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| − | ===Extending the Existential Interpretation to Quantificational Logic=== | + | ==Test Area== |
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| − | The forms commonly viewed as quantified propositions may be viewed again as propositions about propositions, indeed, there is every reason to regard higher order propositions as the genus of quantification under which the more familiar species appear.
| + | {| 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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| − | Let us return to the 2-dimensional case <math>X^\circ = [u, v]</math>. In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers <math>\ell_{ij} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> that have the following characters:
| + | <p>In short, ''error'' appears, and it can be explained only by supposing a ''self'' which is fallible.</p> |
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| − | {| align="center" cellpadding="6"
| + | <p>Ignorance and error are all that distinguish our private selves from the absolute ''ego'' of pure apperception.</p> |
| − | | | + | |- |
| − | <math>\begin{array}{*{11}{l}}
| + | | align="right" | (Peirce, CP 5.233–235) |
| − | \ell_{00} f
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| − | & = & \ell_{\texttt{(} u \texttt{)(} v \texttt{)}} f
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| − | & = & \alpha_{1} f
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| − | & = & \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)}} f
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| − | & = & \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)} ~ \Rightarrow f}
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| − | & = & f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)(} v \texttt{)}
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| − | \\
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| − | \ell_{01} f
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| − | & = & \ell_{\texttt{(} u \texttt{)} ~ v} f
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| − | & = & \alpha_{2} f | |
| − | & = & \Upsilon_{\texttt{(} u \texttt{)} ~ v} f
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| − | & = & \Upsilon_{\texttt{(} u \texttt{)} ~ v ~ \Rightarrow f}
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| − | & = & f ~ \operatorname{likes} ~ \texttt{(} u \texttt{)} ~ v
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| − | \\
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| − | \ell_{10} f
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| − | & = & \ell_{u ~ \texttt{(} v \texttt{)}} f
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| − | & = & \alpha_{4} f
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| − | & = & \Upsilon_{u ~ \texttt{(} v \texttt{)}} f
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| − | & = & \Upsilon_{u ~ \texttt{(} v \texttt{)} ~ \Rightarrow f}
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| − | & = & f ~ \operatorname{likes} ~ u ~ \texttt{(} v \texttt{)}
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| − | \\
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| − | \ell_{11} f
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| − | & = & \ell_{u ~ v} f
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| − | & = & \alpha_{8} f
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| − | & = & \Upsilon_{u ~ v} f
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| − | & = & \Upsilon_{u ~ v ~ \Rightarrow f}
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| − | & = & f ~ \operatorname{likes} ~ u ~ v
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| − | \end{array}</math>
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| | |} | | |} |
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| − | Intuitively, the <math>\ell_{ij}</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 = [u, v],</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] = [[u, v]].</math>. Figure 6 summarizes the action of the <math>\ell_{ij}</math> operators on the <math>f_{i}</math> within <math>X^{\circ 2}.</math>
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| − | <pre>
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| − | <div align="center" style="text-align:center">
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| − | 
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| − | <font size="+2"><math>\texttt{Figure 6.} ~~ \texttt{Higher Order Universe of Discourse} ~ \left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right] \subseteq \left[\left[ u, v \right]\right]</math></font>
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| − | </div>
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| − |
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| − | </pre>
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| − |
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| − | ===Application of Higher Order Propositions to Quantification Theory===
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| − |
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| − | <pre>
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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.
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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.
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| − | With these qualifications in mind we note the following correspondences between classical quantifications and higher order indicator functions:
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| − | <font size="+1">
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| − | <table align="center" cellpadding="10" cellspacing="0" width="80%">
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| − |
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| − | <caption><font size="+2"><math>\texttt{Table 7.} ~~ \texttt{Syllogistic Premisses as Higher Order Indicator Functions}</math></font></caption>
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| − | <tr>
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| − | <td align="center"><math>\operatorname{A}</math></td>
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| − | <td><math>Absolute</math></td>
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| − | <td><math>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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| − | The following Tables develop these ideas in more detail.
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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>\texttt{Table 8.} ~~ \texttt{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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| − |
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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>
| |
| − | <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></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="1" cellspacing="0" style="text-align:center; width:90%">
| |
| − |
| |
| − | <caption><font size="+2"><math>\texttt{Table 9.} ~~ \texttt{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>
| |
| − | <math>v:</math></td>
| |
| − | <td width="6%" style="border-bottom:2px solid black">
| |
| − | <math>1100</math><br>
| |
| − | <math>1010</math></td>
| |
| − | <td width="10%" style="border-bottom:2px solid black; border-right:2px solid black">
| |
| − | <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>
| |
| − |
| |
| − | <font size="+1">
| |
| − | <table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%">
| |
| − |
| |
| − | <caption><font size="+2"><math>\texttt{Table 10.} ~~ \texttt{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>
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| − | <td style="border-bottom:2px solid black"><math>\texttt{No} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td style="border-bottom:2px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr>
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| − |
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| − | <tr>
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| − | <td> </td>
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| − | <td> </td>
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| − | <td><math>\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td> </td>
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| − | <td><math>\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td><math>\ell_{00}</math></td></tr>
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| − |
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| − | <tr>
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| − | <td> </td>
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| − | <td> </td>
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| − | <td><math>\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ v</math></td>
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| − | <td> </td>
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| − | <td><math>\texttt{Some} ~ \texttt{(} u \texttt{)} ~ \texttt{is} ~ v</math></td>
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| − | <td><math>\ell_{01}</math></td></tr>
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| − |
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| − | <tr>
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| − | <td><math>\texttt{O}</math><br><math>\texttt{Obtrusive}</math></td>
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| − | <td><math>\texttt{Particular}</math><br><math>\texttt{Negative}</math></td>
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| − | <td><math>\texttt{Some} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td> </td>
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| − | <td><math>\texttt{Some} ~ u ~ \texttt{is} ~ \texttt{(} v \texttt{)}</math></td>
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| − | <td><math>\ell_{10}</math></td></tr>
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| − |
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| − | <tr>
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| − | <td><math>\texttt{I}</math><br><math>\texttt{Indefinite}</math></td>
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| − | <td><math>\texttt{Particular}</math><br><math>\texttt{Affirmative}</math></td>
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| − | <td><math>\texttt{Some} ~ u ~ \texttt{is} ~ v</math></td>
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| − | <td> </td>
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| − | <td><math>\texttt{Some} ~ u ~ \texttt{is} ~ v</math></td>
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| − | <td><math>\ell_{11}</math></td></tr>
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| − |
| |
| − | </table></font>
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| − |
| |
| − | </pre>
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