Squares of Opposition

1. Squares of Opposition Modern v. Traditional

2. The Traditional Square of Opposition

3. Contradictory • A & O or E & I • Both can’t have the same truth value. • Only one can be true; the other must be false.

4. Contrary • A & E statements only. • Both can’t be true, but both could be false (not guaranteed), unless a necessary truth is involved. • Relationship doesn’t hold in modern square, except with additional, existential premise.

5. Sub-contrary • I & E statements only • Both can’t be false, but both could be true (not guaranteed), unless you’re dealing with a necessary falsehood. • Relationship doesn’t hold in modern square, except with additional, existential premise.

6. Super > Sub-alternate • A to I, E to O • Truth of super implies truth of sub • Note that it doesn’t go the other way. My other car is a super-sub.

7. Sub > Super-alternate • I to A, O to E • Falsity of sub implies falsity of super. • Note that it doesn’t go the other way. Is this making sense?

8. Implications of the Modern Square of Opposition • Don’t freak out! • So, you may have learned about relationships that, unlike Aristotelians, modern logicians assert don’t hold. Get over it. • The important concept to learn here is that of existential import. • Otherwise, one of the most important and easily recognized relationships (contradiction) is preserved. • The other relationships still work, so long as we add a premise asserting that “There is at least one S.” • We will learn more about adding missing premises in the next section, dealing with enthymemes. • FYI, the term enthymeme comes from the Greek en “in” + thymos “mind,” or, literally, a proposition retained “in the mind” rather than made explicit in the argument.

9. Contradictions and the Square of Opposition Only contradiction remains, unless the premise “There is at least one S” is supplied.