I have a pretty simple question regarding basic tautological and contradiction laws. I feel Velleman's book strives to state such them so simply it leaves out an important detail. I'll get to that in a moment with my question.
It is said that $P ∧ (¬P ∨ P)$ (i.e., P and a tautology) is equivalent to $P$.
Likewise, it is said that $P ∨ (¬P ∨ P)$ (i.e., P or a tautology) is equivalent to the tautology itself.
When manipulating logical statements involving the former, the aforementioned set of symbols can be shortened to just $P$. For example: $$P ∧ (¬P ∨ P) ∨ Q ≡ P ∨ Q$$
It is easy to see why this is - if for no other reason, then the absorption laws make it quite clear. In the case of the "P or a tautology" example, the idemptotent laws make it clear why that is true. The contradiction laws work similarly.
So here is my question. What if $P ∧ (¬Q ∨ Q)$ or $P ∨ (¬Q ∨ Q)$? Is there a way to deal with these - to shorten them? The "P and/ or" part is still true, and the "tautology" part is still true... but the tautologies involve a different variable. Velleman's definitions were too broad as to specify what to do here. If anything, they imply the variables do not matter. I have a hard time believing this. Can anybody offer any input?
Yes, you can still apply the same ideas:
For $P \land (\neg Q \lor Q)$: Since $\neg Q \lor Q$ is a tautology, we get '$P$ and a tautology', which is just $P$. So, $P \land (\neg Q \lor Q) \equiv P$
Likewise, $P \lor (\neg Q \lor Q)$ becomes '$P$ or a tautology', which is just a tautology. So: $P \lor (\neg Q \lor Q)$ is a tautology.
It is very useful to have an explicit tautology symbol. Logicians often use $\top$ for this.
So then you have:
$P \land (\neg Q \lor Q) \equiv P \land \top \equiv P$
and
$P \lor (\neg Q \lor Q) \equiv P \lor \top \equiv \top$