The definition of proposition and the principle of the excluded third

Question

A proposition is defined as a statement or assertion that can either be true or false.

The principle of the excluded middle states that any proposition can either be true or false.

Do these two statements overlap? It seems that the definition of proposition already tells us that it is either true or false. What is the purpose of the principle of the excluded middle?

Answer 1

A proposition is defined as a statement or assertion that can either be true or false.

Here's a more direct definition:

• A proposition/sentence is a string of symbols that is well-formed according to the logic's syntactic rules and where no variable is free.

Noting that a proposition's truth—and whether it's even meaningful—is relative to how we interpret it (what context and definitions have been agreed on), leaving ‘statements’, ‘assertions’, ‘truth’, ‘falsity’, and the like out the above definition makes it less ambiguous. This definition tells us that in classical logic,

• if A then or B is not a proposition;
• 45++5=53 and 1=1 (whose truth-functional form is P and Q) is a proposition; as a matter of fact, it is a true statement once x++y is understood to mean add $x$ and $y$ then output the smallest prime that is not smaller than the sum;
• x=x is not technically a proposition, due to the free variable;
• each of the square of every number is nonnegative and I have a pet dog can be both true and false, albeit not simultaneously (the former is meaningful as a statement because of mathematical axioms, and false in the domain of imaginary numbers).

N.B. In the above, just for clarity, by ‘statement’, I specifically mean a proposition in the context of an interpretation. However, this distinction between proposition and statement is not important and indeed not commonly observed.

The principle of the excluded middle states that any proposition can either be true or false.

Wikipedia:

• The principle of the excluded third states that for every proposition, either itself 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.

Answer 2

A better way to state the first definition is that a proposition is the sort of thing that is eligible to be judged true or false. That is, they are the sort of things $P$ about which it at least makes sense to say, "$P$ is true" and "$P$ is false."

This is separate from whether the only possible truth values for $P$ are true and false, and whether the proposition $P ∨ ¬P$ is always true (regardless of $P$). In general propositions needn't satisfy either of these two properties.

Answer 3

Where $p$ is a proposition

The Principle of Bivalence (PB): $p$ is only either true or false. A ball (proposition) is white(true) or black (false)

The Law of the Excluded Middle (LEM): $p \vee \neg p$. Either a proposition is true or its negation is true. Given a ball, the ball is white OR not (the ball) is white. In classical logic negation is defined truth functionally so not simply denies the stated truth value of a proposition i.e. $\neg p$ means change the truth value of $p$. In my analogy, if not meant the same then not (the ball) is white = the ball is not white i.e. the color of the ball has changed from white to not white. In a two-valued (two-colored) system, there's only one truth value/color $p$ (a ball) can be under negation, to wit, false (black).

What exactly has been excluded by LEM? The neither $p$ is true nor $\neg p$ is true. Not (That a ball is neither white nor not white): $\neg(\neg p \wedge \neg \neg p) = \neg(p \wedge \neg p) = p \vee \neg p$

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Rejecting PB: An example would be to float a third truth value (color) as is done with trivalent logic.

Rejecting LEM: Amounts to paraconsistent logic and dialetheism. Contradictions can be true. A proposition $p$ (a ball) can be both true (white) AND not true (not white)

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