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 doesnt
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 cant. 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. Lets 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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