Changes

Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems.  Parentheses <math>(\ldots)</math> may be used to indicate logical negation.  Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>(a_1, \ldots, a_k)</math>, as treated in earlier reports.  This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math>  Together these form a new universe of discourse <math>\underline{X}^\circ</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math>

Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems.  Parentheses <math>(\ldots)</math> may be used to indicate logical negation.  Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>(a_1, \ldots, a_k)</math>, as treated in earlier reports.  This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math>  Together these form a new universe of discourse <math>\underline{X}^\circ</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math>

The square brackets have been chosen to recall the rectangular frame of a venn diagram.  In thinking about a universe of discourse it is a good idea to keep this picture in mind, where we constantly think of the elementary cells <u>''x''</u>, the defining features <u>''x''</u>_''i'', and the potential shadings ''f''&nbsp;:&nbsp;<u>''X''</u>&nbsp;&rarr;&nbsp;'''B''', all at the same time, remaining aware of the arbitrariness of the way that we choose to inscribe our distinctions in the medium of a continuous space.

The square brackets have been chosen to recall the rectangular frame of a venn diagram.  In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline\mathbf{x},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.

Finally, let ''X''* denote the space of linear functions, (hom&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''K'''), which in the finite case has the same dimensionality as ''X'', and let the same notation be extended across the table.

Finally, let ''X''* denote the space of linear functions, (hom&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''K'''), which in the finite case has the same dimensionality as ''X'', and let the same notation be extended across the table.