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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.)

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    $\begingroup$ What is Moore 1980? $\endgroup$ – Todd Trimble Jan 1 at 3:24
  • $\begingroup$ @ToddTrimble probably philpapers.org/rec/GREBFL . $\endgroup$ – none Jan 1 at 10:15
  • $\begingroup$ @none Thanks. Of course the point is really to edit such relevant information into the text, but Malik should confirm or put in the link or fuller citation. $\endgroup$ – Todd Trimble Jan 1 at 14:26
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    $\begingroup$ @AndrejBauer The reference is to p. 100 of Moore 1980 (philpapers.org/rec/GREBFL). I will include the quote in the original question. $\endgroup$ – Mallik Jan 1 at 22:08
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    $\begingroup$ @AndrejBauer Actually, Moore provides a reference to Lowenheim, which is probably what you were after: Lowenheim, "Uber Moglichkeiten im Relativkalkul", Math. Annalen, 76 (1915). pp. 450- 459; translated in van Heijenoort (footnote 6), 228-251. $\endgroup$ – Mallik Jan 1 at 22:13
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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 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, as was discovered by W. F. Lawvere.

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    $\begingroup$ I agree, the Moore quote in the question definitely sounded to me like it was describing someone who was using an "internal language" without realizing it. On the other hand I do think there is something special about finitary syntax: it requires less of the metatheory. The behavior of infinitary logic can be highly dependent on the set-theoretic axioms assumed in the metatheory that determines the meaning of "infinite" (e.g. compact cardinals etc.). $\endgroup$ – Mike Shulman Jan 2 at 17:16
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    $\begingroup$ Of course finitary logic also depends on the (finitary) metatheory (e.g. whether or not Con(PA) is provable is a finitary statement about syntax that is true in some metatheories and false in others), but at least a particular proof in a finitary logic (of a feasible size) can be written down completely and verified by a human or a computer. $\endgroup$ – Mike Shulman Jan 2 at 17:16
  • $\begingroup$ Mathematics must be founded on fintary syntax. It is the only choice that is transparent and unambiguous for humans to read, manipulate, and understand. Everything else is an abstraction that can ultimately be explained only in fintary language. $\endgroup$ – Monroe Eskew Jan 2 at 19:51
  • $\begingroup$ @MonroeEskew: that is precisely the influence of the philosophy of lanugage I mentioned in my answer. Of course, it is difficult to imagine how it could be otherwise (myself included), as we're still in its grips. A similar, but perhaps a little more objective and modern standard is: verification of proofs and mathematics constructions must be doable by computers. $\endgroup$ – Andrej Bauer Jan 3 at 7:10
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    $\begingroup$ @MonroeEskew: (continued) However, the kind of foundation you speak of is only one aspect of foundations of mathematics, the part which focuses on how to reach objectivity in mathematical activity by multiple independent agents. Your remark therefore justifies the importance of finitary syntax, but does not give it a primary role, unless you assume that the primary business of logic is to give mathematics a foundation of the kind that the philosophers of language would appreciate. I myself do not subscribe to that view so easily. $\endgroup$ – Andrej Bauer Jan 3 at 7:12
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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.

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  • $\begingroup$ I'm not familiar with the examples from categorical logic, but Andrej's answer suggests that losing the distinctions between having infinitely many constants in the language and infinitely many elements in the domain is not necessarily a drawback of a theory. Are there also obvious situations where it would it be a problem? $\endgroup$ – Mallik Jan 4 at 3:53
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    $\begingroup$ @Mallik if you find the problems of contemporary model theory interesting, then you certainly lose something by dropping the distinction between structures and languages. $\endgroup$ – Monroe Eskew Jan 4 at 7:43
  • $\begingroup$ @ Mallik: In the “internal language” setups @Andrej is speaking of, you don’t lose those distinctions again; rather, you get a powerful analysis of what the early approach was doing, from a modern point of view on syntax. An example that doesn’t need familiarity with categorical logic is what’s classically known as the complete or elementary diagram of a structure. The general idea is: you’re using a language including names for elements of the structure. So no need to throw out the progress; you can have the best of both worlds. $\endgroup$ – Peter LeFanu Lumsdaine Jan 4 at 9:17

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