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Where's the notion of interpretation (model) originally introduced? I find it used in Skolem's paper "Logico-combinatorial investigations in the satisfiability or provability of mathematical propositions" (1920). However, I do not find any explicit reference to semantical ideas in Frege's Begriffsschrift. Where did the formal idea of semantics come up? What's the relation of this to Tarski's theory of truth? I really have a mess in my head.

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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 "$x \times (1 - x) = 0.$" 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, "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.

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What puzzles me is that Skolem worked on his proof of Löwenheim–Skolem theorem in 1920, which is prior to Tarski's article. Gödel's completeness theorem is prior to Tarski's article too. How did they have such results without having a well-defined notion of semantics? – Matt Mar 7 '10 at 22:26
OK, I see what you're getting at: there must have been some sort of pre-Tarskian notion of semantics. I'm looking at my copy of Jean van Heijenoort's *From Frege to Gödel" right now, hoping that a 1915 article by Löwenheim ("On possibilities in the calculus of relatives") will shed some light on this. – John Goodrick Mar 8 '10 at 0:54

When you say: "A group is a set G together with an operation such that ..." isn't that a notion of a model?

How about when you prove that the parallel postulate cannot be proved from the other axioms of geometry by exhibiting hyperbolic geometry? The "Poincaré model" is 1882, but dates to Beltrami 1868. I wonder if Tarski et. al. got the term "model" from this.

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Cayley did introduce abstract groups in his 1854 paper, but I think that Matt is asking where the idea first came from to connect logical validity with the semantic notion of "truth in a model." There is no discussion of this logical connection in Cayley's work, as far as I know. The point about the "Poincaré model" is interesting, though. – John Goodrick Mar 8 '10 at 2:31
The Poincaré model is used to show that hyperbolic geometry is equiconsistent with Euclidean geometry. Is that the first equiconsistency result? – Gerald Edgar Mar 8 '10 at 12:40
Gerald, couldn't one also view the introduction of zero, or the rational numbers from the integers, etc., as (much earlier) (equi)consistency results of the same type? These were earlier. But it may be anachronistic to speak of any of these as consistency results, rather than merely as mathematical constructions. – Joel David Hamkins Mar 8 '10 at 13:29

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