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Craig · #5 (**permalink**) 02-16-2008, 04:31 PM

It was all very helpful, Brian...I'm scratching my head about this:

Quote:

He followed this up with a bunch of gibberish that looked like he was trying to use truth tables. Well, it was non-sense. The law of excluded middle simply says, "Every proposition is either true or not true." Symbolically, this can be stated as follows: (A v ¬A). This law is not proved, but rather is assumed. However, it's derivation can be stated as follows...

1. Show (A v ¬A)
2. Assume ¬(A v ¬A)
We are assuming the opposite of what we are trying to show. If we can do this and derive a contradiction, then we have shown what we set out to show. This form of derivation is commonly called Indirect Derivation.
3.Show A → (A v ¬A)
4. Assume A
5. (A v ¬A) [Addition]
6. ¬A [Modus Tollens - 2, 3]
7.Show ¬A → (A v ¬A)
8. Assume ¬A
9. (¬A v A) [Addition]
10. (A v ¬A) [Modus Ponens - 6, 7]
11. ¬¬(A v ¬A) [Double Negation - 10]
This constitutes a derivation of the law of excluded middle because by assuming ¬(A v ¬A) I was able to derive its contradiction ¬¬(A v ¬A). The problem with this argument is that the logical laws I used such as Addition assume the law of excluded middle. Again, the law of excluded middle is foundational. It is assumed and by its assumption you can prove other things. If your atheist friend wants to deny it, then he may. But my guess is that he really does not live this way, and his doing so is arbitrary rather than empirically derived.

Now, your atheist friend seemed to be saying that the law of excluded middle is something like this: (A v B v C) → (A v C). This is absurd. I think he is confusing exlcuded middle with the transitive property of mathematics which says, if A=B and B=C, then A=C. By the way, it is not the case that (A v B v C) → (A v C). For example, it is a true proposition that "Either my name is Tom, or my name is Brian or my name is Bob". However, it is not true that "Either my name is Tom or my name is Bob."

I have NO clue about any of those formulas...are you saying his formula only allowed for one of two answers, not both...except he cannot demonstrate either of the options are viable? For instance, you saying "Either my name is Tom or my name is Bob" but in reality it is "Brian"? So he can only demonstrate an either/or, but can't ensure an accurate answer?