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=====2.3.3.1.  Flexible Roles and Suitable Models=====

=====2.3.3.1.  Flexible Roles and Suitable Models=====

When giving names and habitations to things by the use of letters and types, a certain flexibility may be allowed in the roles assigned by interpretation.  For example, in the form "p: U -> B", the name "p" may be taken to denote a proposition or a function, indifferently, and the type U may be associated with a set of interpretations or a set of boolean vectors, correspondingly, whichever makes sense in a given context of use.  One dimension that does matter is drawn through these three beads:  propositions, interpretations, and values.  On the alternate line it is produced by the distinctions among collections, individuals, and values.

When giving names and habitations to things by the use of letters and types, a certain flexibility may be allowed in the roles assigned by interpretation.  For example, in the form "''p'' : ''U'' → '''B'''", the name "''p''" may be taken to denote a proposition or a function, indifferently, and the type ''U'' may be associated with a set of interpretations or a set of boolean vectors, correspondingly, whichever makes sense in a given context of use.  One dimension that does matter is drawn through these three beads:  propositions, interpretations, and values.  On the alternate line it is produced by the distinctions among collections, individuals, and values.

One relation that is of telling importance is the relation of interpretations to the value they give a proposition.  In its full sense and general case this should be recognized as a three-place relation, involving all three types of entities (propositions, interpretations, and values) inextricably.  However, for many applications the substance of the information in the three-place relation is conveyed well enough by the data of its bounding or derivative two-place relations.

One relation that is of telling importance is the relation of interpretations to the value they give a proposition.  In its full sense and general case this should be recognized as a three-place relation, involving all three types of entities (propositions, interpretations, and values) inextricably.  However, for many applications the substance of the information in the three-place relation is conveyed well enough by the data of its bounding or derivative two-place relations.

The interpretations that render a proposition true, i.e. the substitutions for which the proposition evaluates to true, are said to satisfy the proposition and to be its models.  With a doubly modulated sense that is too apt to be purely accidental, the model set is the "content" of the proposition's formal expression (Eulenberg, 1986).  In functional terms the models of a proposition p are the pre-images of truth under the function p.  Collectively, they form the set of vectors in p^–1(1).  In another usage the set of models is called the fiber of truth, in other words, the equivalence class [1]p of the value 1 under the mapping p.

The interpretations that render a proposition true, that is, the substitutions for which the proposition evaluates to true, are said to satisfy the proposition and to be its models.  With a doubly modulated sense that is too apt to be purely accidental, the model set is the "content" of the proposition's formal expression (Eulenberg, 1986).  In functional terms the models of a proposition ''p'' are the pre-images of truth under the function ''p''.  Collectively, they form the set of vectors in ''p''^–1(1).  In another usage the set of models is called the ''fiber'' of truth, in other words, the equivalence class [1]_''p'' of the value 1 under the mapping ''p''.

=====2.3.3.2.  Functional Pragmatism=====

=====2.3.3.2.  Functional Pragmatism=====