contrapositive

Is it try that if P -> Q, then ~Q -> ~P always?

Let's see!

Background: A -> B is an abbreviation for ~A OR B.  See https://mathworld.wolfram.com/Implies.html

| P | Q | ~P | ~Q | ~P OR Q = P->Q | ~~Q | ~~Q OR ~P = ~Q -> ~P | Same?
-----------------------------------------------------------------------
| T | T | F  | F  | F OR T  =  T   | T   | T OR F    = T        |   ✅
| T | F | F  | T  | F OR F  =  F   | F   | F OR F    = F        |   ✅
| F | T | T  | F  | T OR T  =  T   | T   | T OR T    = T        |   ✅
| F | F | T  | T  | T OR F  =  T   | F   | F OR T    = T        |   ✅

Therefore the contrapositive of a proposition holds the same truth value as the proposition itself.

OK, so what's up with cats and dogs?

"No Dogs are Cats"

restated using "if"

"There is no such dog that is also a cat"

or

"if a thing is a dog it is not a cat"

"D -> ~C"

Contrapositive:

"C -> ~D"

"If a thing is a cat, it is not a dog"