I long imagined that the essential flaw of binary logic lay in its foundation axiomsdefinition which exclude middles by asserting everything is EITHER true OR false with no inbetween. (This results in paradoxes which are considered to be unimportant by most people when compared with the overall value of logic.)
Recently discovered that this flaw is resting on a deeper one still, more subtle and even better hidden: There is no "single thing" which is completely false, because falseness is ALWAYS composed of at least two half-truths. One or both of these "half truths" is being interpreted out of its proper context, and therefore being assigned a truth value which it cannot sustain permanently, since eventually "truth will out." (Context is important to truth. For example, a passage of time changes the truth value of some statements.)
So, to assert anything is categorically false converts an underlying duality into a seeming singleton, which is then asserted to be equal in strength to truth, when it is used in the definition of logic. A more precise definition of logic would include this, because it affects how logic is used.
This insight occurred to me during a particularly focused meditation a few weeks ago, where I finally realized there was something wrong with falseness itself, and then traced this further. I am unaware if anyone else has chanced upon this insight, although I know that there are others who see excluded middles as problematic.
[Edit: later, I find the idea is discussed in detail by Korzybski, and given a name "multiordinal." Here's a tip of that iceberg.]