Monthly Archives: March 2017

A Possible Interpretation of Proslogion 2

One of my struggles in trying to understand Proslogion 2 is how Anselm gets to the actual existence of God rather than what he arrives at in Proslogion 3, namely the inconceivability of God’s non-existence.  I’ve also struggled with the notion of using a two-place predicate like “greater than”, since Anselm tells us that if God exists in the mind alone, a greater could be conceived, i.e. to think of God as existing in reality.  Here, we are saying that we could conceive of one and the same concept in greater ways rather than conducting a comparison of the God concept to other items in the world.  The following interpretation approximates what Anselm seems to be arguing, and I would say that it is a sound argument for God’s existence.

D1. God is defined as that which cannot be conceived to admit of more greatness.
P1. For all x, if x exists in intelletu and not in re, then it can be conceived that x exists in intellectu and not in re.
P2. For all x, if it can be conceived that x exists in intellectu and not in re, then it can be conceived that x exists in intellectu and in re.
P3. For all x, if it can be conceived that x exists in intellectu and not in re and it can be conceived that x exists in intellectu and in re, then it is conceivable that x admits of more greatness.
P4. God exists in intellectu.
C. Therefore, God exists in re.


E!x ≝ x exists in re
Ix ≝ x exists in intellectu
Gx ≝ x admits of more greatness
©… ≝ it is conceivable that…

g ≝ (ɿx)~©Gx

1. (∀x)[(Ix ∧ ~E!x) ⊃ ©(Ix ∧~E!x)] (premise)
2. (∀x)[©(Ix ∧ ~E!x) ⊃ ©(Ix ∧ E!x)] (premise)
3. (∀x){[©(Ix ∧ ~E!x) ∧ ©(Ix ∧ E!x)] ⊃ ©Gx} (premise)
4. Ig (premise)
5. ~E!g (IP)
6. Ig ∧ ~E!g (4,5 Conj)
7. (Ig ∧ ~E!g) ⊃ ©(Ig ∧~E!g) (1 UI)
8. ©(Ig ∧~E!g) (6,7 MP)
9. ©(Ig ∧ ~E!g) ⊃ ©(Ig ∧ E!g) (2 UI)
10. ©(Ig ∧ E!g) (8,9 MP)
11. ©(Ig ∧~E!g) ∧ ©(Ig ∧ E!g) (8,10 Conj)
12. ©(Ig ∧ ~E!g) ∧ ©(Ig ∧ E!g)] ⊃ ©Gg (3 UI)
13. ©Gg (11,12 MP)
14. (∃x){{~©Gx ∧ (∀y)[~©Gy ⊃ (y = x)]} ∧ ©Gx} (13 theory of descriptions)
15. {~©Gμ ∧ (∀y)[~©Gy ⊃ (y = μ)]} ∧ ©Gμ (14 EI)
16. {(∀y)[~©Gy ⊃ (y = μ)] ∧ ~©Gμ} ∧ ©Gμ (15 Comm)
17. (∀y)[~©Gy ⊃ (y = μ)] ∧ {~©Gμ ∧ ©Gμ} (16 Assoc)
18. ~©Gμ ∧ ©Gμ (17 Simp)
19. E!g (5-18 IP)


[Edit: My friend, Matt, thinks my argument may be susceptible to parody.  Here is my response]

Generally, I think parodies fail because such supposed objects, like islands of which none greater can be conceived, do not really exist in the intellect for the very same reason round squares are not abstract objects in the mind.  The phrase is nonesense, and so does not pick out any object of the understanding.

Islands just are the sorts of things that admit of degrees of greatness, so are other objects used in parody. For example, islands are present in a specified location that is surrounded by water, but it is unclear how big an island should be when considering its greatness.  It certainly cannot be omnipresent and be an island.  How many trees, island beauties, or sandy beaches ought there to be on the island which cannot be conceivably greater?  

My argument can motivate this response by proving that the greatest conceivable island is not an object that exists in the intellect.  This is because specifying that there is an island than which none greater can be conceived leads to the conclusion that God is an island, and that seems like a good reductio of the idea such a concept can be conceived.

So, if we grant the parody, I could prove that island can be predicated of God, or a being than which a greater cannot be conceived. But since islands are essentially contingent and admit of degrees of greatness, island cannot be a predicate of God, who is the being than which none greater can be conceived. So, we must reject the assumption that a greatest conceivable island exists in intellectu and we can base it on the somewhat reasonable premise that God is not an island. I would argue as follows:


Lx ≝ x is an island

i ≝ (ɿx)(~©Gx ∧ Lx)

20. ~Lg (premise)
21. (∃x){{~©Gx ∧ (∀y)[~©Gy ⊃ (y = x)]} ∧ E!x} (19 theory of descriptions)
22. Ii (IP)
23. (∃x){{(~©Gx ∧ Lx) ∧ (∀y)[(~©Gy ∧ Ly) ⊃ (y = x)]} ∧ Ix} (22 theory of descriptions)
24. {~©Gμ ∧ (∀y)[~©Gy ⊃ (y = μ)]} ∧ E!μ (21 EI)
25. {(~©Gν ∧ Lν) ∧ (∀y)[(~©Gy ∧ Ly) ⊃ (y = ν)]} ∧ Iν (23 ΕΙ)
26. ~©Gμ ∧ (∀y)[~©Gy ⊃ (y = μ)] (24 Simp)
27. (∀y)[~©Gy ⊃ (y = μ)] (26 Simp)
28. (~©Gν ∧ Lν) ∧ (∀y)[(~©Gy ∧ Ly) ⊃ (y = ν)] (25 Simp)
29. ~©Gν ∧ Lν (28 Simp)
30. ~©Gν (29 Simp)
31. ~©Gν ⊃ (ν = μ) (27 UI)
32. ν = μ (30,31 MP)
33. ~©Gμ ∧ Lμ (29,32 ID)
34. (~©Gμ ∧ Lμ) ∧ (∀y)[~©Gy ⊃ (y = μ)] (27,33 Conj)
35. ~©Gμ ∧ {Lμ ∧ (∀y)[~©Gy ⊃ (y = μ)]} (34 Assoc)
36. ~©Gμ ∧ {(∀y)[~©Gy ⊃ (y = μ)] ∧ Lμ} (35 Comm)
37. {~©Gμ ∧ {(∀y)[~©Gy ⊃ (y = μ)]} ∧ Lμ (36 Assoc)
38. (∃x){{~©Gx ∧ {(∀y)[~©Gy ⊃ (y = x)]} ∧ Lx} (37 EG)
39. Lg (38 theory of descriptions)
40. ~Lg ∧ Lg
41. ~Ii (22-40 IP)

So as long as you can provide the premise that God is not an island, not a pizza, etc. the proof works to show that such objects really are not in the intellect.

Absolute and Relative Identity

I have seen some argue that any relative identity claim can be reduce to an absolute identity claim in the following manner:

1) x and y are the same F  ≝ is an Fis an F, and x = y.

However, I don’t think this works.  Part of the motivation for relative identity is that there may be circumstances like:

2) x and y are the same F, but x and y are not the same G.

But (1) and (2) are not compatible, since we would have to affirm and deny absolute identity between x and y.  So the relative identity theorist should reject (1) given his commitment to (2).

Relative identity is not just absolute identity, plus the idea that x and y fall under the same sortal.  Moreover, this would be to suggest that relative identity is derivative, and absolute identity is the more primitive notion.  I would argue that is it the other way around.  So I would define absolute identity in terms of relative identity in the following manner:

4) x = y ≝ for any sortal, S, if x is an S or y is an S, then x and are the same S.

In other words, the absolute identity between x and y is derived from the fact that for any sortal which belongs to either x or y, it is the case that x and y count as the same S.  I say “either x is an S, or y is an S” as opposed to “both x is an S and y is an S” to avoid situations where x can be counted as an S and some y cannot, but they are the same S for any sortal underwhich both can be counted.  For there to be absolute identity, it must be the case that all sortals that belong to x must also belong to y.  I believe (4) captures this.

So to say x and y are absolutely identical is to say that for any sortal underwhich x or y can be counted, x and y are the same sortal.