Add missing tokens for "propositional variable"

content/normal-modal-logic/completeness/truth-lemma.tex

@@ -11,8 +11,8 @@
 \olsection{The Truth Lemma}
 
 The canonical model~$\mModel{M^\Sigma}$ is defined in such a way that
-$\mSat{M^\Sigma}{!A}[\Delta]$ iff $!A \in \Delta$. For propositional
-variables, the definition of $V^\Sigma$ yields this directly. We have
+$\mSat{M^\Sigma}{!A}[\Delta]$ iff $!A \in \Delta$. For !!{propositional
+variable}s, the definition of $V^\Sigma$ yields this directly. We have
 to verify that the equivalence holds for all !!{formula}s, however. We
 do this by induction. The inductive step involves proving the
 equivalence for !!{formula}s involving propositional operators (where

content/normal-modal-logic/filtrations/S5-decidable.tex

@@ -19,7 +19,7 @@
 \end{thm}
 
 \begin{proof}
-  Let $!A$ be given, and suppose the propositional variables
+  Let $!A$ be given, and suppose the !!{propositional variable}s
   occurring in $!A$ are among $p_1$, \dots, $p_k$.  Since for each
   $n$ there are only finitely many models with $n$ worlds assigning a
   value to $p_1$, \dots, $p_k$, we can enumerate, \emph{in parallel}, all

content/normal-modal-logic/syntax-and-semantics/language-modal-logic.tex

@@ -36,7 +36,7 @@
 
 \tagitem{prvTrue}{$\ltrue$ is an atomic !!{formula}.}{}
 
-\item Every propositional variable $\Obj p_i$ is an (atomic) !!{formula}.
+\item Every !!{propositional variable} $\Obj p_i$ is an (atomic) !!{formula}.
 
 \tagitem{prvNot}{If $!A$ is !!a{formula}, then $\lnot !A$ is
   !!a{formula}.}{}

content/normal-modal-logic/syntax-and-semantics/schemas.tex

@@ -18,8 +18,8 @@
   \SSubst{!C}{\subst{!D_1}{p_1}, \dots, \subst{!D_n}{p_n}} \right) }.
   \]
   The !!{formula}~$!C$ is called the \emph{characteristic} !!{formula} of
-  the schema, and it is unique up to a renaming of the propositional
-  variables. !!^a{formula}~$!A$ is an \emph{instance} of a schema if
+  the schema, and it is unique up to a renaming of the !!{propositional
+  variable}s. !!^a{formula}~$!A$ is an \emph{instance} of a schema if
   it is a member of the set.
 \end{defn}
 

content/normal-modal-logic/syntax-and-semantics/tautological-instances.tex

@@ -32,8 +32,8 @@
 \end{defn}
 
 \begin{lem}\ollabel{lem:valid-taut}
-  Suppose $!A$ is a modal-free !!{formula} whose propositional
-  variables are $p_1$, \dots, $p_n$, and let $!D_1$, \dots,
+  Suppose $!A$ is a modal-free !!{formula} whose !!{propositional
+  variable}s are $p_1$, \dots, $p_n$, and let $!D_1$, \dots,
   $!D_n$ be modal !!{formula}s. Then for any assignment $\pAssign{v}$,
   any model $\mModel{M} = \tuple{W, R, V}$, and any $w \in W$ such
   that $\pAssign{v}(p_i) = \True$ if and only if $\mSat{M}{!D_i}[w]$ we have

content/normal-modal-logic/syntax-and-semantics/truth-in-model.tex

@@ -1,5 +1,3 @@
-% Part: normal-modal-logic
-% Chapter: syntax-and-semantics
 % Section: truth-in-model
 
 \documentclass[../../../include/open-logic-section]{subfiles}
@@ -56,7 +54,7 @@
 
 \begin{prob}
   Consider the following model $\mModel{M}$ for the language
-  comprising $p_1$, $p_2$, $p_3$ as the only propositional variables:
+  comprising $p_1$, $p_2$, $p_3$ as the only !!{propositional variable}s:
   \begin{center}
     \begin{tikzpicture}[modal]
       \node[world] (w1) [label={[align=right]left:\mTrue{p_1}\\\mFalse{p_2}\\\mFalse{p_3}}]

content/propositional-logic/syntax-and-semantics/preliminaries.tex

@@ -89,8 +89,8 @@
 
 
 \begin{defn}[Uniform Substitution]
-If $!A$ and $!B$ are !!{formula}s, and $\Obj p_i$ is a propositional
-!!{variable}, then $\Subst{!A}{!B}{\Obj p_i}$ denotes the result of
+If $!A$ and $!B$ are !!{formula}s, and $\Obj p_i$ is a !!{propositional
+variable}, then $\Subst{!A}{!B}{\Obj p_i}$ denotes the result of
 replacing each occurrence of $\Obj p_i$ by an occurrence of $!B$ in $!A$;
 similarly, the simultaneous substitution of $\Obj p_1$, \dots,~$\Obj p_n$ by
 !!{formula}s $!B_1$, \dots,~$!B_n$ is denoted by

content/propositional-logic/syntax-and-semantics/valuations-sat.tex

@@ -132,8 +132,8 @@
 
 \begin{thm}[Local Determination]
 \ollabel{thm:LocalDetermination} Suppose that $\pAssign{v_1}$ and
-$\pAssign{v_2}$ are !!{valuation}s that agree on the propositional
-letters occurring in $!A$, i.e., $\pAssign{v_1}(\Obj p_n) =
+$\pAssign{v_2}$ are !!{valuation}s that agree on the !!{propositional
+variable}s occurring in $!A$, i.e., $\pAssign{v_1}(\Obj p_n) =
 \pAssign{v_2}(\Obj p_n)$ whenever $\Obj p_n$ occurs in some
 !!{formula}~$!A$. Then $\pValue{v_1}$ and $\pValue{v_2}$ also agree
 on~$!A$, i.e., $\pValue{v_1}(!A) = \pValue{v_2}(!A)$.