“If you’re not part of the solution, you’re part of the problem.”
This tired old adage sounds accusing and incriminating, like you’re supposed to feel guilty for not working towards a solution, a better world. It’s a bold, shaming aphorism.
Here’s my version.
“If you’re not part of the problem, you’re part of the solution!”
Optimism. Relief. Exoneration.
Mine sounds so much more optimistic than the original. Yet, they are exactly logically equivalent.
Both of them state that “you” are “part of the solution” or “part of the problem” (and maybe both). There are two sets: “problem”, “solution” and you are a member of at least one.
Don’t believe me? Here’s a rough guide to a formal proof.
Let S = “you are part of the solution” and P = “you are part of the problem”.
The first statement translates to “~S => P”.
The second statement translates to “~P => S”.
We will show that one implies the other. First, we will prove that (~S => P) => (~P => S).
1. ~S => P (assumption) 2. ~P (assumption) 3. ~S (assumption) 4. P (1&3, implication, depends on 1,3) 5. P & ~P (2&4, depends on 1,2,3) 6. ~~S (RAA, 5&3, depends on 1,2) 7. S (double negation, depends on 1,2) 8. ~P=>S (2&7, implication, depends on 1) 9. (~S=>P)=>(~P=>S) (1&8, implication, no dependencies)
The proof that (~P=>S)=>(~S=>P) is virtually identical (just swap the roles of P and S).