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Revision as of 13:28, 17 June 2009
, 13:28, 17 June 2009→Note 24: convert graphics
==Note 24==
==Note 24==
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.
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 f : X -> B, its
enlargement or shift map Ef : EX -> B, and its difference map
To illustrate this possibility, let's return to the differential
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.
analysis of the conjunctive proposition f<p, q> = pq, giving the
development a slightly different twist at the appropriate point.
Figure 24-1 shows the proposition pq once again, which we now view
Figure 24-1 shows the proposition <math>pq\!</math> once again, which we now view as a scalar field — analogous to a ''potential hill'' in physics, but in logic tantamount to a ''potential plateau'' — where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.
as a scalar field, in effect, a potential "plateau" of elevation 1
{| align="center" cellspacing="10" style="text-align:center"
| |
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
|-
| o-------------------o o-------------------o |
| <math>\text{Figure 24-1. Proposition}~ pq : X \to \mathbb{B}</math>
| / \ / \ |
|}
Figure 24-1. Proposition pq : X -> B
Given any proposition f : X -> B, the "tacit extension" of f to EX
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.
is notated !e!f : EX -> B and defined by the equation !e!f = f, so
it's really just the same proposition living in a bigger universe.
Figure 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>
{| align="center" cellspacing="10" style="text-align:center"
| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]
|-
| <math>\text{Figure 24-2. Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}</math>
|-
|
| X |
<math>\begin{array}{rcccccc}
\varepsilon (pq)
| / \ / \ |
& = & p & \cdot & q & \cdot & (\operatorname{d}p)(\operatorname{d}q)
\\[4pt]
& + & p & \cdot & q & \cdot & (\operatorname{d}p)~\operatorname{d}q~
\\[4pt]
& + & p & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q)
| / / \ \ |
\\[4pt]
& + & p & \cdot & q & \cdot & ~\operatorname{d}p~~\operatorname{d}q~
\end{array}</math>
|}
</pre>
==Note 25==
==Note 25==
