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\begin{center}\begin{figure}\begin{tabular}{|l|l|l|}
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Premise & Conclusion & Name \\
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$G \in \mathcal{F}$ & $\mathcal{F} \vdash G$ & Assumption \\
$\mathcal{F} \vdash G$ and $\mathcal{F} \subset \mathcal{F}'$ & $\mathcal{F}' \subset G$ & Monotonicity \\
$\mathcal{F} \vdash G$ & $\mathcal{F} \vdash \neg \neg G$ & Double negation \\
$\mathcal{F} \vdash F$, $\mathcal{F} \vdash G$ & $\mathcal{F} \vdash (F \wedge G)$ & $\wedge$-introduction \\
$\mathcal{F} \vdash (F \wedge G)$ & $\mathcal{F} \vdash F$ & $\wedge$-elimination \\
$\mathcal{F} \vdash (F \wedge G)$ & $\mathcal{F} \vdash (G \wedge F)$ & $\wedge$-symmetry \\
$\mathcal{F} \vdash F$ & $\mathcal{F} \vdash (F \vee G)$ & $\vee$-introduction \\
$\mathcal{F} \vdash (F \vee G),$ && \\
$\mathcal{F} \cup \{F\} \vdash H, \mathcal{F} \cup \{G\} \vdash H$ & $\mathcal{F} \vdash H$ & $\vee$-elimination \\
$\mathcal{F} \vdash (F \vee G)$ & $\mathcal{F} \vdash (G \vee F)$ & $\vee$-symmetry \\
$\mathcal{F} \cup \{F\} \vdash G$ & $\mathcal{F} \vdash F \rightarrow G$ & $\rightarrow$-introduction \\
$\mathcal{F} \vdash (F \rightarrow G), \mathcal{F} \vdash F$ & $\mathcal{F} \vdash G$ & $\rightarrow$-elimination \\
$\mathcal{F} \vdash F$ & $\mathcal{F} \vdash (F)$ & $(,)$-introduction \\
$\mathcal{F} \vdash (F)$ & $\mathcal{F} \vdash F$ & $(,)$-elimination \\
$\mathcal{F} \vdash ((F \wedge G) \wedge H)$ & $\mathcal{F} \vdash (F \wedge G \wedge H)$ & $\wedge$-parentheses rule \\
$\mathcal{F} \vdash ((F \vee G) \vee H)$ & $\mathcal{F} \vdash (F \vee G \vee H)$ & $\vee$-parentheses rule \\ 
\hline
\end{tabular}
\caption{Table 1.5 -- proof system for propositional logic} \end{figure}\end{center}

\begin{center}\begin{figure}\begin{tabular}{|l|l|l|}
\hline
Rule & Name \\
\hline 
$\mathcal{F} \vdash (F \vee G)$ if and only if $\mathcal{F} \vdash \neg (\neg F \wedge \neg G)$ & $\vee$-definition \\
$\mathcal{F} \vdash (F \rightarrow G)$ if and only if $\mathcal{F} \vdash (\neg F \vee G)$ & $\rightarrow$-definition \\
$\mathcal{F} \vdash (F \leftrightarrow G)$ if and only if both $\mathcal{F} \vdash (F \rightarrow G)$ and $\mathcal{F} \vdash (G \rightarrow F)$ & $\leftrightarrow$-definition \\
\hline
\end{tabular}
\caption{Table 1.6  -- proof system for propositional logic}
\end{figure}\end{center} 

\begin{center}\begin{figure}\begin{tabular}{|l|l|l|l|}
\hline
Premise & Conclusion & Name & Source \\
\hline
$\mathcal{F} \vdash (\neg F \vee G), \mathcal{F} \vdash F$ & $\mathcal{F} \vdash G$ & $\vee$-modus ponens & pg. 16 in text \\
$\mathcal{F} \vdash (F \vee G), \mathcal{F} \vdash \neg F$ & $\mathcal{F} \vdash G$ & 2nd $\vee$-modus ponens & 20 Feb 2018 class \\
none & $\mathcal{F} \vdash (\neg G \vee G)$ & Tautology rule & Example 1.32 \\
$\mathcal{F} \vdash (F \wedge \neg F)$ & $\mathcal{F} \vdash G$ & Contradiction rule & Example 1.33 \\
$\mathcal{F} \cup \{F\} \vdash G$ & $\mathcal{F} \cup \{\neg G\} \vdash \neg F$ & Contrapositive & Example 1.34 \\
$\mathcal{F} \cup \{F\} \vdash G, \mathcal{F} \cup \{\neg F\} \vdash G$ & $\mathcal{F} \vdash G$ & Proof by cases & Example 1.35 \\
$\mathcal{F} \cup \{F\} \vdash G, \mathcal{F} \cup \{F\} \vdash \neg G$ & $\mathcal{F} \vdash \neg F$ & Proof by contradiction & Example 1.36 \\
$\mathcal{F} \vdash \neg (F \vee G)$ & $\mathcal{F} \vdash \neg F \wedge \neg G$ & deMorgan & Proposition 1.44 \\
$\mathcal{F} \vdash \neg F \wedge \neg G$ & $\mathcal{F} \vdash \neg (F \vee G)$ & deMorgan & Proposition 1.44 \\
$\mathcal{F} \vdash \neg (F \wedge G)$ & $\mathcal{F} \vdash \neg F \vee \neg G$ & deMorgan & Proposition 1.44 \\
$\mathcal{F} \vdash \neg F \vee \neg G$ & $\mathcal{F} \vdash \neg (F \wedge G)$ & deMorgan & Proposition 1.44 \\
$\mathcal{F} \vdash F \wedge (G \vee H)$ & $\mathcal{F} \vdash ((F \wedge G) \vee (F \wedge H))$ & $\wedge$-distributivity & Proposition 1.45 \\
$\mathcal{F} \vdash ((F \wedge G) \vee (F \wedge H))$ & $\mathcal{F} \vdash F \wedge (G \vee H)$ & $\wedge$-distributivity & Proposition 1.45 \\
$\mathcal{F} \vdash (F \vee (G \wedge H))$ & $\mathcal{F} \vdash ((F \vee G) \wedge (F \vee H))$ & $\vee$-distributivity & Proposition 1.46 \\
$\mathcal{F} \vdash ((F \vee G) \wedge (F \vee H))$ & $\mathcal{F} \vdash (F \vee (G \wedge H))$ & $\vee$-distributivity & Proposition 1.46 \\
\hline
\end{tabular}
\caption{Further rules derived for propositional logic}\end{figure}\end{center}
\end{document}