Problems with bivalent logic

The father of classical logic in the Western tradition, Aristotle, had already noticed that in order to save the law of excluded middle i.e. A or not A one is forced to abandon the assignment of any truth value to A or its alternative in cases when those propositions refer to future events, the so called “future contingents”. That is, neither A nor its negation can be true without inevitably leading to fatalism (a logically deterministic worldview). This conundrum motivated later attempts in solving it within logics endowed with richer semantics e.g. the introduction of an additional (intermediate) truth value.  Such treatments however have notoriously failed to maintain the law of excluded middle as a necessary truth.

The “Liar” paradox, has put in question the law of non contradiction, i.e. A and notA is necessarily false. The “Liar” paradox originally put forward by Epimenides 600 BCE consists in the question concerning the truth value of the sentence L: “L is false”; whether we assume L to be true or false, the immediate conclusion contradicts the original assumption. In consequence, the advocates of dialethism, a school that questions the universality of the law of non contradiction suggested a solution to the “Liar” paradox, by claiming that the liar sentence is both true and false i.e. that it is a true contradiction!

The family of the so-called Sorites paradoxes (sorites derives from the Greek word for heap), has also consistently resisted bivalent logic treatment. It consists in questioning when does an object cease to have its defining/essential property; in the typical example of a heap, one proceeds in the following way: one grain of sand does not make a heap, and adding one grain does not transform a non-heap into a heap, but following this reasoning consistently one eventually arrives at a collection, of say a million grains, yet is committed to refrain from calling it a heap – paradox! Presently the most promising solutions to the paradox have come from treatments with a fuzzy semantics; an infinite set of fractional truth values spanning the divide between truth and falsity.

The above examples suggest that there’s something amiss with our classical conceptual toolkit intended to tame and organize reality in an elegant and complete manner. Consequently statements such as neither A nor not A, A and not A cannot be completely ruled out as absurd and unthinkable. What’s more, our logical dichotomy may just be a practical approximation of an infinite valued semantics which underlies reality, or maybe not (or maybe both, or neither).

Buddhist attitude

The Eastern approach to rules governing reasoning appears to have a less presumptuous character – instead of imposing a conceptual framework on reality only to witness its inevitable failure, its promoters refrain from such exercises. In Buddhism, the non committal attitude to the classical dichotomy and hence the law of excluded middle (or the law of non contradiction) does not spring from a conviction (presumption) that there exists a more general semantics which accurately reflects reality, but rather this attitude it is a kind of a side effect of the more fundamental doctrine which advocates renouncing concepts and abandoning any conceptual confinement in general. The Zen Koans, often endowed with paradoxical content serve as literary glimpses of a world devoid of some of its most fundamental constructs and blindly cherished shackles – bivalent logic being one of them.

Ganto’s Axe

One day Tokusan told his student Ganto, I have two monks who have been here for many years. Go and examine them. Ganto picked up an axe and went to the hut where the two monks were meditating. He raised the axe, saying, If you say a word I will cut off your heads; and if you do not say a word, I will also cut off your heads.

Both monks continued their meditation as if he had not spoken. Ganto dropped the axe and said, You are true Zen students. He returned to Tokusan and related the incident. I see your side well, Tokusan agreed, but tell me, how is their side? “Tozan may admit them, replied Ganto, but they should not be admitted under Tokusan.

(from Zen Koans by Venerable Gyomay M. Kubose)

Ganto’s Axe is a good example of a paradoxical Koan. Although from the premises of Ganto’s threat it necessarily follows, using natural (classical bivalent) deduction that heads will fall, the Zen monks do not engage in such reasoning, having successfully transcended it – a fact master Tokusan fails to realize, and hence Ganto’s surprisingly defiant recommendation.

Incidentally, the concept of an empty set is an integral part of the formalism underlying the tandem of mathematics and classical logic. And precisely all the contradictory (impossible) propositions are said to be elements of the empty set – this unusually sounding phrase rings with ironic dissonance within the rigid realm of bi-valence. It would seem that there is after all a little Zen in modern mathematics, if only nominally.

The name of the eight Jhana “The Sphere of Neither Perception Nor Non-perception”, has a paradoxical ring to it because the label/name attempts to capture/define that which resists such treatment – that which is intended to be free of any conceptual constructs, and hence by definition inexpressible in any language. Hence the emphasis on experience in most Buddhist schools.