You’ve almost certainly heard someone at some point in your life parrot the saying, “If you’re not part of the solution, you’re part of the problem”. It’s usually a prefix to a call to action of some type, with an implied dire imperative meant to preempt any rational discourse on the issue. The idea is implied that the problem is a given, the solution is a given, and any attempt at discourse is an impediment on the solution.
But the statement is never true. Consider the rule of contraposition, wherein saying A implies B would in turn be equivalent to saying that not B implies not A. The pattern is to take the two halves of the statement, invert them, and then negate them both. In a system of logic, if the initial statement is true, then the contrapositive is also true. If the initial statement is false, then the contrapositive is also false. If there is conflict between the two, then they are both false (as we ere on the side of consistent logic, rather than complete logic)
But the contrapositive statement, “If you’re not part of the problem, you’re part of the solution” is clearly not the sentiment of the original statement, therefore a contradiction proves the original statement to be false. If you’re not part of the solution, you have no involvement with the problem that we can as of yet ascertain.
There are two other logical operations that we can apply as well, that tell us a few things. They are the Inverse and the Negation, and they are completely separate statements from the original statement that actually need to be proved on their own.
The inverse of a statement is just swapping the two parts: “A implies B” inverts to “B implies A”. We cannot tell if the inverse is true or false just from knowing that the original statement is true or false. Specifically, when we state that a specific quality implies a general property, the relationship cannot be inverted. For example, for the statement “If George ate cyanide, then George is dead”, it is not the case that “If George is dead, then George ate cyanide.” Maybe he got shot.
The negation of a statement is just taking each of the parts and negating them without moving them. So “A implies B” negates to “not A implies not B”. If you make connections quickly, you’ll see that the negation is actually the contrapositive of the inverse; whatever applies to the inverse applies to the negation. Going back to the George example, “if George did not eat cyanide, then George is not dead”. In this particular case, it’s pretty easy to tell that George could be dead for a variety of reasons not limited eating cyanide.
Knowing all of this, let’s put it all together:
- Given: “If you’re not part of the solution, then you’re part of the problem.”
- Negation: “If you’re part of the solution, then you’re not part of the problem.” This is implied by the users of this phrase. You must become part of the solution to no longer be part of the problem.
- Inverse: “If you’re part of the problem, then you’re not part of the solution.” We’ll see in a moment how this can be false.
- Contrapositive: “If you’re not part of the problem, you’re part of the solution.” This would go expressly against the concept of the original statement.
Only the first and last statements are logically equivalent, but the speakers of such phrases often imply that the negation is true and the contrapositive is specifically not true. For example: by not donating to green energy research, you’re part of the problem of massive corporate pollution. By not campaigning against gay marriage, you implicitly support it (NOTE: these issues are chosen for strongest reaction, not as indications of any opinions of mine. Specifically, I do support gay marriage and green energy research).
And the implied sentiment of the negation is also suspect. We can easily imagine scenarios where it is false (thus also proving the inverse false). Going back to the pollution example, many corporations in fact do donate to green energy research, or buy carbon offset credits, or dump their waste at approved dumping grounds, so they are making themselves a part of the solution. But they are still part of the problem, because they should be not generating so much pollution to begin with.
So there you have it, next time someone gets in your face about needing to act now and not let the enemy win, while they are taking breaths in between bouts of attempts to shout down dissent, you can appeal to their waning rationality with a soliloquy on logic!