Changes

We now return to our consideration of the effects of various differential operators on propositions.  This time around we have enough exact terminology that we shall be able to explain what is actually going on here in a rather more articulate fashion.

We now return to our consideration of the effects of various differential operators on propositions.  This time around we have enough exact terminology that we shall be able to explain what is actually going on here in a rather more articulate fashion.

The first transformation of the source proposition <math>q\!</math> that we may wish to stop and examine, though it is not unusual to skip right over this stage of analysis, frequently regarding it as a purely intermediary stage, holding scarcely even so much as the passing interest, is the work of the ''enlargement'' or ''shift'' operator <math>\operatorname{E}.</math>

The first transformation of the source proposition q that we may

wish to stop and examine, though it is not unusual to skip right

over this stage of analysis, frequently regarding it as a purely

intermediary stage, holding scarcely even so much as the passing

interest, is the work of the "enlargement" or "shift" operator E.

Applying the operator E to the operand proposition q yields:

Applying the operator <math>\operatorname{E}</math> to the operand proposition <math>q\!</math> yields:

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

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

| Eq                                              |

| Eq                                              |

|                                                |

|                                                |

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

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

The enlarged proposition Eq is a minimally interpretable as

The enlarged proposition <math>\operatorname{E}q</math> is minimally interpretable as a function on the six variables of <math>\{ u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}.</math>  In other words, <math>\operatorname{E}q : \operatorname{E}X \to \mathbb{B},</math> or <math>\operatorname{E}q : X \times \operatorname{d}X \to \mathbb{B}.</math>

as a function on the six variables of {u, v, w, du, dv, dw}.

In other words, Eq : EX -> B, or Eq : X x dX -> B.

Conjoining a query on the center cell, c = uvw, yields:

Conjoining a query on the center cell, <math>c = u\ v\ w\!</math>, yields:

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

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

| Eq.c                                            |

| Eq.c                                            |