\documentclass{article}
\usepackage[margin=1.25in]{geometry}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{complexity}
\usepackage{graphicx}
\usepackage{bussproofs}
\usepackage{bm}

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{defn}[theorem]{Definition}

\def\limplies{\rightarrow}
\def\liff{\leftrightarrow}
\def\land{\wedge}
\def\lor{\vee}
\newcommand\bigcdot{\bm{\cdot}}
\def\sequent{\Rightarrow}
\def\fCenter{\sequent}
\def\inftype#1#2{{\mbox{\it #1:#2}}}

\def\lectureNumber{XX}
\def\scribeName{YOUR NAME}
%% The lecture date is set below

\title{Math 260B: Mathematical Logic, Winter 2020, UC San Diego \\ Lecture \lectureNumber}
\author{Instructor: Sam Buss \\ {Scribe: \scribeName}}

% Date of the lecture:
\date{January 7, 2020}  %% THIS IS THE DATE OF THE LECTURE

\begin{document}
	
\maketitle

Scribe note text goes here.

INFERENCES RULES FOR PROPOSITIONAL LOGIC INCLUDE:
(See documentation for bussproofs.sty.)
The Cut rule is missing below.
% Also available \LeftLabel commands.

\subparagraph*{}
{\bf Weak Structural Rules}
\hfill \\[2ex]
\begin{tabular}{c@{}c@{\hspace*{.25in}}c@{}c}
\inftype{Exchange}{left} &
\Axiom$\Gamma,A,B,\Pi\fCenter\Delta$
\UnaryInf$\Gamma,B,A,\Pi\fCenter\Delta$
\DisplayProof
&
\inftype{Exchange}{right} &
\Axiom$\Gamma\fCenter\Delta,A,B,\Lambda$
\UnaryInf$\Gamma\fCenter\Delta,B,A,\Lambda$
\DisplayProof
\\[3ex]
\inftype{Contraction}{left} &
\Axiom$A,A,\Gamma\fCenter\Delta$
\UnaryInf$A,\Gamma\fCenter\Delta$
\DisplayProof
\glossary{exchange rule}
&
\inftype{Contraction}{right} &
\Axiom$\Gamma\fCenter\Delta,A,A$
\UnaryInf$\Gamma\fCenter\Delta,A$
\DisplayProof
\glossary{contraction rule}
\\[3ex]
\inftype{Weakening}{left} &
\Axiom$\Gamma\fCenter\Delta$
\UnaryInf$A,\Gamma\fCenter\Delta$
\DisplayProof
&
\inftype{Weakening}{right} &
\Axiom$\Gamma\fCenter\Delta$
\UnaryInf$\Gamma\fCenter\Delta,A$
\DisplayProof
\end{tabular}

\vbox{
\subparagraph*{}
{\bf The Propositional Rules}
\hfill \\[2ex]
\hspace*{.05in}\begin{tabular}{r@{}c@{\hspace*{.5in}}r@{}c}
\inftype{$\lnot$}{left} &
\Axiom$\Gamma\fCenter\Delta,A$
\UnaryInf$\lnot A,\Gamma\fCenter\Delta$
\DisplayProof
&
\inftype{$\lnot$}{right} &
\Axiom$A,\Gamma\fCenter\Delta$
\UnaryInf$\Gamma\fCenter\Delta,\lnot A$
\DisplayProof
\\[3ex]
\inftype{$\land$}{left} &
\Axiom$A,B,\Gamma\fCenter\Delta$
\UnaryInf$A\land B,\Gamma\fCenter\Delta$
\DisplayProof
&
\inftype{$\land$}{right} &
\Axiom$\Gamma\fCenter\Delta,A$
\Axiom$\Gamma\fCenter\Delta,B$
\BinaryInf$\Gamma\fCenter\Delta,A\land B$
\DisplayProof
\\[3ex]
\inftype{$\lor$}{left} &
\Axiom$A,\Gamma\fCenter\Delta$
\Axiom$B,\Gamma\fCenter\Delta$
\BinaryInf$A\lor B,\Gamma\fCenter\Delta$
\DisplayProof
&
\inftype{$\lor$}{right} &
\Axiom$\Gamma\fCenter\Delta,A,B$
\UnaryInf$\Gamma\fCenter\Delta,A\lor B$
\DisplayProof
\\[3ex]
\inftype{$\limplies$}{left} &
\Axiom$\Gamma\fCenter\Delta,A$
\Axiom$B,\Gamma\fCenter\Delta$
\BinaryInf$A\limplies B,\Gamma\fCenter\Delta$
\DisplayProof
&
\inftype{$\limplies$}{right} &
\Axiom$A,\Gamma\fCenter\Delta,B$
\UnaryInf$\Gamma\fCenter\Delta,A\limplies B$
\DisplayProof
\end{tabular}
}
	
\end{document}
