The Catuṣkoṭi, the Saptabhaṇgī, and “Non-Classical” Logic

Abstract

The Principles of Excluded Middle and Non-Contradiction are highly orthodox in Western philosophy. They are much less so in Indian philosophy. Indeed, there are logical/metaphysical positions that clearly violate them. One of these is the Buddhist catuṣkoṭi; another is the Jain saptabhaṇgī. Contemporary Western logicians have, however, investigated systems of “non-classical” logic in which these principles fail, and some of these bear important relationships to the catuṣkoṭi and the saptabhaṇgī. In this chapter, we will look at these two principles and see how these may inform and be informed by those systems.

Appendix: Technical and Historical Details Concerning Some Paraconsistent and Paracomplete Logics

A logic where Explosion fails is called paraconsistent. A logic where Implosion fails is often now called paracomplete. Classical logic is neither paraconsistent nor paracomplete. Not all paraconsistent and paracomplete logics are many-valued logics. In this appendix, I will discuss a few points of technical and historical interest concerning some that are.

FDE

The logic FDE is the core of a family of logics called relevant logics. It is both paraconsistent and paracomplete. It was invented/discovered by the US logicians A. R. Anderson and N. D. Belnap in 1962. The main concern of relevant logic is that if A entails B, A should be relevant to B. If a logic satisfies Explosion or Implosion, this is obviously not the case. The 4-valued semantics was invented/discovered a little later, by J. M. Dunn. (For discussion and references, see Anderson and Belnap (1975), Chap. 3.)

One way of setting up the semantics of FDE is as follows. The language contains a set of propositional parameters, P, and the connectives, ∧, ∨, and ¬. An interpretation is a binary relation ρ ⊆ P × {0, 1}. Given an interpretation, truth and falsity are assigned independently to all formulas as follows. ⊩⁺ A means that A is true; ⊩⁻ A means that A is false. If p ∈ P:

• ⊩⁺ p iff pρ1

• ⊩⁻ p iff pρ0

Then:

• ⊩⁺ ¬A iff ⊩⁻ A

• ⊩⁻ ¬A iff ⊩⁺ A

• ⊩⁺ A ∧ B iff ⊩⁺ A and ⊩⁺ B

• ⊩⁻ A ∧ B iff ⊩⁻ A or ⊩⁻ B

• ⊩⁺ A ∨ B iff ⊩⁺ A or ⊩⁺ B

• ⊩⁻ A ∨ B iff ⊩⁻ A and ⊩⁻ B

If Σ is a set of formulas, then Σ ⊨ A iff for all ρ: if ⊩⁺ B, for all B ∈ Σ, then ⊩⁺ A.

We may define a conditional, A ⊃ B, as usual in classical logic. As is easy to check, both ⊨ A ⊃ A and A, A ⊃ B ⊨ B fail – the first, because A may have the value n; the second, because A may have the value b. Full relevant logics add a new conditional, →, to the language and give it an appropriate (and more complex) semantics. These inferences hold for →.

As is clear, the relational FDE truth/falsity conditions are exactly those of classical logic (though, in the case of classical logic, the falsity conditions are redundant). The definition of validity is also exactly the same as that of classical logic. If an interpretation is a total function (that is, it relates every p to exactly one member of {0, 1}), then it is an interpretation of classical logic. Hence, FDE expands the possibilities (interpretations) countenanced by classical logic.

Given a relational FDE interpretation, there are obviously four possibilities for a formula, A:

• ⊩⁺ A and ⊮⁺ A

• ⊮⁺ A and ⊩⁻ A

• ⊩⁺ A and ⊩⁻ A

• ⊮⁺ A and ⊮⁻ A

If we write these four possibilities as t, f, b, and n, then the relational truth/falsity conditions deliver the Diamond Lattice and its operators, as may easily be checked. And relational validity is equivalent to preserving the values t and b. Hence, the relational semantics and the 4-valued semantics are equivalent (see Priest (2008: 8.4)).

The system FDEe was introduced in Priest (2018) and, unlike the other logics mentioned in this essay, was motivated by Buddhist considerations.

K₃ and B₃

If in the relational semantics one requires that for no p, pρ1 and pρ0, then, as is easy to check, this is so for all formulas. The semantics is then one for the logic K₃. K₃ is paracomplete, but not paraconsistent. As indicated, the 3-valued version of the semantics is obtained by taking the right-hand side of the Diamond Lattice of 2.3.

K₃ was invented/discovered by the US mathematician S. C. Kleene in 1938 (see also his book (1952: §64)). Kleene was concerned with partial recursive functions. The value of such a function may not be defined. Hence, if f is such a function, the equation f(i) = j may be neither true nor false. Hence, Kleene calls the value n “undefined.”

If we replace the value n by the value e of 2.5, the resulting logic is often called “weak Kleene logic,” but it is better called Bochvar Logic (B₃), since it was invented by the Russian logician D. A. Bochvar in a paper in Russian in 1938. (An English translation appears as Bochvar and Bergmann (1981).) Like K₃, B₃ is paracomplete, but not paraconsistent. Bockhvar interprets the value e as nonsense. (So, for the connectives: nonsense-in, nonsense-out.) As the title of the paper indicates, he takes sentences involved in paradoxes of self-reference, such as that involved in Russell’s paradox, {x: x ∉ x} ∈ {x: x ∉ x}, to be nonsensical.

A rule system that is sound and complete with respect to B₃ can be obtained from that of K₃ by removing ∨-introduction, and adding:

$$ \frac{A^{\dagger }}{A\vee \neg A} $$

(Recall that A^† is any formula containing all the propositional parameters of A.) See Priest (2019).

LP and H₃

If in the relational semantics one requires that, for every p, pρ1 or pρ0, then, as is easy to check, this holds for all formulas. The semantics is then one for the logic LP. LP is paraconsistent, but not paracomplete. As indicated, the 3-valued version of the semantics is obtained by taking the left-hand side of the Diamond Lattice of 2.3.

The logic LP was invented/discovered by G. Priest (1979). Like Bochvar, he thought of the value b as applying to paradoxical sentences. (He calls the value paradoxical.) But unlike Bochvar, he read it as both true and false – and so as a species of truth.

To round out the picture: B₃ may equally be obtained from LP by replacing the value b with e – since e is not designated. However, if we then take e to be designated, we obtain a logic usually now often called “Paraconsistent Weak Kleene,” though it would be better called Halldén logic (H₃), since it was invented/discovered by the Swedish logician Sören Halldén in 1949. Like LP, H₃ is paraconsistent, but not paracomplete. As the title of Halldén’s work indicates, he interprets the middle value as nonsensical, like Bochvar. Why he takes the value to be designated is somewhat opaque, however.

A sound and complete system of rules for H₃ can be obtained by taking the rules for LP, deleting the rule for ∧-elimination, and replacing it with:

$$ \frac{A\kern1em \neg A}{A^{\dagger }}\kern0.5em \frac{A\wedge B}{A\vee {B}^{\dagger }}\kern0.5em \frac{A\wedge B}{A^{\dagger}\vee B} $$

See Priest (2019).