149 lines
4.3 KiB
TeX
149 lines
4.3 KiB
TeX
% File: quiz9.tex
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% Created: 14:56:39 Sun, 14 Nov 2021 EST
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% Last Change: 14:56:39 Sun, 14 Nov 2021 EST
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%
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\documentclass[letterpaper]{article}
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\usepackage{amsmath}
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\usepackage{graphicx}
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\usepackage{cancel}
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\usepackage{amssymb}
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\usepackage{listings}
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\usepackage[shortlabels]{enumitem}
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\usepackage{lipsum}
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\usepackage{soul}
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\usepackage{geometry}
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\usepackage{lplfitch}
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\geometry{portrait, margin=1in}
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\date{11/14/2021}
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\title{%
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Quiz \# 9\\
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\large PHIL--205--01:Symbolic Logic}
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\author{Blizzard MacDougall}
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\begin{document}
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\maketitle
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\pagenumbering{arabic}
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\section{Section 1}
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Annotate the following proof :\\
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\fitchprf{}
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{
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\subproof
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{
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\pline[1.]{\lnot\forall x(Fx\lif Gx)}
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}
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{
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\pline[2.]{\exists x\lnot(Fx\lif Gx)}[\textbf{CQ:} 1]
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\subproof
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{
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\pline[3.]{\lnot(Fa\lif Ga)}
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}
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{
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\subproof
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{
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\pline[4.]{\lnot\exists x(Fx\land\lnot Gx)}
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}
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{
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\pline[5.]{\forall x\lnot(Fx\land \lnot Gx)}[\textbf{CQ:} 4]\\
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\pline[6.]{\lnot(Fa\land\lnot Ga)}[\forall\textbf{Elim: }5]\\
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\pline[7.]{\lnot Fa\lor \lnot\lnot Ga}[\textbf{DeM: }6]\\
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\subproof
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{
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\pline[8.]{Fa}
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}
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{
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\pline[9.]{\lnot\lnot Ga}[\textbf{DS: }7, 8]\\
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\pline[10.]{Ga}[\textbf{DNE: }9]
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}
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\pline[11.]{Fa\lif Ga}[\lifi{8--10}]\\
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\pline[12.]{\lfalse}[\lfalsei{3}{11}]
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}
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\pline[13.]{\lnot\lnot\exists x(Fx\land\lnot Gx)}[\lnoti{4--12}]\\
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\pline[14.]{\exists x(Fx\land\lnot Gx)}[\textbf{DNE: }13]
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}
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\pline[15.]{\exists x(Fx\land\lnot Gx)}[\exists\textbf{Elim: }2, 3-14]
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}
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\subproof
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{
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\pline[16.]{\exists x(Fx\land\lnot Gx)}
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}
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{
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\subproof
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{
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\pline[17.]{Fa\land\lnot Ga}
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}
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{
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\subproof
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{
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\pline[18.]{Fa\lif Ga}
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}
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{
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\pline[19.]{Fa}[\lande{17}]\\
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\pline[20.]{Ga}[\life{18}{19}]\\
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\pline[21.]{\lnot Ga}[\lande{17}]\\
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\pline[22.]{\lfalse}[\lfalsei{20}{21}]
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}
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\pline[23.]{\lnot(Fa\lif Ga)}[\lnoti{18-22}]\\
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\pline[24.]{\exists x\lnot(Fx\lif Gx)}[\exists\textbf{Intro: }23]
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}
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\pline[25.]{\exists x\lnot(Fx\lif Gx)}[\exists\textbf{Elim: }16, 17-24]\\
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\pline[26.]{\lnot\forall x(Fx\lif Gx)}[\textbf{CQ: }25]
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}
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\pline[27.]{\lnot\forall x(Fx\lif Gx)\liff \exists x(Fx\land\lnot Gx)}[\liffi{1--15}{16--26}]
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}
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\newpage
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\section{Section 2}
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Construct a formal proof of one (or both) of the following argument(s).
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\begin{enumerate}[A.]
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\item $[\forall x(Fx\lor Gx);\ \neg\forall xGx]\vdash\exists xFx$\\
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\fitchprf
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{
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\pline[1.]{\forall x(Fx\lor Gx)}\\
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\pline[2.]{\lnot\forall xGx}
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}
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{
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\pline[3.]{\exists x\lnot Gx}[\textbf{CQ:} 2]\\
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\subproof
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{
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\pline[4.]{\lnot Ga}
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}
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{
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\pline[5.]{Fa\lor Ga}[\forall\textbf{Elim: }1]\\
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\pline[6.]{Fa}[\textbf{DS: }5, 4]\\
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\pline[7.]{\exists xFx}[\exists\textbf{Intro: }6]
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}
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\pline[8.]{\exists xFx}[\exists\textbf{Elim: }3,4-7]
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}
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\item $\vdash [\forall x(Fx\rightarrow Gx)\land\exists x(Fx\land Hx)]\rightarrow\exists x(Gx\rightarrow Hx)$\\
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\fitchprf{}
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{
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\subproof
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{
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\pline[1.]{\forall x(Fx\lif Gx)\land\exists x(Fx\land Hx)}
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}
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{
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\pline[2.]{\exists x(Fx\land Hx)}[\lande{1}]\\
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\subproof
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{
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\pline[3.]{Fa\land Ha}
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}
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{
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\subproof
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{
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\pline[4.]{Ga}
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}
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{
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\pline[5.]{Ha}[\lande{3}]
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}
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\pline[6.]{Ga\lif Ha}[\lifi{4--5}]\\
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\pline[7.]{\exists x(Gx\lif Hx)}[\exists\textbf{Intro: }6]
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}
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\pline[8.]{\exists x(Gx\lif Hx)}[\exists\textbf{Elim: }2,3-7]
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}
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\pline[9.]{[\forall x(Fx\lif Gx)\land\exists x(Fx\land Hx)]\lif\exists x(Gx\lif Hx)}[\lifi{1--8}]
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}
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\end{enumerate}
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\end{document}
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