Changes

The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math>  The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus.  (I plan to use this notation in the logical analysis of neural network systems.)  The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math>

The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math>  The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus.  (I plan to use this notation in the logical analysis of neural network systems.)  The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math>

I now proceed to explain these concepts in more detail.  The two most important ideas developed in the table are:

I now proceed to explain these concepts in more detail.  The most important ideas developed in Table&nbsp;5 are these:

* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.

* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.

For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math>  The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''.  Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension.  One strategy that is general enough for our present purposes is as follows.  For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\operatorname{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:

For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math>  The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''.  Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension.  One strategy that is general enough for our present purposes is as follows.  For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\operatorname{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:

: <u>''x''</u>_''i'' : '''R'''^''n'' &rarr; '''B''' such that:

: <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \mbox{such that:}</math>

: <u>''x''</u>_''i''(''x'') = 1 if ''x'' &isin; ''L''_''i'',

: <math>\begin{matrix}

 

\underline{x}_i(\mathbf{x}) = 1 & \mbox{if} & \mathbf{x} \in L_i, \\

: <u>''x''</u>_''i''(''x'') = 0 if otherwise.

\underline{x}_i(\mathbf{x}) = 0 & \mbox{if} & \mathbf{x} \not\in L_i.

In other notations that are sometimes used, the operator <math>\chi (\ )</math> or the corner brackets <math>\lceil \ldots \rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values, given as elements of '''B'''.  Finally, it is not uncommon to use the name of the relation itself as a predicate that maps ''n''-tuples into truth values.  In each of these notations, the above definition could be expressed as follows:

In other notations that are sometimes used, the operator <math>\chi (\ )</math> or the corner brackets <math>\lceil \ldots \rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values, given as elements of '''B'''.  Finally, it is not uncommon to use the name of the relation itself as a predicate that maps ''n''-tuples into truth values.  In each of these notations, the above definition could be expressed as follows: