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In logic, the law of excluded middle (or the principle of excluded middle) states that for any proposition, either that proposition is true or its negation is true.

This principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false.

If I say that

-p is true whenever p is false

do I use the the law of excluded middle or the principle of bivalence? (It is not clear in my mind.)

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"B when (whenerver) A" means: "if A, then B".

Thus

"not-p is true whenever p is false"

means: "if p is false, then not-p is true", which is another way to state the Law of Exclude Middle.


Bivalence and Excluded Middle are obviously related, but they are not the same.

See Bivalence: Relationship to the law of the excluded middle:

The difference between the principle [of bivalence] and the law [of excluded middle] is important because there are logics which validate the law but which do not validate the principle. For example, the three-valued Logic of Paradox (LP) validates the law of excluded middle, but not the law of non-contradiction, ¬(P ∧ ¬P), and its intended semantics is not bivalent. In classical two-valued logic both the law of excluded middle and the law of non-contradiction hold.

Bivalence is a semantical principle: it states that the semantic has only two truth values.

Exclude middle is about the negation "opeartion".

We may have a logic with more than two truth values and we may have a logic without negation.

In classic logic, where we have a bivalent semantic and the negation sign, the two interact via the truth table.

The negation "swaps" the truth value:

p is TRUE iff not-p is FALSE,

and this is LEM.

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  • Thanks @MauroAllegranza, but can't we say that "not-p is true whenever p is false" is a consequence of the principle of bivalence? – Beginner Feb 7 '20 at 16:07
  • But isn't that last step from an implication p -> q to a disjunction -p v q again only possible using LEM itself? – lemontree Feb 7 '20 at 16:20
  • @lemontree perhaps it should be rephrased as “either p is not false or not-p is true”. You don’t need LEM to state it as it is in the answer though, only the fact that not-false is true. The rest of it is from how the material conditional is defined. – Era Feb 8 '20 at 3:35

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