Beyond A=A

[S.R. Prozak]

Multivalent Logic - Beyond A=A

The concept of truth has great significance not only to philosophy and mathematics, but to human existence in general. While the average person may be content with the everyday notion of truth as something factual, philosophers and mathematicians tackle the concept of truth more systematically. Truth has been thoroughly discussed on Genius. The topic is truly an evergreen. I can recall endless discussions that usually resulted in endless confusion. Ironically, truth seems the more elusive the more intensely it is pursued.

The arguments brought forward here encountered several difficulties. The first of these difficulties is that we generally think of a truth as bivalent. An assertion is either right or wrong. A truth variable can take on only two possible values, “true” or “false”. This notion goes back to antiquity, to the Greeks in particular, who investigated the relations of propositions in view of their form. Aristotelian logic lays out a method of deductive reasoning that makes use of bivalent logic.

The principle of bivalence is that for any proposition P, either P is true or P is false. This is closely related to, though distinct from, the law of excluded middle and the law of non-contradiction. The law of excluded middle says that that for any proposition, either it or its contradiction is true; for any proposition P, either P or not P. The law of non-contradiction states that for any proposition P, it is not both the case that P and not-P. The difference between the law of excluded middle and the law of non-contradiction is fairly subtle. This comparison should make the distinction clear:

Law of the excluded middle: P or (not P) is true
Law of non-contradiction: Not (P and (not P)) is true

Obviously, both laws hold for any bivalent truth system. If we remove the law of excluded middle from a formal logical system, the result will be a system called intuitionistic logic. In logical calculus it is allowed to argue P or not P without knowing which one specifically is true, but intuitionistic bivalent logic doesn’t allow that. The introduction of (P or ~P) is considered a logical flaw.

The question that was discussed most extensively on Genius is whether classical logic is adequate for philosophical purposes. Its usefulness in mathematics, digital circuits, and ordinary language is certainly undoubted. But, can a bivalent truth system as defined by propositional calculus describe reality? Can it lighten the path to wisdom? In my opinion, it can’t. Classical logic is too limited. We need something better than bivalent logic. Fortunately, today we can draw on a choice of existing logical systems. There is a three-valued logic that includes one indeterminate state besides the true and false values. There is fuzzy logic, which is probabilistic. In fuzzy logic, truth values represent the continuum between 0 and 1 expressed by a fraction 1/x, whereas values closer to 1 are truer than values close to 0. In addition there are a number of other more exotic logical systems of which most people haven’t heard.

The Buddhist logic is multivalent. If you have read some of the longer sutras, you will probably be familiar with it. Buddhist logic rises above the common bivalent notion by adding two more relations, resulting in a set of four possible relations between two given proposition in the form of “either
or
both
neither”.

In the Aggi-Vacchagotta Sutta (Majjhima Nikaya 72), the wanderer Vacchagotta asks the Buddha whether the cosmos is eternal. The Buddha answers:

“Eternal doesn't apply. Not eternal doesn't apply. Both eternal and not eternal doesn't apply. Neither eternal nor does not reappear doesn't apply.”

Similar arguments appear throughout the Pali canon. The above dialogue can be formalized easily. Let’s assume that X is a truth placeholder and that P and Q are contradictive propositions with Q = not P and P = not Q. Buddhist logic allows four possible equivalences:

(a1) X = P
(a2) X = Q
(a3) X = P and Q
(a4) X = neither P nor Q,

or respectively their negation, as phrased in the argument with Vacchagotta:

(b1) X = not (P)
(b2) X = not (Q)
(b3) X = not (P and Q)
(b4) X = not (neither P nor Q),

By replacing Q with Not (P) in (a1-4), we get:

(c1) X = P
(c2) X = ~P
(c3) X = P & ~P
(c4) X = ~P & ~(~P)

According to the existing rules of inference (double negation, commutative and associative laws), (c4) can be reduced to (c3). However, it is obvious that (c3) contradicts classical logic, because it states the opposite of the law of non-contradiction. In other words, contradictions are allowed. We must realize that this defines a whole new system of logic with three elements {P, ~P, P & ~P}. It is obvious that the rules of inference need to be modified in order to accommodate for P & ~P.

Logicians call this a paraconsistent logic. The fundamental idea of paraconsistent logic is to challenge the logical principle that anything follows from contradictory premises, ex contradictione quodlibet (ECQ). It is a well-known theorem of classical logic that a formula A is a theorem of a theory T if and only if the theory T with the negation of the closure of A as an additional axiom is inconsistent. In fact, it is a routine argument that the closure of A is a theorem of T whenever A itself is a theorem of T.

If you are familiar with 19th century German philosophy, you will realize that the P & ~P, in the form X = P & Q with Q = ~P essentially represents the idea of Hegelian dialectic, where P is the thesis, Q is the antithesis and X = P & Q is the synthesis. Hegel was convinced that the understanding of reality is an unfolding process that proceeds historically in a dialectical fashion. According to Hegel, human knowledge -and the enterprise of philosophy in particular- is geared towards understanding reality, or the “absolute spirit”. In contrast, Buddhist logic uses the dialectical method to invalidate points of view. This is amplified by negation, by which P & ~P becomes ~P & ~(~P), or double negation by which the argument becomes ~(~P & ~(~P)). The negation merely renders the paradox in a different form, hence, the strategy is not employed to arrive at a new conclusion, but to frustrate, disappoint, and disillusion the logician, the theoretician, the audience, or whoever is intent on arriving at any point of view. In fact, it negates all views. The further discourse between Vacchagotta and the Buddha illustrates this:

Vacchagotta: “How is it, Master Gotama, when Master Gotama is asked if he holds the view the cosmos is eternal..., he says ...no... in each case. Seeing what drawback, then, is Master Gotama thus entirely dissociated from each of these ten positions?”

Buddha: “Vaccha, the position that the cosmos is eternal is a thicket of views, a wilderness of views, a contortion of views, a writhing of views, a fetter of views. [
] The position that the cosmos is not eternal is a thicket of views, a wilderness of views, a contortion of views, a writhing of views, a fetter of views. [
]”

Vacchagotta: “Does Master Gotama have any position at all?”

Buddha: “A position, Vaccha, is something that a Tathagata has done away with. What a Tathagata sees is this: 'such is form, such its origin, such its disappearance; such is feeling, such its origin, such its disappearance; such is perception... such are mental fabrications... such is consciousness, such its origin, such its disappearance.' Because of this, I say, a Tathagata -- with the ending, fading out, cessation, renunciation, and relinquishment of all construings, all excogitations, all I-making and mine-making and obsession with conceit -- is, through lack of clinging/sustenance, released.”

Cheers, Thomas 

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           -I-
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