Abstract
In this paper, we outline and critically evaluate Thomas Hofweber’s solution to a semantic puzzle he calls Frege’s Other Puzzle. After sketching the Puzzle and two traditional responses to it—the Substantival Strategy and the Adjectival Strategy—we outline Hofweber’s proposed version of Adjectivalism. We argue that two key components—the syntactic and semantic components—of Hofweber’s analysis both suffer from serious empirical difficulties. Ultimately, this suggests that an altogether different solution to Frege’s Other Puzzle is required.
This publication was funded by LMU Munich’s Institutional Strategy LMU excellent within the framework of the German Excellence Initiative.
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Notes
- 1.
This follows a referentiality test due originally to Kratzer and Heim (1998).
- 2.
The label “Adjectivalism” is due to Dummett (1991). It is somewhat unfortunate, however, because it suggests that what Frege calls “attributive uses” like (1a) must be adjectives. However, the intended view is that “attributive uses” are non-referential expressions, and this is consistent with ‘four’ in (1a) being an adjective or a determiner. Despite this, we follow the literature in retaining the label “Adjectivalism”.
- 3.
(17) is in fact the denotation of ‘four’ assumed by Breheny (2008). In GQT, cardinal determiners are actually given lower-bounded truth conditions, so that ‘four’ denotes a relation between sets whose intersection has a cardinality of at least four:
(i)
[[four]] = {<S,S’>: S,S’ ⊆ U and |S ∩ S| ≥ 4}
The reason for adopting a two-sided analysis instead will become apparent in the next section, when we consider paraphrases of basic arithmetic equations like ‘three and two is five’.
- 4.
See Landman (2003).
- 5.
- 6.
- 7.
The details are complex and beyond the scope of a single paper. But see Hofweber (2016).
- 8.
- 9.
- 10.
See Snyder (2017).
- 11.
- 12.
See Mikkelsen (2005).
- 13.
See Schlenker (2003).
- 14.
- 15.
To a first approximation, intransitive counting consists in reciting the numerals in their canonical order –“1, 2, 3,...” In contrast, transitive counting consists in the counting of things. That is, when transitively counting we use the numerals to answer ‘how many’-questions, roughly by establishing a one-to-one correspondence between an initial segment of those numerals and a collection of objects being counted.
- 16.
See especially Partee (1986a).
- 17.
See Snyder et al. (2021).
- 18.
- 19.
- 20.
See Snyder (2017).
- 21.
For one thing, unlike all prototypical determiners, ‘many’ has a comparative and superlative form – ‘more’ and ‘most’, respectively – and is gradable – cf. ‘very/so/how many’.
- 22.
- 23.
See Barwise and Cooper (1981).
- 24.
An anonymous reviewer observes that the same argument would extend to sets, which should be just as objectionable from a nominalist perspective, but that this kind of commitment might be avoided by appealing to a pluralist metalanguage, perhaps following Boolos (1985). As far as we know, whether all of GQT can be recovered within a pluralist metalanguage is an open question, though McKay (2006) makes progress in this direction. Even so, the question would remain as to whether an empirically adequate, nominalist-friendly pluralist semantics for number expressions could be formulated, something which some of us have cast doubt on in other work (e.g Snyder and Shapiro 2021).
- 25.
- 26.
Contra Moltmann (2013).
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Snyder, E., Samuels, R., Shapiro, S. (2022). Hofweber’s Nominalist Naturalism. In: Oliveri, G., Ternullo, C., Boscolo, S. (eds) Objects, Structures, and Logics. Boston Studies in the Philosophy and History of Science, vol 339. Springer, Cham. https://doi.org/10.1007/978-3-030-84706-7_3
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