"Logics come from dynamics" (D.R. Finkelstein)
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth" ((Sherlock Holmes) A.C. Doyle)Mister Spock is the Vulcan logic-head par excellence for all you Trekkies out there, and he speaks of logic as if it's something universal, as if it were the very logos itself of the Cosmos. If there was some such universal logic per se then it would have to be λόγος itself, the ontologically a priori, foundational hypostasis upon which all logics are based. In that case, it would be the logic-as-such of logics, the latter of which are instances of one of the fundamental principles of reality. So logos cannot be a logic proper, it would be a metalogic of all metalogics; it would be hyperlogic.
Therefore λόγος is, in essence, the hyperformal meta-structure supra-framework, if you will, the structuring-ordering principle-as-such of reality-as-such. You could even say that it's one of the hyper-attractors in universal Chaos, the ur-bifurcator that bifurcates Chaotic bifurcations, the Pink Noise in the backdrop of universal White Noise.
So when someone speaks of logic, they speak of a logic out of many possible logics. All possible logics are extrapolations of λόγος, and it is impossible to formulate λόγος formally in terms of propositional or mathematical logic, because it is that which makes all logics possible. It is the underlying hyperlogic of all logics.
(You can skip the following argument to the last paragraph, if this kind of stuff is a pain in the ass to you, which is very understandable)
Okay, so what, you say. First of all, consider. What is logic? According to Sherlock, when the impossible is eliminated, the remainder is the fact, which is a matter of default. This is a reformulation of the law of the excluded middle, the tertium no datur of Aristotle. Either-or; P ⋁ Q (P or Q). In terms of truthfunctional logic's "Boolean" values (0, 1), without applying strict disjunction:
P ⋁ Q = 1 (if P = 1, Q = 1)P ⋁ Q = 1 (if P = 1, Q = 0)P ⋁ Q = 1 (if P = 0, Q = 1)P ⋁ Q = 0 (if P = 0, Q = 0).Now conjunctive values (P ⋀ Q (P and Q)) are such that P ⋀ Q = 1 if and only if P and Q are both = 1. In the classical propositional calculus, a distributive law is formulable over (P, ⋀, ⋁), such that P ⋀ (Q ⋁ R) = (P ⋀ Q) ⋁ (Q ⋀ R), which fails in quantum logic over measurements of position and momentum with respect to the uncertainty principle.
There are varieties of non-bivalent, non-truthfunctional, non-Boolean, non-classical multivalued logics and quantum logics, such as those of Reichenbach, von Neumann, Finkelstein, Kochen-Specker, Feynman, and so on (back in the day I even invented a five-valued logic over the real continuum (0, 1)).
Consider the following schematic: Let 2-v denote the Boolean/ Aristotlean / Venn-diagrammatic / truthfunctional first order logic, let 3-v denote multivalued or quantum logic of three values (e.g. Reichenbach's three valued logic), let א-v (Aleph-v) denote a multi-valued logic with cardinality of values mapped injectively into the set of integers, and let P(א))-v (P(Aleph)-v) denote the uncountably transfinite multi-valued logic (fuzzy logic with cardinality of values mapped injectively into the set of real numbers, i.e., probabilistic logics, quantum logics, and logics over Hilbert spaces, etc.). Let ⊂ denote 'is a proper subset of' :
∅ ⊂ 2-v ⊂ 3-v ⊂ n-v (n > 3) ⊂ א-v ⊂ P(א-v)I include ∅ the empty set because a logic over the empty set (a logic over a single value) is still a logic, though trivial. It is necessary to generate the Boolean universe of two values, as well as the rest of the logics over which are founded upon the theory of the construction of the field of real numbers (algebraically and set-theoretically). But at every step of the inclusions there is no falsification nor abrogation of the cruder logic; rather, the cruder logics get more and more refined. It becomes more subtle. But the hyperlogical λόγος is the through line, from the trivial to the transfinite.
However, the tertium non datur stops at 2-v; it simply doesn't apply to the rest. Language and reality are not the same thing, logic(s) and reality are not the same thing. Reality does not preclude language, which goes without saying (so to speak), and there is no language without λόγος; there is no reality without λόγος, but λόγος is not all of reality. It is a structuring principle of reality-as-such. There are more than two points to the classical logic stick. In fact, the logic stick is the real line unit interval [0,1], endpoints included. It has an uncountably infinite number of points. This ain't no shit, Sherlock.
