Changes

==Note 5==

==Note 5==

We have been studying the action of the difference operator <math>\operatorname{D}</math> on propositions of the form <math>f : P \times Q \to \mathbb{B},</math> as illustrated by the example <math>f(p, q) = pq\!</math> that is known in logic as the conjunction of <math>p\!</math> and <math>q.\!</math>  The resulting difference map <math>\operatorname{D}f</math> is a (first order) differential proposition, that is, a proposition of the form <math>\operatorname{D}f : P \times Q \times \operatorname{d}P \times \operatorname{d}Q \to \mathbb{B}.</math>

We have been studying the action of the difference operator D,

also known as the "localization operator", on the proposition

f : !P! x !Q! -> B that is commonly called the conjunction pq.

We categorized Df as a (first order) differential proposition,

a proposition of the type Df : !P! x !Q! x d!P! x d!Q! -> B.

Abstracting from the augmented venn diagram that shows how the

Abstracting from the augmented venn diagram that shows how the ''models'' or ''satisfying interpretations'' of <math>\operatorname{D}f</math> distribute over the extended universe of discourse <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q,</math> the difference map <math>\operatorname{D}f</math> can be represented in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>X = P \times Q</math> and whose arrows are labeled with the elements of <math>\operatorname{d}X = \operatorname{d}P \times \operatorname{d}Q,</math> as shown in the following Figure.

models or the satisfying interpretations of Df distribute over

the (first order) extended space EX = !P! x !Q! x d!P! x d!Q!,

we can represent Df in the form of a digraph or directed graph,

one whose points are labeled with the elements of X = !P! x !Q!

and whose arcs are labeled with the elements of dX = d!P! x d!Q!.

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

| f =                 p q                      |

| [[Image:Directed Graph PQ Difference Conj.jpg|500px]]

|}

| Df =             p  q  ((dp)(dq))              |

|                                                 |

|                                                 |

Any proposition worth its salt has many equivalent ways to view it,

any one of which may reveal some unsuspected aspect of its meaning.

We will encounter more and more of these variant readings as we go.

 

Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal an unsuspected aspect of the proposition's meaning. We will encounter more and more of these alternative readings as we go.

==Note 6==

==Note 6==