determining free and bounded variable occurrences in a WFF

Question

show all variable occurrences that are free and ones that are bounded and indicate the quantifier that binds them

∀a[∃b(P(a,b,c,d)) ∧ ∀c(∃a(R(b,a,c,d)))]

my attempt:

free: P(c), P(d), P(a), R(b), R(d)

bounded: R(a) by ∃a R(c) by ∀a P(b) by ∃b

I was wondering if P(a) should be considered a free variable since ∃a is nested within another Quantifier. Any insights?

Answer 1

Your ideas are correct. I don't like how you wrote it, but you're thinking correctly.

You're writing things such $P(c)$ and $R(a)$, but $P$ and $R$ are $4$-ary (is there a word for this?) predicates, it doesn't make sense to write what you wrote.

You can really just write something like what's below.

The first $a$ is bounded by the first universal quantifier.

The first $b$ is bounded by the first existential quantifier.

The first $c$ is free.

The first $d$ is free.

The second $a$ is bounded by the second existential quantifier.

The second $b$ is free.

The second $c$ is bounded by the second universal quantifier.

The second $d$ is free.

As for your question, no, the first $a$ is not free, it is bounded. Perhaps it's easier for you to convince yourself of this if you distribute the universal quantifier over the conjunction (this can be done), then the part of the conjunction on the right of $\land$ is independent from the one on the left. This is just a trick though, it's not correct because you'll be looking at a different (albeit in a very lax sense equivalent) formula.