The Square of Opposition: Four Colours Sufficient for the “Map” of Logic from the “Four-Colours Theorem” to the “Four-Letters Theorem”

Title:The Square of Opposition: Four Colours Sufficient for the “Map” of Logic from the “Four-Colours Theorem” to the “Four-Letters Theorem”

Authors:Vasil Penchev

Tags:four-color theorem, four-letter theorem and square of opposition

Abstract:

One can ask the fllowing questions questions to the title. 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.
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.
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.