Changes

Previously I introduced a calculus for propositional logic, fixing its meaning according to what C.S. Peirce called the ''existential interpretation''.  As far as it concerns propositional calculus this interpretation settles the meanings that are associated with merely the most basic symbols and logical connectives.  Now we must extend and refine the existential interpretation to comprehend the analysis of ''quantifications'', that is, quantified propositions.  In doing so we recognize two additional aspects of logic that need to be developed, over and above the material of propositional logic.  At the formal extreme there is the aspect of higher order functional types, into which we have already ventured a little above.  At the level of the fundamental content of the available propositions we have to introduce a different interpretation for what we may call ''elemental'' or ''singular'' propositions.

Previously I introduced a calculus for propositional logic, fixing its meaning according to what C.S. Peirce called the ''existential interpretation''.  As far as it concerns propositional calculus this interpretation settles the meanings that are associated with merely the most basic symbols and logical connectives.  Now we must extend and refine the existential interpretation to comprehend the analysis of ''quantifications'', that is, quantified propositions.  In doing so we recognize two additional aspects of logic that need to be developed, over and above the material of propositional logic.  At the formal extreme there is the aspect of higher order functional types, into which we have already ventured a little above.  At the level of the fundamental content of the available propositions we have to introduce a different interpretation for what we may call ''elemental'' or ''singular'' propositions.

Let us return to the 2-dimensional case <math>X^\circ = [x, y].</math>  In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers <math>L_{uv} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> that have the following characters:

Let us return to the 2-dimensional case <math>X^\circ = [x, y].</math>  In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers <math>\ell_{uv} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> that have the following characters:

{| cellpadding="4"

{| cellpadding="4"

| width="36" | &nbsp; || <math>L_{00} f\!</math>

| width="36" | &nbsp; || <math>\ell_{00} f\!</math>

|-

|-

| &nbsp; || = || <math>L_{(\!| x |\!)(\!| y |\!)} f</math>

| &nbsp; || = || <math>\ell_{(\!| x |\!)(\!| y |\!)} f</math>

|-

|-

| &nbsp; || = || <math>\alpha_1 f\!</math>

| &nbsp; || = || <math>\alpha_1 f\!</math>

{| cellpadding="4"

{| cellpadding="4"

| width="36" | &nbsp; || <math>L_{01} f\!</math>

| width="36" | &nbsp; || <math>\ell_{01} f\!</math>

|-

|-

| &nbsp; || = || <math>L_{(\!| x |\!)\ y} f</math>

| &nbsp; || = || <math>\ell_{(\!| x |\!)\ y} f</math>

|-

|-

| &nbsp; || = || <math>\alpha_2 f\!</math>

| &nbsp; || = || <math>\alpha_2 f\!</math>

{| cellpadding="4"

{| cellpadding="4"

| width="36" | &nbsp; || <math>L_{10} f\!</math>

| width="36" | &nbsp; || <math>\ell_{10} f\!</math>

|-

|-

| &nbsp; || = || <math>L_{x\ (\!| y |\!)} f</math>

| &nbsp; || = || <math>\ell_{x\ (\!| y |\!)} f</math>

|-

|-

| &nbsp; || = || <math>\alpha_4 f\!</math>

| &nbsp; || = || <math>\alpha_4 f\!</math>

{| cellpadding="4"

{| cellpadding="4"

| width="36" | &nbsp; || <math>L_{11} f\!</math>

| width="36" | &nbsp; || <math>\ell_{11} f\!</math>

|-

|-

| &nbsp; || = || <math>L_{x\ y} f</math>

| &nbsp; || = || <math>\ell_{x\ y} f</math>

|-

|-

| &nbsp; || = || <math>\alpha_8 f\!</math>

| &nbsp; || = || <math>\alpha_8 f\!</math>

|}

|}

Intuitively, the <math>L_{uv}\!</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values.  Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\circ = [x, y],</math> and so they mediate a subtext <math>[L_{00}, L_{01}, L_{10}, L_{11}]\!</math> that takes place within the higher order universe of discourse <math>X^{\circ 2} = [X^\circ] = [[x, y]].\!</math>  Figure&nbsp;6 summarizes the action of the <math>L_{uv}\!</math> operators on the <math>f_i\!</math> within <math>X^{\circ 2}.\!</math>

Intuitively, the <math>\ell_{uv}\!</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values.  Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\circ = [x, y],</math> and so they mediate a subtext <math>[\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]\!</math> that takes place within the higher order universe of discourse <math>X^{\circ 2} = [X^\circ] = [[x, y]].\!</math>  Figure&nbsp;6 summarizes the action of the <math>\ell_{uv}\!</math> operators on the <math>f_i\!</math> within <math>X^{\circ 2}.\!</math>

<pre>

<pre>

| align=left | &nbsp;

| align=left | &nbsp;

| align=left | No x is y

| align=left | No x is y

| <math>(\!| L_{11} |\!)</math>

| <math>(\!| \ell_{11} |\!)</math>

|-

|-

| <math>\mathrm{A}\!</math><br>Absolute

| <math>\mathrm{A}\!</math><br>Absolute

| align=left | &nbsp;

| align=left | &nbsp;

| align=left | No x is (y)

| align=left | No x is (y)

| <math>(\!| L_{10} |\!)</math>

| <math>(\!| \ell_{10} |\!)</math>

|-

|-

| &nbsp;

| &nbsp;

| align=left | No y is (x)

| align=left | No y is (x)

| align=left | No (x) is y

| align=left | No (x) is y

| <math>(\!| L_{01} |\!)</math>

| <math>(\!| \ell_{01} |\!)</math>

|-

|-

| &nbsp;

| &nbsp;

| align=left | No (y) is (x)

| align=left | No (y) is (x)

| align=left | No (x) is (y)

| align=left | No (x) is (y)

| <math>(\!| L_{00} |\!)</math>

| <math>(\!| \ell_{00} |\!)</math>

|-

|-

| &nbsp;

| &nbsp;

| align=left | &nbsp;

| align=left | &nbsp;

| align=left | Some (x) is (y)

| align=left | Some (x) is (y)

| <math>L_{00}\!</math>

| <math>\ell_{00}\!</math>

|-

|-

| &nbsp;

| &nbsp;

| align=left | &nbsp;

| align=left | &nbsp;

| align=left | Some (x) is y

| align=left | Some (x) is y

| <math>L_{01}\!</math>

| <math>\ell_{01}\!</math>

|-

|-

| <math>\mathrm{O}\!</math><br>Obtrusive

| <math>\mathrm{O}\!</math><br>Obtrusive

| align=left | &nbsp;

| align=left | &nbsp;

| align=left | Some x is (y)

| align=left | Some x is (y)

| <math>L_{10}\!</math>

| <math>\ell_{10}\!</math>

|-

|-

| <math>\mathrm{I}\!</math><br>Indefinite

| <math>\mathrm{I}\!</math><br>Indefinite

| align=left | &nbsp;

| align=left | &nbsp;

| align=left | Some x is y

| align=left | Some x is y

| <math>L_{11}\!</math>

| <math>\ell_{11}\!</math>

|}<br>

|}<br>

|- style="background:ghostwhite"

|- style="background:ghostwhite"

| align="right" | ''x'' : || 1100 || ''f''

| align="right" | ''x'' : || 1100 || ''f''

| (''L''₁₁)

| <math>(\!| \ell_{11} |\!)</math>

| (''L''₁₀)

| <math>(\!| \ell_{10} |\!)</math>

| (''L''₀₁)

| <math>(\!| \ell_{01} |\!)</math>

| (''L''₀₀)

| <math>(\!| \ell_{00} |\!)</math>

| ''L''₀₀

| <math>\ell_{00}</math>

| ''L''₀₁

| <math>\ell_{01}</math>

| ''L''₁₀

| <math>\ell_{10}</math>

| ''L''₁₁

| <math>\ell_{11}</math>

|- style="background:ghostwhite"

|- style="background:ghostwhite"

| align="right" | ''y'' : || 1010 || &nbsp;

| align="right" | ''y'' : || 1010 || &nbsp;