Accepted for Reflections on Paraconsistency, 6th UNILOG - World Congress and School on Universal Logic, June 16-26, 2018, Vichy, France. But I'll be giving two other talks there instead.
Divisions of Everything:
Cognitive Frameworks Which Ground Contradiction and Noncontradiction
We describe eight cognitive frameworks, "divisions of everything", which frame the issues that our minds address. We may consider these frameworks as interpolations which vary from a state of contradiction (the Nullsome) to a state of noncontradiction (the Sevensome).
The issue of existence requires that our mind support two perspectives. On the one hand, we need to be able to ask, Does this chair exist or not? On the other hand, we need to be able to definitively answer, This chair exists (if it exists) or not (if not). Thus we need one perspective where opposites coexists (as with free will) and another perspective where all is the same (as with fate). And our mind slides readily from a perspective of free will to that of fate, but not the other way around. Similarly, when we think of the "outside" of a cup, we also have the "inside", but if we fall inside a cup, or a world, then there is only inside. Thus we can describe an underlying cognitive framework (a Twosome), a division of Everything into two perspectives ("opposites coexist" and "all is the same").
Similarly, our minds work with three perspectives in considering participation. We "take a stand", "follow through" and "reflect". Such a three-cycle grounds the scientific method.
For questions of knowledge we need four perspectives: Whether, What, How and Why. Five perspectives ground decision making and six perspectives ground morality.
These cognitive frameworks are not explicitly defined but rather we experience them implicitly as framing the particular issues that our minds entertain. We may think of them as resulting from our mind dividing Everything, where the latter is given by these four properties:
Seven perspectives establish the logical square which is needed for a self-standing, explicit system, where we may distinguish a privileged opposite - what is true, or what is good - from what is false, or what is bad. A self-standing system includes the four corners of the logical square: "All are true", "There exists at least one that is true", "There exists at least one that is not true", "All are not true", but also three sides: "All are true and at least one is true", "There exists at least one true and at least one not true", "All are not true and at least one is not true".
Now adding an eight perspective - the last side of the logical square - "All are true and all are not true" - yields a system that must be empty. We can identify this system with a state of contradiction, a division of everything into no perspectives, which we can identify with God. Indeed, we can imagine God who asks himself, Am I necessary? Would I be if I was not? and who engenders, from a state of contradiction, through a proof by contradiction, various divisions of everything. We describe in more detail how our minds shift from one cognitive framework to another by operations +1, +2 and +3 mod 8.