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, or something in accordance with common experience and reason, philosophers and mathematicians tackle the concept of truth more systematically.
We generally think of a truth as bivalent, which means that an assertion or a truth variable can take on only two possible values, “true” or “false”. The bivalent notion of truth goes back to antiquity, more precisely to the Greeks, who investigated the relations of propositions in view of their form rather than their content. Aristoteles was the first Western philosopher who laid out a formal method of logical analysis. According to Aristotle, we can make valid (=true) conclusions by applying deductive reasoning to statements. Such statements come in the form of syllogisms, which consist of premises and conclusions. An example of a valid syllogism is: “All human beings are mortal ( = premise 1), I am a human being (= premise 2), therefore, I am mortal, ( = conclusion). This form of reasoning presupposes classical 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 doesnÂ’t allow that. The introduction of (P or ~P) is considered a logical flaw.
The question that is perhaps of most interest is whether the classical system is adequate for ontological questions. Can a bivalent truth system succeed in describing reality? Its usefulness in mathematics, digital circuits, and ordinary language is certainly undoubted, but can it lighten the path to wisdom? In my opinion, it canÂ’t. Classical logic is too limited. We need something different. Fortunately, we can draw on a choice of existing logical systems, including three-valued logic (true, false, and one indeterminate state), fuzzy logic (probabilistic, i.e. truth values representing the continuum between 0 and 1 expressed by a fraction 1/x), as well as a number of other more exotic systems.
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 an example from the Aggi-Vacchagotta Sutta (Majjhima Nikaya 72), the wanderer Vacchagotta asks the Buddha whether the cosmos, the soul, and the Buddha are eternal. The Buddha answers:
“Even so, Vaccha, any physical form by which one describing the Tathagata would describe him: That the Tathagata has abandoned, its root destroyed, like an uprooted palm tree, deprived of the conditions of existence, not destined for future arising. Freed from the classification of form, Vaccha, the Tathagata is deep, boundless, hard-to-fathom, like the sea. Reappears doesn't apply. Does not reappear doesn't apply. Both does & does not reappear doesn't apply. Neither reappears nor does not reappear doesn't apply.”
Similar arguments appear throughout the Pali canon. We can formalize this easily. LetÂ’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 conventional 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.
The development of paraconsistent logic was initiated in order to challenge the logical principle that anything follows from contradictory premises, ex contradictione quodlibet (ECQ). Let => be a relation of logical consequence, defined either semantically or proof-theoretically. Let us say that => is explosive if for every formula A and B, {A , ~A} => B. Classical logic, intuitionistic logic, and most other standard logics are explosive. A logic is said to be paraconsistent if its relation of logical consequence is not explosive.
Vacchagotta: “How is it, Master Gotama, when Master Gotama is asked if he holds the view the cosmos is eternal... after death a Tathagata neither exists nor does not exist: only this is true, anything otherwise is worthless, 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. It is accompanied by suffering, distress, despair, & fever, and it does not lead to disenchantment, dispassion, cessation; to calm, direct knowledge, full Awakening, Unbinding.”
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