I just wrote the following in a footnote about how to argue that ternary logic is more fundamental than binary:
The "Law of Excluded Middle" is equal in strength to the assertion that there is "true" and "false." This means there are three elements required for binary logic to work: True, False, and LEM. The evidence for this is clear: In every discussion of binary logic fundamentals, this "law" is mentioned. It is assumed to be self-evident -- at least as strongly as Euclid's fifth postulate used to be. However, despite its name, the LEM is not a law, it's simply an assumption. Once it is replaced with another equally-reasonable assumption, the nature of "true" and "false" change. Although people initially have a hard time understanding true and false without polarity, it's quite possible, and plain-to-see within ternary logic. As you can imagine, ternary logic handles the trinity of True-LEM-False natively. From a ternary perspective, pure polarization is never true: no two things are ever exactly equal. Within this so-called "polarity," it can be shown that true is more fundamental than false. Here is how that works: False is a couplet of half-truths which appears as a singularity. Hence, anything we call false can be shown to have an intrinsic structural duality. Since the inner structure of false is composed of two half-truths, Occam's Razor tells us "false" cannot be fundamental in the way "true" is. Like a house built on sand, the trinity ("true"+("halftrue"+"halftrue")) falls apart, revealing a true singularity: truth.
It's kind of densely worded, but I've been working on this argument for years and never wrote it so succinctly. It's worth repeating, with a headline of its own, so here it is.
I recognize that I have provided no evidence for the "false = two halftruths" concept, and there are other statements in here which need to be unpacked and supported.
This all links to a broader theory I've been working on via thought experiments for the past year. I'm currently calling it "Context Theory" but it may end up having a different name by the time it's finished, since I'm beginning to see that there is an overlap with the concept of AI Attention. "Attention" is the most-studied concept1 in Artificial General Intelligence research these days. However, I'm mostly just pleased that I got it all in one place concisely. A more detailed unravelling is for a future day.
On a related note, C.S. Peirce long ago observed how logic cannot be broken into fewer than three elements. I haven't studied his material enough to know if he touches on this particular approach or not. A Wikipedia article on his Semiotic Theory says most concisely:
Here is Peirce's definition of the triadic sign relation that formed the core of his definition of logic. "Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C." (Peirce 1902, NEM 4, 20–21). This definition, together with Peirce's definitions of correspondence and determination, is sufficient to derive all of the statements that are necessarily true for all sign relations.
The article goes into much greater detail in the section on Sign, Object, and Interpretant.
Footnote:
1. The language model approach now known as GPT was first revealed in a 2017 paper from Google researchers called "Attention is all you need." That paper currently has an astounding 38,000 citations. [Update only six months later, we're up to 64,934 citations.]