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3 Tried fixing the LaTeX formulas again (why are they not displaying correctly?)

As best as I have been able to figure out, the pre-Tarskian notions of "semantics" in mathematical logic grew out of the "algebra of logic" introduced by Boole ("An investigation into the laws of thought," 1854, and some earlier papers) and elaborated by Charles Peirce, Schröder, and others.

It's difficult for me to follow all the arguments in the old papers, but roughly the idea behind Boole's logic was to study logical equations such as ''$x "$x + (1 - x) = 1$'' 1$" or ''$x "$x \cdot times (1 - x) = 0.$'' 0.$" Here, + means exclusive or, multiplication is and,'' "and," and subtracting $x$ means taking a conjunction with not-$x$. The number 1 should be interpreted as an always-true proposition, 0 as an always-false proposition, and $x$ as a propositional variable (or as Boole might say, a proposition with ''indeterminate "indeterminate truth value'')value").

Boole was interested in this analogy between logic an algebra, and here maybe we see the beginnings of the notion of interpretation in logic: we can check the validity of these formulas by considering whether they are true for all possible propositions $x$. (At least, I think this is what Boole meant -- you should track down the Dover reprinting of The Laws of Thought if you want to do some more historical investigation.)

Peirce considered the possibility of different "domains of individuals," which could be finite, infinite, or even uncountable. I think it was Peirce who first generalized Boole's calculus of logical propositions to the "calculus of relatives," where a "relative" is the interpretation of some $n$-ary predicate in a domain of individuals. To track down the beginnings of this, I would try Peirce's 1870 "Description of a notation for the logic of relatives," which unfortunately I cannot access right now from where I am.

Peirce, Schröder, and even Löweinheim in his 1915 "On possibilities in the calculus of relatives" continued to use algebraic notation along the lines of Boole, with many $0$'s and $1$'s. Even the domain "domain of individuals'' was denoted by $1^1$!

One thing in particular that is confusing about reading Löweinheim's paper is that, while he is clearly aware that the domain of individuals $1^1$ could be one of any number of possible collections of things (some finite, others infinite), he seems to insist on talking about ''"the domain of individuals $1^1$'' 1^1$" and referring to every possible domain by the same name$1^1$! Obviously, this is confusing if you want to think about comparing two different such domains, and maybe one of Tarski's key contributions here was simply to introduce a notation ''$\mathfrak{A} \models \varphi$'' which explicitly names the universe$\mathfrak{A}$and suggests comparison with other universes$\mathfrak{B}, \mathfrak{C}, \ldots$. My original answer (missing the key point): My knee-jerk answer to this question was going to be, "Tarski!" But you seem to already be aware of Tarski's work, so maybe you're looking for something different? In particular: Alfred Tarski's 1933 article "The concept of truth in formalized languages" (in Polish, unfortunately) seems to be generally regarded as the first place where the concept of "logical satisfaction" (in the modern sense) was first defined. There was already an "application" of semantic methods in logic by 1940: Gödel's proof of that Con(ZFC) implies Con(ZFC + GCH + AC). (It might be fun to try to find an even earlier application of semantic methods to prove a syntactic result.) Certainly by the 1960's the field of model theory was coming into its own with the work of A. Robinson, Vaught, Morley, and others. 2 Added a new discussion above my old answer (but kept my original answer) Updated answer: As best as I have been able to figure out, the pre-Tarskian notions of "semantics" in mathematical logic grew out of the "algebra of logic" introduced by Boole ("An investigation into the laws of thought," 1854, and some earlier papers) and elaborated by Charles Peirce, Schröder, and others. It's difficult for me to follow all the arguments in the old papers, but roughly the idea behind Boole's logic was to study logical equations such as ''$x + (1 - x) = 1$'' or ''$x \cdot (1 - x) = 0.$'' Here, + means exclusive or, multiplication is and,'' and subtracting$x$means taking a conjunction with not-$x$. The number 1 should be interpreted as an always-true proposition, 0 as an always-false proposition, and$x$as a propositional variable (or as Boole might say, a proposition with ''indeterminate truth value''). Boole was interested in this analogy between logic an algebra, and here maybe we see the beginnings of the notion of interpretation in logic: we can check the validity of these formulas by considering whether they are true for all possible propositions$x$. (At least, I think this is what Boole meant -- you should track down the Dover reprinting of The Laws of Thought if you want to do some more historical investigation.) Peirce considered the possibility of different "domains of individuals," which could be finite, infinite, or even uncountable. I think it was Peirce who first generalized Boole's calculus of logical propositions to the "calculus of relatives," where a "relative" is the interpretation of some$n$-ary predicate in a domain of individuals. To track down the beginnings of this, I would try Peirce's 1870 "Description of a notation for the logic of relatives," which unfortunately I cannot access right now from where I am. Peirce, Schröder, and even Löweinheim in his 1915 "On possibilities in the calculus of relatives" continued to use algebraic notation along the lines of Boole, with many$0$'s and$1$'s. Even the domain of individuals'' was denoted by$1^1$! One thing in particular that is confusing about reading Löweinheim's paper is that, while he is clearly aware that the domain of individuals$1^1$could be one of any number of possible collections of things (some finite, others infinite), he seems to insist on talking about ''the domain of individuals$1^1$'' and referring to every possible domain by the same name$1^1$! Obviously, this is confusing if you want to think about comparing two different such domains, and maybe one of Tarski's key contributions here was simply to introduce a notation ''$\mathfrak{A} \models \varphi$'' which explicitly names the universe$\mathfrak{A}$and suggests comparison with other universes$\mathfrak{B}, \mathfrak{C}, \ldots\$.

My original answer (missing the key point):

My knee-jerk answer to this question was going to be, "Tarksi!" Tarski!" But you seem to already be aware of Tarski's work, so maybe you're looking for something different?

1

My knee-jerk answer to this question was going to be, "Tarksi!" But you seem to already be aware of Tarski's work, so maybe you're looking for something different?

In particular: Alfred Tarski's 1933 article "The concept of truth in formalized languages" (in Polish, unfortunately) seems to be generally regarded as the first place where the concept of "logical satisfaction" (in the modern sense) was first defined.

There was already an "application" of semantic methods in logic by 1940: Gödel's proof of that Con(ZFC) implies Con(ZFC + GCH + AC). (It might be fun to try to find an even earlier application of semantic methods to prove a syntactic result.) Certainly by the 1960's the field of model theory was coming into its own with the work of A. Robinson, Vaught, Morley, and others.