Changes

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.

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 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 change the notation for ''partial functions'', or ''pre-functions'', to one that is less likely to be confounded.  This gives the scheme:

: ''q'' : ''X'' &rarr; ''Y'' means that ''q'' is functional at ''X''.

 

| <math>q : X \to Y</math> means that <math>~q~</math> is functional at <math>~X.~</math>

: ''q'' : ''X'' &larr; ''Y'' means that ''q'' is functional at ''Y''.

 

| <math>q : X \leftarrow Y</math> means that <math>~q~</math> is functional at <math>~X.~</math>

: ''q'' : ''X'' ~> ''Y'' means that ''q'' is pre-functional at ''X''.

 

| <math>q : X \rightharpoonup Y</math> means that <math>~q~</math> is pre-functional at <math>~X.~</math>

: ''q'' : ''X'' <~ ''Y'' means that ''q'' is pre-functional at ''Y''.

| <math>q : X \leftharpoonup Y</math> means that <math>~q~</math> is pre-functional at <math>~Y.~</math>

For now, I will pretend that ''v'' is a function in ''R'' of ''S'', ''v'' : ''R'' &larr; ''S'', amounting to the functional alias of the 2-adic relation ''V''&nbsp;&sube;&nbsp;''R''&nbsp;&times;&nbsp;''S'', and associated with the 2-adic relative term ''v'' whose relate lies in the set ''R'' of real numbers and whose correlate lies in the set ''S'' of syntactic terms.

For now, I will pretend that ''v'' is a function in ''R'' of ''S'', ''v'' : ''R'' &larr; ''S'', amounting to the functional alias of the 2-adic relation ''V''&nbsp;&sube;&nbsp;''R''&nbsp;&times;&nbsp;''S'', and associated with the 2-adic relative term ''v'' whose relate lies in the set ''R'' of real numbers and whose correlate lies in the set ''S'' of syntactic terms.