On "Scope"

A while ago (Jan '26), I tried to find where the word "scope" - in the sense of "quantifier scope" came from. Historians of logic may already know all this, but village semanticists like me don't. Overview articles in semantics tend not to get into this, so I thought I'd put it here where other linguists can see it. Let me thank Ellen Lau and Alexander Williams for spurring this, and a helpful conversation about it after the fact.

I was interested in the issue because I was curious why linguists talk about the scope of quantifiers, while computer scientists talk about the scope of variables (sort of like dynamic semanticists talk about discourse referents). The idea of scope goes back at least to Frege. He uses Gebiet ('region', 'territory') to name how far a quantifier may bind a variable, in the Begriffsschrift (Section 11). But he uses the term to talk about where a variable keeps its meaning.

He says, when considering a formula we would now write as ∀a. Fa, "Thus, one cannot put something arbitrary in the place of a without jeopardizing the judgment's validity. This explains the reason the concavity with the German letter* written within is necessary: it marks off the region [Gebiet] that the Generality [universal quantification] concerns, through the letter indicated. Only within its region does the German letter retain its meaning." (trans. MPS; original German in [1], another translation in [2]).

*He says "German letter" because the convention he uses is Gothic letters for bound variables.

The gender morphology makes it unambiguous that Frege does not talk about the region of the quantifier (or what would be the region of Generality, in his terms), but rather, the region of the "German letter", i.e., the place where the variable keeps its meaning fixed. "Generality" (Allgemeinheit) would go with the feminine genitive pronoun ihr but Frege talks about seines Gebietes (the masc/neut possessive pronoun), like the masculine "letter" (Buchstabe).

Wilfrid Hodges has a presentation about related issues. According to him, the English term "scope" seems to be introduced by Russell in 1908, and Russell uses this term to mean what Frege meant by Gebiet: the region in which a variable is used in a certain way.

Here is one of the first uses of the term scope in the paper of Russell's that Hodges mentions:  "(1) When a man says " I am lying," we may interpret his statement as: "There is a proposition which I am affirming and which is false." All statements that "there is so-and-so" may be regarded as denying that the opposite is always true; thus "I am lying" becomes: "It is not true of all propositions that either I am not affirming them or they are true;" in other words, "It is not true for all propositions p that if I affirm p, p is true." The paradox results from regarding this statement as affirming a proposition, which must therefore come within the scope of the statement."

Hodges also notes the use of the term Wirkungsbereich (translated by dictionaries as 'scope', 'purview', 'sphere of action', etc.) by Hilbert and Ackermann: "To each universal or existential quantifier occurring in a formula, there belongs a definite constituent part of the formula to which it relates. We shall call this part the scope [Wirkungsbereich] of the symbol in question" (quoted in Hodges' presentation, slide 7). Note that their use talks about the Wirkungsbereich of the operator, giving us the notion we use in linguistics. I actually quite like the translation 'sphere of action'.

So, it seems that computer scientists actually retained the original Frege/Russell sense of the term "scope" and "scope" in the sense of "quantifier scope" is a distortion. But (as Hodges notes) talking about the scope of a quantifier as opposed to the scope of a variable has advantages when two quantifiers bind the same variables. However, vacuous binding of this sort is never used in linguistics.

[1] For the enthusiasts, here is Frege's original German:
"Also auch hier kann man nicht etwas Beliebiges an die Stelle von a setzen, ohne die Richtigkeit des Urtheils zu gefährden. Dies erklärt, weshalb die Höhlung mit dem hineingeschriebenen deutschen Buchstaben nöthig ist: sie grenzt das Gebiet ab, auf welches sich die durch den Buchstaben bezeichnete Allgemeinheit bezieht. Nur innerhalb seines Gebietes hält der deutsche Buchstabe seine Bedeutung fest;"

[2] Just in case you want to compare with another translation, which uses "scope" for "Gebiet": "Hence we cannot put something arbitrary in place of a here either without endangering the truth of the judgment. This explains why the concavity with the German letter written into it is necessary: it delimits the scope that the generality indicated by the letter covers. The German letter retains a fixed meaning only within its own scope." [Stefan Bauer-Mengelberg's translation, 1967]