fix error in goedel.tex reported by Sean Ebels-Duggan; add a couple m…

…ore examples

content/many-valued-logic/infinite-valued-logics/goedel.tex

@@ -103,8 +103,8 @@
 are non-tautologies of~$\LogGod[\infty]$:
 \begin{align*}
   & p \lor \lnot p && (p \lif q) \lif (\lnot p \lor q) \\
-  & \lnot\lnot p \lif p && \lnot(p \land q) \lif (\lnot p \lor \lnot q) \\
-  &&& ((p \lif q) \lif p) \lif p
+  & \lnot\lnot p \lif p && \lnot(\lnot p \land \lnot q) \lif (p \lor q) \\
+  & ((p \lif q) \lif p) \lif p && \lnot(p \lif q) \lif (p \land \lnot q)
 \end{align*}
 The example of an intuitionistically invalid !!{formula} that is
 nevertheless a tautology of~$\LogGod[3]$, $(p \lif q) \lor (q \lif
@@ -119,4 +119,9 @@
   tautology of~$\LogGod[\infty]$.
 \end{prob}
 
+\begin{prob}
+  Show that $(p \lif q) \lor (q \lif r) \lor (r \lif s)$, which is 
+  a tautology of $\LogGod[3]$, is not a tautology of~$\LogGod[\infty]$.
+\end{prob}
+
 \end{document}

content/many-valued-logic/three-valued-logics/goedel.tex

@@ -75,24 +75,32 @@
 in~$\LogGod[3]$. For instance, the following are not tautologies:
 \begin{align*}
   & p \lor \lnot p && (p \lif q) \lif (\lnot p \lor q) \\
-  & \lnot\lnot p \lif p && \lnot(p \land q) \lif (\lnot p \lor \lnot q) \\
-  &&& ((p \lif q) \lif p) \lif p
+  & \lnot\lnot p \lif p && \lnot(\lnot p \land \lnot q) \lif (p \lor q) \\
+  & ((p \lif q) \lif p) \lif p && \lnot(p \lif q) \lif (p \land \lnot q)
 \end{align*}
-However, not every tautology of $\LogGod[3]$ is also intuitionistically
-valid, e.g., $(p \lif q) \lor (q \lif p)$.
+However, not every tautology of $\LogGod[3]$ is also
+intuitionistically valid, e.g., $\lnot\lnot p \lor \lnot p$ or $(p
+\lif q) \lor (q \lif p)$.
 
 \begin{prob}
-  Give a truth table to show that $(p \lif q) \lor (q \lif p)$ is a
-  tautology of~$\LogGod[3]$.
+  Give truth tables to show that the following are tautologies
+  of~$\LogGod[3]$:
+  \begin{align*}
+    & \lnot\lnot p \lor \lnot p\\
+    & (p \lif q) \lor (q \lif p) \\
+    & \lnot(p \land q) \lif (\lnot p \lor \lnot q) \\
+    & (p \lif q) \lor (q \lif r) \lor (r \lif s)
+  \end{align*}
 \end{prob}
 
 \begin{prob}
   Give truth tables that show that the following are not tautologies
   of~$\LogGod[3]$
   \begin{align*}
     & (p \lif q) \lif (\lnot p \lor q) \\
-    & \lnot(p \land q) \lif (\lnot p \lor \lnot q) \\
-    & ((p \lif q) \lif p) \lif p
+    & \lnot(\lnot p \land \lnot q) \lif (p \lor q) \\
+    & ((p \lif q) \lif p) \lif p \\
+    & \lnot(p \lif q) \lif (p \land \lnot q)
   \end{align*}
 \end{prob}