I understand that to prove distributive law in elementary set theory we rely on the property of the logical $\land$ and $\lor$ in propositional logic (proof of distributive law in elementary set theory).
I also understand this property of the logical connectors can be proven in propositional logic using truth table but then my question is do we need to prove the truth table?
If we do not need to prove the truth table then can we say truth table is an axiom of propositional logic?
Finally, if a proof for truth table is indeed needed and relies on the elementary set theory (I suspect it does) doesn't this make the proof of distributive law in elementary set theory a circular logic? This is because the proof would be as follow:
distributive law in elementary set theory —> depends on the distributive property of logical connectors —> depends on the truth table —> depends on the elementary set theory.
Best Answer
The first thing I want you to keep in mind is that there are really two somewhat distincts uses of a truth-table.
First, truth-tables are a way to display the semantics of truth-functional operators. For example, we define the material conditional $P \to Q$ to be a certain truth-function: a function that tables in truth-values, and that outputs a truth-value. We could describe that function as follow:
$\to$ is a function that takes in 2 arguments, and each argument is a truth-value: something that is either True or False. $\to$ will output one if those two truth-values as well. Specifically, $P \to Q$ is False if $P$ is True and $Q$ is False, and it is True otherwise.
Now, that is a perfectly good mathematical definition. But we typically display ('specify' if you want) the function by this truth-table:
\begin{array}{cc|c} P&Q&P \to Q\\ \hline T&T&T\\ T&F&F\\ F&T&T\\ F&F&T\\ \end{array}
So this is just a little more organized way of showing how the $\to$ works as a truth-function.
Now, you ask: do we prove that the table for the $\to$ looks like this? No, we don't. We simply stipulate it. We define the $\to$ to work this way. Of course, the intent is for $\to$ to try and capture certain aspects of English 'if ... then ...' statements, and we can debate how good of a good it does in doing so. However, as far as pure logic is concerned: it's just something we define ... and we'll leave it up to you to see if there is some kind of useful application for it. But no, we don't prove our definitions.
So, is it an axiom, you then ask? Well, I don;t know if 'axiom' is quite the right word here, as within the context of logic we typically view axioms as statements that express what is considered some elementary logical or mathematical truth. Again, it is more of a definition, or maybe an assumption: it is part of laying out a system of truth-functional logic.
A second use of truth-table is this: once we have defined our basic operators like $\neg$, $\to$, $\land$, and $\lor$, we can also use truth-tables to investigate the truth-functional properties and relationships of more complex statements. We can, for example, 'work out' the truth-conditions of the statement $P \land (Q \lor R)$, and compare those to the truth-conditions of $(P \land Q) \lor (P \land R)$. And, lo and behold, we find that these two expressions have the exact same truth-conditions, and thus we say that the two statements are logically equivalent.
Again, though, we don't prove this table either. You could say that the table proves the distributive property, and the table itself is not proven. Rather, the table shows what happens to the truth-conditions of these two sentences once we accept the operators that are involved to work a certain way, and as explained in the first half of the post, we already assumed that they work a certain way, like it or not.