When it comes to first order logic over words, what is the difference between $Q_a(x)$ and $\exists x \ Q_a(x)$?
The signature is the usual one: $\tau = \{S, <, Q_a\} $. The universe is the set of positions in the word, $S$ is the successor and $<$ is the usual less-than relation.
$Q_a(x)$ is an atomic formula which is satisfied by all words having an '$a$' in any position of the word.
Similarly, $\exists x \ Q_a(x)$ is also a formula which is satisfied by all words having an '$a$' in any position of the word. Both the formulae are satisfied by the same set of words.
The second formula with the $\exists$ adds just one information to the first one, that there should exist a position $x$ where an '$a$' is present, but with respect to satisfiability, both the formula seem equivalent to me as the first one too means that on the word the letter '$a$' is present.
Is there a word which is not accepted by $Q_a(x)$ but is accepted by $\exists x \ Q_a(x)$ (or the reverse)?