Changes

</pre>

</pre>

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|}

====Note 2====

====Note 2====

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It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.

It will be necessary to develop a more refined analysis of

this statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of DeMorgan's rule,

If the form of the above statement reminds you of De&nbsp;Morgan's rule, it is no accident, as differentiation and negation turn out to be closely related operations.  Indeed, one can find discussions of logical difference calculus in the Boole&ndash;De&nbsp;Morgan correspondence and Peirce also made use of differential operators in a logical context, but the exploration of these ideas has been hampered by a number of factors, not the least of which being a syntax adequate to handle the complexity of expressions that evolve.

it is no accident, as differentiation and negation turn out to be

closely related operations.  Indeed, one can find discussions of

logical difference calculus in the Boole-DeMorgan correspondence

and Peirce also made use of differential operators in a logical

context, but the exploration of these ideas has been hampered

by a number of factors, not the least of which being a syntax

adequate to handle the complexity of expressions that evolve.

For my part, it was definitely a case of the calculus being smarter

For my part, it was definitely a case of the calculus being smarter than the calculator thereof.  The graphical pictures were catalytic in their power over my thinking process, leading me so quickly past so many obstructions that I did not have time to think about all of the difficulties that would otherwise have inhibited the derivation. It did eventually became necessary to write all this up in a linear script, and to deal with the various problems of interpretation and justification that I could imagine, but that took another 120 pages, and so, if you don't like this intuitive approach, then let that be your sufficient notice.

than the calculator thereof.  The graphical pictures were catalytic

in their power over my thinking process, leading me so quickly past

so many obstructions that I did not have time to think about all of

the difficulties that would otherwise have inhibited the derivation.

It did eventually became necessary to write all this up in a linear

script, and to deal with the various problems of interpretation and

justification that I could imagine, but that took another 120 pages,

and so, if you don't like this intuitive approach, then let that be

your sufficient notice.

Let us run through the initial example again, this time attempting

Let us run through the initial example again, this time attempting to interpret the formulas that develop at each stage along the way.

to interpret the formulas that develop at each stage along the way.

We begin with a proposition or a boolean function f(x, y) = xy.

We begin with a proposition or a boolean function <math>f(x, y) = xy.\!</math>

{| align="center" cellpadding="10" style="text-align:center"

|                                       |

| [[Image:Venn_Diagram_F_=_X_And_Y.jpg|500px]]

|-

| [[Image:Cactus_Graph_F_=_X_And_Y.jpg|500px]]

o---------------------------------------o

|                                       |

|                  @                  |

|                                       |

o---------------------------------------o

| f =             x y                  |

A function like this has an abstract type and a concrete type.

A function like this has an abstract type and a concrete type.

The abstract type is what we invoke when we write things like

The abstract type is what we invoke when we write things like