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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 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–De 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 has been the lack of a syntax that was 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 has been the

lack of a syntax that was up to handling the complexity of

the expressions that evolve.

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<p, q> = pq.

We begin with a proposition or a boolean function <math>f(p, q) = pq.\!</math>

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

|                                                |

| [[Image:Venn Diagram F = P And Q.jpg|500px]]

|                                                 |

|-

|         o-----------o  o-----------o          |

| [[Image:Cactus Graph F = P And Q.jpg|500px]]

|         /            \ /            \        |

|}

|    |      P       |% F %|      Q       |    |

|     |              |%%%%%|              |    |

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