Trilemmas on Equality and Constancy

Trilemmas on Equality and Constancy

Consider these two trilemmas:

A has property P;

B does not have property P;

A equals B.

Some things have property P;

Some things do not have property P;

Everything has property P equally.

For instance:

Superman can fly;

Clark Kent can’t fly;

Clark Kent is Superman.

Some men are good;

Some men are not good;

All men are equally good.

         

In each trilemma, any two of the statements imply the negation of the third. Therefore:

If Superman can fly and Clark Kent can’t fly,

          then Clark Kent is not Superman.

If Clark Kent can’t fly and Clark Kent is Superman

          then Superman can’t fly.

If Clark Kent is Superman and Superman can fly,

          then Clark Kent can fly.

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.

So the trilemmas defy syllogistic reasoning, yet also summarize it!

These trilemmas apply to two dual concepts; object equality and predicate constancy. Objects are equal when they have equal properties:

(x = y)   =    For any property P,   P(x) if and only if P(y).

A predicate is constant when it applies equally to all objects. Constancy is also when the predicate is always true, or never:

Con(P)   =    For all objects x and y, P(x) if and only if P(y).

Con(P)   =    P is true for all x, or P is true for no x.

Constancy is the “all or no” quantifier. It is to the “all” quantifier as “if and only if” is to “and”.

Now compare and contrast:

(x = y)   =  For any property P,   P(x) if and only if P(y).

Con(P)  =  For any objects x and y,  P(x) if and only if P(y).

Equality and constancy are complementary. Both start from objects having a property equally; equality generalizes the property, constancy generalizes the objects.

Equality defines identities; constancy defines laws. 

Here are two Penrose Trilemmas: impossible logic triads inscribed upon impossible figures: