Changes

==Note 24==

==Note 24==

Now that we've introduced the field picture for thinking about propositions and their analytic series, a very pleasing way of picturing the relationship among a proposition <math>f : X \to \mathbb{B},</math> its enlargement or shift map <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B},</math> and its difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B}</math> can now be drawn.

Now that we've introduced the field picture as an aid to thinking about propositions and their analytic series, a very pleasing way of picturing the relationships among a proposition <math>f : X \to \mathbb{B},</math> its enlargement or shift map <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B},</math> and its difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B}</math> can now be drawn.

To illustrate this possibility, let's return to the differential analysis of the conjunctive proposition <math>f(p, q) = pq,\!</math> giving the development a slightly different twist at the appropriate point.

To illustrate this possibility, let's return to the differential analysis of the conjunctive proposition <math>f(p, q) = pq,\!</math> giving the development a slightly different twist at the appropriate point.

Figure&nbsp;24-1 shows the proposition <math>pq\!</math> once again, which we now view as a scalar field, in effect, a potential "plateau" of elevation 1 over the shaded region, with an elevation of 0 everywhere else.

Figure&nbsp;24-1 shows the proposition <math>pq\!</math> once again, which we now view as a scalar field &mdash; analogous to a ''potential hill'' in physics, but in logic tantamount to a ''potential plateau'' &mdash; where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.

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Given any proposition <math>f : X \to \mathbb{B},</math> the ''tacit extension'' of <math>f\!</math> to <math>\operatorname{E}X</math> is notated <math>\varepsilon f : \operatorname{E}X \to \mathbb{B}</math> and defined by the equation <math>\varepsilon f = f,</math> so it's really just the same proposition living in a bigger universe.

Given a proposition <math>f : X \to \mathbb{B},</math> the ''tacit extension'' of <math>f\!</math> to <math>\operatorname{E}X</math> is denoted <math>\varepsilon f : \operatorname{E}X \to \mathbb{B}</math> and defined by the equation <math>\varepsilon f = f,</math> so it's really just the same proposition residing in a bigger universe.  Tacit extensions formalize the intuitive idea that a function on a particular set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a "don't care" condition on the new variables.

Figure&nbsp;24-2 shows the tacit extension of the scalar field <math>pq : X \to \mathbb{B}</math> to the differential field <math>\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>

 

Figure&nbsp;24-2 illustrates the tacit extension of the proposition or scalar field <math>f = pq : X \to \mathbb{B}</math> to give the extended proposition or differential field that we notate as <math>\varepsilon f = \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>

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