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=====1.3.10.6.  Stretching Principles=====

=====1.3.10.6.  Stretching Principles=====

There is a principle, of constant use in this work, that needs to be made explicit.  In order to give it a name, I refer to this idea as the ''stretching principle''.  Expressed in different ways, it says that:

There is a principle, of constant use in this work, that needs to be made explicit.  In order to give it a name, I refer to this idea as the "stretching principle".  Expressed in different ways, it says that:

1. Any relation of values extends to a relation of what is valued.

# Any relation of values extends to a relation of what is valued.

# For every operation on values, there is a corresponding operation on the actions, conducts, functions, procedures, or processes that lead to these values, as well as there being analogous operations on the objects that instigate all of these various proceedings.

Nothing about the application of the stretching principle guarantees that the analogues it generates will be as useful as the material it works on.  It is another question entirely whether the links that are forged in this fashion are equal in their strength and apposite in their bearing to the tried and true utilities of the original ties, but in principle they are always there.

Nothing about the application of the stretching principle guarantees that the analogues it generates will be as useful as the material it works on.  It is another question entirely whether the links that are forged in this fashion are equal in their strength and apposite in their bearing to the tried and true utilities of the original ties, but in principle they are always there.

In particular, a connection F : Bk �> B can be understood to indicate a relation among boolean values, namely, the k�ary relation F�1(1) c Bk.  If these k values are values of things in a universe U, that is, if one imagines each value in a k�tuple of values to be the functional image that results from evaluating an element of U under one of its possible aspects of value, then one has in mind the k propositions fj : U �> B, for j = 1 to k, in sum, one embodies the imagination f = <f1, ..., fk>.  Together, the imagination f ? (U �> B)k and the connection F : Bk �> B stretch each other to cover the universe U, yielding a new proposition P : U �> B.

In particular, a connection F : Bk �> B can be understood to indicate a relation among boolean values, namely, the k�ary relation F�1(1) c Bk.  If these k values are values of things in a universe U, that is, if one imagines each value in a k�tuple of values to be the functional image that results from evaluating an element of U under one of its possible aspects of value, then one has in mind the k propositions fj : U �> B, for j = 1 to k, in sum, one embodies the imagination f = <f1, ..., fk>.  Together, the imagination f ? (U �> B)k and the connection F : Bk �> B stretch each other to cover the universe U, yielding a new proposition P : U �> B.

To encapsulate the form of this general result, I define a composition that takes an imagination f = <f1, ..., fk> C (U �> B)k and a boolean connection F : Bk �> B and gives a proposition P : U �> B.  Depending on the situation, specifically, according to whether many F and many f, a single F and many f, or many F and a single f are being considered, respectively, I refer to this P under one of three descriptions:

To encapsulate the form of this general result, I define a composition that takes an imagination f = <f1, ..., fk> C (U �> B)k and a boolean connection F : Bk �> B and gives a proposition P : U �> B.  Depending on the situation, specifically, according to whether many F and many f, a single F and many f, or many F and a single f are being considered, respectively, I refer to this P under one of three descriptions: