by Brian Bosse, Copyright April 10, 2010, all rights reserved.
Definition: A declarative sentence is a sentence that asserts something or makes a statement.
The propositional calculus concerns itself with declarative sentences only as opposed to interrogatives, exclamatory, and imperative statements. Also, we will make an important distinction between a proposition and a declarative sentence. Consider the following three sentences…
1. ‘4’ is an even number.
2. ‘2+2’ is an even number.
3. The square root of 16 is an even number.
Even though sentences 1 through 3 above are different sentences, they all express the same thing. What these three different sentences express is the same proposition. Consider the following three sentences…
4. God is good.
5. Dios es bueno.
6. θεος εστιν αγαθος.
Once again, sentences 4 through 6 are different sentences all expressing the same thing. As such, we make a distinction between a sentence and that which it expresses. This is the motivation for our next definition…
Definition: A proposition is the referent of a declarative sentence.
Within the propositional calculus we will use certain capital letters called ‘sentence letters’ to stand for sentences. For example, we might assign the letter ‘P’ to sentence 4 above. As such, in the propositional calculus, when we see the letter ‘P’ we know it represents the sentence ‘God is good.’ So, if we say, “P is a true statement,” then we are really saying, “‘God is good’ is a true statement.” This is the motivation for the next definition and convention.
Definition: Sentence letters are symbols representing declarative sentences.
Convention: Capital letters ‘P’ through ‘Z’ are sentence letters.
1. Sentence letters are symbolic sentences.
2. If φ is a symbolic sentence, then so is ¬φ.
Informally, ‘¬’ is an abbreviation for “it is not the case that.” It can be thought of as the negation of the symbolic sentence following it.
3. If φ and ψ are symbolic sentences, then so is (φ → ψ).
Informally, ‘→’ indicates a conditional, an “if…then…” sentence. Given the symbolic sentence (φ → ψ), ‘φ’ is called the antecedent of the conditional (the ‘if’ part), and ‘ψ’ is called the consequent of the conditional (the ‘then’ part).
4. If φ and ψ are symbolic sentences, then so is (φ ∧ ψ).
Informally, ‘∧’ indicates a conjunction (“and”).
5. If φ and ψ are symbolic sentences, then so is (φ ∨ ψ).
Informally, ‘∨’ indicates a disjunction (“or”).
6. If φ and ψ are symbolic sentences, then so is (φ ↔ ψ).
Informally, ‘↔’ indicates a bi-conditional, an “if and only if” sentence.
7. Nothing else is a symbolic sentence within the propositional calculus.
by Brian Bosse, Copyright April 05, 2010, all rights reserved.
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.
□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:
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)
An Informal Presentation of the Modal Ontological ArgumentPremise 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 ObjectionsIt 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.
by Brian Bosse, Copyright July 07, 2009, all rights reserved.
The following paper is the recreation of a formal debate that took place on this website a number of years ago. It is a debate with an atheist utilizing the presuppositional apologetic method.
I hope you enjoy the debate.
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