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Let me now address one last question that may have occurred to some.  What has happened, in this suggested scheme of functional reasoning, to the distinction that is quite pointedly made by careful logicians between (1) the connectives called ''conditionals'' and symbolized by the signs <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called ''implications'' and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, and, in a related question:  What has happened to the distinction that is equally insistently made between (3) the connective called the ''biconditional'' and signified by the sign <math>(\leftrightarrow)</math> and (4) the assertion that is called an ''equivalence'' and signified by the sign <math>(\Leftrightarrow)</math>?  My answer is this:  For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.
 
Let me now address one last question that may have occurred to some.  What has happened, in this suggested scheme of functional reasoning, to the distinction that is quite pointedly made by careful logicians between (1) the connectives called ''conditionals'' and symbolized by the signs <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called ''implications'' and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, and, in a related question:  What has happened to the distinction that is equally insistently made between (3) the connective called the ''biconditional'' and signified by the sign <math>(\leftrightarrow)</math> and (4) the assertion that is called an ''equivalence'' and signified by the sign <math>(\Leftrightarrow)</math>?  My answer is this:  For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.
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<pre>
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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In this Subsection, I finally bring together many of what may
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have appeared to be wholly independent threads of development,
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in the hope of paying off a percentage of my promissory notes,
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even if a goodly number my creditors have no doubt long since
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forgotten, if not exactly forgiven the debentures in question.
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For ease of reference, I repeat here a couple of the
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definitions that are needed again in this discussion.
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| A "boolean connection" of degree k, also known as a "boolean function"
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| on k variables, is a map of the form F : %B%^k -> %B%.  In other words,
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| a boolean connection of degree k is a proposition about things in the
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| universe of discourse X = %B%^k.
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|
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| An "imagination" of degree k on X is a k-tuple of propositions
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| about things in the universe X.  By way of displaying the kinds
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| of notation that are used to express this idea, the imagination
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| #f# = <f_1, ..., f_k> is can be given as a sequence of indicator
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| functions f_j : X -> %B%, for j = 1 to k.  All of these features
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| of the typical imagination #f# can be summed up in either one of
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| two ways:  either in the form of a membership statement, stating
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| words to the effect that #f# belongs to the space (X -> %B%)^k,
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| or in the form of the type declaration that #f# : (X -> %B%)^k,
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| though perhaps the latter specification is slightly more precise
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| than the former.
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The definition of the "stretch" operation and the uses of the
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various brands of denotational operators can be reviewed here:
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055.  http://suo.ieee.org/email/msg07466.html
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057.  http://suo.ieee.org/email/msg07469.html
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070.  http://suo.ieee.org/ontology/msg03473.html
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071.  http://suo.ieee.org/ontology/msg03479.html
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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</pre>
      
==Stretching Exercises==
 
==Stretching Exercises==
Jon Awbrey
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