The Buddhist logic is multivalent
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 doesn’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 can’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. 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 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.”
I should perhaps mention that the four basic propositions (a1) to (a4) -sometimes called tetralemma- are not really a unique feature of Budhist philosophy. They appear in ancient Indian philosophy and it may be supposed that they have been taught to the Buddha by his early teachers. You are right in saying that the tetralemma may be stated in terms of propositional logic. That is what I did in the previous post. It is important to keep in mind that P and Q are antithetic, i.e. P = ~Q. It follows that only the first two propositions (X=Q or X=P) concur with Aristotelian logic, wheras the latter two (X=P & X=Q), (X=~P & X=~Q) are contradictory in bivalent logic.
I suggested that by replacing Q with ~P and applying the rules of inference of classical logic, it is possible to reduce the tetralemma to three statements, although I must confess that I doubt whether the replacement of the variable is a permissible operation, since we are dealing with a paraconsistent logic.
Most people don't see too many real-life applications for multivalent logic, since day-to-day problems can usually be be solved with conventional logic. Multivalent logic, however, has quite a number of applications in science and engineering. For example, computer scientists should be familiar with fuzzy logic, linear logic, Belnap's 4-valued system, and Gödel's logic. But let's distinguish between multivalent and paraconsistent. I think the important property of Buddhist logic is paraconsistency, not multivalence. Paraconsistent logics are a fairly new topic in mathematics. The systematic investigation of paraconsistent systems has only begun recently in the 20th century, so it can be expected that there are not too many applications yet, since research is still in its infancy. Perhaps one day we will have machines that do not only distinguish between 0 and 1, but that will be capable of internal states that are neither 0 nor 1, or both 0 and 1.
1. Agree with mindset [A].
2. Agree with mindset [b].
3.Come up with their own[C].
The first three lines roughly coincide with the Buddhist tetralemma. If mindset A and B are antithetic then the solution of the conflict is provided by C, a higher-order mindset or idea. The German philosopher Friedrich Hegel formulated the very same thing at the beginning of the 19th century:
"Hegel accomplished what Kant had declared impossible. According to Hegel, mind is capable of arriving at full knowledge about things in themselves. He formulated a dialectical method, according to which knowledge pushes forwards to greater certainty, and ultimately towards knowledge of the noumenal world. He said that ultimate reality is absolute mind, reason, or spirit, which manifests itself in history and in the universe. Hegel set forth the proposition, "what is real is rational and what is rational is real," and from this he concluded that everything that is, is knowable. The world mind (Weltgeist) is universal; the rational activities of individuals are therefore instances of the Absolute. The self-development of mind is the result of evolving idea systems, a process that he called the dialectical processes of thesis and antithesis. According to Hegel, an idea, a thesis, always contains incompleteness, and thus, yields a conflicting idea, an antithesis. In a higher-level theory, a third point of view, the synthesis, arises that provides the solution. The synthesis overcomes the conflict between thesis and antithesis by reconciling the truth contained in both at a higher level of insight. The synthesis then becomes a new thesis that is subsequently confronted by another antithesis, and so forth. By this dialectical method, the collective mind, namely that of a group, society, nation and ultimately the world, advances towards the perfection of its knowledge." (thebigview.com)
So, it seems that the third proposition of the Buddhist tetralemma (a3) X = P and Q is indeed the Hegelian synthesis. What is the fourth proposition,,,
