Changes

# The points of the space <math>X\!</math> that have the conjunctive descriptions:<br><code>(u) v w</code> or <code>u (v) w</code> or <code>u v (w)</code> or <code>u v w</code>, where "<code>(x)</code>" is "not&nbsp;<code>x</code>".

# The points of the space <math>X\!</math> that have the conjunctive descriptions:<br><code>(u) v w</code> or <code>u (v) w</code> or <code>u v (w)</code> or <code>u v w</code>, where "<code>(x)</code>" is "not&nbsp;<code>x</code>".

The next thing that one typically does is to consider the effects of various ''operators'' on the proposition of interest, which may be called the ''operand'' or the ''source'' proposition, leaving the corresponding terms ''opus'' or ''target'' as names for the result.

The next thing that one typically does is to consider the effects

of various "operators" on the proposition of interest, which may

be called the "operand" or the "source" proposition, leaving the

corresponding terms "opus" or "target" as names for the result.

In our initial consideration of the proposition q, we naturally

In our initial consideration of the proposition <math>q\!</math>, we naturally interpret it as a function of the three variables that it wears on its sleeve, as it were, namely, those that we find contained in the basis <math>\{ u, v, w \}</math>.  As we begin to regard this proposition from the standpoint of a differential analysis, however, we may need to regard it as ''tacitly embedded'' in any number of higher dimensional spaces.  Just by way of starting out, our immediate interest is with the ''first order differential analysis'' (FODA), and this requires us to regard all of the propositions in sight as functions of the variables in the first order extended basis, specifically, those in the set <math>\{ u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>.  Now this does not change the expression of any proposition, like <math>q\!</math>, that does not mention the extra variables, only changing how it gets interpreted as a function.  A level of interpretive flexibility of this order is very useful, and it is quite common throughout mathematics.  In this discussion, I will invoke its application under the name of the ''[[tacit extension]]'' of a proposition to any universe of discourse based on a superset of its original basis.

interpret it as a function of the three variables that it wears

on its sleeve, as it were, namely, those that we find contained

in the basis {u, v, w}.  As we begin to regard this proposition

from the standpoint of a differential analysis, however, we may

need to regard it as "tacitly embedded" in any number of higher

dimensional spaces.  Just by way of starting out, our immediate

interest is with the "first order differential analysis" (FODA),

and this requires us to regard all of the propositions in sight

as functions of the variables in the first order extended basis,

specifically, those in the set {u, v, w, du, dv, dw}.  Now this

does not change the expression of any proposition, like q, that

does not mention the extra variables, only changing how it gets

interpreted as a function.  A level of interpretive flexibility

of this order is very useful, and it is quite common throughout

mathematics.  In this discussion, I will invoke its application

under the name of the "tacit extension" of a proposition to any

universe of discourse based on a superset of its original basis.

===Note 6===

===Note 6===