**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

**: impossible logic triads inscribed upon impossible figures:***Penrose Trilemmas*
## No comments:

## Post a Comment