Proving What We Already "Know"

[Belacqua]

Other than one's existence, unless a proposed truth is falsifiable, it is a subjective proposition, of which there are an infinite number.

IMHO the principle of noncontradiction counts as a non-subjective truth precisely because it cannot be falsified. But your point is well taken with respect to empirical truthes.

This is from Britannica:

the three fundamental laws of logic: 

(1) the law of contradiction, (2) the law of excluded middle (or third), and (3) the principle of identity. 

The three laws can be stated symbolically as follows. (1) For all propositions p, it is impossible for both p and not p to be true, or: ∼(p · ∼p), in which ∼ means “not” and · means “and.” (2) Either p or ∼p must be true, there being no third or middle true proposition between them, or: p ∨ ∼p, in which ∨ means “or.” (3) If a propositional function F is true of an individual variable x, then F is true of x, or: F(x) ⊃ F(x), in which ⊃ means “formally implies.” Another formulation of the principle of identity asserts that a thing is identical with itself, or (∀x) (x = x), in which ∀ means “for every”; or simply that x is x.

https://www.britannica.com/topic/laws-of-thought

Are these considered falsifiable? They certainly wouldn't be subjective. 

I think we have to be careful not to blend categories here. As you say, beliefs we have about the world that are empirically supported don't operate the same as these purely logical rules. 

I would also insist that it's an error to apply the laws of logic to the phenomenological, internal, experiential world (what used to be called, poetically, the heart). Our desires, loves, etc., may be contradictory. Our personal beliefs may be both "I believe" and "I don't believe," simultaneously.