What Does V Mean in the Notation of Vaught, Denumerable models of complete theories(1959)?

Question

I have been working through the aforementioned piece by R. L. Vaught, and I am curious as to what this particular piece of notation means.

$$\bigvee v_i\phi$$

and similarly

$$\bigwedge v_i\phi$$

In my vocabulary, it is a set disjunction (and similarly a set conjunction), however $\phi$ here is a logical formula. It makes a bit of sense if you look at the sets of tuples defined by those that realise $\phi$, however I can't quite wrap my head around it. It appears to be a historical piece of notation, and the only thing I can think that would make sense is perhaps they mean For All, and There Exists? However I've no idea which way around this is.

Here is a link to the paper for anyone who would like to examine it in situ.

Answer 1

Yes, $\bigvee v_i \phi$ is Vaught's notation for $\exists v_i \phi$, and $\bigwedge v_i \phi$ is his notation for $\forall v_i \phi$.

The idea is that an existential quantifier is like a disjunction over all possible values of the variable $v_i$, while a universal quantifier is like a conjunction.

You're right that this notation is "historical" - the notation $\exists$ and $\forall$ for the quantifiers is completely standard now.

Answer 2

Existential and Universal quantifiers respectively.

See Alfred Tarski & Robert Vaught, Arithmetical extensions of relational systems (1956), item [18] in the bibliography, page 84, for the specification of the language:

... and the existential quantification $\vee v_k \phi$ of an expression $\phi$ under a variable $v_k$.