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→Application of Higher Order Propositions to Quantification Theory
With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:
With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:
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|+ <math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math>
|+ <math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math>
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|
<math>\begin{array}{clcl}
<math>\begin{array}{clcl}
\mathrm{A} &
\mathrm{A}
\mathrm{Universal~Affirmative} &
& \mathrm{Universal~Affirmative}
\mathrm{All}\ u\ \mathrm{is}\ v &
& \mathrm{All} ~ u ~ \mathrm{is} ~ v
\mathrm{Indicator~of}\ u (v) = 0 \\
& \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 0
\mathrm{E} &
\\
\mathrm{Universal~Negative} &
\mathrm{E}
\mathrm{All}\ u\ \mathrm{is}\ (v) &
& \mathrm{Universal~Negative}
\mathrm{Indicator~of}\ u \cdot v = 0 \\
& \mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}
\mathrm{I} &
& \mathrm{Indicator~of} ~ u \cdot v = 0
\mathrm{Particular~Affirmative} &
\\
\mathrm{Some}\ u\ \mathrm{is}\ v &
\mathrm{I}
\mathrm{Indicator~of}\ u \cdot v = 1 \\
& \mathrm{Particular~Affirmative}
\mathrm{O} &
& \mathrm{Some} ~ u ~ \mathrm{is} ~ v
\mathrm{Particular~Negative} &
& \mathrm{Indicator~of} ~ u \cdot v = 1
\mathrm{Some}\ u\ \mathrm{is}\ (v) &
\\
\mathrm{Indicator~of}\ u (v) = 1 \\
\mathrm{O}
& \mathrm{Particular~Negative}
& \mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}
& \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 1
\end{array}</math>
\end{array}</math>
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