Changes

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The interpretations of these new symbols can be diverse, but the easiest

The interpretations of these new symbols can be diverse,

option for now is just to say that <math>\operatorname{d}p</math> means "change <math>p\!</math>" and <math>\operatorname{d}q</math> means "change <math>q\!</math>".

but the easiest interpretation for now is just to say

that "dp" means "change p" and "dq" means "change q".

Drawing a venn diagram for the differential extension EX = X x dX

Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.

requires four logical dimensions, !P!, !Q!, d!P!, d!Q!, but it is

possible to project a suggestion of what the differential features

dp and dq are about on the 2-dimensional base space X = !P! x !Q!

by drawing arrows that cross the boundaries of the basic circles

in the venn diagram for X, reading an arrow as dp if it crosses

the boundary between p and (p) in either direction and reading

an arrow as dq if it crosses the boundary between q and (q)

in either direction.

{| align="center" cellpadding="10"

| [[Image:Venn Diagram P Q dP dQ.jpg|500px]]

|}

|    |              |%%%%%|              |    |

|     o              o%%%%%o              o    |

|     \              \%%%/              /      |

We can form propositions from these differential variables

Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in all the ways that propositions are formed on ordinary logical variables alone.  For example, the proposition <math>\texttt{(} \operatorname{d}p \texttt{(} \operatorname{d}q \texttt{))}</math> says the same thing as <math>\operatorname{d}p \Rightarrow \operatorname{d}q,</math> in other words, that there is no change in <math>p\!</math> without a change in <math>q\!</math>.

in the same way that we would any other logical variables,

for example, taking the differential proposition (dp (dq))

as saying that dp implies dq, in other words, that there

is "no change in p without a change in q".

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==Note 3==

==Note 3==