Monday, June 17, 2019

Beginning the Fifth Corner of Four

I have begun reading the new book by Graham Priest called The Fifth Corner of Four. This book covers both Buddhist Logic and Metaphysics, and particularly the philosophical import of the Catuskoti. The Catuskoti, for those who don't know, is a principle within Buddhist logic that allows for any proposition to have one of four values: True only, False only, Both True and False, and Neither True nor False.

Of course, in all the Aristotle-related systems of Logic (such as Classical Logic) only the first two cotis would be considered live options. This means that any attempt to interpret the Catuskoti from such a framework will be doomed to failure. However, we can very easily capture the logical structure of the Catuskoti by appealing to the simple Relevant Logic called FDE ('First Degree Entailment').

Now, where the title of the book comes in is to explain certain rather odd passages in the Buddhist literature which seem to indicate that the Catuskoti is not applicable to certain questions concerning the nature of ultimate reality. These passages can obviously not be captured by using FDE, but as Graham Priest has alluded to in previous work, we can adequately capture them by adding a fifth, 'Bochvar-style' truth-value to our system. The main effect that this has on the deductive capacity of the apparatus is to invalidate the rule of Addition (although there is a way that we can regain a variant of this rule with some suitable refinements).

I am very interested to see how this work will turn out. However, what I am particularly interested to know is what is the correct system of Relevant Logic to capture the higher-degree inferences of Buddhist philosophy. We can see on the Stanford Encyclopedia of Philosophy article for Logic in Classical Indian Philosophy that:

"Many of the arguments formulated in these texts correspond to such well recognized rules of inference as modus ponens (i.e., from α and α→β, one infers β), modus tollens (i.e., from ¬β and α→β, one infers ¬α), disjunctive syllogism (i.e., from ¬α and α∨β, one infers β), constructive dilemma (i.e., from α∨β, α→γ and β→γ, one infers γ), categorical syllogism (i.e., from α→β and β→γ, one infers α→γ) and reductio ad absurdum (i.e., if something false follows from an assumption, then the assumption is false)."

Of course, the minimal system needed to capture these rules of inference is N* coupled both with a Reflexive accessibility relation on at least the actual world, and with an Intensional Disjunction to accommodate Disjunctive Syllogism. I am particularly interested to know whether the Buddhist logicians would have accepted the principles of Prefixing and Suffixing, and thus the Relevant Logic B.

However, it really remains to be seen whether they would have accepted the Exported form of Categorical Syllogism. One would hope not, since that would lead to the problem of incompleteness in the Routley-Meyer semantics.

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