What is "mathematical logic"? Is it the logic of mathematical reasoning, or is it the claim that mathematics and logic are identical?
Also, is "quantificational logic" a particular type of "mathematical logic"?
(cf. "What is the philosophical ground for distinguishing logic and mathematics?")
The modern term is due to George Boole, with his booklet of 1847:
aiming at the application of mathematical tools and methods to the study of logic.
For a recent historical overview, see:
In the same booklet, Boole set also a link with language:
[page 5] The theory of Logic is thus intimately connected with that of Language.
This link was "already there" from the beginnig: see Aristotle's Categories and Leibniz's calculus ratiocinator and lingua characteristica.
The link with language supports the basic distinction between propositional calculus, where the level of "analysis" of logical structure of language is very rough, considering only the sentential connectives, and quantification theory, where we have a more "fine-grained" (but still very far from describing the "real life" functioning of natural language) level of detail, based on quantification.
The modern codification of logical languages has many "fathers", but the official birth is Gottlob Frege's Begriffsschrift (1879).
Modern mathematical logic is "mathematical" also in another sense: some branches of it are simply mathematical disciplines, like e.g. Computability theory and Constructive mathematics, or they are
"the logic of mathematical reasoning",
like Model theory and Proof theory.
Edit
It seems that the name "mathematical logic" has been used first by Augustus De Morgan in On the syllogism no.III (1858), reprinted into:
@Mauro ALLEGRANZA beat me to it with an excellent answer but since i had already drafted this here's another take:
The answer is "neither".
It is definitely not the claim that mathematics and logic are the same thing. The central concern of logic, on most accounts, is the notion of logical consequence and/or valid inference. You commonly find words to this effect at the beginning of introductory texts. It's a lot harder to say what the central concern of math is. Historically notions like quantity, magnitude, number, etc have been offered but math covers so much conceptual ground it's hard to pick out one thing common to all but not trivial. Logical consequence by contrast is not only non-trivial, it remains somewhat mysterious. Compare the notion of computability, which remained mysterious until Turing came along and provided a model that is universally considered to be satisfactory both formally and intuitively. Nobody has yet managed to do that for the informal, intuitive concept of logical consequence.
Note that there is no prima facie reason to think that our intuitive notion of logical consequence has anything to do with mathematics.
What about "the logic of mathematical reasoning"? Mathematical reasoning, by definition, is what mathematicians do when they "do" mathematics, and it is informal. It is now possible to publish formal proofs of mathematical results, but almost nobody does this and some think it is a bad idea. Mathematical argumentation may be highly disciplined, but it is not formal. However mathematical logic is formal. You could argue that ML is an attempt to formally express the informal reasoning of working mathematicians, but that's something very different than offering a "logic of mathematics". The latter suggests ( to me at least) an attempt to give an explanation (in some formal system, which is itself a mathematical object) of how informal mathematical reasoning works, which is really a philosophical project. ML does not explain anything, it's just a technology.
It also depends on what you mean by "mathematics". Classical or intuitionistic? Most texts on ML that I've come across implicitly assume that Classical Mathematics (which relies on truth-conditional semantics) is the only game in town, which is not the case.
I personally think of mathematical logic as mathematicized logic: treatment of logical forms as mathematical objects, without claiming that logic is reducible to mathematics. So it's more of a methodological concept.
Józef Maria Bocheński, O.P.'s 1959 A Precis of Mathematical Logic defines "mathematical logic" this way on p. 1:
0.2. Logic and mathematics. Mathematical logic is called 'mathematical' because of its origin, since it has been developed particularly with the aim of examining the foundations of this science. There is moreover a certain external resemblance between its formulas and those of mathematics. Certain logicians also claim that mathematics is only a part of logic, although this opinion is far from receiving general approval. However, mathematical logic does not consider either numbers or quantities as such, but any objects whatsoever.
Mathematical Logic and Computation, Jeremy Avigad(2022):
In the phrase mathematical logic, the word “mathematical” is ambiguous. It can be taken to specify the methods used, so that the phrase refers to the mathematical study of the principles of reasoning. It can be taken to demarcate the type of reasoning considered, so that the phrase refers to the study of specifically mathematical reasoning. Or it can be taken to indicate the purpose, so that the phrase refers to the study of logic with an eye toward mathematical applications.
