PPT Slide
It would be simpler to just say that I(p) is a subset of IR x (IR union LV), and write
if E is an asserted triple with the form s p o
then I(E) = true iff <I(s),I(o)> is in I(p), …… rather than IEXT(I(p))…..
Because we might want to interpret a triple like [a a b].
Suppose I(a) was a set of pairs, then how could that set itself be inside one of the pairs in the set? That would violate the axiom of foundation (a basic axiom of Zermelo-Fraenkel set theory).
We could use a nonstandard set theory that allows non-well-founded sets, but that would be a radical move….The use of IEXT is a less controversial alternative.
a appears in its own extension, which is fine.