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on January 20 at 08:21PM

Contrapositives

I am a little confused as to when a condition also implies a contrapositive. Is it when the condition states either "any" or "every"?

1 Reply

on January 20 at 10:18PM

Hello @rosemarie,

I think I can explain that. There isn't anything that implies a contrapositive. Rather, words like "any" and "every" are indicators that we are looking at sufficient and necessary reasoning. A contrapositive is simply another way to diagram a conditional statement.

For example:
Every reptile lays eggs.
r - - - - - - - - -> lays eggs
(sufficient) (necessary)

A contrapositive demonstrates what happens when the necessary condition fails. The sufficient condition also fails.

does not lay eggs - - - > is not a reptile.

This is the contrapositive of our previous diagram. They both carry the exact same logic. You may prefer to use one or the other, but they have the same meaning.

I'll give you another example.

Any student in the history class (H) also takes geometry (G).
H - - - - - - - -> G

If a student is not taking geometry, what can we conclude? That they are not in the history class.

not G - - - - - - - > not H

This is what the contrapositive represents. If you meant to say that "any" and "every" introduce sufficient conditions, then that would be correct.