I was looking up free logic on wikipedia
https://en.wikipedia.org/wiki/Free_logic
and reading it posed more questions than it explained.
The "explanation" paragraph lists three theorems of predicate logic, for example:
$\forall xA \Rightarrow \exists xA$
and its corresponding free logic version
$\forall xA \land E!t \rightarrow \exists xA$
I have a hard time understanding these statements as there are no variables in the scope of $\forall$ or $\exists$. I could read something like
$\forall xA(x) \rightarrow \exists xA(x)$
but that is not what is written.
Similarly,
$Fy \rightarrow (E!y \rightarrow \exists xFx)$
just does not make sense to me. I just cannot make sense of $Fy$ which I take to be equal to $f(y)$, so there appears to be some sloppiness in notation. However, $f(y)$ is still cryptic. I guess part of my problem is with free variables but I cannot for the life of me come up with a statement in mathematics that contains a free variable (don't shoot me, I am a noob).
Side question: What does the introduction of an existence predicate achieve? I thought that existence is to be expressed through a quantifier and this helped me to make sense of Quine's slogan "to be is to be the value of a variable". The whole wiki article is rather cryptic for a beginner, so where can one read an easy summary of the topics touched upon here?
I have made several corrections in the Wikipedia article; apparently, there was a confusion about the translation from Karel Lambert's dot notation of Principia Mathematica in his paper Free Logic and the Concept of Existence to the contemporary notation. Thus,
$\forall xA(x) \rightarrow \exists xA(x)$ is now changed to $\forall xA \rightarrow (E!t \rightarrow A(t/x))$
$E!$ is not a quantifier, though it may look like one at first sight. It can be taken as a primitive logical predicate whereby $E!t$ says of a singular term $t$ that it is in the inner (i.e., seeking existential import) domain of quantification. Alternatively, $E!t$ can be taken as an open formula defined as $\exists x(x=t)$.
It may be helpful to remark that, when possible, reading a formula in the subject-predicate form may provide a quicker intuitive idea than the function-argument form will do. Hence, one can read $Fy$ as saying that 'some $y$ is $F$'. The formula $Fy \rightarrow (E!y \rightarrow \exists xFx)$ introduces an additional assumption of existence (denotatum), $E!y$, for the singular term $y$ of which $F$ is predicated.
Lambert's open access paper mentioned above conveys the basic ideas quite clearly. John Nolt's article Free Logic offers a broader sweep, but not much more demanding.