\documentclass[landscape]{article} \usepackage[a4paper,margin=0.5in,landscape]{geometry} \newcommand{\Tarrowintro}{$\rightarrow$I} \newcommand{\Tarrowelim}{$\rightarrow$E} \newcommand{\Tconjintro}{$\land$I} \newcommand{\Tconjelim}{$\land$E} \newcommand{\Tdisjintro}{$\lor$I} \newcommand{\Tdisjelim}{$\lor$E} \newcommand{\Tunivintro}{$\forall$I} \newcommand{\Tunivelim}{$\forall$E} \newcommand{\Texistintro}{$\exists$I} \newcommand{\Texistelim}{$\exists$E} \newcommand{\Tarrow}{\rightarrow} \newcommand{\Tand}{\land} \newcommand{\Tor}{\lor} \newcommand{\Tforall}{\forall} \newcommand{\Texists}{\exists} \newcommand{\Tneg}{\lnot} \usepackage{bussproofs} \begin{document} \begin{prooftree} \AxiomC{} \RightLabel{DNE} \UnaryInfC{$\Tneg{\Tneg{\left(A \Tor \Tneg{A}\right)}} \Tarrow \left(A \Tor \Tneg{A}\right)$} \AxiomC{[$\Tneg{\left(A \Tor \Tneg{A}\right)}$]} \AxiomC{[$\Tneg{\left(A \Tor \Tneg{A}\right)}$]} \AxiomC{[$A$]} \RightLabel{\Tdisjintro} \UnaryInfC{$A \Tor \Tneg{A}$} \RightLabel{\Tarrowelim} \BinaryInfC{$\bot$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tneg{A}$} \RightLabel{\Tdisjintro} \UnaryInfC{$A \Tor \Tneg{A}$} \RightLabel{\Tarrowelim} \BinaryInfC{$\bot$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tneg{\Tneg{\left(A \Tor \Tneg{A}\right)}}$} \RightLabel{\Tarrowelim} \BinaryInfC{$A \Tor \Tneg{A}$} \end{prooftree} \vspace{1cm} \begin{prooftree} \AxiomC{} \RightLabel{DP} \UnaryInfC{$\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$} \AxiomC{[$\Tneg{\Tforall_{x} P{x}}$]} \AxiomC{[$P{x} \Tarrow \Tforall_{x} P{x}$]} \AxiomC{[$P{x}$]} \RightLabel{\Tarrowelim} \BinaryInfC{$\Tforall_{x} P{x}$} \RightLabel{\Tarrowelim} \BinaryInfC{$\bot$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tneg{P{x}}$} \RightLabel{\Texistintro} \UnaryInfC{$\Texists_{x} \Tneg{P{x}}$} \RightLabel{\Texistelim} \BinaryInfC{$\Texists_{x} \Tneg{P{x}}$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tneg{\Tforall_{x} P{x}} \Tarrow \Texists_{x} \Tneg{P{x}}$} \end{prooftree} \vspace{1cm} \begin{prooftree} \AxiomC{} \RightLabel{DP} \UnaryInfC{$\Texists_{x} \left(\left(P{x} \Tarrow Q{x}\right) \Tarrow \Tforall_{x} \left(P{x} \Tarrow Q{x}\right)\right)$} \AxiomC{[$\Tneg{\Tforall_{x} \left(P{x} \Tarrow Q{x}\right)}$]} \AxiomC{[$\left(P{x} \Tarrow Q{x}\right) \Tarrow \Tforall_{x} \left(P{x} \Tarrow Q{x}\right)$]} \AxiomC{[$P{x} \Tarrow Q{x}$]} \RightLabel{\Tarrowelim} \BinaryInfC{$\Tforall_{x} \left(P{x} \Tarrow Q{x}\right)$} \RightLabel{\Tarrowelim} \BinaryInfC{$\bot$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tneg{\left(P{x} \Tarrow Q{x}\right)}$} \RightLabel{\Texistintro} \UnaryInfC{$\Texists_{x} \Tneg{\left(P{x} \Tarrow Q{x}\right)}$} \RightLabel{\Texistelim} \BinaryInfC{$\Texists_{x} \Tneg{\left(P{x} \Tarrow Q{x}\right)}$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tneg{\Tforall_{x} \left(P{x} \Tarrow Q{x}\right)} \Tarrow \Texists_{x} \Tneg{\left(P{x} \Tarrow Q{x}\right)}$} \end{prooftree} \vspace{1cm} \begin{prooftree} \AxiomC{} \RightLabel{DP} \UnaryInfC{$\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$} \AxiomC{[$\Tforall_{x} \left(P{x} \Tor Q{x}\right)$]} \RightLabel{\Tunivelim} \UnaryInfC{$P{x} \Tor Q{x}$} \AxiomC{[$P{x} \Tarrow \Tforall_{x} P{x}$]} \AxiomC{[$P{x}$]} \RightLabel{\Tarrowelim} \BinaryInfC{$\Tforall_{x} P{x}$} \RightLabel{\Tdisjintro} \UnaryInfC{$\Tforall_{x} P{x} \Tor \Texists_{x} Q{x}$} \AxiomC{[$Q{x}$]} \RightLabel{\Texistintro} \UnaryInfC{$\Texists_{x} Q{x}$} \RightLabel{\Tdisjintro} \UnaryInfC{$\Tforall_{x} P{x} \Tor \Texists_{x} Q{x}$} \RightLabel{\Tdisjelim} \TrinaryInfC{$\Tforall_{x} P{x} \Tor \Texists_{x} Q{x}$} \RightLabel{\Texistelim} \BinaryInfC{$\Tforall_{x} P{x} \Tor \Texists_{x} Q{x}$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tforall_{x} \left(P{x} \Tor Q{x}\right) \Tarrow \left(\Tforall_{x} P{x} \Tor \Texists_{x} Q{x}\right)$} \end{prooftree} \clearpage \begin{prooftree} \AxiomC{} \RightLabel{GLPO} \UnaryInfC{$\Tforall_{x} \Tneg{A} \Tor \Texists_{x} A$} \AxiomC{[$\Tforall_{x} \Tneg{A}$]} \RightLabel{\Tunivelim} \UnaryInfC{$\Tneg{A}$} \RightLabel{\Tdisjintro} \UnaryInfC{$A \Tor \Tneg{A}$} \AxiomC{[$\Texists_{x} A$]} \AxiomC{[$A$]} \RightLabel{\Texistelim} \BinaryInfC{$A$} \RightLabel{\Tdisjintro} \UnaryInfC{$A \Tor \Tneg{A}$} \RightLabel{\Tdisjelim} \TrinaryInfC{$A \Tor \Tneg{A}$} \end{prooftree} \vspace{1cm} \begin{prooftree} \AxiomC{} \RightLabel{GLPO} \UnaryInfC{$\Tforall_{x} \Tneg{P{y}} \Tor \Texists_{x} P{y}$} \AxiomC{[$\Tforall_{x} \Tneg{P{y}}$]} \RightLabel{\Tunivelim} \UnaryInfC{$\Tneg{P{y}}$} \RightLabel{\Tdisjintro} \UnaryInfC{$P{y} \Tor \Tneg{P{y}}$} \AxiomC{[$\Texists_{x} P{y}$]} \AxiomC{[$P{y}$]} \RightLabel{\Texistelim} \BinaryInfC{$P{y}$} \RightLabel{\Tdisjintro} \UnaryInfC{$P{y} \Tor \Tneg{P{y}}$} \RightLabel{\Tdisjelim} \TrinaryInfC{$P{y} \Tor \Tneg{P{y}}$} \RightLabel{\Tunivintro} \UnaryInfC{$\Tforall_{y} \left(P{y} \Tor \Tneg{P{y}}\right)$} \RightLabel{\Tunivelim} \UnaryInfC{$P{x} \Tor \Tneg{P{x}}$} \end{prooftree} \vspace{1cm} \begin{prooftree} \AxiomC{} \RightLabel{GLPO} \UnaryInfC{$\Tforall_{x} \Tneg{P{y}} \Tor \Texists_{x} P{y}$} \AxiomC{[$\Tforall_{x} \Tneg{P{y}}$]} \RightLabel{\Tunivelim} \UnaryInfC{$\Tneg{P{y}}$} \RightLabel{\Tdisjintro} \UnaryInfC{$P{y} \Tor \Tneg{P{y}}$} \AxiomC{[$\Texists_{x} P{y}$]} \AxiomC{[$P{y}$]} \RightLabel{\Texistelim} \BinaryInfC{$P{y}$} \RightLabel{\Tdisjintro} \UnaryInfC{$P{y} \Tor \Tneg{P{y}}$} \RightLabel{\Tdisjelim} \TrinaryInfC{$P{y} \Tor \Tneg{P{y}}$} \end{prooftree} \vspace{1cm} \begin{prooftree} \AxiomC{} \RightLabel{GLPO} \UnaryInfC{$\Tforall_{x} \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)} \Tor \Texists_{x} \left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)$} \AxiomC{[$\Tforall_{x} \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$]} \RightLabel{\Tunivelim} \UnaryInfC{$\Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$} \RightLabel{\Tdisjintro} \UnaryInfC{$\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right) \Tor \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$} \AxiomC{[$\Texists_{x} \left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)$]} \AxiomC{[$P{y} \Tand \Tneg{\Tforall_{x} P{x}}$]} \RightLabel{\Texistelim} \BinaryInfC{$P{y} \Tand \Tneg{\Tforall_{x} P{x}}$} \RightLabel{\Tdisjintro} \UnaryInfC{$\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right) \Tor \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$} \RightLabel{\Tdisjelim} \TrinaryInfC{$\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right) \Tor \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$} \RightLabel{\Tunivintro} \UnaryInfC{$\Tforall_{y} \left(\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right) \Tor \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}\right)$} \RightLabel{\Tunivelim} \UnaryInfC{$\left(P{x} \Tand \Tneg{\Tforall_{x} P{x}}\right) \Tor \Tneg{\left(P{x} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$} \end{prooftree} \clearpage \begin{prooftree} \AxiomC{} \RightLabel{DNE} \UnaryInfC{$\Tneg{\Tneg{\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)}} \Tarrow \Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$} \AxiomC{[$\Tneg{\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)}$]} \AxiomC{} \RightLabel{DNE} \UnaryInfC{$\Tneg{\Tneg{P{x}}} \Tarrow P{x}$} \AxiomC{[$\Tneg{\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)}$]} \AxiomC{} \RightLabel{DNE} \UnaryInfC{$\Tneg{\Tneg{\Tforall_{x} P{x}}} \Tarrow \Tforall_{x} P{x}$} \AxiomC{[$\Tneg{P{x}}$]} \AxiomC{[$P{x}$]} \RightLabel{\Tarrowelim} \BinaryInfC{$\bot$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tneg{\Tneg{\Tforall_{x} P{x}}}$} \RightLabel{\Tarrowelim} \BinaryInfC{$\Tforall_{x} P{x}$} \RightLabel{\Tarrowintro} \UnaryInfC{$P{x} \Tarrow \Tforall_{x} P{x}$} \RightLabel{\Texistintro} \UnaryInfC{$\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$} \RightLabel{\Tarrowelim} \BinaryInfC{$\bot$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tneg{\Tneg{P{x}}}$} \RightLabel{\Tarrowelim} \BinaryInfC{$P{x}$} \RightLabel{\Tunivintro} \UnaryInfC{$\Tforall_{x} P{x}$} \RightLabel{\Tarrowintro} \UnaryInfC{$P{x} \Tarrow \Tforall_{x} P{x}$} \RightLabel{\Texistintro} \UnaryInfC{$\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$} \RightLabel{\Tarrowelim} \BinaryInfC{$\bot$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tneg{\Tneg{\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)}}$} \RightLabel{\Tarrowelim} \BinaryInfC{$\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$} \end{prooftree} \begin{prooftree} \AxiomC{} \RightLabel{DNE} \UnaryInfC{$\Tneg{\Tneg{\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)}} \Tarrow \Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)$} \AxiomC{[$\Tneg{\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)}$]} \AxiomC{} \RightLabel{DNE} \UnaryInfC{$\Tneg{\Tneg{P{x}}} \Tarrow P{x}$} \AxiomC{[$\Texists_{x} P{x}$]} \AxiomC{[$\Tneg{\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)}$]} \AxiomC{[$P{x}$]} \RightLabel{\Tarrowintro} \UnaryInfC{$\Texists_{x} P{x} \Tarrow P{x}$} \RightLabel{\Texistintro} \UnaryInfC{$\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)$} \RightLabel{\Tarrowelim} \BinaryInfC{$\bot$} \RightLabel{\Texistelim} \BinaryInfC{$\bot$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tneg{\Tneg{P{x}}}$} \RightLabel{\Tarrowelim} \BinaryInfC{$P{x}$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Texists_{x} P{x} \Tarrow P{x}$} \RightLabel{\Texistintro} \UnaryInfC{$\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)$} \RightLabel{\Tarrowelim} \BinaryInfC{$\bot$} \RightLabel{\Tarrowintro} \UnaryInfC{$\Tneg{\Tneg{\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)}}$} \RightLabel{\Tarrowelim} \BinaryInfC{$\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)$} \end{prooftree} \end{document}