A Formal Logic Proof of Atheism (2000)
(1.) An extensional analytic sentence is one that, through substitution of synonyms for synonyms, results in a narrowly logical truth, e.g., a truth in standard propositional logic. The theorems of propositional logic are not sentences, but propositions that are expressible by sentences. Proof of (1): see my LANGUAGE AND TIME book.
(2.) The sentence-scheme "c causes e" analytically entails (but is not synonymous with) "c and e exist; e's existence stands to c in the relation of being the result of c's existence, such that this relation is not that of e being narrowly, logically necessitated by c." The reason there is no synonymy is that there are other features of the causal relation, features not mentioned in the entailed sentence. Proof of (2). See pp. 176-187 of my article "Causation and the Logical Impossibility of a Divine Cause," in PHILOSOPHICAL TOPICS, Volume 24 (Spring, 1996)
(3.) The sentence-schema "x is omnipotent" analytically implies "for any possible existent y, necessarily, if x wills that y exist, y exists."
(4.) The relation expressed by "x wills that y exist and y exists as a consequence of this willing" is a species of the relation expressed by "x causes y to exist."
(5.) If God exists, God is omnipotent and the cause of the universe that exists.
(6) If the universe is willed to exist by God and the universe does not exist, then it is the case that [by (3), (4) and (5)] (a) God wills the universe to exist and the universe exists and (b) God wills the universe to exist and the universe does not exist.
(7) The proposition expressed by the sentence, "God wills the universe to exist and the universe exists, and God wills the universe to exist and the universe does not exist," is a negation of a theorem of standard propositional logic, namely, that it is not the case that both p and not-p.
(8) Therefore, God narrowly logically necessitates whatever possibility he causes to exist.
(9) Therefore, it is not the case that the universe is caused to exist by God [from (2)].
(10) Therefore, God does not exist [from (5) and .
Question on a different topic: In "Causation and the Logical Impossibility of a Divine Cause," does "logical impossibility" mean broadly logical impossibility?
Answer: No (a) There is no such thing as broadly logical truths or falsehoods. A logical truth, by definition, is true solely by virtue of its form, and any truth that is not a narrowly logical truth but a so-called broadly logical truth, is a truth by virtue of its content. Such truths are metaphysically necessary. Nowhere in my article did I say that it is merely metaphysically impossible for God to be a cause of the universe's existence.
(b) It is evident that I am talking about narrowly logical truths,e.g., the truths of standard propositional logic, for I regularly use the criteria for identifying such truths. For example, on page 186, I write "The concept expressed by 'is a living organism' analytically includes the concept expressed by 'is mortal,' and the relevant logical truths (e.g., "if x is an organism that dies, then x dies) can be obtained by substituting synonyms for synonyms."