The square of opposition: Four “colors” enough for the “map” of logic

Intrigue (a few questions):
Why square? Why four? What is the common in the following facts?
1) The square of opposition.
2) The “letters” of DNA.
3) The number of colors enough for any map.
4) The minimal number of points, which allows of them not be always well-ordered.

The number of entities in each of the above cases is four though the nature of each entity seems to be quite different in each one.

Prehistory:

The first three share (1-3) being great problems and thus generating scientific traditions correspondingly in logic, genetics and mathematical topology. However, the fourth one (4) is obvious: triangle has not diagonals, quadrangle is just what allows of its vertices not to be well-ordered in general just for its diagonals. Thus the limit of three as well as its transcendence by four seems to be privileged philosophically, ontologically, and even theologically: It is sufficient to mention Hegel’s triad, Peirce’s or Saussure’s sign, Trinity in Christianity, or Carl Gustav Jung’s discussion about the transition from Three to Four in the archetypes in “the collective unconscious” in our age.

Thesis:

The base of all cited absolutely different problems and scientific traditions is just (4). Thus, the square of opposition can be related to those problems and interpreted both ontologically and differently in terms of the cited scientific areas and in a few others.

Arguments in favor of the thesis:

(1) Logic can be discussed as a formal doctrine about correct conclusion, which is necessarily a well-ordering from premise(s) to conclusion(s). To be meaningful, that, to which logic is applied, should not be initially well-ordered just for being able to be well-ordered as a result of the application of logical tools. 

(2) Consequently the initial “map”, to which logic is to be applied, should be “colored” at least by four different types of propositions, e.g. those kinds in the square of opposition. They are generated by two absolutely independent binary oppositions: “are – are not” and “all – some” thus resulting exactly in the four types of the “square”. 

(3) Five or more types of propositions would be redundant from the discussed viewpoint since they would necessary iff the set of four entities would be always well-orderable, which is not true in general. 

(4) Logic can be discussed as a special kind of encoding namely that by a single “word” thus representing a well-ordered sequence of its elementary symbols, i.e. the letters in its alphabet. The absence of well-ordering needs at least four letters to be relevantly encoded just as many (namely four) as the “letters” in DNA or the minimal number of colors necessary for a geographical map. 

(5) The alphabet of four letters is able to encode any set, which is neither well-ordered nor even well-orderable in general, just to be well-ordered as a result eventually involving the axiom of choice in the form of the well-ordering principle (theorem). It can encode the absence of well-ordering as the gap between two bits, i.e. the independence of two fundamental binary oppositions (such as both “are – are not” and “all – some” in the square of opposition). [Fifth World Congress on the Square of Opposition, November 11-15, 2016, Easter Island, Handbook of abstracts (edited by Jean-Yves Beziau, Arthur Buchsbaum, Manuel Correia), pp. 35-36]

The presentation also as a PDF, a video, or as slides @ EasyChair

A related paper (the above one) as a PDF, or @ repositories: @ EasyChair  

The draft paper as a PDF or a video:

From the four-colors theorem to a generalizing “four-letters theorem”: A sketch for “human proof” and the philosophical interpretation

Abstract. The “four-color” theorem seems to be generalizable as follows. The four-letters alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA (RNA) plan(s) of any (all) alive being(s).
Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters.
That admits to be formulated as a “four-letters theorem”, and thus one can search for a properly mathematical proof of the statement.
It would imply the “four colour theorem”, the proof of which many philosophers and mathematicians believe not to be entirely satisfactory for it is not a “human proof”, but intermediated by computers unavoidably since the necessary calculations exceed the human capabilities fundamentally. It is furthermore rather unsatisfactory because it consists in enumerating and proving all cases one by one.
Sometimes, a more general theorem turns out to be much easier for proving including a general “human” method, and the particular and too difficult for proving theorem to be implied as a corollary in certain simple conditions.
The same approach will be followed as to the four colour theorem, i.e. to be deduced more or less trivially from the “four-letters theorem” if the latter is proved. References are only classical and thus very well-known papers: their complete bibliographic description is omitted.

The paper at EasyChair or @ SocArxiv, or @ EasciChair. or @ SSRN, or @ PhilPapers