diff --git a/Bookshelf/Enderton/Logic.tex b/Bookshelf/Enderton/Logic.tex index 1132037..acc7ef6 100644 --- a/Bookshelf/Enderton/Logic.tex +++ b/Bookshelf/Enderton/Logic.tex @@ -458,6 +458,32 @@ TODO \end{proof} +\section{A Parsing Algorithm}% +\hyperlabel{sec:parsing-algorithmm} + +\subsection{\unverified{Lemma 13A}}% +\hyperlabel{sub:lemma-13a} + + \begin{lemma}[13A] + Every wff has the same number of left as right parentheses. + \end{lemma} + + \begin{proof} + Refer to \nameref{sub:balanced-parentheses}. + \end{proof} + +\subsection{\sorry{Lemma 13B}}% +\hyperlabel{sub:lemma-13b} + + \begin{lemma}[13B] + Any proper initial segment of a wff contains an excess of left parentheses. + Thus no proper initial segment of a wff can itself be a wff. + \end{lemma} + + \begin{proof} + TODO + \end{proof} + \section{Exercises 1}% \hyperlabel{sec:exercises-1} @@ -1930,4 +1956,91 @@ \end{proof} +\subsection{\sorry{Exercise 1.3.1}}% +\hyperlabel{sub:exercise-1.3.1} + + Rewrite the tautologies in the "selected list" at the end of Section 1.2, but + using the conventions of the present section to minimize the number of + parentheses. + + \begin{answer} + TODO + \end{answer} + +\subsection{\sorry{Exercise 1.3.2}}% +\hyperlabel{sub:exercise-1.3.2} + + Give an example of wffs $\alpha$ and $\beta$ and expressions $\gamma$ and + $\delta$ such that $(\alpha \land \beta) = (\gamma \land \delta)$ but + $\alpha \neq \gamma$. + + \begin{answer} + TODO + \end{answer} + +\subsection{\sorry{Exercise 1.3.3}}% +\hyperlabel{sub:exercise-1.3.3} + + Carry out the argument for \nameref{sub:lemma-13b} for the case of the + operation $\mathcal{E}_{\neg}$. + + \begin{answer} + TODO + \end{answer} + +\subsection{\sorry{Exercise 1.3.4}}% +\hyperlabel{sub:exercise-1.3.4} + + Suppose that we modify our definition of wff by omitting all \textit{right} + parentheses. + Thus instead of + $$((A \land (\neg B)) \Rightarrow (C \lor D))$$ + we use $$((A \land (\neg B \Rightarrow (C \land D.$$ + Show that we still have unique readability (i.e., each wff still has only one + possible decomposition). + \textit{Suggestion}: These expressions have the same number of parentheses as + connective symbols. + + \begin{answer} + TODO + \end{answer} + +\subsection{\sorry{Exercise 1.3.5}}% +\hyperlabel{sub:exercise-1.3.5} + + The English language has a tendency to use two-part connectives: "both + $\ldots$ and $\ldots$" "either $\ldots$ or $\ldots$" "if $\ldots$, then + $\ldots$." + How does this affect unique readability in English? + + \begin{answer} + TODO + \end{answer} + +\subsection{\sorry{Exercise 1.3.6}}% +\hyperlabel{sub:exercise-1.3.6} + + We have given an algorithm for analyzing a wff by constructing its tree from + the top down. + There are also ways of constructing the tree from the bottom up. + This can be done by looking through the formula for innermost pairs of + parentheses. + Give a complete description of an algorithm of this sort. + + \begin{answer} + TODO + \end{answer} + +\subsection{\sorry{Exercise 1.3.7}}% +\hyperlabel{sub:exercise-1.3.7} + + Suppose that left and right parentheses are indistinguishable. + Thus, instead of $\alpha \lor (\beta \land \gamma))$ we have + $|\alpha \lor |\beta \land \gamma||$. + Do formulas still have unique decomposition? + + \begin{answer} + TODO + \end{answer} + \end{document}