7. Diagonal Quantifier Troikas and Trilemmas
Here is a Diagonal Quantifier Troika:
Moe: No frogs are princes.
Larry: Some but not all frogs are princes.
Curly: All frogs are princes.
Moe, Larry and Curly all agree that frogs exist.
When the Stooges vote, each of the following propositions passes by 2/3 majorities each:
LK: Some frogs are princes.
ML: Some frogs are not princes.
KM: All or no frogs are princes.
The last can be read, “All frogs are equally princes.”
Call this a Diagonal Quantifier Trilemma.
Any two of a trilemma imply the negation of the third. Therefore:
If some frogs are princes, and some frogs are not princes, then sumbunal frogs are princes.
If some frogs are not princes, and ollerno frogs are princes, then no frogs are princes.
If ollerno frogs are princes, and some frogs are princes, then all frogs are princes.
In general a diagonal-quantifier trilemma has the form:
Some A have property P;
Some A do not have property P;
All As have property P equally;
- choose two!
Some men are good;
Some men are not good;
All men are equally good;
- choose two!
The trilemma implies these three deduction rules:
If some men are good, and some men are not good,
then not all men are equally good.
If some men are not good, and all men are equally good,
then no men are good.
If all men are equally good, and some men are good,
then all men are good.