceivably greater than z.

P2. There is some x such that x is conceivable and it is not the case that there is some y such that y is conceivably greater than x.

P3. For all x and y, either x is conceivably greater than y or y is conceivably greater than x, or if it is not the case that either x is conceivably greater than y or that y is conceivably greater than x, there is some z such that z is the mereological sum of x and y, and either z is conceivably greater than x or z is conceivably greater than y.

C. God exists.^{1}

E!x ≝ x exists

Cx ≝ x is conceivable

Gxy ≝ x is conceivably greater than y

σ<x,y> ≝ the mereological sum of x and y

g ≝ (ɿx)~(∃y)Gyx

1. (∀x){Cx ⊃ (∃y){[(y = x) ∧ E!y] ∨ (∃z)[(z = x) ∧ (~E!z ∧ Gyz)]}} (premise)

2. (∃x)(Cx ∧ ~(∃y)Gyx) (premise)

3. (∀x)(∀y){[Gxy ∨ Gyx] ∨ {~(Gxy ∨ Gyx) ⊃ (∃z)[(z = σ<x,y>) ∧ (Gzx ∨ Gzy)]}} (premise)

4. Cμ ∧ ~(∃y)Gyμ (2 EI)

5. ~(∃y)Gyμ (4 Simp)

6. (∃z)[~(∃z_{1})Gz_{1}z ∧ ~(z = μ)] (IP)

7. ~(∃z_{1})Gz_{1}ν ∧ ~(ν = μ) (6 EI)

8. (∀y){[Gνy ∨ Gyν] ∨ {~(Gνy ∨ Gyν) ⊃ (∃z)[(z = σ<ν,y>) ∧ (Gzν ∨ Gzy)]}} (3 UI)

9. [Gνμ ∨ Gμν] ∨ {~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)]} (8 UI)

10. (∀y)~Gyμ (5 QN)

11. ~Gνμ (10 UI)

12. ~(∃z_{1})Gz_{1}ν (7 Simp)

13. (∀z_{1})~Gz_{1}ν (12 QN)

14. ~Gμν (13 UI)

15. Gνμ ∨ [Gμν ∨ {~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)]}] (9 Assoc)

16. Gμν ∨ {~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)]} (11,15 DS)

17. ~(Gνμ ∨ Gμν) ⊃ (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)] (14,16 DS)

18. ~Gνμ ∧ ~Gμν (11,14 Conj)

19. ~(Gνμ ∨ Gμν) (18 DeM)

20. (∃z)[(z = σ<ν,μ>) ∧ (Gzν ∨ Gzμ)] (17,19 MP)

21. (ζ = σ<ν,μ>) ∧ (Gζν ∨ Gζμ) (20 EI)

22. Gζν ∨ Gζμ (21 Simp)

23. ~Gζμ (10 UI)

24. Gζν (22,23 DS)

25. ~Gζν (13 UI)

26. Gζν ∧ ~Gζν (24,25 Conj)

24. ~(∃z)[~(∃z_{1})Gz_{1}z ∧ ~(z = μ)] (6-23 IP)

25. (∀z)~[~(∃z_{1})Gz_{1}z ∧ ~(z = μ)] (24 QN)

26. (∀z)[~~(∃z_{1})Gz_{1}z ∨ ~~(z = μ)] (25 DeM)

27. (∀z)[~(∃z_{1})Gz_{1}z ⊃ ~~(z = μ)] (26 Impl)

28. (∀z)[~(∃z_{1})Gz_{1}z ⊃ (z = μ)] (27 DN)

29. {Cμ ∧ ~(∃y)Gyμ} ∧ (∀z)[~(∃z_{1})Gz_{1}z ⊃ (z = μ)] (4,28 Conj)

30. Cμ ∧ {~(∃y)Gyμ ∧ (∀z)[~(∃z_{1})Gz_{1}z ⊃ (z = μ)]} (29 Assoc)

31. {~(∃y)Gyμ ∧ (∀z)[~(∃z_{1})Gz_{1}z ⊃ (z = μ)]} ∧ Cμ (30 Comm)

32. (∃x){~(∃y)Gyx ∧ (∀z)[~(∃z_{1})Gz_{1}z ⊃ (z =x)]} ∧ Cx} (31 EG)

33. Cg (32 theory of descriptions)

34. Cg ⊃ (∃y){[(y = g) ∧ E!y] ∨ (∃z)[(z = g) ∧ (~E!z ∧ Gyz)]} (1 UI)

35. (∃y){[(y = g) ∧ E!y] ∨ (∃z)[(z = g) ∧ (~E!z ∧ Gyz)]} (33,34 MP)

36. [(ξ = g) ∧ E!ξ] ∨ (∃z)[(z = g) ∧ (~E!z ∧ Gξz)] (35 EI)

37. (∃z)[(z = g) ∧ (~E!z ∧ Gξz)] (IP)

38. (ν = g) ∧ (~E!ν ∧ Gξν) (37 EI)

39. ~E!ν ∧ Gξν (38 Simp)

40. Gξν (39 Simp)

41. (ν = g) (38 Simp)

42. Gξg (40,41 ID)

43. (∃x){~(∃y)Gyx ∧ (∀z)[~(∃z_{1})Gz_{1}z ⊃ (z = x)]} ∧ Gξx} (42 theory of descriptions)

44. {~(∃y)Gyζ ∧ (∀z)[~(∃z_{1})Gz_{1}z ⊃ (z = ζ)]} ∧ Gξζ (43 EI)

45. ~(∃y)Gyζ ∧ (∀z)[~(∃z_{1})Gz_{1}z ⊃ (z = ζ)](44 Simp)

46. ~(∃y)Gyζ (45 Simp)

47. (∀y)~Gyζ (46 QN)

48. ~Gξζ (47 UI)

49. Gξζ (44 Simp)

50. Gξζ ∧ ~Gξζ (48,49 Conj)

51. ~(∃z)[(z = g) ∧ (~E!z ∧ Gξz)] (37-50 IP)

52. (ξ = g) ∧ E!ξ (36,51 DS)

53. (ξ = g) (52 Simp)

54. E!ξ (52 Simp)

55. E!g (53,54 ID)

QED

^{1} Some aspects of this argument are influenced by Oppenheimer & Zalta (1991), i.e. the existential quantifier carries no existential import and is analogous to Anselm’s existence *in intellectu* whereas E! is a predicate that indicates existence *in re*. One weakness of Oppenheimer & Zalta’s argument is that it depends on a non-logical axiom regarding Gxy such that it is connected. In other words, either Gxy or Gyx or (x = y). This requires all individuals to stand in a greater than relationship. It is plausible, though, that two non-identical individuals could share equal greatness. I am able to derive the uniqueness of the being than which none greater can be conceived by appealing to the notion that the merelogical composite of two equally great individuals is at least greater than one of its proper parts, which I take to be a modest premise. The interesting thing about my formulation is the first premise, which distinguishes *in intellectu *from *in re *existence, and captures Anselm’s claim that a greater could be conceived than a being that exists in the understanding alone without begging the question that this greater thing actually exists—it is merely conceptually greater. See P.E Oppenheimer & E.N. Zalta. (1991). “On the Logic of the Ontological Argument.” In *Philosophical Perspectives*. Vol. 5. 509-529.

## Domino Logic

Here are some really interesting videos in which dominos simulate logical reasoning and computing: