Changes

For example, the annnotation <math>X_1 : A_1 :: X_2 : A_2\!</math> may be read to say that <math>X_1\!</math> is to <math>A_1\!</math> as <math>X_2\!</math> is to <math>A_2,\!</math> where the step from <math>A_1\!</math> to <math>A_2\!</math> is permitted by a previously accepted rule.

For example, the annnotation <math>X_1 : A_1 :: X_2 : A_2\!</math> may be read to say that <math>X_1\!</math> is to <math>A_1\!</math> as <math>X_2\!</math> is to <math>A_2,\!</math> where the step from <math>A_1\!</math> to <math>A_2\!</math> is permitted by a previously accepted rule.

This can be illustrated by considering the derivation of Rule&nbsp;3 in detail.

This can be illustrated by considering the derivation of Rule&nbsp;3 in the following augmented form:

<pre>

R3a. "u C X" is to R1a, namely, "u C X",

as

<math>\begin{array}{lcclc}

\text{R3a.}

& \text{is to}

& \text{R1a.}

& & \text{as} & &

R3b. "{X}(u)" is to R1b, namely, "{X}(u)",

Notice how the sequence of analogies pivots on the term <math>\text{R3b},\!</math> viewing it first under the aegis of <math>\text{R1b},\!</math> as the second term of the first analogy, and then turning to view it again under the guise of <math>\text{R2a},\!</math> as the first term of the second analogy.

and

"{X}(u)" is to R2a, namely, "f(u)",

as

By way of convention, rules that are tailored to a particular application, case, or subject, and rules that are adapted to a particular goal, object, or purpose, I frequently refer to as ''Facts''.

 

 

By way of convention, rules that are tailored to a particular application, case, or subject, and rules that are adapted to a particular goal, object, or purpose, I frequently refer to as "Facts".

Besides linking rules together into extended sequences of equivalents, there is one other way that is commonly used to get new rules from old.  Novel starting points for rules can be obtained by extracting pairs of equivalent expressions from a sequence that falls under an established rule, and then by stating their equality in the proper form of equation.  For example, by extracting the equivalent expressions that are annotated as "R3a" and "R3c" in Rule 3 and by explictly stating their equivalence, on obtains the specialized result that is recorded in Corollary 1.

Besides linking rules together into extended sequences of equivalents, there is one other way that is commonly used to get new rules from old.  Novel starting points for rules can be obtained by extracting pairs of equivalent expressions from a sequence that falls under an established rule, and then by stating their equality in the proper form of equation.  For example, by extracting the equivalent expressions that are annotated as "R3a" and "R3c" in Rule 3 and by explictly stating their equivalence, on obtains the specialized result that is recorded in Corollary 1.