Roots of construing universal (existential) quantification as `long' conjunctions (disjunctions)

Question

I'm interested in the origins of such construals, where, for example, a statement like $\forall x\phi$ is taken to be the same as $\phi(a)\land\phi(b)\land...$. I know logical atomism has to do something with it, but I seem unable to determine which works exactly, or if the idea goes further back. Some reference names (preferably but not necessarily with some relevant pages included) will be much appreciated.

Answer 1

I think that the source of this view is Ludwig Wittgenstein's Tractatus Logico-Philosophicus (1921):

5 A proposition is a truth-function of elementary propositions.

5.3 All propositions are results of truth-operations on elementary propositions.

5.5 Every truth-function is a result of successive applications to elementary propositions of the operation ‘$N(\overline ξ)$’ [using directly the new symbol introduce by W in 5.502]. This operation negates all the propositions in the right-hand pair of brackets, and I call it the negation of those propositions.

The $N$ operator is simply the Logical NOR (aka: Peirce's arrow) with the basic difference that the argument-list $\overline ξ$ can be infinite.

Thus:

5.52 If $ξ$ has as its values all the values of a function $fx$ for all values of $x$, then $N(\overline ξ) = \lnot (∃x).fx$.

This truth-functional approach to logic, according to which "every proposition is a truth-function of elementary propositions" [Ramsey's review (1923) of the Tractatus] was followed by Frank Plumpton Ramsey in The Foundations of Mathematics (1925):

[...] by writing "$(x).fx$" we assert the logical product of all propositions of the form "$fx$"; by writing "$(\exists x).fx$" we assert their logical sum.

The concepts of "logical product" and "logical sum" were common in The Algebra of Logic Tradition; see e.g. Ivor Grattan-Guinness, Wiener on the logics of Russell and Schröder, Ann.Sci (1975):

Schröder wrote '$A_u$' to represent the proposition that '$A$ concerns $u$' [$A$ über $u$], where '$A$' denotes a proposition, '$\Pi_u A_u$' that $A$ is true for all $u$, and '$\Sigma_u A_u$' that $A$ is true for at least one $u$. '$A_u$' is the analogue of Russell's propositional function, and the three properties above correspond in Russell's system to '$fx$', '$(x) . fx$' and '$(\exists x) . fx$' respectively.

The symbols $\Pi$ and $\Sigma$ for the quantifiers were still used by Löwenheim (1915) and Skolem (1920).

If we "reason semantically" (as in Löwenheim (1915), page 232), the approach is quite obvious:

Each $\Sigma$ or $\Pi$ ranging over the subscripts - that is, over all individuals of the domain of the first degree, which, following Schröder, we call $1$ [...]

A relative expressions [formula] in which every $\Sigma$ and $\Pi$ ranges over the subscripts, that is, over the individuals of $1$ [the "universe"], will be called a first-order expression.