Changes

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The first and last items on this list, namely, the sentence <math>\text{R4a}\!</math> stating <math>x \in Q</math> and the sentence <math>\text{R4e}\!</math> stating <math>\upharpoonleft Q \upharpoonright (x) = \underline{1},</math> are just the pair of sentences from Rule&nbsp;3 whose equivalence for all <math>x \in X</math> is usually taken to define the idea of an indicator function <math>\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}.</math>  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 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 example, the expression <math>^{\backprime\backprime} \downharpoonleft x \in Q \downharpoonright \, ^{\prime\prime}</math> 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 at most be 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.

The use of the basic connectives can be expressed in the form of a STR as follows:

The use of the basic connectives can be expressed in the form of a STR as follows:

Logical Translation Rule 0

Logical Translation Rule 0