Changes

| [[Image:Cactus Graph ((dP)(dQ)).jpg|500px]]

| [[Image:Cactus Graph ((dP)(dQ)).jpg|500px]]

|}

|}

We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.

We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.

Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same 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>

Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same 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>

Given the proposition <math>f(p, q)\!</math> over the space <math>X = P \times Q,</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f</math> over the differential extension <math>\operatorname{E}X</math> that is defined by the

Given the proposition <math>f(p, q)\!</math> over the space <math>X = P \times Q,</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f</math> over the differential extension <math>\operatorname{E}X</math> that is defined by the

The differential proposition that results may be interpreted to say "change <math>p\!</math> or change <math>q\!</math> or both".  And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it.

The differential proposition that results may be interpreted to say "change <math>p\!</math> or change <math>q\!</math> or both".  And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it.

Last time we computed what is variously called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\operatorname{D}f_x</math> of the proposition <math>f(p, q) = pq\!</math> at the point <math>x\!</math> where <math>p = 1\!</math> and <math>q = 1.\!</math>

Last time we computed what is variously called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\operatorname{D}f_x</math> of the proposition <math>f(p, q) = pq\!</math> at the point <math>x\!</math> where <math>p = 1\!</math> and <math>q = 1.\!</math>

The information borne by <math>\operatorname{D}f</math> should be clear enough from a survey of these six points &mdash; they tell you what you have to do from each point of <math>X\!</math> in order to change the value borne by <math>f(p, q),\!</math> that is, the move you have to make in order to reach a point where the value of the proposition <math>f(p, q)\!</math> is different from what it is where you started.

The information borne by <math>\operatorname{D}f</math> should be clear enough from a survey of these six points &mdash; they tell you what you have to do from each point of <math>X\!</math> in order to change the value borne by <math>f(p, q),\!</math> that is, the move you have to make in order to reach a point where the value of the proposition <math>f(p, q)\!</math> is different from what it is where you started.

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 <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>

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.

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.

The ''enlargement'' or ''shift'' operator <math>\operatorname{E}</math> exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features that play out on the surface of our initial example, <math>f(p, q) = pq.\!</math>

The ''enlargement'' or ''shift'' operator <math>\operatorname{E}</math> exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features that play out on the surface of our initial example, <math>f(p, q) = pq.\!</math>