TITLE: Syntax/semantics conflation leads to infinitary logic
QUESTION [8 upvotes]: It is written (cf. Moore 1980, page 100) that mathematical logicians (e.g. Peirce, Schröder, Hilbert) at the turn of the last century did not yet distinguish between syntax and semantics when formulating logical and logico-mathematical theories. 
I am trying to understand how conflating syntax and semantics could (as Moore claims) encourage a move to infinitary logics.
One straightforward example concerns how to understand quantifiers. A purely syntactic understanding of the quantifiers would regard them as certain formal expressions governed by certain syntactical rules (eg. that $\exists$ distributes over logical disjunctions while $\forall$ does not, intro and elim rules in proof). 
However, Peirce is said to have understood existential and universal quantifiers as identical with (possibly) infinite disjunctions and conjunctions. This makes the quantification dependent on the domain in a way that the syntactic version does not seem to be. 
Another case is Lowenheim's requirement that when introducing a first-order expression one must also specify the domain of individuals, where the quantifier ranges over the name of each individual in the domain. Here again Moore claims that the conflation of syntax and semantics encouraged Lowenheim's move to a language allowing infinite conjunctions, disjunctions, and even transfinitely-many quantifiers: 

Lowenheim’s expressions and quantifiers involved semantics as well as syntax.
  That is, he considered a first-order expression to require that one specify a
  domain of individuals, where the quantifier ranged over the name of each
  individual in the domain. Thus the domain (a semantic concept) was built into
  the syntactic expression. Likewise he did not explicitly distinguish the names of
  individuals from the individuals themselves, nor did he introduce relation-symbols
  in a fashion distinct from relations. Once again the partial conflation of syntax and semantics encouraged the use of infinitely long expressions. (Moore 1980, p. 100)

In what ways does Lowenheim's method diverge from a "pure" syntactical approach? Is the main point that by understanding the quantifier as ranging over names of individuals rather the individuals themselves, Lowenheim commits himself to a language with infinitely many constants in order for any of the quantifiers to have infinite domains? (This would seem to be a different point from the way that Peirce was supposed to have been led to infinite language as described above.)

REPLY [11 votes]: Andrej gives a good general answer, but here is a more specific elaboration of what I think may have been in Moore’s mind here.
Work in the language of PA, or some similar language intended to be interpreted in the natural numbers.  Consider the two formulas (in modern notation):
$$ \exists x\ \varphi(x) \qquad \qquad \bigvee_{n \in \mathbb{N}} \varphi(\bar{n}) $$
where $\bar{n}$ is the numeral $S^n(0)$, as usual.
What’s the difference?  Obvious answer: the first formula is an existential quantifier; the second is an (infinitary) disjunction. But that terminological answer is a bit unsatisfying, since (as you know) existential quantification can be analysed as a kind of disjunction, and vice versa.  The real essential difference is that the former is quantifying over the domain of individuals under consideration, while the latter is quantifying over the actual natural numbers.
We can exhibit this difference concretely by looking at their interpretations in a nonstandard model of PA.  However, as long as they are just interpreted in $\mathbb{N}$, there’s no difference — they’re completely equivalent, since the domain of individuals is the actual natural numbers.  So to articulate the difference, we have to see the formulas as part of a syntax that can be interpreted in many different structures, not simply as a way of describing the single structure $\mathbb{N}$.
In summary, if syntax is not set up as independent from semantics, but just for a specific fixed structure, then one loses the distinction between existential quantifiers and infinitary disjunctions indexed by the domain.  Similarly, one loses the distinction between having infinitely many constants in the language and infinitely many elements in the domain, and so on.  In a modern setup, where the language is defined independently of the interpretation, these distinctions are clear, but in early setups, it was not at all so.
When one first articulates the distinction, therefore, one can analyse the ambiguous existential quantifications of the older language either as internally indexed (i.e. existential quantifications in the modern sense), or externally (i.e. as infinitary disjunctions in the modern sense).  With a modern eye, we do the former without thinking; but for Löwenheim, it was just as reasonable to understand them in the latter sense, and be led thereby to infinitary logics.