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In the SEP entry on Wittgenstein, the description of the logic utilised in the Tractatus Logico-Philosophicus to describe his logical atomism appears to be at least formally classical propositional calculus (I mean this in the modern mathematical sense as inspired by Frege). Further, the SEP doesn't mention the use of qualifiers (so unless its an over-sight, or it is already subsumed in what Wittgenstein means by a proposition) it can't be first-order logic.
Is this correct?
You might check out Michael Potter's "The Logic of the Tractatus" in the Handbook of the History of Logic Vol. 5: From Russell to Church.
In that he discusses Wittgenstein's logical atomism, his presentation of truth-tables for Fregean logic, among other things.
In it, Potter claims that Wittgenstein's system is not only quantificational, but admits of higher-order quantification. This is not surprising, since the Tractatus was heavily influenced by Frege who developed a higher-ordered logic in his Begriffschrift.
Potter compares the Tractatus's utilization of ever more inclusive metalanguages to a Russell-Ramsey type theory, where each level of metalanguage represents an ascent to a new order within the type theory.
There is a discussion of quantification as well as an exposition of Wittgenstein's class theory.
In short, Potter is able to pull out an impressive amount of technical material from what is an undoubtedly quite obscure work.
