Understanding how contrapositive work

A usefull distinction to be made here is the distinction between : sufficient condition and necessary condition.

The sentence "Sunny --> Light" says that

the fact "it is sunny" is a SUFFICIENT condition in order " there is light" to be true. ( Saying that "X is sufficient for Y" does not exclude the possibility of obtaining Y without X. If I am a pianist, I am a musician. This is perfectily true, and it does not deny the possibility of being a musician without being a pianist).

One cannot object to this that " one can make light with a flashlight".

Such an objection amounts to saying " the sun is not a necessary condition in order " there is light" to be true".

Such an objection is directed at an assertion that was NOT contained in the original sentence.

Both the original sentence and its contrapositive say exactly the same thing under various points of view, namely

(1) sun is sufficient for light

(2) light is necessary for sun

(3) absence of light is sufficient for absence of sun

(4) absence of sun is necessary for absence of light.

NOTE : Saying " X --> Y " is equivalent to

(1) X is sufficient for Y

or, equivalently

(2) Y is necessary for X


The statement "if $p$ then $q$" has truth value true if it's true or false otherwise. The contrapositive is "if not $q$ then not $p$". The two statements are equivalent, i.e. always the same truth value. This is unrelated to the question of what that common truth value is. The point is the logically possible universes in which "If it's sunny there is light" are true are precisely those in which "if there's no light it's not sunny" are true.

You may have confused the contrapositive with the converse, whose truth value can be different. Confusing "if $p$ then $q$" with its converse, "if $q$ then $p$", is a fallacy called affirming the consequent. Confusing "if not $q$ then if not $p$" with its converse, "if not $p$ then if not $q$", is a fallacy called denying the antecedent.


Read your correct contrapositive again:

If there is no light then it is not sunny.

That's true - it's equivalent to the first statement. You could create light with a flashlght, but then there would be light.

What

you could create light with a flashlight.

tells you is that the original true statement is not "if and only if". The converse

If there is light then it is sunny

is false.

Tags:

Logic