Everything in the indented section, except for the outline numbers, is a direct quote from Ilion's post. I am trying to sort out axioms (A) from logically proven propositions based on those axioms (P). When statements are basically repetitions of other statements, they may be given the same outline number, or deleted.

A1) When an entity reasons, it chooses to move from one thought or concept to another based on (its understanding of) the content of the concepts and of the logical relationship between them.

A2) GIVEN the reality of the natural/physical/material world, IF atheism were indeed the truth about the nature of reality, THEN everything which exists and/or transpires must be wholly reducible, without remainder, to purely physical/material states and causes.

A3) This "everything" (which exists and must be wholly

P1) IF atheism were indeed the truth about the nature of reality, THEN this movement from (what we call) thought to though (which activity or change-of-mental-state we call 'reasoning') *has* to be caused by, and must be wholly explicable in terms of, state-changes of matter. That is, it is not the content of, and logical relationship between, two thoughts which prompts a reasoning entity to move from the one thought to the other, but rather it is some change-of-state of some matter which determines that an entity "thinks" any particular "thought" when it does.

P2) ... there exist entities and events in the world which are not wholly reducible, without remainder, to purely physical/material states and causes,

P3) ... the denial that 'God is' is a false proposition.

Well, this is somewhat incomplete, but the completion seems straightforward. Let's put this in a prepositional calculus form. First, I'll lay out the bare argument.

Z = "atheism is true"

C(x) = "x changes based on the content of concepts and logical relationships"

R(x) = "x reasons"

F(x) = "x exists or changes solely on the basis of material causes"

T(x) = "x is a mind"

E(x) = "x exists"

Then, I'll rewrite Ilion's statements above.

A1) R(x) ⇒ C(x)

A2) Z ⇒ ∀x(E(x)⇒F(x))

A3) T(x) ⇒ E(x)

P1) Z ⇒ ∀x(R(x)⇒F(x))

P2) ~∀xF(x)

P3) ~Z

Let's add a couple of axioms needed to fill this out, which I suspect were meant to be implied.

B1) ∃x(T(x) & R(x))

B2) C(x) ⇒ ~F(x)

The (shortened) proof is in the table below. Note that this proof does not work without B1 and B2.

1 | Z | Assumed for contradiction |

2 | ∀x(E(x)⇒F(x)) | 1, A2 |

3 | T(c) & R(c) | B1 |

4 | T(c) | B1 |

5 | E(c) | B1, A3 |

6 | F(c) | 1, B1, A2, A3 |

7 | R(c) | B1 |

8 | C(c) | B1, A1 |

9 | ~F(c) | B1, A1, B2 |

10 | F(c) & ~F(c) | 1, A1, A2, A3, B1, B2 |

11 | ~Z | A1, A2, A3, B1, B2 |

This proof is valid. The soundness of this proof is questionable on more than one front (Elizabeth Liddle questioned a different axiom); I want to look at B2. If a change is based in part on concepts and/or logical relationships (CLR, for short), does that imply it is not based solely on material causes? I disagree. I would say that changes based on CLR are actually based solely on material causes.

My position is that CLR are patterned-yet-material reactions in the brain to material stimuli. We react with the same pattern of brain reactions to similar stimuli, and name these reactions the process of reasoning. Different people will likely store different physical patterns, but they will create the same behavior when reasoning.

So, as opposed to C(x) ⇒ ~F(x), I would say C(x) ⇒ F(x), rendering the proof unworkable. Naturally, should Ilion offer alternative versions of B1/B2, I'll take another look.

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