Changes

As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand.  The lexical operators <math>\operatorname{p}^k</math> and <math>\operatorname{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.

As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand.  The lexical operators <math>\operatorname{p}^k</math> and <math>\operatorname{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 10.  A Realm of Intentional Features'''

|+ '''Table 10.  A Realm of Intentional Features'''

| width=50% |

|

<p><math>\begin{array}{lllll}

| p⁰<font face="lucida calligraphy">A</font>

\operatorname{p}^0 \mathcal{A}

& = & \{ a_1, \ldots, a_n \}

| <font face="lucida calligraphy">A</font>

& = & \mathcal{A}               \\

| =

\operatorname{p}^1 \mathcal{A}

| {''a''₁&nbsp;,

& = & \{ a_1^\prime, \ldots, a_n^\prime \}

| &hellip;,

& = & \mathcal{A}^\prime         \\

| ''a''_''n''&nbsp;}

\operatorname{p}^2 \mathcal{A}

& = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \}

| p¹<font face="lucida calligraphy">A</font>

& = & \mathcal{A}^{\prime\prime} \\

| =

\end{array}</math></p>

| <font face="lucida calligraphy">A</font>&prime;

<p><math>\begin{array}{lll}

\ldots & & \ldots \\

| {''a''₁&prime;,

\operatorname{p}^k \mathcal{A}

| &hellip;,

& = & \{\operatorname{p}^k a_1, \ldots, \operatorname{p}^k a_n\} \\

| ''a''_''n''&prime;}

\end{array}</math></p>

|

| p²<font face="lucida calligraphy">A</font>

<p><math>\begin{array}{lll}

| =

\operatorname{Q}^0 \mathcal{A}

| <font face="lucida calligraphy">A</font>&Prime;

& = & \mathcal{A}  \\

\operatorname{Q}^1 \mathcal{A}

| {''a''₁&Prime;,

& = & \mathcal{A}

\cup \mathcal{A}'  \\

| ''a''_''n''&Prime;}

\operatorname{Q}^2 \mathcal{A}

& = & \mathcal{A}

\cup \mathcal{A}'

| &nbsp;

\cup \mathcal{A}'' \\

| &nbsp;

\ldots & & \ldots  \\

\operatorname{Q}^k \mathcal{A}

& = & \mathcal{A}

\cup \mathcal{A}'

| p^''k''<font face="lucida calligraphy">A</font>

\cup \ldots

| =

\cup \operatorname{p}^k \mathcal{A} \\

| &nbsp;

\end{array}</math></p>

|}<br>

| {p^''k''''a''₁,

| &hellip;,

| p^''k''''a''_''n''}

|}

{| cellpadding="4"

| Q⁰<font face="lucida calligraphy">A</font>

| =

| <font face="lucida calligraphy">A</font>

| Q¹<font face="lucida calligraphy">A</font>

| =

| <font face="lucida calligraphy">A</font> &cup; <font face="lucida calligraphy">A</font>&prime;

| Q²<font face="lucida calligraphy">A</font>

| =

| <font face="lucida calligraphy">A</font> &cup; <font face="lucida calligraphy">A</font>&prime; &cup; <font face="lucida calligraphy">A</font>&Prime;

| &nbsp;

| Q^''k''<font face="lucida calligraphy">A</font>

| =

| <font face="lucida calligraphy">A</font> &cup; <font face="lucida calligraphy">A</font>&prime; &cup; &hellip; &cup; p^''k''<font face="lucida calligraphy">A</font>

|}

</font><br>

The resulting augmentations of our logical basis found a series of discursive universes that may be called the ''intentional extension'' of propositional calculus.  The pattern of this extension is analogous to that of the differential extension, which was developed in terms of the operators d^''k'' and E^''k'', and there is an obvious and natural relation between these two extensions that falls within our purview to explore.  In contexts displaying this regular pattern, where a series of domains stretches up from an anchoring domain ''X'' through an indefinite number of higher reaches, I refer to a particular collection of domains based on ''X'' as a ''realm'' of ''X'', and when the succession exhibits a temporal aspect, as a ''reign'' of ''X''.

The resulting augmentations of our logical basis found a series of discursive universes that may be called the ''intentional extension'' of propositional calculus.  The pattern of this extension is analogous to that of the differential extension, which was developed in terms of the operators d^''k'' and E^''k'', and there is an obvious and natural relation between these two extensions that falls within our purview to explore.  In contexts displaying this regular pattern, where a series of domains stretches up from an anchoring domain ''X'' through an indefinite number of higher reaches, I refer to a particular collection of domains based on ''X'' as a ''realm'' of ''X'', and when the succession exhibits a temporal aspect, as a ''reign'' of ''X''.