I want to know when I have to introduce a new name when constructing a truth tree for predicate logic. For example:
Determine whether this is valid or not using a truth tree.
($\exists$xPx & $\exists$xQx) $\models$ $\exists$x(Px & Qx)
Do I simplify the first part as Pa | Qa OR Pa | Qb?
On
A new name must be used when "unpacking" an existential quantifier and a negated universal quantifier.
From $\exists x Px$ to $Pa$, the parameter $a$ must be "new".
The same for de derivation of $\lnot Pa$ from $\lnot \forall x Pa$.
In your example, if we start with the prermise: $∃xPx \land ∃xQx$, we have to apply the $\land$-rule in order to derive: $∃xPx$ and $∃xQx$.
In the next steps, using the $∃$-rule, we have to introduce two new names to get: $Pa$ and $Qb$.
Well, if you think about this logically:
$\exists x \ P(x) \land \exists x \ Q(x)$ says that there is something that is a $P$, and that there is something that is a $Q$. Can we assume that is the same thing? Of course not! For example, suppose we are talking about numbers, and $P$ is 'Even', and $Q$ is 'Odd'. So, if you use $a$ to denote 'some object that has property $P$', then you will need to use a different constant $b$ to denote 'some object that has property $Q$'.
And by the way, just because we use different constants $a$ and $b$ does not mean that $a$ and $b$ denote different objects; they can be different labels for the same object. That is, it can turn out later that $a = b$. But by using $a$ and $b$ we leave it open as to whether 'the thing with property $P$' is the same 'thing with property $Q$' or not. And that is exactly what we want for the truth-tree method, since in the truth tree method we are trying to systematically explore all the possible ways to satisfy some set of sentences. Whereas if you set $P(a) \land Q(a)$ in order to satisfy $\exists x \ P(x) \land \exists x \ Q(x)$, you will have found a way to satisfy that statement, but you do not explore all possible ways to satisfy that statement.
Long story short: Anytime you drop an existential, you need to introduce a new constant!