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===Commentary Note 11.11===

===Commentary Note 11.11===

The preceding exercises were intended to beef-up our "functional" literacy skills to the point where we can read our functional alphabets backwards and forwards and to ferret out the local functionalites that may be immanent in relative terms no matter where they locate themselves within the domains of relations.  I am hopeful that these skills will serve us in good stead as we work to build a catwalk from Peirce's platform to contemporary scenes on the logic of relatives, and back again.

The preceding exercises were intended to beef-up our "functional" literacy skills to the point where we can read our functional alphabets backwards and forwards and recognize the local functionalities that may be immanent in relative terms no matter where they locate themselves within the domains of relations.  These skills will serve us in good stead as we work to build a catwalk from Peirce's platform of 1870 to contemporary scenes on the logic of relatives, and back again.

By way of extending a few very tentative plancks, let us experiment with the following definitions:

By way of extending a few very tentative planks, let us experiment with the following definitions:

# A relative term "''p''" and the corresponding relation ''P'' ⊆ ''X'' × ''Y'' are both called "functional on relates" if and only if ''P'' is a function at ''X'', in symbols, ''P'' : ''X'' → ''Y''.

 

# A relative term "''p''" and the corresponding relation ''P'' ⊆ ''X'' × ''Y'' are both called "functional on correlates" if and only if ''P'' is function at ''Y'', in symbols, ''P'' : ''X'' ← ''Y''.

<p>A relative term <math>^{\backprime\backprime} p ^{\prime\prime}</math> and the corresponding relation <math>P \subseteq X \times Y</math> are both called ''functional on relates'' if and only if <math>P\!</math> is a function at <math>X,\!</math> in symbols, <math>P : X \to Y.</math></p>

<p>A relative term <math>^{\backprime\backprime} p ^{\prime\prime}</math> and the corresponding relation <math>P \subseteq X \times Y</math> are both called ''functional on correlates'' if and only if <math>P\!</math> is a function at <math>Y,\!</math> in symbols, <math>P : X \leftarrow Y.</math></p>

When a relation happens to be a function, it may be excusable to use the same name for it in both applications, writing out explicit type markers like ''P''&nbsp;:&nbsp;''X''&nbsp;&times;&nbsp;''Y'', ''P''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''Y'', ''P''&nbsp;:&nbsp;''X''&nbsp;&larr;&nbsp;''Y'', as the case may be, when and if it serves to clarify matters.

When a relation happens to be a function, it may be excusable to use the same name for it in both applications, writing out explicit type markers like ''P''&nbsp;:&nbsp;''X''&nbsp;&times;&nbsp;''Y'', ''P''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''Y'', ''P''&nbsp;:&nbsp;''X''&nbsp;&larr;&nbsp;''Y'', as the case may be, when and if it serves to clarify matters.