Abstract
Plural Logic is an extension of First-Order Logic which has, as well as singular terms and quantifiers, their plural counterparts. Analogously, Higher-Level Plural Logic is an extension of Plural Logic which has, as well as plural terms and quantifiers, higher-level plural ones. Roughly speaking, higher-level plurals stand to plurals like plurals stand to singulars; they are pluralised plurals. Allegedly, Higher-Level Plural Logic enjoys the expressive power of a simple type theory while committing us to nothing more than the austere ontology of First-Order Logic. Were this true, Higher-Level Plural Logic would be a useful tool, with various applications in philosophy and linguistics. However, while the notions of plural reference and quantification enjoy widespread acceptance today, their higher-level counterparts have been received with a lot of scepticism. In this paper, I argue for the legitimacy of Higher-Level Plural Logic by providing evidence to the effect that natural languages contain higher-level plural expressions and showing that it is likely that they do so in an indispensable manner. Since the arguments I put forward are of the same sort advocates of Plural Logic have employed to defend their position, I conclude that the commonly held view that Plural Logic is legitimate, but not so its higher-level plural extensions is untenable.
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Notes
For example, Black (1971) uses it to support his eliminativist view about sets, Boolos (1984), Boolos (1985) and Uzquiano (2003) use it in the framework of an eliminativist view of classes and Hossack (2000) uses it in an eliminative project about complex objects. Moreover, a notable eliminativist project that employs plurals is that of Lewis (1991). He endorses a theory of sets where set membership is reduced to singleton-set membership plus mereological fusion of singletons and where polyadic relations are reduced to plural reference to ordered pairs. Thus his aim is not to eliminate sets altogether, but rather to replace the (in his view) mysterious notion of set membership with a stock of allegedly more transparent notions.
For a survey of the philosophical debate on plurals and a description of a simple logic of plurals, see Linnebo (2017).
I will occasionally speak of ‘pluralities’ in this paper. However, note that I will use the term ‘plurality’ as a convenient shorthand for ‘objects’. This is common practice in this debate and is necessary due to the expressive limitations of English.
Even though they ultimately argue against the claim that the higher-level plural reading carries a substantial advantage.
For instance, one could assume the existence of ordered pairs, but this would be at odds with the purpose of the neo-Fregean programme by not only presupposing the existence of a plethora of entities without providing a further story as to why this is legitimate, but also by presupposing the existence of entities which are suspiciously set-theoretic in nature. Another option would be to use a pairing function, but this strategy would be problematic too, since the existence of a pairing function makes specific demands on the size of one’s background ontology, as argued by Shapiro and Weir (2000).
The idea would be to express a definition analogous to that of, for instance, Hausdorff as follows:
\(\langle a, b\rangle := ((a,1),(b,2))\), where ((a, 1), (b, 2)) is meant to represent a second-level plurality which consists of the plurality of a and 1, on the one hand, and the plurality of b and 2, on the other. See Grimau (2018, Chap. 7) for a proposal along these lines.
In the literature on PL it is often assumed that singular reference is a limiting case of plural reference, since plural terms may happen to denote a single thing. On this basis, formal plural terms are often interpreted as possibly denoting a single thing. In this article, for the sake of simplicity, I will not adhere to this assumption at the formal level. For us, a plural term will denote more than one thing and thus will serve to regiment the natural language phenomenon of strict plural reference. Moreover, Oliver and Smiley (2016) further liberalise the notion by allowing formal plural terms to fail to denote. Since allowing for vacuous reference would also significantly complicate the formalism and none of my arguments hinge on this choice, I discard this possibility as well.
Properties can be seen as collectivising entities in the sense that they gather together all of their instances. One such proposal can be found in Florio (2010).
A survey can be found in Florio (2014).
See, for example, Boolos (1984, p. 65).
See Linnebo (2006) for a development of this view with respect to type theory.
Even though I do not have a strong preference for either option, I take the typed route since it captures better the intuition that the ascent from the singular to the plural and from the basic plural to higher levels is a matter of expressive rather than ontological expansion.
I limit the hierarchy to finite levels for simplicity.
Note that we do not have any mechanism of complex term formation, such as a rule to form definite descriptions out of non-logical predicates or to form lists out of terms. Although these would allow HLPL to regiment natural language more accurately (and thus might be necessary for applications of HLPL in natural language or ordinary reasoning analysis), I leave them out for the sake of simplicity, since nothing that I discuss in this article turns on the availability or lack thereof of these complex terms. (see Oliver and Smiley (2016) for a language which allows for definite description formation).
I will remain neutral as to the specific nature of these entities, since this issue is irrelevant in the present context.
The ordered triples acting as models are not ordinary ones: they require that some of their members be pluralities. However, this is unproblematic in light of the fact that we can code these unorthodox ordered pairs using techniques already available to us, as shown in Linnebo and Rayo (2012, pp. 304–306).
In this article I rule out the possibility of referring to mixed higher-level pluralities, that is, pluralities consisting of, say, a single individual on the one hand and a first-level plurality on the other (intuitively the denotation of e.g. Rafa Nadal and the Williams sisters). Given that none of the arguments put forward in the present article depend on this and keeping in mind that they could be accounted for by complicating the formalism (i.e. adopting a logic analogous to cumulative type theory; see Linnebo and Rayo (2012) for details), I leave mixed higher-level pluralities aside.
An \(x^{k}\)-variant of s is an assignment that only differs from s at most in what it assigns to \(x^{k}\).
Authors have used different terminology to refer to higher-level plurals. These are some of the terms that have been used in the literature: ‘perplurals’ (Hazen 1997; McKay 2006), ‘pluplurals’ (Rosen and Dorr 2002; Simons 2016), ‘plurally plurals’ (Hossack 2000; McKay 2006; Rumfitt 2005; Uzquiano 2004), ‘hyperplurals’ (Cotnoir 2013), ‘superplurals’ (Oliver and Smiley 2016; Rayo 2006).
Rayo only claims that English, in particular, does not contain such devices.
In its set-theoretic version, a cover of a set a (such as the denotation of a plural term, under set-theoretic singularism) is a set of non-empty subsets of a, where every member of a belongs to some such subset. In its sum-based version, a cover of a sum s (such as the denotation of a plural term, under sum-based singularism) is a set of sums whose fusion is s.
Although neither Gillon nor Schwarzschild endorse their semantics in the context of HLPL, Linnebo and Nicolas (2008) note its potential relevance for the present topic.
In this article I will mostly be using the expression ‘group’ in its non-technical informal sense. Whenever I use it in the sense just described, I will make it explicit.
I use commas (possibly followed by a conjunction) in order to indicate where one nested list ends and another one begins. This rather artificial notation is more naturally captured by intonation in spoken language.
According to this understanding of lists, nested lists of plurals would be third-level plurals. E.g. the cat lovers and their cats, and the dog lovers and their dogs.
See Oliver and Smiley (2016, Chap. 8), who call them ‘plurally exhaustive descriptions’.
The notion of pseudo-singularity comes from Oliver and Smiley’s work on plurals. See Oliver and Smiley (2016, pp. 305–306).
Some languages only display the anaphoric form of plural override. For instance, in French and Latvian that is the only way in which we find the phenomenon.
I am grateful to an anonymous referee for this example and, more generally, for pointing out this issue.
This was also noted, although not specifically in relation with HLPL, by Jespersen (1924, p. 189).
For instance, Corbett (2000) talks of ‘semantically composing plural on plural’.
Ben-Yami (2013, pp. 85–86) raises the objection that ‘the translations of the Icelandic phrases in fact disagree with the use Linnebo would like to make of them’. This is because they do make use of an expression like pair. However, this objection appears to put the cart before the horse. English translations cannot play the role Ben-Yami intends them to play here, since they cannot help us identify a non-English expression as higher-level plural, given the expressive limitations of English. If this were a valid criterion of identification, the outcome of the investigation would be decided from the start.
According to Hurford (2003), Estonian is ‘to a large extent’ similar to Finnish in this respect. I focus on Finnish for simplicity.
While the following examples were reported in Appleyard (1987, p. 252), they were originally recorded a whole century before, in Reinisch (1884). Appleyard reports that even though in his study he found similar forms, they had evolved into mere alternative first-level plural forms. It would not be surprising if the distinction had been lost today.
In this example, the first plural suffix (e) is irregular and the second one (où) is regular.
I wish to thank an anonymous referee for pointing this out to me.
See Black (1971, p. 633) for one such proposal.
This has been proposed by McKay (2006, p. 138).
To see that this reading is available imagine (6) as being uttered in a context in which some lecturers met to discuss the results of the final evaluation, while some students met to plan an end-of-the-year party, and the person who utters (6) is explaining why the lecturers complained that the students were too loud.
This is reported in Quine (1950, sec. 38).
See Boolos (1984, p. 57, fn7) for details.
And see also the discussion of pseudo-singularity in general in Sect. 5.1.1.
An analogous example is given in Linnebo and Nicolas (2008) and I follow their discussion. However, they do not tackle the issue of non-set-sized denotations.
In what follows, I use ‘group’ in its technical sense.
See Ben-Yami (2013, p. 89).
An analogous example was originally proposed in Linnebo and Nicolas (2008).
See Oliver and Smiley (2016, Chap. 10) for more details on these two analyses of lists.
Moreover, the data involving failures of substitutivity is compatible with articulation falling on the side of the semantic value of the terms which have it (and a fortiori with treating them as referring expressions). I believe more needs to be said to justify placing articulation outside of their semantic interpretation.
See Ben-Yami (2013, p. 97).
See, for instance, Resnik (1988, p. 77).
Naturally, the possibility of endorsing PL for reasons not considered in this article cannot be ruled out, but, to the best of my knowledge, the main reasons appealed to in the literature in defence of PL have been examined.
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Acknowledgements
This paper profited from feedback by Joan Bertran-San Millán, Stefan Krämer, Stephan Leuenberger, Øystein Linnebo and Adam Rieger, and by audiences in Glasgow, Leeds, Madrid and Prague. Moreover, I wish to especially thank Michele Palmira and Bruno Whittle, whose comments led to a substantial improvement of the manuscript. Finally, I am grateful to the anonymous referees who reviewed this work for their thorough and insightful feedback.
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The research for this paper was funded by the Grant AH/L503915/1 of the Arts and Humanities Research Council and the Grant GA18-00113S of the Czech Science Foundation.
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Grimau, B. In defence of Higher-Level Plural Logic: drawing conclusions from natural language. Synthese 198, 5253–5280 (2021). https://doi.org/10.1007/s11229-019-02399-z
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DOI: https://doi.org/10.1007/s11229-019-02399-z