Full Idea
One reason for wanting a three-valued logic is to act as a basis of a theory of presupposition.
Gist of Idea
Three-valued logic is useful for a theory of presupposition
Source
Edwin D. Mares (Negation [2014], 3.1)
Book Reference
'Bloomsbury Companion to Philosophical Logic', ed/tr. Horsten,L/Pettigrew,R [Bloomsbury 2014], p.185
A Reaction
[He cites Strawson 1950] The point is that you can get a result when the presupposition does not apply, as in talk of the 'present King of France'.