In this post, I have formulated Anselm’s argument for the necessary existence of a being than which none greater can be conceived. However, I have noted that the argument depends upon a two-place “greater than” predicate that functions with something like the Neo-Platonic “Great Chain of Being” in mind. Some thing, x, is conceived to be greater than y in the sense that x is understood to have more capacities or has an essence that can be actualized to a greater degree. For example, a plant is understood to contingently exists, grows, takes in nutrients, and reproduces. An animal is understood to be greater in the sense that it too contingently exists, grows, takes in nutrients, and reproduces, but it also has capacities like sentience, and can self-move, etc. So the greater something is, the more powers/more capacities it is understood to have. If God exists, then God would be that being which none more powerful could be conceived, which is just to say “none greater”. I find the metaphysics where a two-place “conceivably greater than” predicate can be objectively exemplified to be extremely plausible. There is an objective sense in which I have greater capacities and abilities than a flea.

The argument is as follows:

D1. Some x is an Anselmian God if and only if x is conceivable, it is not the case that there is something that is conceivably greater than x, and x necessarily exists.

P1. There is some x conceivable such that there is nothing conceivably greater than x.

P2. For all x, if the possibility of failing to conceive of x implies the possibility that x doesn’t exist, x is mentally dependent (premise).

P3. For all x, if x is mentally dependent, there is some y such that y is conceivably greater than x (premise).

P4. If there is some x such that necessarily there is some z and z is identical to x, and x is an Anselmian God, then necessarily there exists an Anselmian God.

Therefore,

C1. Necessarily, there is an Anselmian God.

That is the argument in ordinary language. To show that it is a formally valid syllogism, I offer the following formal deduction:

Let,

Cx ≝ x is conceived

Mx ≝ x is mentally dependent

Gxy ≝ x is conceived to be greater than y

Θx ≝ (∃x){[♢Cx & ~(∃y)♢Gyx]& ☐(∃z)(z=x)} (Def Θx)

1. (∃x)[♢Cx & ~(∃y)♢Gyx] (premise)

2. (∀x){[♢~Cx ⊃ ♢~(∃z)(z=x)] ⊃ Mx} (premise)

3. (∀x)[Mx ⊃ (∃y)♢Gyx] (premise)

4. (∃x)[☐(∃z)(z=x)& Θx] ⊃ ☐(∃x)Θx (premise)

5. (∀x){[♢Cx & ~(∃y)♢Gyx] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (IP)

6. ♢Cμ & ~(∃y)♢Gyμ (1 EI)

7. [♢~Cμ ⊃ ♢~(∃z)(z=μ)] ⊃ Mμ (2 UI)

8. Mμ ⊃ (∃y)(♢Gyμ) (3 UI)

9. [♢~Cμ ⊃ ♢~(∃z)(z=μ)] ⊃ (♢Gyμ)(7,8 HS)

10. ♢Cμ & ~(∃y)♢Gyμ] ⊃ [♢~Cμ ⊃ ♢~(∃z)(z=μ)] (5 UI)

11. ♢~Cμ ⊃ ♢~(∃z)(z=μ) (6,10 MP)

12. (∃y)♢Gyμ (7,9 MP)

13. ♢Gνμ (12 EI)

14. ~(∃y)♢Gyμ (6 Simp)

15. (∀y)~(♢Gyμ) (14 QN)

16. ~♢Gνμ (15 UI)

17. ♢Gνμ & ~♢Gνμ (13,16 Conj)

18. ~(∀x){[♢Cx & ~(∃y)♢Gyx] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (5-17 IP)

19. (∃x)~{[♢Cx & ~(∃y)♢Gyx] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (18 QN)

20. (∃x) ~{~[♢Cx & ~(∃y)♢Gyx] ∨ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (19 Impl)

21. (∃x){~~[♢Cx & ~(∃y)♢Gyx] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (20 DeM)

22. (∃x){[♢Cx & ~(∃y)♢Gyx] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (21 DN)

23. (∃x){[♢Cx & ~(∃y)♢Gyx] & ~[~♢~Cx ∨ ♢~(∃z)(z=x)]} (22 Impl)

24. (∃x){[♢Cx & ~(∃y)♢Gyx] & ~[☐Cx ∨ ♢~(∃z)(z=x)]} (23 ME)

25. (∃x){[♢Cx & ~(∃y)♢Gyx] & [~☐Cx & ~♢~(∃z)(z=x)]} (24 DeM)

26. (∃x){[♢Cx & ~(∃y)♢Gyx] & [~☐Cx & ☐(∃z)(z=x)]} (25 ME)

27. [♢Cμ & ~(∃y)♢Gyμ] & [~☐Cμ & ☐(∃z)(z=μ)] (26 EI)

28. ~☐Cμ & ☐(∃z)(z=μ) (27 Simp)

29. ☐(∃z)(z=μ) (28 Simp)

30. [♢Cμ & ~(∃y)♢Gyμ] (27 Simp)

31. [♢Cμ & ~(∃y)♢Gyμ] & ☐(∃z)(z=μ) (29,30 Conj)

32. Θμ (31 Def “Θx”)

33. ☐(∃z)(z=μ) & Θμ (29,32 Conj)

34 (∃x)[☐(∃z)(z=x) & Θx] (33 EG)

35. ☐(∃x)Θx (4,34 MP)

QED

Indeed, I find the above argument very persuasive. However, there may be some who are resistant to the notion that the two-place “conceivably greater-than” predicate can actually and objectively be exemplified. For such a person, I propose a more modest version of the argument. The more modest version is that, since C1, i.e. “☐(∃x)Θx”, is provable given P1-P4,one can argue that if P1-P4 are jointly possible, C1 is possible, and so an Anselmian God necessarily exists. This follows given S5 in modal logic, which says that ◊☐P entails ☐P. The argument can be formally proved as follows:

Let, also:

P1 ≝ (∃x)[♢Cx & ~(∃y)♢Gyx]

P2 ≝ (∀x){[♢~Cx ⊃ ♢~(∃z)(z=x)] ⊃ Mx}

P3 ≝ (∀x)[Mx ⊃ (∃y)♢Gyx]

P4 ≝ (∃x)[☐(∃z)(z=x) & Θx] ⊃ ☐(∃x)Θx

C1 ≝ ☐(∃x)Θx

36. ◊[(P1 & P2) & (P3 & P4)] (premise)

37. [(P1 & P2) & (P3 & P4)] ⊢ C1 (premise; proved by 1-35)

38. [◊[(P1 & P2) & (P3 & P4)]& {[(P1 & P2) & (P3 & P4)]⊢ C1}] ⊃ ◊C1 (premise)

39. ◊[(P1 & P2) & (P3 & P4)] & {[(P1 & P2) & (P3 & P4)] ⊢ C1} (36,37 Conj)

40. ◊C1 (38,39 MP)

41. ◊☐(∃x)Θx (40 Def “C1”)

42. ☐(∃x)Θx (41 by “S5”)

QED (again)

Since (37) is established, and (38) merely argues that if premises are jointly possible, and those premises prove some conclusion, then the conclusion is possible, (38) is relatively uncontroversial. So, if one objects that P1-P4 are not actually true, but admits that they are at least broadly logically, or metaphysically compossible, then one ought to agree that, necessarily, an Anselmian God exists.