The Ontological Argument

Introduction

The ontological argument I am presenting is a formal modal argument based on possible world semantics. For those uncomfortable with all of the formal logical notation, I will provide an informal summation of the argument at the end of the post. To begin, I would like to explain what I mean by possible world semantics and modal logic.

In propositional logic, there are functions that assign truth-values to atomic sentences, and functions that assign truth-values to more complex sentences built up from these atomic sentences using the sentential connectives: ¬, →, ↔, Λ, V. In modal semantics, a set W of possible worlds is introduced where these truth-value functions assign a truth-value to each sentence for each of the possible worlds in W. It is possible for particular sentences to be assigned different truth-values in different possible worlds. For instance, in some possible world it is true that Germany won World War II; whereas, in another possible world, it false that Germany won World War II. This makes truth-value relative to a particular possible world. We can now introduce the modal operators of 'necessity' and 'possibility' that make up modal logic.

Modal Operators

□p = 'p' is necessarily true. For 'p' to be necessary (□p), then 'p' is true in all possible worlds.
◊p = 'p' is possibly true. For 'p' to possible (◊p), then 'p' is true in at least one possible world.
p = 'p' is actually true. For 'p' to be actual (p), then 'p' is true in the real world. It is my burden to prove the actuality of the existence of the Christian God.

It should be noted that we can define both □ and ◊ in terms of each other.

Rule N: □p ↔ ¬◊¬p. That is to say, 'p' is necessarily true if and only if it is not the case that 'p' is false in at least one world.
Rule P: ◊p ↔ ¬□¬p. That is to say, 'p' is possibly true if and only if it is not the case that 'p' is false in all possible worlds.

Modal logic is essentially propositional logic combined with the modal operators □ and ◊ as defined above. The logical rules I will be using in my proof are as follows:

Logical Rules

A. Disjunctive Syllogism: [(a V b) Λ ¬b] → a
B. Modus Ponens: [(a → b) Λ a] → b
C. Modal Modus Tollens: [□(a → b) Λ □¬b] → □¬a
D. Substitution: [(a V b) Λ (b → c)] → (a V c)
E. Becker's Postulate: □a → □□a; ◊a → □◊a (Modal status is always necessary.)
F. Excluded Middle: a V ¬a
G. Modal Axiom: □a → a

A Formal Presentation of the Ontological Argument

Let 'p' stand for the proposition: "God exists."

1. □(p → □p)
2. ◊p
3. ◊p ↔ ¬□¬p        (Rule P)
4. ◊p → ¬□¬p         (Definition of '↔' - 3)
5. ¬□¬p                  (Rule B - 2 and 4)
6. □p → p              (Rule G - 1)
7. □p V ¬□p           (Rule F)
8. ¬□p → □¬□p     (Rule E)
9. □p V □¬□p         (Rule D – 7 and 8)
       10. Assume □¬□p
       11. □(p → □p)         (Repeat - 1)
       12. □¬p                    (Rule C - 10 and 11)
13. □¬□p → □¬p   (Conditional Derivation - 10 and 12)
14. □p V □¬p         (Rule D – 9 and 13)
15. □p                    (Rule A – 5 and 14)
16. p                      (Rule B – 6 and 15)

An Informal Presentation of the Modal Ontological Argument

Premise 1 asserts that if in any possible world God exists, then He exists in every possible world. Premise 2 asserts that it is possible for God to exist, which is the same things as saying that God exists in at least one possible world. From this it follows that God exists in every possible world including the real world. This is the essence of the sixteen step argument given above. (And they say that logic is supposed to clarify things for us!)

Possible Objections

It is clear that premises (1) and (2) are key, and they need to be established. Also, Becker's postulate (Rule E) is something the skeptic might take aim at.

The Ontological Argument

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