The Ontological Argument Revisited

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Here is an exposition of Anselm’s ontological argument. One premise is adapted from Zalta and Oppenheimer (1991) On the Logic of the Ontological Argument

However, Zalta and Oppenheimer are more concerned with arguing for an historically accurate version of the argument. They are convinced by Barnes that it is anachronistic to import contemporary modal notions into the argument. Instead, they offer an exposition that purports to infer God’s existence from God’s being. Though their argument is more simple, it involves a Meinongian interpretation of existential claims such that the existential quantifier needn’t imply existential import. My version aims to have many of the strengths that Zalta and Oppenheimer’s version has without the drawbacks of a)relying upon a Meinongian interpretation, or b) using existence as a predicate.

Note: I believe, like Zalta and Oppenheimer’s argument, that premise (2) can be used to derive God’s uniqueness. I am not sure if I can directly derive each of God’s perfections, as Zalta and Oppenheimer can using their version. That will be a future project. Also, while I use modal operators in the argument, I don’t really make use of many modal axioms to make inferences (aside from equivalency inferences). I make use of modal operators, rather, because my argument has the extremely interesting feature of deriving the necessary existence of the Anselmian God from a couple of modest premises. So the operators are their so as to track a key divine attribute Anselm claims to be able to derive from the proof. I believe the derivation of this attribute makes my version immune to Gaunilo-style parodies. That is, it shows that the Anselmian premises entail a necessarily existing being, and since islands, pizzas, and pencils are not metaphysically necessary, no object within the argument’s domain of discourse can satisfy the premises and also be contingent. Again, since this argument does not use existence as a predicate, it is immune to any Kantian objection. Finally, the argument does not illicitly move from conceivability to possibility, nor does it depend upon the S5 axiom of modal logic.

The argument is as follows:

1. Something is an Anselmian God if and only if it is conceivable but not necessarily conceived, necessarily exists, and nothing can be conceived of which is greater (definition).
2. There is something conceivable such that nothing can be conceived of which is greater (premise).
3. If the possibility of failing to conceive of x implies the possibility that x doesn’t exist, there is something conceivable that is greater than x (premise).
4. Therefore, an Anselmian God exists.

The deduction is as follows:


Cx – x is conceived
Gxy – x is greater than y
Θx – x is an Anselmian God

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

Posted on March 28, 2014, in Arguments for God and tagged , , , . Bookmark the permalink. 1 Comment.

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