Is the following true in propositional logic? If so, is it a law/theorem and does it have a name? How could I make the statement more precise? If it's not true, can I change the conditions so that I can get a meaningful result?
If $P\Leftrightarrow Q$ is true for all "singletons" $x_1,x_2,x_3,...$, where:
- $P,Q$ are propositions about $x_1,x_2,x_3,...$,
- The only logical connectives used in $P,Q$ are $\wedge$(and) and $\vee$(or).
For example $P$ may be something like $x_1\wedge x_2\vee x_3$.
Then it seems that $P'\Leftrightarrow Q'$ is also true, where $P',Q'$ are obtained by interchanging every $\wedge $ and $\vee$ in $P,Q$ respectively, for example, if $P$ is $x_1 \wedge x_2 \vee x_3$ then $P'$ is $x_1 \vee x_2 \wedge x_3$. Here is my reasoning why $P'\Leftrightarrow Q'$ should be true.
Since $P\Leftrightarrow Q$ is true for all $x_1,x_2,...$, I can replace all the $x_1,x_2,...$ by $\neg x_1,\neg x_2,...$ to get statements $P_1,Q_1$ and I know $P_1 \Leftrightarrow Q_1$ is true, then by De Morgan's laws, $P_1$ is equivalent to $\neg P'$ and $Q_1$ is equivalent to $\neg Q'$, so $\neg P' \Leftrightarrow \neg Q'$ is true and so $P'\Leftrightarrow Q'$ is true.