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\author{Tom Cuchta}
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pg. 96 \# 1: ``Some of Aristotle's followers like all of Aquinas' followers. None of Aristotle's followers like any idealist. Therefore, none of Aquinas' followers are idealists." \\
\begin{tabular}{|lll|}
\hline
$\{1\}$ & (1) $(\exists x)(Ax \wedge (\forall y)(Qy \rightarrow Lxy))$ & Premise \\
$\{2\}$ & (2) $(\forall x)(Ax \rightarrow (\forall y)(Iy \rightarrow \neg Lxy))$ & Premise \\
$\{1\}$ & (3) $A\alpha \wedge (\forall y)(Qy \rightarrow L\alpha y)$ & 1 EG \\
$\{2\}$ & (4) $A\alpha \rightarrow (\forall y)(Iy \rightarrow \neg L\alpha y)$ & 2 US \\
$\{1\}$ & (5) $A\alpha$ & 3 Law of Simplification \\
$\{2\}$ & (6) $(\forall y)(Iy \rightarrow \neg L\alpha y)$ & 4 5 Law of Detachment \\
$\{1\}$ & (7) $(\forall y)(Qy \rightarrow L\alpha y) \wedge A\alpha$ & 3 Commutative Law for $\wedge$ \\
$\{1\}$ & (8) $(\forall y)(Qy \rightarrow L\alpha y)$ & 7 Law of Simplification \\
$\{1\}$ & (9) $Qy \rightarrow L\alpha y$ & 8 US \\
$\{2\}$ & (10) $Iy \rightarrow \neg L\alpha y$ & 6 US \\
$\{2\}$ & (11) $\neg (\neg L\alpha y) \rightarrow \neg Iy$ & 10 Law of Contraposition \\
$\{2\}$ & (12) $L\alpha y \rightarrow \neg Iy$ & 11 TE (Double Negation) \\
$\{1,2\}$ & (13) $Qy \rightarrow \neg Iy$ & 9 12 Syllogism \\
\hline
\end{tabular}

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pg. 96 \# 3: ``None of the paintings is valuable, except the battle pieces. All the battle pieces are painted in oils. Some of the paintings are not painted in oil. Some paintings are not framed. Therefore, none of the paintings not painted in oils is valuable."\\
\begin{tabular}{|lll|}
\hline
$\{1\}$ & (1) $(\forall x)(Px \rightarrow (\neg Vx \vee Bx))$ & Premise \\
$\{2\}$ & (2) $(\forall x)(Bx \rightarrow Ox)$ & Premise \\
$\{3\}$ & (3) $(\exists x)(Px \wedge \neg Ox)$ & Premise \\
$\{4\}$ & (4) $(\exists x)(Px \wedge \neg Fx)$ & Premise \\
\hline
\end{tabular} \\
Conclusion: $(\forall x)((Px \wedge \neg Ox) \rightarrow \neg Vx)$ \\
This argument is not valid... try to find an interpretation.
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pg. 96 \# 8: ``Every philosophical empiricist admires Hume. Some philosophical idealists like no one who admires Hume. Therefore, some philosophical idealists like no philosophical empiricist." \\
\begin{tabular}{|lll|}
\hline
$\{1\}$ & (1) $(\forall x)(Ex \rightarrow Hx)$ & Premise \\
$\{2\}$ & (2) $(\exists x)(Ix \wedge (\forall y)( Hy \rightarrow \neg Lxy))$ & Premise \\
$\{2\}$ & (3) $I \alpha \wedge (\forall y)(Hy \rightarrow \neg L\alpha y)$ & 2 ES \\
$\{2\}$ & (4) $(\forall y)(Hy \rightarrow \neg L\alpha y) \wedge I \alpha$ & 3 Commutative Law for $\wedge$ \\
$\{2\}$ & (5) $(\forall y)(Hy \rightarrow \neg L\alpha y)$ & 4 Law of Simplification \\
$\{2\}$ & (6) $Hy \rightarrow \neg L\alpha y$ & 5 US \\
$\{1\}$ & (7) $Ey \rightarrow Hy$ & 1 US \\
$\{1,2\}$ & (8) $Ey \rightarrow \neg L\alpha y$ & 6 7 Syllogism \\
$\{1,2\}$ & (9) $(\forall y)(Ey \rightarrow \neg L\alpha y)$ & 8 UG \\
$\{1,2\}$ & (10) $I\alpha$ & 3 Law of Simplification \\
$\{1,2\}$ & (11) $I\alpha \wedge (\forall y)(Ey \rightarrow \neg L\alpha y)$ & 9 10 Law of Adjunction \\
$\{1,2\}$ & (12) $(\exists x)(Ix \wedge (\forall y)(Ey \rightarrow \neg Lxy))$ & 11 EG \\
\hline
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pg. 96 \# 4: ``Some psychologists admire Freud. Some psychologists like no one who admires Freud. Therefore, some psychologists are not liked by all psychologists."
\begin{tabular}{|lll|}
\hline
$\{1\}$ & (1) $(\exists x)(Px \wedge Fx)$ & Premise \\
$\{2\}$ & (2) $(\exists x)(Px \wedge (\forall y)(Fy \rightarrow \neg Lxy))$ & Premise \\
\hline
\end{tabular}\\
Conclusion: $(\exists x)(Px \wedge (\forall y)(Py \wedge \neg Lxy))$ \\
Find an interpretation!
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
$x$ & $y$& $Px$ & $Py$ & $Fx$ & $Fy$ & $Lxy$ & $Px \wedge Fx$ & $Fy \rightarrow \neg Lxy$ & $Py \wedge \neg Lxy$ \\
$0$ & $0$& $T$ & $F$ & $T$ & $T$ & $F$ & $T$ & $T$ & $F$ \\
\hline
\end{tabular}
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