Changes

Going back to Rule 1, we see that it lists a pair of concrete sentences and authorizes exchanges in either direction between the syntactic structures that have these two forms.  But a sentence is any sign that denotes a proposition, and so there are any number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged.  For example, a larger collection of equivalent sentences is recorded in Rule 4.

Going back to Rule 1, we see that it lists a pair of concrete sentences and authorizes exchanges in either direction between the syntactic structures that have these two forms.  But a sentence is any sign that denotes a proposition, and so there are any number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged.  For example, a larger collection of equivalent sentences is recorded in Rule 4.

<pre>

<br>

If X c U is fixed

| style="border-top:1px solid black" | <math>\text{If}\!</math>

| style="border-top:1px solid black" | <math>Q \subseteq X ~\text{is fixed}</math>

 

 

 

 

 

 

The first and last items on this list, namely, the sentences "u C X" and "{X}(u) = 1" that are annotated as "R4a" and "R4e", respectively, are just the pair of sentences from Rule 3 whose equivalence for all u C U is usually taken to define the idea of an indicator function {X} : U -> B.  At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their own ostensible types and the ruling type of a sentence.  On reflection, and taken in context, these problems are not as serious as they initially seem.  For instance, the expression "[u C X]" ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence.  As a general rule, if one can see it on the page, then it cannot be a proposition, but can be, at best, a sign of one.

The first and last items on this list, namely, the sentences "u C X" and "{X}(u) = 1" that are annotated as "R4a" and "R4e", respectively, are just the pair of sentences from Rule 3 whose equivalence for all u C U is usually taken to define the idea of an indicator function {X} : U -> B.  At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their own ostensible types and the ruling type of a sentence.  On reflection, and taken in context, these problems are not as serious as they initially seem.  For instance, the expression "[u C X]" ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence.  As a general rule, if one can see it on the page, then it cannot be a proposition, but can be, at best, a sign of one.