excluded middle vs. bivalence.

For clarity and brevity let’s just stick with propositions with singular terms. According to the Principle of Excluded Middle (PEM), for any proposition p (e.g., ‘Socrates is wise’ and ‘Socrates is not wise’), there’s *only* two kinds: there’s the ones that say something of something and there’s the ones that deny something of something. Let’s call the former ‘p’ and the latter ‘~p’. To put PEM in symbols we could scribble the following:

PEM: p v ~p

Notice what’s absent in PEM. PEM doesn’t say anything about whether the p’s and ~p’s are true or false. For all PEM might care it might be the case that both p’s and ~p’s are true, or both false, or one half true and the other half neither true nor false, or one half true and false but the other half only false, and so on. The point is that PEM says *nothing* about truth and falsity, so from a logical point of view, PEM implies *nothing* about whether a single proposition is anything at all, much less true or false or pink or blue or all (or none) of the above.

So that’s why we need (among other things) Bivalence, which says that all propositions p (e.g., ‘Socrates is wise’ and ‘Socrates is not wise’) are only either true or false. In other symbols:

B: Tp v Fp

Assuming that ‘truth’ and ‘falsity’ are contraries (that they cannot apply to the same subject (e.g., a proposition) at the same time), we get a semantic version of what Aristotle called the Principle of Non-Contradiction:

PNC: ~(Tp ^ Fp)

PEM’s being true implies that there are only p’s and ~p’s. B’s being true implies that all the p’s and ~p’s are true or false. PNC says that the p’s and ~p’s that are true are not false and vice-versa.

My hunch is that Aristotle thought that PEM and B are really the same since he thought of a proposition as an assertion that something is the case, which is tantamount to saying that the proposition one is asserting is true.

Aristotle also thought that truth depends upon being, but not vice-versa. That is, that we are entitled to infer Socrates’ sitting from knowing that ‘Socrates is sitting’ is true, and vice-versa. But the truth of the latter depends on the former in a way that that the former does not depend on the latter. That is, because Socrates’ sitting obtains, the proposition ‘Socrates is sitting’ is true, but Socrates’ sitting does not obtain because the proposition ‘Socrates is sitting’ is true, but rather the opposite is true, that Socrates’ sitting makes the proposition ‘Socrates is sitting’ true. The idea is that the world *makes* propositions true or false, but propositions don’t make the world do anything.

For this reason (on my view) Aristotle rejected B (or something like it) about contingent propositions regarding the future. Since tomorrow’s sea battle doesn’t exist (it’s TOMORROW!) there’s nothing to make the proposition ‘there is a sea battle tomorrow’ either true or false. And hence, so much for B.