Friday, September 16, 2005

The law of excluded middle and self-refutation

The law of excluded-middle is a close cousin to the law of non-contradiction. While the law of non-contradiction tell us that a proposition and it’s negation cannot both be true, the law of excluded-middle tells us that either a proposition or its negation is true. So either my sister is pregnant, or my sister is not pregnant. One of those is true and the other is false. The negation of the law of excluded-middle is that a statement and its proposition are both true. In that case, my sister is both pregnant and not pregnant.

The problem with those who reject logic is that their arguments against logic are necessarily self-refuting, so before I show how that is so, I’ll explain what self-refutation is all about. A self-refuting statement is a statement that fails to meet its own standard for being true. All meaningful statements are about something. For example, the statement, “Steve teaches philosophy,” is about Steve. Some statements include themselves in their field of reference. For example, the statement, “All statements represent propositions,” is about all statements including the statement itself. A statement that is self-referential and that denies what it asserts is self-refuting. Here are some examples:

"I cannot speak a word of English," is self-refuting when spoken in English.

"All statements over five words long are false," is self-refuting because it is over five words long.

Self-refuting statements are necessarily false. Now the reason all arguments against logic are self-refuting is because they assume logic in their argument against logic. Take the denial of the law of non-contradiction for example. If a person says, “The law of non-contradiction is false,” then they are assuming that the person who believes in the law of non-contradiction is mistaken. But the only way to assume that is to use the law of non-contradiction, because otherwise, one could say that the law of non-contradiction is both true and false. But if the law of non-contradiction is true (which it must be if we are to deny it), then it is necessarily a mistake to deny the law of non-contradiction. So the law of non-contradiction is necessarily true, and denying it is self-refuting.

Denying the law of excluded middle is self-refuting for the same reason. Those who deny the law of excluded-middle say that either/or thinking is mistaken, and both/and thinking is correct. But in doing so, they are actually using either/or thinking. They are unwilling to say that BOTH either/or AND both/and thinking are correct. Instead, they are assuming that EITHER either/or OR both/and is correct, but not both, and they deny either/or thinking while accepting both/and thinking.

The only way to be consistent is to say that BOTH either/or AND both/and thinking are correct. But in that case, one is not denying the validity of the law of excluded-middle. Instead, one is affirming it. Since the law of excluded-middle excludes both/and thinking, then both/and thinking cannot be correct. If EITHER either/or thinking OR both/and thinking is correct, and if EITHER/OR thinking is correct, then both/and thinking is not correct. What we can see is that either/or thinking (the law of excluded-middle) is necessarily true, and both/and thinking is necessarily false because both/and thinking must include either/or thinking, and either/or thinking must exclude both/and thinking. Whether you begin by assuming either/or thinking or both/and thinking, you arrive at the conclusion that either/or thinking is correct, and both/and thinking is incorrect.

[Sorry I couldn't be more clear in this last paragraph, but if you'll just read it slowly and carefully, it make good sense. :-) Trust me.]


At 9/22/2005 3:46 PM , Blogger David Talcott said...

You might want to be careful about how you're using the term "necessity", since that term often carries philosophical baggage. Otherwise, great series. Very enjoyable.

At 9/23/2005 3:53 AM , Blogger ephphatha said...

I agree, David, and thanks for pointing that out. In this context, necessary means impossible to be otherwise. I suppose "impossible" needs to be defined, too, but I'm just going to cross my fingers and hope everybody understands what I'm saying.

At 5/26/2015 3:52 AM , Blogger Paul Bustion said...

I agree with the law of noncontradiction. But I don't know about the law of the excluded middle, because some statements are just nonsense, so they cannot really be either true or false. For example, "this sentence is false" seems to me to be nonsense, not true or false. I'm against Graham Priest's ideas, so I'm not agreeing with his view that the statement is a dialethia.


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