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.)  
 A: 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, $P$ :

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