Changes

Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math>  Then we may comprehend the action of the linear and the positive propositions in the following terms:

Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math>  Then we may comprehend the action of the linear and the positive propositions in the following terms:

* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math>  Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even.  Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.

:* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math>  Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even.  Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.

* The positive proposition ''p''_''J'' : '''B'''^''n'' &rarr; '''B''' evaluates each cell ''x'' of '''B'''^''n'' by looking at the coefficients of ''x'' with regard to the features that ''p''_''J'' "likes", namely those in <font face="lucida calligraphy">A</font>_''J'', and then takes their product in ''B''.  Thus, ''p''_''J''(''x'') assesses the unanimity of the multitude of features that ''x'' has in <font face="lucida calligraphy">A</font>_''J'', yielding one for all and aught for else.  In these consensual or contractual terms, ''p''_''J''(''x'') = 1 means that x is ''AOK'' or congruent with all of the conditions of <font face="lucida calligraphy">A</font>_''J'', while ''p''_''J''(''x'') = 0 means that ''x'' defaults or dissents from some condition of <font face="lucida calligraphy">A</font>_''J''.

:* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J\!</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else.  In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math>

===Basis Relativity and Type Ambiguity===

===Basis Relativity and Type Ambiguity===