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| − | ===Application of Higher Order Propositions to Quantification Theory=== | + | ==Test Area== |
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| − | Our excursion into the vastening landscape of higher order propositions has finally come round to the stage where we can bring its returns to bear on opening up new perspectives for quantificational logic.
| + | {| align="center" cellpadding="0" cellspacing="0" width="90%" |
| | + | | |
| | + | <p>A child hears it said that the stove is hot. But it is not, he says; and, indeed, that central body is not touching it, and only what that touches is hot or cold. But he touches it, and finds the testimony confirmed in a striking way. Thus, he becomes aware of ignorance, and it is necessary to suppose a ''self'' in which this ignorance can inhere. …</p> |
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| − | It's hard to tell if it makes any difference from a purely formal point of view, but it serves intuition to devise a slightly different interpretation for the two-valued space that we use as the target of our basic indicator functions. Therefore, let us declare the type of ''existential-valued functions'' <math>f : \mathbb{B}^k \to \mathbb{E},</math> where <math>\mathbb{E} = \{ -e, +e \} = \{ \operatorname{empty}, \operatorname{exist} \}</math> is a pair of values that indicate whether or not anything exists in the cells of the underlying universe of discourse. As usual, let's not be too fussy about the coding of these functions, reverting to binary codes whenever the intended interpretation is clear enough.
| + | <p>In short, ''error'' appears, and it can be explained only by supposing a ''self'' which is fallible.</p> |
| | | | |
| − | With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:
| + | <p>Ignorance and error are all that distinguish our private selves from the absolute ''ego'' of pure apperception.</p> |
| − | | + | |- |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
| + | | align="right" | (Peirce, CP 5.233–235) |
| − | |+ <math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math>
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| − | | | |
| − | <math>\begin{array}{clcl}
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| − | \mathrm{A} &
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| − | \mathrm{Universal~Affirmative} &
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| − | \mathrm{All}\ u\ \mathrm{is}\ v &
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| − | \mathrm{Indicator~of}\ u (v) = 0 \\
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| − | \mathrm{E} &
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| − | \mathrm{Universal~Negative} &
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| − | \mathrm{All}\ u\ \mathrm{is}\ (v) &
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| − | \mathrm{Indicator~of}\ u \cdot v = 0 \\
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| − | \mathrm{I} &
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| − | \mathrm{Particular~Affirmative} &
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| − | \mathrm{Some}\ u\ \mathrm{is}\ v &
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| − | \mathrm{Indicator~of}\ u \cdot v = 1 \\
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| − | \mathrm{O} &
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| − | \mathrm{Particular~Negative} &
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| − | \mathrm{Some}\ u\ \mathrm{is}\ (v) &
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| − | \mathrm{Indicator~of}\ u (v) = 1 \\
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| − | \end{array}</math>
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| | |} | | |} |
| − |
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| − | <br>
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| − |
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| − | <pre>
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| − | <table align="center" border="1" cellpadding="8" cellspacing="0" width="80%">
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| − |
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| − | <caption><font size="+2"><math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math></font></caption>
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| − |
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| − | <tr>
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| − | <td align="center"><math>\operatorname{A}</math></td>
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| − | <td><math>\text{Absolute}</math></td>
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| − | <td><math>\text{Universal Affirmative}</math></td>
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| − | <td align="center"><math>All ~ u ~ is ~ v</math></td>
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| − | <td><math>Indicator of u ~ \texttt{(} v \texttt{)} = 0</math></td></tr>
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| − |
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| − | <tr>
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| − | <td align="center"><math>\operatorname{E}</math></td>
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| − | <td><math>Exclusive</math></td>
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| − | <td><math>Universal Negative</math></td>
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| − | <td align="center"><math>All ~ u ~ is ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td><math>Indicator of ~ u ~ \cdot ~ v = 0</math></td></tr>
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| − |
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| − | <tr>
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| − | <td align="center"><math>\operatorname{I}</math></td>
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| − | <td><math>Indefinite</math></td>
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| − | <td><math>Particular Affirmative</math></td>
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| − | <td align="center"><math>Some ~ u ~ is ~ v</math></td>
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| − | <td><math>Indicator of ~ u ~ \cdot ~ v = 1</math></td></tr>
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| − |
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| − | <tr>
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| − | <td align="center"><math>\operatorname{O}</math></td>
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| − | <td><math>Obtrusive</math></td>
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| − | <td><math>Particular Negative</math></td>
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| − | <td align="center"><math>Some ~ u ~ is ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td><math>Indicator of ~ u ~ \texttt{(} v \texttt{)} = 1</math></td></tr>
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| − |
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| − | </table></font>
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| − |
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| − | <br>
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| − |
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| − | The following Tables develop these ideas in more detail.
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| − |
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| − | <br>
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| − |
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| − | <table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
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| − |
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| − | <caption><font size="+2"><math>\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}</math></font></caption>
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| − |
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| − | <tr>
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| − | <td width="4%" style="border-bottom:2px solid black" align="right">
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| − | <math>u:</math><br>
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| − | <math>v:</math></td>
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| − | <td width="6%" style="border-bottom:2px solid black">
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| − | <math>1100</math><br>
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| − | <math>1010</math></td>
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| − | <td width="10%" style="border-bottom:2px solid black; border-right:2px solid black">
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| − | <math>f</math></td>
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| − | <td width="10%" style="border-bottom:2px solid black">
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| − | <math>\texttt{(} \ell_{11} \texttt{)}</math><br>
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| − | <math>No ~ u</math><br>
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| − | <math>is ~ v</math></td>
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| − | <td width="10%" style="border-bottom:2px solid black">
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| − | <math>\texttt{(} \ell_{10} \texttt{)}</math><br>
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| − | <math>No ~ u</math><br>
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| − | <math>is ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td width="10%" style="border-bottom:2px solid black">
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| − | <math>\texttt{(} \ell_{01} \texttt{)}</math><br>
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| − | <math>No ~ \texttt{(} u \texttt{)}</math><br>
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| − | <math>is ~ v</math></td>
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| − | <td width="10%" style="border-bottom:2px solid black">
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| − | <math>\texttt{(} \ell_{00} \texttt{)}</math><br>
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| − | <math>No ~ \texttt{(} u \texttt{)}</math><br>
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| − | <math>is ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td width="10%" style="border-bottom:2px solid black">
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| − | <math>\ell_{00}</math><br>
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| − | <math>Some ~ \texttt{(} u \texttt{)}</math><br>
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| − | <math>is ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td width="10%" style="border-bottom:2px solid black">
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| − | <math>\ell_{01}</math><br>
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| − | <math>Some ~ \texttt{(} u \texttt{)}</math><br>
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| − | <math>is ~ v</math></td>
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| − | <td width="10%" style="border-bottom:2px solid black">
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| − | <math>\ell_{10}</math><br>
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| − | <math>Some ~ u</math><br>
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| − | <math>is ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td width="10%" style="border-bottom:2px solid black">
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| − | <math>\ell_{11}</math><br>
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| − | <math>Some ~ u</math><br>
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| − | <math>is ~ v</math></td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{0}</math></td>
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| − | <td><math>0000</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{(~)}</math></td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{1}</math></td>
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| − | <td><math>0001</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{2}</math></td>
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| − | <td><math>0010</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{3}</math></td>
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| − | <td><math>0011</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{4}</math></td>
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| − | <td><math>0100</math></td>
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| − | <td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{5}</math></td>
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| − | <td><math>0101</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{6}</math></td>
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| − | <td><math>0110</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{7}</math></td>
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| − | <td><math>0111</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{8}</math></td>
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| − | <td><math>1000</math></td>
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| − | <td style="border-right:2px solid black"><math>u ~ v</math></td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{9}</math></td>
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| − | <td><math>1001</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{10}</math></td>
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| − | <td><math>1010</math></td>
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| − | <td style="border-right:2px solid black"><math>v</math></td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{11}</math></td>
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| − | <td><math>1011</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td></tr>
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| − |
| |
| − | <tr>
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| − | <td><math>f_{12}</math></td>
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| − | <td><math>1100</math></td>
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| − | <td style="border-right:2px solid black"><math>u</math></td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{13}</math></td>
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| − | <td><math>1101</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td></tr>
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| − |
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| − | <tr>
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| − | <td><math>f_{14}</math></td>
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| − | <td><math>1110</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td></tr>
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| − |
| |
| − | <tr>
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| − | <td><math>f_{15}</math></td>
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| − | <td><math>1111</math></td>
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| − | <td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:white; color:black">0</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td>
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| − | <td style="background:black; color:white">1</td></tr>
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| − |
| |
| − | </table>
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| − |
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| − | <br>
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| − |
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| − | <table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%">
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| − |
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| − | <caption><font size="+2"><math>\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}</math></font></caption>
| |
| − |
| |
| − | <tr>
| |
| − | <td width="4%" style="border-bottom:2px solid black" align="right">
| |
| − | <math>u:</math><br>
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| − | <math>v:</math></td>
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| − | <td width="6%" style="border-bottom:2px solid black">
| |
| − | <math>1100</math><br>
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| − | <math>1010</math></td>
| |
| − | <td width="10%" style="border-bottom:2px solid black; border-right:2px solid black">
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| − | <math>f</math></td>
| |
| − | <td width="10%" style="border-bottom:2px solid black">
| |
| − | <math>\texttt{(} \ell_{11} \texttt{)}</math><br>
| |
| − | <math>No ~ u</math><br>
| |
| − | <math>is ~ v</math></td>
| |
| − | <td width="10%" style="border-bottom:2px solid black">
| |
| − | <math>\texttt{(} \ell_{10} \texttt{)}</math><br>
| |
| − | <math>No ~ u</math><br>
| |
| − | <math>is ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td width="10%" style="border-bottom:2px solid black">
| |
| − | <math>\texttt{(} \ell_{01} \texttt{)}</math><br>
| |
| − | <math>No ~ \texttt{(} u \texttt{)}</math><br>
| |
| − | <math>is ~ v</math></td>
| |
| − | <td width="10%" style="border-bottom:2px solid black">
| |
| − | <math>\texttt{(} \ell_{00} \texttt{)}</math><br>
| |
| − | <math>No ~ \texttt{(} u \texttt{)}</math><br>
| |
| − | <math>is ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td width="10%" style="border-bottom:2px solid black">
| |
| − | <math>\ell_{00}</math><br>
| |
| − | <math>Some ~ \texttt{(} u \texttt{)}</math><br>
| |
| − | <math>is ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td width="10%" style="border-bottom:2px solid black">
| |
| − | <math>\ell_{01}</math><br>
| |
| − | <math>Some ~ \texttt{(} u \texttt{)}</math><br>
| |
| − | <math>is ~ v</math></td>
| |
| − | <td width="10%" style="border-bottom:2px solid black">
| |
| − | <math>\ell_{10}</math><br>
| |
| − | <math>Some ~ u</math><br>
| |
| − | <math>is ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td width="10%" style="border-bottom:2px solid black">
| |
| − | <math>\ell_{11}</math><br>
| |
| − | <math>Some ~ u</math><br>
| |
| − | <math>is ~ v</math></td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td style="border-bottom:2px solid black"><math>f_{0}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>0000</math></td>
| |
| − | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>\texttt{(~)}</math></td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>f_{1}</math></td>
| |
| − | <td><math>0001</math></td>
| |
| − | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>f_{2}</math></td>
| |
| − | <td><math>0010</math></td>
| |
| − | <td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>f_{4}</math></td>
| |
| − | <td><math>0100</math></td>
| |
| − | <td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td style="border-bottom:2px solid black"><math>f_{8}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>1000</math></td>
| |
| − | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>u ~ v</math></td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>f_{3}</math></td>
| |
| − | <td><math>0011</math></td>
| |
| − | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td style="border-bottom:2px solid black"><math>f_{12}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>1100</math></td>
| |
| − | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>u</math></td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>f_{6}</math></td>
| |
| − | <td><math>0110</math></td>
| |
| − | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td style="border-bottom:2px solid black"><math>f_{9}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>1001</math></td>
| |
| − | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>f_{5}</math></td>
| |
| − | <td><math>0101</math></td>
| |
| − | <td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td style="border-bottom:2px solid black"><math>f_{10}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>1010</math></td>
| |
| − | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>v</math></td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>f_{7}</math></td>
| |
| − | <td><math>0111</math></td>
| |
| − | <td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>f_{11}</math></td>
| |
| − | <td><math>1011</math></td>
| |
| − | <td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>f_{13}</math></td>
| |
| − | <td><math>1101</math></td>
| |
| − | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td style="border-bottom:2px solid black"><math>f_{14}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>1110</math></td>
| |
| − | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:white; color:black">0</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td>
| |
| − | <td style="border-bottom:2px solid black; background:black; color:white">1</td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>f_{15}</math></td>
| |
| − | <td><math>1111</math></td>
| |
| − | <td style="border-right:2px solid black"><math>\texttt{((~))}</math></td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:white; color:black">0</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td>
| |
| − | <td style="background:black; color:white">1</td></tr>
| |
| − |
| |
| − | </table>
| |
| − |
| |
| − | <br>
| |
| − |
| |
| − | <table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
| |
| − |
| |
| − | <caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption>
| |
| − |
| |
| − | <tr>
| |
| − | <td style="border-bottom:2px solid black"><math>\texttt{Mnemonic}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>\texttt{Category}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>\texttt{Classical Form}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>\texttt{Alternate Form}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>\texttt{Symmetric Form}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>\texttt{Operator}</math></td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>\texttt{E}</math><br><math>\texttt{Exclusive}</math></td>
| |
| − | <td><math>\texttt{Universal}</math><br><math>\texttt{Negative}</math></td>
| |
| − | <td><math>\texttt{All} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td> </td>
| |
| − | <td><math>\texttt{No} ~ u ~ \texttt{is} ~ v</math></td>
| |
| − | <td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>\texttt{A}</math><br><math>\texttt{Absolute}</math></td>
| |
| − | <td><math>\texttt{Universal}</math><br><math>\texttt{Affirmative}</math></td>
| |
| − | <td><math>\texttt{All} ~ u ~ \texttt{is} ~ v</math></td>
| |
| − | <td> </td>
| |
| − | <td><math>\texttt{No} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td> </td>
| |
| − | <td> </td>
| |
| − | <td><math>\texttt{All} ~ v ~ \texttt{is} ~ u</math></td>
| |
| − | <td><math>\texttt{No} ~ v ~ \texttt{is} ~ \texttt{(} u \texttt{)}</math></td>
| |
| − | <td><math>\texttt{No} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ v</math></td>
| |
| − | <td><math>\texttt{(} \ell_{01} \texttt{)}</math></td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td style="border-bottom:2px solid black"> </td>
| |
| − | <td style="border-bottom:2px solid black"> </td>
| |
| − | <td style="border-bottom:2px solid black"><math>\texttt{All} ~ \texttt{(} v \texttt{)} ~ \texttt{is} ~ u</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>\texttt{No} ~ \texttt{(} v \texttt{)} ~ \texttt{is} ~ \texttt{(} u \texttt{)}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>\texttt{No} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td style="border-bottom:2px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td> </td>
| |
| − | <td> </td>
| |
| − | <td><math>\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td> </td>
| |
| − | <td><math>\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td><math>\ell_{00}</math></td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td> </td>
| |
| − | <td> </td>
| |
| − | <td><math>\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ v</math></td>
| |
| − | <td> </td>
| |
| − | <td><math>\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ v</math></td>
| |
| − | <td><math>\ell_{01}</math></td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>\texttt{O}</math><br><math>\texttt{Obtrusive}</math></td>
| |
| − | <td><math>\texttt{Particular}</math><br><math>\texttt{Negative}</math></td>
| |
| − | <td><math>\texttt{Some} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td> </td>
| |
| − | <td><math>\texttt{Some} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
| |
| − | <td><math>\ell_{10}</math></td></tr>
| |
| − |
| |
| − | <tr>
| |
| − | <td><math>\texttt{I}</math><br><math>\texttt{Indefinite}</math></td>
| |
| − | <td><math>\texttt{Particular}</math><br><math>\texttt{Affirmative}</math></td>
| |
| − | <td><math>\texttt{Some} ~ u ~ \texttt{is} ~ v</math></td>
| |
| − | <td> </td>
| |
| − | <td><math>\texttt{Some} ~ u ~ \texttt{is} ~ v</math></td>
| |
| − | <td><math>\ell_{11}</math></td></tr>
| |
| − |
| |
| − | </table></font>
| |
| − |
| |
| − | <br>
| |
| − |
| |
| − | <pre>
| |
