An Ontological Argument from Actuality

Here is a refinement on my ontological argument from actuality:

1. Something is an Anselmian God if and only if it is conceivable, nothing can be conceived of which is more actual, and it necessarily exists (definition Θ).

2. There is something conceivable such that nothing can be conceived of which is more actual (premise).

3. 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).

4. For all x, if x is mentally dependent, there is something conceivable that is more actual than x (premise).

Therefore,

5. An Anselmian God exists.

I start out with a definition of an Anselmian God, which is a stipulation, but is rooted in the idea that a Being of Pure Actuality is arguably perfect and possesses a good number of divine attributes.  

As I noted in a previous post, the traditional argument uses a “greater than” relation, which some find suspect.  “Greatness” would have been understood by Anselm as something that can be evaluated objectively on a scale, as in the Neo-Platonic notion of the Great Chain of Being.  To the contemporary ear, “greatness” seems subjective and vague.  I think “actual” in the Thomistic-Aristotelian sense is a fair approximation of greatness, but we can have a better sense of what “actual” means.  Thomas is able to derive the divine attributes from a being of Pure Actuality, so “most actual” is plausibly a divinely loaded superlative.  Moreover, it seems to me that the act-potency distinction is not something the contemporary ear would take to be dependent on subjective opinions.  So, I think (2) is fairly impeccable.  

I think (3) is a bit clunky, but it basically means that if something is merely a concept, then it is mentally dependent.  So, in the case of God, if God is merely a concept in the mind, then the possibility that God could fail to be conceived by all minds that exist implies that God, as a mere concept, could fail to exist, and so depends upon minds to continue to exist.  Put another, if God is merely a concept, then there was no God in the Jurassic period, as William Lane Craig once suggested to John Dominic Crossan.

Finally, (4) says that if something is mentally dependent, then something is conceivable that is more actual than it.  Some people think, for instance, that moral values are mind-dependent.  So, for instance, the actuality of the value of human life, VHL, depends on there being an actual community of minds that actually conceive of human life as valuable.  Were such a community to cease to exist, the VHL would only potentially exist, even if humans existed.  If the VHL were an objective fact grounded in human nature, then the actuality of the VHL would obtain whenever humans actual exist.  There is a certain assymetry that suggests that grounding the VHL in human nature is to view VHL as more actual than grounding VHL in the subjective opinions of a community of minds.  For the VHL to be actual in one case, there need only be actual humans exemplifying human nature, where as in the latter, there needs to be actual humans exemplifying human nature and an actual community of minds that actually is of the opinion that human life is valuable.  For, without the humans, a community of minds that endorses the VHL would really just be saying that VHL potentially exists and would be actual upon the occassion of human life.  We could say, then, that x is more actual than y iff the existence of x depends upon the actualization of fewer potentials than y depends upon.  VHL grounded in the actuality of human nature depends upon the actualization of fewer potentials than VHL grounded in subjective opinions about humans. So (4) just tells us that for any x that depends upon the mental for its actuality, it is conceivable that there is something that is more actual (and less dependent) than x, e.g. to conceive that x can actually exist independent of mentally conceiving of x.

Let

Cx – x is conceived
Mx – x is mentally dependent
Axy – x is more actual than y
Θx- x is an Anselmian God, 

that is: 

1. (∀x){Θx ≝ ([♢Cx & ~(∃y)(Ayx & ♢Cy)] & ☐(∃z)(z=x))} (Def Θ)
2. (∃x)[♢Cx & ~(∃y)(Ayx & ♢Cy)] (premise)
3. (∀x){[♢~Cx ⊃ ♢~(∃z)(z=x)] ⊃ Mx} (premise)
4. (∀x){Mx ⊃ [(∃y)(Ayx & ♢Cy)]} (premise)
5. (∀x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (IP)
6. ♢Cu & ~(∃y)(Ayu & ♢Cy) (2 EI)
7. [♢~Cu ⊃ ♢~(∃z)(z=u)] ⊃ Mu (3 UI)
8. Mu ⊃ [(∃y)(Ayu & ♢Cy)] (4 UI)
9. [♢~Cu ⊃ ♢~(∃z)(z=u)] ⊃ [(∃y)(Ayu & ♢Cy)] (7,8 HS)
10. [♢Cu & ~(∃y)(Ayu & ♢Cy)] ⊃ [♢~Cu ⊃ ♢~(∃z)(z=u)] (5 UI)
11. ♢~Cu ⊃ ♢~(∃z)(z=u) (6,10 MP)
12. (∃y)(Ayu & ♢Cy) (9,11 MP)
13. Avu & ♢Cv (12 EI)
14. ~(∃y)(Ayu & ♢Cy) (6 Simp)
15. (∀y)~(Ayu & ♢Cy) (14 QN)
16. ~(Avu & ♢Cv) (15 UI)
17. (Avu & ♢Cv) & ~(Avu & ♢Cv) (13,16 Conj)
18. ~(∀x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (5-17 IP)
19. (∃x)~{[♢Cx & ~(∃y)(Ayx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (18 QN)
20. (∃x) ~{~[♢Cx & ~(∃y)(Ayx & ♢Cy)] ∨ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (19 Impl)
21. (∃x){~~[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (20 DeM)
22. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (21 DN)
23. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[~♢~Cx ∨ ♢~(∃z)(z=x)]} (22 Impl)
24. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & ~[☐Cx ∨ ♢~(∃z)(z=x)]} (23 ME)
25. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & [~☐Cx & ~♢~(∃z)(z=x)]} (24 DeM)
26. (∃x){[♢Cx & ~(∃y)(Ayx & ♢Cy)] & [~☐Cx & ☐(∃z)(z=x)]} (25 ME)
27. [♢Cu & ~(∃y)(Ayu & ♢Cy)] & [~☐Cu & ☐(∃z)(z=u)] (26 EI)
28. ~☐Cu & ☐(∃z)(z=u) (27 Simp)
29. ☐(∃z)(z=u) (28 Simp)
30. [♢Cu & ~(∃y)(Ayu & ♢Cy)] (27 Simp)
31. [♢Cu & ~(∃y)(Ayu & ♢Cy)] & ☐(∃z)(z=u) (29,30 Conj)
32. Θu (1,31 “Def Θ”)
33. (∃x)Θx (32 EG)

Posted on April 16, 2015, in Arguments for God and tagged , , . Bookmark the permalink. 1 Comment.

  1. Anselm’s argument is a little different, and I explained the argument to someone on Facebook this way: Anselm wasn’t thinking of greatness in some subjective “what would be great for me” terms. Rather, he had the Neo-Platonic great chain of being in mind http://en.wikipedia.org/wiki/Great_chain_of_being. X is greater than Y in the sense that X has more capacities or has an essence that can be actualized to a greater degree. For example, a plant contingently exists, grows, takes in nutrients, and reproduces. An animal is 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 are attributed to it. If God exists, then God would be that being which none more powerful could be conceived, none greater…

    The argument, then, is as follows:

    1. Something is an Anselmian God if and only if it is conceivable, nothing can be conceived of which is greater, and it necessarily exists (definition Θ).

    2. There is something conceivable such that nothing can be conceived of which is greater (premise).

    3. 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).

    4. For all x, if x is mentally dependent, there is something conceivable that is greater than x (premise).

    Therefore,

    5. An Anselmian God exists.

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

    Cx – x is conceived
    Mx – x is mentally dependent
    Gxy – x is greater than y
    Θx- x is an Anselmian God,

    that is:

    1. (∀x){Θx ≝ ([♢Cx & ~(∃y)(Gyx & ♢Cy)] & ☐(∃z)(z=x))} (Def Θ)
    2. (∃x)[♢Cx & ~(∃y)(Gyx & ♢Cy)] (premise)
    3. (∀x){[♢~Cx ⊃ ♢~(∃z)(z=x)] ⊃ Mx} (premise)
    4. (∀x){Mx ⊃ [(∃y)(Gyx & ♢Cy)]} (premise)
    5. (∀x){[♢Cx & ~(∃y)(Gyx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (IP)
    6. ♢Cu & ~(∃y)(Gyu & ♢Cy) (2 EI)
    7. [♢~Cu ⊃ ♢~(∃z)(z=u)] ⊃ Mu (3 UI)
    8. Mu ⊃ [(∃y)(Gyu & ♢Cy)] (4 UI)
    9. [♢~Cu ⊃ ♢~(∃z)(z=u)] ⊃ [(∃y)(Gyu & ♢Cy)] (7,8 HS)
    10. [♢Cu & ~(∃y)(Gyu & ♢Cy)] ⊃ [♢~Cu ⊃ ♢~(∃z)(z=u)] (5 UI)
    11. ♢~Cu ⊃ ♢~(∃z)(z=u) (6,10 MP)
    12. (∃y)(Gyu & ♢Cy) (9,11 MP)
    13. Gvu & ♢Cv (12 EI)
    14. ~(∃y)(Gyu & ♢Cy) (6 Simp)
    15. (∀y)~(Gyu & ♢Cy) (14 QN)
    16. ~(Gvu & ♢Cv) (15 UI)
    17. (Gvu & ♢Cv) & ~(Gvu & ♢Cv) (13,16 Conj)
    18. ~(∀x){[♢Cx & ~(∃y)(Gyx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (5-17 IP)
    19. (∃x)~{[♢Cx & ~(∃y)(Gyx & ♢Cy)] ⊃ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (18 QN)
    20. (∃x) ~{~[♢Cx & ~(∃y)(Gyx & ♢Cy)] ∨ [♢~Cx ⊃ ♢~(∃z)(z=x)]} (19 Impl)
    21. (∃x){~~[♢Cx & ~(∃y)(Gyx & ♢Cy)] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (20 DeM)
    22. (∃x){[♢Cx & ~(∃y)(Gyx & ♢Cy)] & ~[♢~Cx ⊃ ♢~(∃z)(z=x)]} (21 DN)
    23. (∃x){[♢Cx & ~(∃y)(Gyx & ♢Cy)] & ~[~♢~Cx ∨ ♢~(∃z)(z=x)]} (22 Impl)
    24. (∃x){[♢Cx & ~(∃y)(Gyx & ♢Cy)] & ~[☐Cx ∨ ♢~(∃z)(z=x)]} (23 ME)
    25. (∃x){[♢Cx & ~(∃y)(Gyx & ♢Cy)] & [~☐Cx & ~♢~(∃z)(z=x)]} (24 DeM)
    26. (∃x){[♢Cx & ~(∃y)(Gyx & ♢Cy)] & [~☐Cx & ☐(∃z)(z=x)]} (25 ME)
    27. [♢Cu & ~(∃y)(Gyu & ♢Cy)] & [~☐Cu & ☐(∃z)(z=u)] (26 EI)
    28. ~☐Cu & ☐(∃z)(z=u) (27 Simp)
    29. ☐(∃z)(z=u) (28 Simp)
    30. [♢Cu & ~(∃y)(Gyu & ♢Cy)] (27 Simp)
    31. [♢Cu & ~(∃y)(Gyu & ♢Cy)] & ☐(∃z)(z=u) (29,30 Conj)
    32. Θu (1,31 Def Θ)
    33. (∃x)Θx (32 EG)

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