Changes

I am optimistic that the some of the tethering material that I spun along the "Relations In General" thread will help us to track the equivalential and functional properties of special relations in a way that will not weigh too heavily on the embedding of syntax in 1-dimensional strings on 2-dimensional pages.  But I cannot see far enough ahead to foresee all the consequences of trying this tack, and so it must remain a bit experimental.

I am optimistic that the some of the tethering material that I spun along the "Relations In General" thread will help us to track the equivalential and functional properties of special relations in a way that will not weigh too heavily on the embedding of syntax in 1-dimensional strings on 2-dimensional pages.  But I cannot see far enough ahead to foresee all the consequences of trying this tack, and so it must remain a bit experimental.

The first obstacle to get past is the order convention that Peirce's orientation to relative terms causes him to use for functions.  To focus on a concrete example of immedeiate use in this discussion, let's take the "number of" function that Peirce dneotes by means of square brackets and re-formulate it as a 2-adic relative term &mdash; say <math>~v~</math> &mdash; where:

The first obstacle to get past is the order convention that Peirce's orientation to relative terms causes him to use for functions.  To focus on a concrete example of immedeiate use in this discussion, let's take the "number of" function that Peirce denotes by means of square brackets and re-formulate it as a 2-adic relative term &mdash; say <math>~v~</math> &mdash; where:

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To set the 2-adic relative term 'v' within a suitable context of interpretation, let us suppose that 'v' corresponds to a relation ''V''&nbsp;&sube;&nbsp;''R''&nbsp;&times;&nbsp;''S'', where ''R'' is the set of real numbers and ''S'' is a suitable syntactic domain, here described as "terms".  Then the 2-adic relation ''V'' is evidently a function from ''S'' to ''R''.  We might think to use the plain letter "''v''" to denote this function, as ''v''&nbsp;:&nbsp;''S''&nbsp;&rarr;&nbsp;''R'', but I worry this may be a chaos waiting to happen.  Also, I think that we should anticipate the very great likelihood that we cannot always assign numbers to every term in whatever syntactic domain S that we choose, so it is probably better to account the 2-adic relation ''V'' as a partial function from ''S'' to ''R''.  All things considered, then, let me try out the following impedimentaria of strategies and compromises.

To set the 2-adic relative term <math>~v~</math> within a suitable context of interpretation, let us suppose that <math>~v~</math> corresponds to a relation <math>V \subseteq R \times S,</math> where <math>~R~</math> is the set of real numbers and <math>~S~</math> is a suitable syntactic domain, here described as "terms".  Then the 2-adic relation <math>~V~</math> is evidently a function from <math>~S~</math> to <math>~R.~</math> We might think to use the plain letter <math>{}^{\backprime\backprime} v {}^{\prime\prime}</math> to denote this function, as <math>v : S \to R,</math> but I worry this may be a chaos waiting to happen.  Also, I think we should anticipate the very great likelihood that we cannot always assign numbers to every term in whatever syntactic domain <math>~S~</math> that we choose, so it is probably better to account the 2-adic relation <math>~V~</math> as a partial function from <math>~S~</math> to <math>~R.~</math> All things considered, then, let me try out the following impedimentaria of strategies and compromises.

First, I will adapt the functional arrow notation so that it allows us to detach the functional orientation from the order in which the names of domains are written on the page.  Second, I will need to change the notation for "pre-functions", or "partial functions", from one likely confound to a slightly less likely confound.  This gives the scheme:

First, I will adapt the functional arrow notation so that it allows us to detach the functional orientation from the order in which the names of domains are written on the page.  Second, I will need to change the notation for "pre-functions", or "partial functions", from one likely confound to a slightly less likely confound.  This gives the scheme: