Changes

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 <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> can be understood to indicate a relation among boolean values, namely, the <math>k\!</math>-ary relation <math>F^{-1} (\underline{1}) \subseteq \underline\mathbb{B}^k.</math>  If these <math>k\!</math> values are values of things in a universe <math>X,</math> that is, if one imagines each value in a <math>k\!</math>-tuple of values to be the functional image that results from evaluating an element of <math>X\!</math> under one of its possible aspects of value, then one has in mind the <math>k\!</math> propositions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> in sum, one embodies the imagination <math>\underline{f} = (f_1, \ldots, f_k).</math>  Together, the imagination <math>\underline{f} \in (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> stretch each other to cover the universe <math>X,\!</math> yielding a new proposition <math>p : X \to \underline\mathbb{B}.</math>

In particular, a connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> can be understood to indicate a relation among boolean values, namely, the <math>k\!</math>-ary relation <math>F^{-1} (\underline{1}) \subseteq \underline\mathbb{B}^k.</math>  If these <math>k\!</math> values are values of things in a universe <math>X,\!</math> that is, if one imagines each value in a <math>k\!</math>-tuple of values to be the functional image that results from evaluating an element of <math>X\!</math> under one of its possible aspects of value, then one has in mind the <math>k\!</math> propositions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> in sum, one embodies the imagination <math>\underline{f} = (f_1, \ldots, f_k).</math>  Together, the imagination <math>\underline{f} \in (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> stretch each other to cover the universe <math>X,\!</math> yielding a new proposition <math>p : X \to \underline\mathbb{B}.</math>

To encapsulate the form of this general result, I define a composition that takes an imagination <math>\underline{f} = (f_1, \ldots, f_k) \in (X \to \underline\mathbb{B})^k</math> and a boolean connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> and gives a proposition <math>p : X \to \underline\mathbb{B}.</math>  Depending on the situation, specifically, according to whether many <math>F\!</math> and many <math>\underline{f},</math> a single <math>F\!</math> and many <math>\underline{f},</math> or many <math>F\!</math> and a single <math>\underline{f}</math> are being considered, respectively, the proposition <math>p\!</math> thus constructed may be referred to under one of three descriptions:

To encapsulate the form of this general result, I define a composition that takes an imagination <math>\underline{f} = (f_1, \ldots, f_k) \in (X \to \underline\mathbb{B})^k</math> and a boolean connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> and gives a proposition <math>p : X \to \underline\mathbb{B}.</math>  Depending on the situation, specifically, according to whether many <math>F\!</math> and many <math>\underline{f},</math> a single <math>F\!</math> and many <math>\underline{f},</math> or many <math>F\!</math> and a single <math>\underline{f}</math> are being considered, respectively, the proposition <math>p\!</math> thus constructed may be referred to under one of three descriptions:

# An imagination of degree <math>k\!</math> on <math>X,\!</math> in other words, a <math>k\!</math>-tuple of propositions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> or an object of the form <math>\underline{f} = (f_1, \ldots, f_k) : (X \to \underline\mathbb{B})^k.</math>

# An imagination of degree <math>k\!</math> on <math>X,\!</math> in other words, a <math>k\!</math>-tuple of propositions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> or an object of the form <math>\underline{f} = (f_1, \ldots, f_k) : (X \to \underline\mathbb{B})^k.</math>

# A connection of degree <math>k,\!</math> in other words, a proposition about things in <math>\underline\mathbb{B}^k,</math> or a boolean function of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.</math>

# A connection of degree <math>k,\!</math> in other words, a proposition about things in <math>\underline\mathbb{B}^k,</math> or a boolean function of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.</math>

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The cartesian power Bk, as a cartesian product, is characterized by the possession of a "projective imagination" p = <p1, ..., pk> of degree k on Bk, along with the property that any imagination f = <f1, ..., fk> of degree k on an arbitrary set W determines a unique map f! : W -> Bk, the play of whose projective images <p1(f!(w), ..., pk(f!(w)) on the functional image f!(w) matches the play of images <f1(w), ..., fk(w)> under f, term for term and at every element w in W.

The cartesian power Bk, as a cartesian product, is characterized by the possession of a "projective imagination" p = <p1, ..., pk> of degree k on Bk, along with the property that any imagination f = <f1, ..., fk> of degree k on an arbitrary set W determines a unique map f! : W -> Bk, the play of whose projective images <p1(f!(w), ..., pk(f!(w)) on the functional image f!(w) matches the play of images <f1(w), ..., fk(w)> under f, term for term and at every element w in W.