The Principle of Bivalence is a fundamental principle of propositions, or any sentences capable of being true. It says:
The Principle of Bivalence: Any sentence capable of being true (i.e. any proposition) is either true or false, and is not both true and false.
P = T v F & ~(T & F)
The principle itself comes from a naturalistic conception of truth. Aristotle gave the best account for such a notion of truth for its succinctness and clarity: “To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true.” (Metaphysics IV 7, 1011b27). Alfred Tarski aimed to follow such a notion of truth in his “T-Schema” account that goes that for any proposition P, “P” is true if and only if (iff) P, and “P” is false iff ~P. For example, the sentence “The chestnut is on the mat” is true iff the chestnut is on the mat and false if it is not the case that the chestnut is on the mat.
Some implications of the Principle of Bivalence are:
~T → F & ~F → T
This says that if some proposition is not true, then it will be false. And if some proposition is not false, then it will be true. This would seem to be true in any case where we are dealing with some sentence capable of being true. If it’s not false to say that “This area of sand is a heap” then that must be because it is true to say “This area of sand is a heap”. (I may examine vague predicates such as “heap”, “baldness” and the like in a future post. The issue with vague predicates seems to be an issue about how much to expand truth conditions in borderline cases.)
So far we have said a bit about propositions, and how the Principle of Bivalence is to apply to them. Is there anything to be said via the Principle of Bivalence about non-propositions, i.e. those things that aren’t capable of being true (e.g. a chair)? I think a Principle of Nonsense is in order.
The Principle of Nonsense: for any non-proposition, it is true that it is not true and not false (or, equivalently, it is not the case that it is either true or false, i.e. it is not the case that The Principle of Bivalence holds).
NP = ~T & ~F
Some implications (or equivalences) of the Principle of Nonsense:
~(T v F)
This just shows us explicitly that it is a rejection of the Principle of Bivalence in play.
~(~T → F) & ~(~F → T)
Obviously, the truth that a chair is not true does not indicate that a chair is false. Likewise, the truth that a chair is not false does not indicate that a chair is true. A chair is neither true nor false. Another way to say it: a chair lacks truth-values. It is truth-functionally meaningless.