# “If you are not part of the problem, you are part of the solution” contraposition law in logic

I came across this SMBC comic today. In short what it does is trying to apply the law of contraposition to the famous adage:

“If you're not part of the solution, you are part of the problem.”

Reversing it like so:

``````¬S → P = ¬P → S
``````

Making it a logical conclusion that:

“If you are not part of the problem, you are part of the solution.”

Is that argument valid according to the rules of classical logic? I believe there is something fallacious, perhaps in the way the first statement is notated, but I cannot figure out what.

• It's a false dichotomy, to say the least... Mar 4, 2013 at 0:11
• So the fallacy would be just informal, but is impeccable from a formal point of view? Mar 4, 2013 at 0:15
• Well, that's basically right, I think -- starting from absurd premises, strictly anything "formally" follows. I'm curious what others might have to say here, though. Mar 4, 2013 at 0:19
• It is classically valid. Contraposition is a valid inference in classical logic. Some forms of relevance logic and other logics that deviate from the material conditional invalidate this inference. What Joseph is hitting on, I imagine, is that the conditional is probably false (because you could, for instance, be neither a part of the problem or solution). Mar 4, 2013 at 2:50
• I understand what @JosephWeissman meant, I understand that the premises may be flawed. And I also am aware that contraposition is classically valid. I was wondering, rather, mostly if there was a flaw in the way the statement is rendered and the logical proposition executed. Mar 4, 2013 at 3:06