The four outcomes of the tetralemma

In the tetralemma system of logic ( https://en.m.wikipedia.org/wiki/Tetralemma ) each theory can be evaluated to following four outcomes True, False,Neither true nor false , both true and false.

What is the necessity of having the last two values as separate?

ie In this system what is the logical necessity of keeping the 'neither' and 'both' values distinct.)

• I'm not familiar with the tetralemma system of logic, but when you change up the rules of logic, the basic facts like DeMorgan's law cannot be assumed. For this reason, I think it's rather misleading to write the four posssible outcomes using the same logical symbols that we are used to in standard logic ($X \land \neg X$ and $\neg(X \lor \neg X$). – Dustan Levenstein Dec 9 '15 at 17:48
• @DustanLevenstein Of course De Morgan's Laws are based on the axiom of excluded middle which the tetralemma does not incorporate... Otherwise both the clauses are same... but does this automatacilly imply that they are distinct? – ARi Dec 9 '15 at 17:56
• Classically, an assignment of truth values is a function $v:S\to\{0,1\}$, where $S$ is the set of statements. It might be the case that in tetralemma, this is relaxed so that an assignment of truth values is only assumed to be a relation $v\subseteq S\times\{0,1\}$ (i.e. instead of a single truth value, each statement is assigned a subset of $\{0,1\}$. The sets $\emptyset$ and $\{0,1\}$ are clearly distinct, so this would explain the need to keep them "separate".) I don't know anything about tetralemma though, so this is just a guess. – Dejan Govc Dec 9 '15 at 18:56

I'm not aware about a "modern" tetralemma system of logic... But if you are referring to Catuṣkoṭi :

the catuṣkoṭi is a "four-cornered" system of argumentation that involves the systematic examination and rejection of each of the 4 possibilities of a proposition, $P$ :

1. $P$; that is, being.

2. not $P$; that is, not being.

3. $P$ and not $P$; that is, being and not being.

4. not ($P$ or not $P$); that is, neither being nor not being.

It is interesting to note that under propositional logic, De Morgan's laws imply that the fourth case (neither $P$ nor not $P$) is equivalent to the third case ($P$ and not $P$), and is therefore superfluous.

In other words, in a such system of logic, De Morgan's laws must be rejected.

This must be taken into account when we try to reconcile this doctrine with modern Paraconsistent Logic or Dialetheism; in e.g. $\mathsf{LP}$ ("Logic of Paradox") system of logic (proposed by logician F.G.Asenjo in 1966 and later popularized by G.Priest and others) De Morgan's laws holds.

Thus, in the above form of argument, the negation is not truth-functional.

For a modern analysis of this Buddhist argument, see : Indian Logic - Ch.3 : Buddhist Logic, by Amita Chatterjee, into Leila Haaparanta (editor), The Development of Modern Logic, page 927.