Changes

To denote lists of propositions and to detail their components, we use notations like:

To denote lists of propositions and to detail their components, we use notations like:

: <math>\mathbf{a} = (a, b, c),\ \mathbf{p} = (p, q, r),\ \mathbf{x} = (x, y, z),\!</math>

: <math>\mathbf{a} = (a, b, c),\ \mathbf{p} = (p, q, r),\ \mathbf{x} = (x, y, z),\!</math>

or, in more complicated situations:

or, in more complicated situations:

: <math>x = (x_1, x_2, x_3),\ y = (y_1, y_2, y_3),\ z = (z_1, z_2, z_3).\!</math>

: <math>x = (x_1, x_2, x_3),\ y = (y_1, y_2, y_3),\ z = (z_1, z_2, z_3).\!</math>

In a universe where some region is ruled by a proposition, it is natural to ask whether we can change the value of that proposition by changing the features of our current state.

In a universe where some region is ruled by a proposition, it is natural to ask whether we can change the value of that proposition by changing the features of our current state.

A typical operator <math>\operatorname{F}</math> takes us from thinking about a given function <math>f\!</math> to thinking about another function <math>g\!</math>.  To express the fact that <math>g\!</math> can be obtained by applying the operator <math>\operatorname{F}</math> to <math>f\!</math>, we write <math>g = \operatorname{F}f.</math>

A typical operator <math>\operatorname{F}</math> takes us from thinking about a given function <math>f\!</math> to thinking about another function <math>g\!</math>.  To express the fact that <math>g\!</math> can be obtained by applying the operator <math>\operatorname{F}</math> to <math>f\!</math>, we write <math>g = \operatorname{F}f.</math>

The first operator, <math>\operatorname{E}</math>, associates with a function <math>f : X \to Y</math> another function <math>\operatorname{E}f</math>, where <math>\operatorname{E}f : X \times X \to Y</math> is defined by the following equation:

The first operator, E, associates with a function f : X -> Y

another function Ef, where Ef : X x X -> Y is defined by the

following equation:

  Ef(x, y) = f(x + y).

: <math>\operatorname{E}f(x, y) = f(x + y).</math>

E is called a "shift operator" because it takes us from contemplating the

<math>\operatorname{E}</math> is called a "shift operator" because it takes us from contemplating the value of <math>f\!</math> at a place <math>x\!</math> to considering the value of <math>f\!</math> at a shift of <math>y\!</math> away. Thus, <math>\operatorname{E}</math> tells us the absolute effect on <math>f\!</math> that is obtained by changing its argument from <math>x\!</math> by an amount that is equal to <math>y\!</math>.

value of f at a place x to considering the value of f at a shift of y away.

Thus, E tells us the absolute effect on f that is obtained by changing its

argument from x by an amount that is equal to y.

Historical Note.  The protean "shift operator" E was originally called

Historical Note.  The protean "shift operator" E was originally called

the "enlargement operator", hence the initial "E" of the usual notation.

the "enlargement operator", hence the initial "E" of the usual notation.