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. 

on Leibniz on equilibrious things.

“When two incompatible things are equally good, and neither in themselves, nor by their combination with other things, has the one any advantage over the other, God will produce neither of them.”

G. W. Leibniz, Letters to Clark

Suppose that a thing x is equilibrious if and only if there are two things such that one or the other is x but not both and neither one is better than the other. Letting ‘E’ abbreviate ‘equilibrious’ and letting ‘B’ abbreviate ‘better’, we can formalize the nature of the equilibrious as such:

(x)(Ex ↔︎ (∃y)(∃z)(((y = x v z = x) • ~(y = x • z = x)) • (~Byz • ~Bzy))

Now, according to Leibniz, there are no equilibrious things (see the proof below; the premise 2 in the proof is a formalization of Leibniz’s claim above).  But how does Leibniz know this? Why couldn’t it be the case that the best possible world (which, for Leibniz, is the actual world) requires at least one equilibrious thing?

Logical Vocabulary

Predicates: A, B, C, …

Things: a, b, c, …

Statement placeholders: φ, ψ, ω …

Logical equivalence: ⇔

Special abbreviations: W: wills   B: better   E: equilibrious   g: God

Propositional attitude abbreviation schema:

[constant]_{[PROPOSITIONAL ATTITUDE]}([PROPOSITION])

e.g.: ‘r_W(φ)’ reads ‘r wills that φ’, and  ‘g_W(φ)’ reads ‘g wills that φ’

Special inference rule:          Divine Decree (DD)

                                                        g_W(φ) ⇔ φ

i.e. If God wills that φ, then infer that φ, and vice versa.