and a few more typos

thanks Matt
OpenLogicProject · Apr 25, 2018 · 7553196 · 7553196
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diff --git a/content/counterfactuals/minimal-change-semantics/sphere-models.tex b/content/counterfactuals/minimal-change-semantics/sphere-models.tex
@@ -81,9 +81,13 @@
   $\mSat{M}{!B \cif !C}[w]$ iff either
   \begin{enumerate}
   \item\ollabel{sphere-vac} For all $u \in \bigcup O_w$, $\mSat/{M}{!C}[u]$, or
-  \item\ollabel{sphere-nonvac} For some $S \in O_w$ and some $u \in
-    S$, $\mSat{M}{!B}[u]$ and for all $v \in S$, either
-    $\mSat/{M}{!B}[u]$ or $\mSat{M}{!C}[u]$.
+  \item\ollabel{sphere-nonvac} For some $S \in O_w$,
+    \begin{enumerate}
+      \item $\mSat{M}{!B}[u]$ for some $u \in
+        S$, and
+      \item for all $v \in S$, either
+        $\mSat/{M}{!B}[v]$ or $\mSat{M}{!C}[v]$.
+    \end{enumerate}
   \end{enumerate}
 \end{defn}
 

diff --git a/content/normal-modal-logic/syntax-and-semantics/tautological-instances.tex b/content/normal-modal-logic/syntax-and-semantics/tautological-instances.tex
@@ -36,7 +36,7 @@
   variables 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 $v(p_i) = \True$ if and only if $\mSat{M}{!D_i}[w]$ we have
+  that $\pAssign{v}(p_i) = \True$ if and only if $\mSat{M}{!D_i}[w]$ we have
   that $\pSat{v}{!A}$ if and only if
   $\mSat{M}{\SSubst{!A}{\subst{!D_1}{p_1}, \dots,
       \subst{!D_n}{p_n}}}[w]$.
@@ -51,7 +51,7 @@
       and $\mSat{M}{\ltrue}[w]$}.}{}
   \item \indcase{!A}{p_i}{%
     \begin{align*}
-      \pSat{v}{p_i} \Leftrightarrow {} & v(p_i) = \True &&
+      \pSat{v}{p_i} \Leftrightarrow {} & \pAssign{v}(p_i) = \True &&
       \text{by definition of $\pSat{v}{p_i}$};\\
       \Leftrightarrow {} & \mSat{M}{!D_i}[w] && \text{by assumption}\\
       \Leftrightarrow {} & \mSat{M}{\SSubst{p_i}{\subst{!D_1}{p_1}, \dots,
@@ -61,7 +61,7 @@
       \end{align*}}
   \tagitem{prvFalse}{\indcase{!A}{\lnot !B}{%
       \begin{align*}
-        \pSat{v}{\lnot !B} \Leftrightarrow & \pSat/{v}{!B}
+        \pSat{v}{\lnot !B} \Leftrightarrow {} & \pSat/{v}{!B}
         && \text{by definition of $\pSat{v}{}$};\\
         \Leftrightarrow {} & \mSat/{M}{\SSubst{!B}{\subst{!D_1}{p_1}, \dots,
           \subst{!D_n}{p_n}}}[w] && \text{by induction hypothesis};\\