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According to the Law of Excluded Middle, a statement is either true or false, but in the case of these paradoxes, if the statement is true, then it's false, and if it's false then it's true.

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I love it! I don't know much about the Law of Excluded Middle but I love the logic you've presented -- Of course I like things being BLACK/WHITE and TRUE/FALSE (the EITHER/OR argument). It's when the gray area (the middle) is introduced that I get all frazzled!


The argument sounds to me like a double-bind!

trosie

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I want to know people's thoughts on how what they call the "Liar's Paradox" relates to the logical Law of Excluded Middle. I had a professor try to use such semantic paradoxes once to demonstrate that Aristotelian logic doesn't apply universally.

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One of the primary principles of logic is the law of (non-) contradiction. Basically it states that no statement (proposition, assertion, etc.) can be both true and not true--false--(e.g., A can not be non-A) at the same time and in the same sense. For example, it can not both be true and not true (in the same time and sense) that a person is and is not a Christian. All such statements are false. It is a first principle (it is axiomatic or self-evident) of thought or epistemology. (Thus, one cannot not use it. To deny it is like saying "I cannot utter a word in English.") The distinction between true and false is based on this principle.

The second primary law of logic is the principle of excluded middle. The law of (non-)contradiction simply states that A cannot equal or be non-A (or p & non-p). But both could be "quip," that is, neither true or false--simply not both--but not necessarily true or false either. But excluded middle states A or non-A, that is, a proposition is either true or false--it must be one or the other (and not quip!). Thus, a proposition or statement must be true or false. (See, e.g., Matthew 12:30).

The third primary law of logic is called the law of identity. It states that A=A or that "if any statement is true, then it is true."(8) An example would be Christ is Christ and not non-Christ (e.g., the Christ of the Bible is not the "Christ" of the cults or the occult.

The fourth primary law of logic is the law of logical or rational inference. An example of this principle is: if A=B, and B=C, then A=C. It's importance can be seen relevant to all discursive or non-axiomatic knowledge or reasoning. For example, the word trinity is not found in the Bible, but the concept is found in the Bible (see, e.g., Deut. 6:4; Eph. 1:3; John 1:1; 20:28; Acts 5:3-4; and Mark 12:29-30).

"Evolution is the substance of fossils hoped for, the evidence of links not seen."

“The problem with creation is not that it has been tried and found wanting, but that it has been found difficult, and left untried.”

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You and your professor are confusing the Principle of Bivalence with the Law of the Excluded Middle. Aristotle accepted the Law of the Excluded Middle, but rejected the Principle of Bivalence, at least in regards to statements about the future.

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Chuck (or anyone else for that matter), can you explain how the two principles do differ. The Cambridge Dictionary of Philosophy goes into probability to demonstrate the point, but I'm not sure I follow it. It concludes its entry on bivalence by noting that not all philosophers accept the distinction between bivalence and excluded middle. I can see why. If it's there, it's a pretty fine line.

Can anyone throw me some clarity?

Anyone's comments on how they would explain these paradoxes would make for some good conversation as well.

"The longer I live, the more I read, the more patiently I think, and the more anxiously I inquire, the less I seem to know." ~President John Adams

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in multi-valued logics there are three truth values: true, false, and indeterminate. Bivalence is denied in favor of trivalence. But excluded middle still holds: If A is false, then not-A is true; if not-A is false, then A is true. The difference that denying bivalence makes is that IF A is NOT TRUE then we cannot infer that A is false (it could be indeterminate). To say that A is false is not the same thing as saying that not-A is true, unless we assume bivalence

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I appreciate your response, Alan. I feel I'm getting closer. Tell me about this indeterminance, though. Is it simply due to our ignorance in a particular situation regarding a particular subject or what?

"The longer I live, the more I read, the more patiently I think, and the more anxiously I inquire, the less I seem to know." ~President John Adams

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Tell me about this indeterminance, though. Is it simply due to our ignorance in a particular situation regarding a particular subject or what?

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So on what basis or bases do some philosophers deny the distinction between bivalence and excluded middle

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I'm sorry. I didn't make my question clear. I understand what you've said. I took the Cambridge Dictionary's comment on some philosophers rejecting the distinction to mean that these philosophers understand it but reject it for some reason anyway. I was just wondering if they have arguments for rejecting the distinction or if they simply reject it because they don't understand it.

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My logic text, possibly seeking to avoid having to face this issue, possibly legitimately, simply responded that statements such as “this statement is false” are not propositions. --SeaKayaker

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