Abstract
David Lewis’s “Adverbs of Quantification” provided an account of words such as always and usually in examples where they occur in construction with restrictive clauses and indefinite descriptions. According to Lewis, adverbs of quantification quantify “cases”, which are tuples of witnesses for indefinite descriptions. This paper is an elementary introduction to Lewis’s approach. In a formalization, clauses contribute properties of cases, and adverbs of quantificaton contribute relations between properties of cases.
Thanks to Zoltán Gendler Szabó, Dorit Abusch, Tim Fernando, and a reviewer for comments. Thanks also to students in the 2019 and 2021 editions of Semantics II at Cornell, who read drafts of the paper, reacted to class presentations of it, and worked with a computational implementation.
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Notes
- 1.
As pointed out by Zoltán Gendler Szabó, similar paraphrases are found in Russell (1905), where they go in the other direction. Russell says that C(everything) means “C(x) is always true”.
- 2.
This leaves open why the determiners in the paraphrases are to be considered quantificational, and Lewis does not try to give an account of this. As worked out in sections “Semantics of the LF” and “Dynamic Case Quantification”, the terminology is justified in Lewis’s theory, where the semantics of AQs is isomorphic to the semantics of determiners, as analyzed in generalized quantifier theory.
- 3.
This point is made by Lewis. I’m not going to say this every time I could.
- 4.
- 5.
Then in section “Dynamic Case Quantification”, the subscripts are eliminated, by encoding the variables that are quantified in the semantics.
- 6.
Lewis encodes the same information by introducing variables such as x and y, with the understanding that all such variables in the restriction are quantified. This is (10) in the notation used by Lewis: always if x is a woman and y is a cat and x owns y, y loves x.
- 7.
Adjunction is the tree transformation that maps [A ... X ... ] to [AXi [A ... ei ... ]].
- 8.
A node A c-commands a node B if and only if the node dominating A dominates B, and A does not dominate B.
- 9.
This last step, which is not found in Heim, is included here in order to produce something that has the syntactic shape of an atomic formula.
- 10.
∃ is not an adverb of quantification, because it has one argument rather than two.
- 11.
See Chap. 3 in this volume.
- 12.
While Lewis theorized about quantifying cases, the noun case hardly figured in his database of examples. Moltmann (2017) is a study of uses and interpretations of this noun.
- 13.
So on Lewis’s analysis, AQs quantify cases, and cases are tuples of individuals. Or if the analysis is generalized to pronouns of other types, cases are tuples of model-theoretic objects of various types. The principal competing account of cases is that they are situations (Berman, 1987; Von Fintel, 1994). This results in a really different theory. Notably, the account of pronouns has to be different, because situations do not provide antecedents for pronouns in the same direct way as tuples. An approach to this is that pronouns in AQ constructions are elliptical definite descriptions (Elbourne, 2001). A paraphrase of the semantics of (2b) is roughly “in almosted situation of a dart hitting a unit square, does the dart in that situation hit the unit square in that situation on the diagonal”.
- 14.
The notation “λ 1 2” indicates that the indices on the AQ are used to form cases by lambda-binding of variables. This notation is not part of the LF.
- 15.
This follows the plot of Heim (1982), where first an analysis using indexing and existential closure is stated, and then the analysis is re-formulated in dynamic terms. Rather than following Heim’s file change semantics, the second part as stated here is close to Paul Dekker’s formulation of dynamic semantics, see Dekker (1994, 2012).
- 16.
The choice of i in [iti] depends on details of the sub-clausal semantics. (34) switches to [it3], on the assumption that 1 and 2 are discourse referents for the woman and the cat.
- 17.
To explain the terminology, Do(A) is the domain of the relation that holds between cases d and e iff e is an element of dA.
- 18.
See Chap. 4 in this volume.
- 19.
See Chap. 13 in this volume.
- 20.
Khoo (2016) is a recent discussion of this empirical picture and counterarguments to it.
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Rooth, M. (2022). On Lewis’s “Adverbs of Quantification”. In: McNally, L., Szabó, Z.G. (eds) A Reader's Guide to Classic Papers in Formal Semantics. Studies in Linguistics and Philosophy, vol 100. Springer, Cham. https://doi.org/10.1007/978-3-030-85308-2_16
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