Aristotle's On Interpretation Ch. 8. segment 18a27: A look into the relations of truth and falsity in contradictory pairs of compound assertions
(18a27) of Ch. 8: A look into the relations of truth and falsity in contradictory pairs of compound assertions
Two assertions contradict when the one affirms and the other denies the same thing of the same thing. For example, “Socrates is wise” contradicts with “Socrates is not wise” because the one affirms and the other denies wisdom of Socrates. Similarly, “Meno is not noble” contradicts with “Meno is noble” because the one denies and the other affirms nobility of Meno.
Now, as we have established in our review of Ch. 7, there is no set of circumstances under which (a) “Socrates is wise” and “Socrates is not wise” or (b) “Meno is noble” and “Meno is not noble” are both true or both false at the same time. In each pair of contradictories, when one assertion happens to be true, the other is definitely false. Correspondingly, when one of the two assertions happens to be false, the other is definitely true. This we may illustrate as follows:
No matter the underlying circumstances, as long as we oppose an affirmation to a negation in which we assert the same thing, i.e. wisdom or nobility, of the same thing, i.e. Socrates or Meno, one of our assertions will be true and the other false. So far, we have found this to necessarily be the case for (i) pairs of contradictory assertions with a particular subject (e.g. Meno is noble - Meno is not noble), as well as (ii) those with a universal subject when one assertion applies its subject universally (e.g. every philosopher is wise - not every philosopher is wise). This topic we have extensively explored in Ch. 7 17b17-17b26.
on the relations of truth and falsity in contradictory compound assertions
Once we recognise that some contradictory simple assertions cannot both be true or false at the same time, we may then raise the question of whether this is so for some pairs of contradictory compound assertions as well. To figure this out, we join each assertion in pair (a) with one from pair (b). We thereby derive the pair of contradictory compound assertions (ab) “Socrates is wise and Meno is not noble - Socrates is not wise and Meno is noble”. We then pursue to determine whether there is any set of circumstances under which both compound assertions can be true or false at the same time.
In our present example, we set two contradictory compound assertions side by side. We do this in order to examine whether any set of circumstances exists under which both assertions can be true or false together. Each of our compound assertions comprises two simple assertions and the conjunction “and”. In turn, each simple assertion originates from two contradictory pairs. The subject of both pairs is a particular, i.e. a single, one of a kind, thing.
As we have observed, there are only three sets of circumstances possible for each of the simple assertions when they are standalone. Once we join two of them together, however, we find that the sets of circumstances possible for our resultant compound assertion are not three but nine.
Now, out of nine sets of circumstances, there are only two under which the compound assertion is true. In both of these cases its contradictory compound comes out false. Be that as it may, it is not the case that when our compound assertion happens to be false, its contradictory is true. In fact, there are five sets of circumstances in which both compound assertions are false together.
Based on our example then, we may conclude that where two compound contradictories cannot be true together, they can definitely be false together. Consequently, whether two contradictory assertions are compound by virtue of a conjunction or through conflation (see Ch. 8 18a18-18a26), it is not necessary for one to be true when the other is false.
Key points: (i) An assertion may be compound by virtue of a conjunction (e.g. “and”) or due to conflation. (ii) When a compound assertion happens to be true, its contradictory will be false. It is not necessary, however, for a compound assertion to be true when its contradictory is false.