# The definition of proposition and the principle of the excluded third

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?

• The concept of "proposition" lies at the intersection of a number of disciplines and areas within those. People from different areas will have different definitions/terminology/ideas about these things! I'd expect a text written by a mathematician about mathematical logic and the law of the excluded middle to define a proposition as something like "a well-formed string of symbols built inductively according to some syntax", and to talk about valuations and semantic/syntactic truth. In that context it's clearer why you need these definitions. Where have you been learning about these things? Sep 20 at 20:41
• Actually, I learn them from a math course. Sep 20 at 20:48
• I think your definition of a proposition is more of an intuition given to people first learning the subject. Unfortunately, I don't myself know a better way to describe a proposition! Sep 20 at 23:07
• Sep 21 at 5:50
• The concept of "proposition" in logic assumes that the meanings and the relationships that hold among them is based only on their truth values. This assumptions is formalized trough the semantical definitions of logical operators (not, and, etc) with truth tables. This is the gist of "a proposition is either true or false", meaning that what counts in the logical relations between them is only their truth value. Sep 21 at 6:32

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.

• 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.
• Can I say that the property that a proposition can either be true or false, is a consequence of the principle of the excluded third, but not the definition of proposition ? Sep 21 at 11:48
• @mnmn1993 1. No, it's the other way round: the bivalence law (a clearer statement of it is, "given a proposition and an interpretation, the former is exclusively true or false") logically implies the excluded-middle law (a clearer statement of it is, "given a proposition and an interpretation, exactly one of the former and its negation is true"), but not vice versa. Sep 21 at 12:21
• @mnmn1993 The definition of proposition is a meta statement, so it's not instrutive to treat it as a proposition (otherwise, isn't it defining itself?) or to discuss whether it implies or logically implies propositions/theorems/laws. Incidentally, given theorems (true propositions) S and T, isn't it technically always true that S implies T, hehe. Sep 21 at 12:26

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.

• I think this is a great answer, because it adds nuance to the original idea ("eligible" is the perfect word here!) while remaining accessible and non-technical. +1 Sep 21 at 13:36

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): $$\neg(\neg p \wedge \neg \neg p) = \neg(p \wedge \neg p) = p \vee \neg p$$

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)