### 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*)(S*x*→ (F*x*↔︎ ~T*x*))
(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’) (

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: *x*)(F*x*↔︎ ~T*x*)
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.