Changes

By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse <math>X^\circ = [\mathcal{X}] = [x_1, x_2] = [x, y],</math> based on two logical features or boolean variables <math>x\!</math> and <math>y.\!</math>

By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse <math>X^\circ = [\mathcal{X}] = [x_1, x_2] = [x, y],</math> based on two logical features or boolean variables <math>x\!</math> and <math>y.\!</math>

 

| The points of <math>X^\circ</math> are collected in the space:

: The points of <math>X^\circ</math> form the space <math>X = \langle \mathcal{X} \rangle = \langle x, y \rangle = \{ (x, y) \} \cong \mathbb{B}^2.</math>

 

: <math>X = \{ ( (\!| x |\!), (\!| y |\!) ), ( (\!| x |\!), y ), ( x, (\!| y |\!) ), ( x, y ) \} \cong \mathbb{B}^2.</math>

| <math>X = \langle x, y \rangle = \{ (x, y) \} \cong \mathbb{B}^2.</math>

 

: <math>X \cong \{ (0, 0), (0, 1), (1, 0), (1, 1) \} \cong \mathbb{B}^2.</math>

 

: The propositions of <math>X^\circ</math> form the space <math>X^\uparrow = (X \to \mathbb{B}) = \{ f : X \to \mathbb{B} \} \cong (\mathbb{B}^2 \to \mathbb{B}).</math>

| In other words, written out in full:

| <math>X = \{ (\lnot x, \lnot y), (\lnot x, y), (x, \lnot y), (x, y) \}.</math>

| <math>X \cong \{ (0, 0), (0, 1), (1, 0), (1, 1) \}.</math>

| <math>X \cong \mathbb{B}^2.</math>

| The propositions of <math>X^\circ</math> make up the space:

| <math>X^\uparrow = (X \to \mathbb{B}) = \{ f : X \to \mathbb{B} \} \cong (\mathbb{B}^2 \to \mathbb{B}).</math>

|}<br>

As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again.

As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again.