some things are both not true and not false.

It seems plausible to suppose that for any statement (S: statement), it is false (F: false) if and only if it is not true (T: true). Formally:

(f)      (x)(Sx → (Fx ↔︎ ~Tx))

(f) seems to be above reproach. But some might confuse (f) with the claim that a thing is false if and only if it is not true, or

(f’)      (x)(Fx ↔︎ ~Tx)

But (f’) isn’t true. To see this, consider b: b is a banana (B: banana) and b is neither true nor false. Thus, it follows that (f’) is false. Here’s the proof: 

The upshot is this: if someone tries to tell you that you must think that a certain sentence is false because you think it is not true, you can tell him that he should consider a banana.