It is written (cf. [Moore 1980](https://philpapers.org/rec/GREBFL), 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.) 



Syntax/semantics conflation leads to infinitary logic
It sounds to me like this question is asking us to divine about the thinking of mathematicians in the early 20th century. Obviously, I can do no such thing, but perhaps I can explain how the early views of logic make very good sense from the point of view of modern logic.


In classical treatments of first-order logic and model theory, the dichotomy between syntax and semantics is quite pronounced, and one easily gets the impression that it is necessary. Students are taught that finitary syntax must be the norm. However, this really is just a design choice. For instance, in categorical logic one customarily considers [internal languages and logics](https://ncatlab.org/nlab/show/internal+logic) of various kinds of categories, without insisting that finitary syntax is paramount or primary.


Let us give an example. The internal language of a *particular* topos $\mathcal{E}$ has as types all (names of) the objects, and as term formers all the (names of) morphisms of the topos, of which there may be arbitrarily many. If there is a formal distinction between an individuum and its name it is certainly not dwelled on or considered essential. This point of view is close to the one of Löwenheim, who "did not explicitly distinguish the names of individuals from the individuals themselves". Because the distinction was not there it could not have lead to infinitary logic.


Furthermore, we take as the axioms all equations of the form $x : A \vdash f(g(x)) = h(x)$ whenever $h = g \circ f$ in $\mathcal{E}$. (I shall address infinitary operations shortly.) The usual finitary syntax is special because of its universal *mathematical* property: it is the internal language of the *initial* topos (where of course "initial" has to be appropriately interpreted).


My point is that it looks as if the mindsets of the early 20th century logicians were closer to that of categorical logic than first-order logic and model theory. Or to put it another way, the conflation of syntax and semantics was *not a mistake*. Using the word "conflation" to describe what they did betrays a very syntactically minded view of logic rooted in philosophy of (human!?) language–which came later to dominate logic for several decades. It is easy to argue in the opposite direction and present the dichotomy as a flaw: the preoccupation with pure syntax by logicians of the mid 20th century demonstrates their inability to think abstractly, and can be likened to the historic period of mathematical analysis during which "function" was equated with "expression".


The question invites primarily opinion-based answers, which is why I voted to close it.


Let me address the question of infinitary languages. Here again we first need to realize that "finite vs. infinitary" is not a political or a philosophical question, or if it is, it should be wrestled from politicians and philosophers and made into a proper mathematical question. The language should be precisely as infinitary as the situation demands. (To my geometrically minded self a language without a situation is meaningless. In fact, a language will *create* a situation for itself if none is given in advance.) Thus, the internal language of elementary toposes (and logical morphisms) is not infinitary because not all such toposes have sufficient infinitary structure, but the internal language of Grothendieck toposes (and geometric morphisms) is infinitary because Grothendieck toposes are cocomplete.


Under the "conflation" of syntax and semantics the semantics naturally suggests the infinitary operations. Contrary to what the question seems to presuppose, one does not even start with a finitary language, and there is no step under which we are "lead" to an infinitary language. I find it odd that one would explain the early 20th century logic as if there was some sort of passage from finitary to infinitary logic. With a century of hindsight (but perhaps not with only half a century of hindsight) it seems more natural that the finite/infinitary distinction appears *afterwards*, when one has already had some success in formulating a logical language appropriate to the semantic domain under consideration. (Of course, a semantic domain incorporates not only an extension of individuals but also the appropriate structure carried by the extension.)


One last remark: early logicians, particularly those who studied complete Boolean algebras, were naturally inclined to interpret $\forall$ and $\exists$ as very large conjunctions and disjunctions. In fact, many mathematicians today still naively think of quantifiers as infinitary versions of connectives, and it takes a bit of logic training to understand that the difference between $\exists$ and $\bigvee$ is a bit like the difference between uniform and pointwise continuity. Quantifiers are infinite suprema and infima only in certain situations, whereas in general they are [adjoints](https://ncatlab.org/nlab/show/quantification#lawvere_quantifiers_quantification_as_adjunction), as was discovered by W. F. Lawvere.


