“In regard to things present or past, propositions, whether positive or negative, are true of necessity or false. And of those contradictorily opposed one, again, must be true and one false […]”
Aristotle, On Interpretation 9 (Harold P. Cook, 1962)
Previously, we discussed a particular kind of proposition pair: contradictories. There are two interesting basic principles that characterize certain relations between such propositions: the Principle of Excluded Middle (PEM, for short) and the Principle of Non-Contradiction (PNC, for short). We have already talked about the latter. Let’s turn to the former.
This basic logical principle expresses a fairly intuitive fact – for any pair of contradictory statements, at least one must be true. For any proposition, P, and its contradictory, not-P, the formal statement of PEM is this:
To make this more specific, recall our earlier lesson on the square of opposition. PEM justifies our statement of the contradictory opposition between Universal Affirmative and Particular Negative propositions, or between Universal Negative and Particular Affirmative propositions. By our earlier definitions, the truth of a Universal Affirmative proposition implies the falsity of a Particular Negative proposition. We can stress this as a conditional statement to make the PEM more precise: if a Universal Affirmative proposition is true, then its correlated Particular Negative proposition is false. Note that the PEM does not target specific individual propositions, but pairs of contradictory opposites on the square of opposition.
[Activity] On the face of it, PEM is very intuitively reasonable. To illustrate, suppose we ask someone the question “Is chalk white?” If they answer, “Either chalk is white, or it isn’t,” we would probably roll our eyes. This answer is worthy of an eye-roll precisely because the PEM is so obvious to us: of course it is one or the other! Can you think of what chalk would look like if it were neither white, nor not white? Is this even possible? Discuss with a partner![Activity] Pair up with a classmate and discuss the following “proof” – Either the Earth is flat, or it isn’t. If the Earth were flat, a large amount of our verified and independently valid scientific observations (and the theories to which they belong) would completely fall apart. But our scientific observations and the theories to which they belong are sound and reliable. Therefore, the Earth isn’t flat. How is PEM being used here?