Changes

Our excursion into the vastening landscape of higher order propositions has finally come round to the stage where we can bring its returns to bear on opening up new perspectives for quantificational logic.

Our excursion into the vastening landscape of higher order propositions has finally come round to the stage where we can bring its returns to bear on opening up new perspectives for quantificational logic.

There is a question arising next that is still experimental in my mind.  Whether it makes much difference from a purely formal point of view is not a question I can answer yet, but it does seem to aid the intuition to invent a slightly different interpretation for the two-valued space that we use as the target of our basic indicator functions.  Therefore, let us declare a type of "existential-valued" functions ''f'' : '''B'''^''k'' &rarr; <font face="lucida calligraphy">E</font>, where <font face="lucida calligraphy">E</font> = {–e, +e} = {"empty", "exist"} is a couple of values that we interpret as indicating whether of not anything exists in the cells of the underlying universe of discourse, venn diagram, or other domain.  As usual, let us not be too strict about the coding of these functions, reverting to binary codes whenever the interpretation is clear enough.

There is a question arising next that is still experimental in my mind.  Whether it makes much difference from a purely formal point of view is not a question I can answer yet, but it does seem to aid the intuition to invent a slightly different interpretation for the two-valued space that we use as the target of our basic indicator functions.  Therefore, let us declare a type of ''existential-valued'' functions <math>f : \mathbb{B}^k \to \mathbb{E},</math> where <math>\mathbb{E} = \{ -e, +e \} = \{ \operatorname{empty}, \operatorname{exist} \}</math> is a couple of values that we interpret as indicating whether of not anything exists in the cells of the underlying universe of discourse, venn diagram, or other domain.  As usual, let us not be too strict about the coding of these functions, reverting to binary codes whenever the interpretation is clear enough.

With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:

With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions: