Can $p_2$ be recovered from $p_1$ and $p_1 \land p_2$ using boolean operations?

Question

Consider the propositional logic formulas $p_1$ and $p_1 \land p_2$, where $p_1$ and $p_2$ are propositional atoms. Can $p_2$ be recovered from $p_1$ and $p_1 \land p_2$ using boolean operations? That is, is there a binary Boolean operation $b$ such that $b(p_1,p_1 \land p_2)$ is equivalent to $p_2$?

Answer 1

We might look at what we want our new expressions truth table to be, from which we can easily find a formula using ordinary boolean operations. So we might check if such a truth table exists:

• If $p_1$ is true and $p_1\land p_2$ is true, then $p_2$ must be true.
• If $p_1$ is true and $p_1\land p_2$ is false, then $p_2$ must be false.
• If $p_1$ is false and $p_1\land p_2$ is true, then we have a contradiction, so this cannot occur.
• If $p_1$ is false and $p_1\land p_2$ is false, then $p_1$ could be either true or false.

Oh dear. So we can't always recover $p_2$, because if $p_1$ is false then $p_1\land p_2$ will be false whatever value $p_1$ takes.