Changes

|}

|}

Notice the distinction, that I continue to maintain at this point, between the logical values <math>\{ \operatorname{falsehood}, \operatorname{truth} \}</math> and the algebraic values <math>\{ 0, 1 \}.\!</math>  This makes it legitimate to write a sentence directly into the right side of the set-builder expression, for instance, weaving the sentence <math>s\!</math> or the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}</math> into the context <math>^{\backprime\backprime} \, \{ (x, y) \in \underline{B}^2 : \ldots \} \, ^{\prime\prime},</math> thereby obtaining the corresponding expressions listed above, while the proposition <math>F(x, y)\!</math> can also be asserted more directly without equating it to <math>\underline{1},</math> since it already has a value in <math>\{ \operatorname{false}, \operatorname{true} \},</math> and thus can be taken as tantamount to an actual sentence.

Notice the distinction, that I continue to maintain at this point, between the logical values <math>\{ \operatorname{falsehood}, \operatorname{truth} \}</math> and the algebraic values <math>\{ 0, 1 \}.\!</math>  This makes it legitimate to write a sentence directly into the righthand side of a set-builder expression, for instance, weaving the sentence <math>s\!</math> or the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}</math> into the context <math>^{\backprime\backprime} \, \{ (x, y) \in \underline{B}^2 : \ldots \} \, ^{\prime\prime},</math> thereby obtaining the corresponding expressions listed above.  It also allows us to assert the proposition <math>F(x, y)\!</math> in a more direct way, without detouring through the equation <math>F(x, y) = \underline{1},</math> since it already has a value in <math>\{ \operatorname{falsehood}, \operatorname{true} \},</math> and thus can be taken as tantamount to an actual sentence.

If the appropriate safeguards can be kept in mind, avoiding all danger of confusing propositions with sentences and sentences with assertions, then the marks of these distinctions need not be forced to clutter the account of the more substantive indications, that is, the ones that really matter.  If this level of understanding can be achieved, then it may be possible to relax these restrictions, along with the absolute dichotomy between algebraic and logical values, which tends to inhibit the flexibility of interpretation.

If the appropriate safeguards can be kept in mind, avoiding all danger of confusing propositions with sentences and sentences with assertions, then the marks of these distinctions need not be forced to clutter the account of the more substantive indications, that is, the ones that really matter.  If this level of understanding can be achieved, then it may be possible to relax these restrictions, along with the absolute dichotomy between algebraic and logical values, which tends to inhibit the flexibility of interpretation.

<pre>

<pre>

To continue the exercise, let p and q be arbitrary propositions about

To continue the exercise, let p and q be arbitrary propositions about

things in the universe X, that is, maps of the form p, q : X -> %B%,

things in the universe X, that is, maps of the form p, q : X -> %B%,