Syllogisms, Necessary Conditions and Counterexamples

Translate before judging

“All A are B” places every A inside B. It does not mean all B are A. “Some A are B” asserts at least one common member. “No A are B” means the sets do not overlap.

Use only the information given. In standard logical interpretation, a universal statement by itself does not assert that the set has members. If a test specifies a different convention, follow that instruction.

Worked syllogism

Statements: All researchers are readers. Some readers are artists. Conclusion 1: Some researchers are artists. Not necessarily: the artist-readers may all lie outside the researchers. Conclusion 2: All researchers are readers. Follows directly. A quick counterexample is stronger than a guessed diagram: imagine two researchers who are not artists and a third reader who is an artist. Both statements hold, while conclusion 1 fails.

Necessary and sufficient conditions

“If P, then Q” means P is sufficient for Q and Q is necessary for P. It does not establish “if Q, then P”. Its contrapositive, “if not Q, then not P”, is logically equivalent. Example: Every approved application has a verification record. If a particular application has no verification record, it cannot be approved under this rule. Having a record alone does not prove approval.

Worked ordering problem

P is before Q; R is after Q; S is before P. The order must be S, P, Q, R when these are the only four people. If a fifth person T is introduced without a constraint, T's position is not determined. Do not silently assume the list is complete.

Practice with explanations

1. All roses are flowers. Some flowers fade quickly. Must some roses fade quickly? No; the quickly fading flowers need not include roses.
2. Some books are manuals. No manuals are novels. Must some books be non-novels? Yes; the books that are manuals cannot be novels.
3. If a number is divisible by 4, it is even. Is every even number divisible by 4? No; 6 is a counterexample.
4. A is taller than B; B is taller than C. Is A taller than C? Yes, using transitivity of this relation.

Common traps

Reversing all-statements, importing real-world beliefs, assuming existence without support, treating “may be true” as “must be true”, or adding an unstated ordering constraint.

Revision routine

For every disputed conclusion, try to construct one situation in which the statements remain true but the conclusion is false. If you can, the conclusion is not necessary. If you cannot, prove it from the premises rather than relying on lack of imagination.