Semantic values in higher-order semantics

Abstract

Recently, some philosophers have argued that we should take quantification of any (finite) order to be a legitimate and irreducible, sui generis kind of quantification. In particular, they hold that a semantic theory for higher-order quantification must itself be couched in higher-order terms. Øystein Linnebo has criticized such views on the grounds that they are committed to general claims about the semantic values of expressions that are by their own lights inexpressible. I show that Linnebo’s objection rests on the assumption of a notion of semantic value or contribution which both applies to expressions of any order, and picks out, for each expression, an extra-linguistic correlate of that expression. I go on to argue that higher-orderists can plausibly reject this assumption, by means of a hierarchy of notions they can use to describe the extra-lingustic correlates of expressions of different orders.

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Notes

  1. 1.

    For simplicity, I ignore operators which form expressions other than sentences, such as singular term-forming functors on singular terms.

  2. 2.

    For some discussion of how far “up” we may be able to get using only natural language resources, see Rayo and Yablo (2001), Sect. X, Linnebo and Nicolas (2008), the ‘Appendix on Pairing’ by Lewis, Burgess, and Hazen in Lewis (1990), as well as Hazen (1997).

  3. 3.

    The assumption of monadicity, i.e. that every predicate of L 1 (other than ‘=’) is one-place, is purely for ease of exposition.

  4. 4.

    The argument to follow is Williamson’s (cf. 2003, p. 426).

  5. 5.

    There may be independent, informal reasons for regimenting talk of interpretations of predicates in terms of second-order vocabulary (cf. Williamson 2003, p. 453). For discussion and development of the higher-order approach to the problem of absolute generality, see also Boolos 1985; Rayo and Uzquiano 1999; Rayo 2006.

  6. 6.

    As is customary, I use uppercase letters for higher-order variables. For orders greater than 2, the order of the variable is indicated with a superscript. Note that I also use uppercase letters such as ‘P’ and ‘S’ as first-order meta-lingustic variables ranging over object language predicates and sentences, respectively.

  7. 7.

    I set aside here (and throughout the paper) the possibility of a substitutional semantics for second-order quantification.

  8. 8.

    The same conclusion can also be reached through a related but slightly simpler argument from cardinality considerations; see (Linnebo and Rayo 2012, Appendix B). Thanks here to Øystein Linnebo.

  9. 9.

    Strictly speaking, the situation is slightly more complicated, but the crucial point remains: as long as we allow for the possibility of giving an adequate semantics for whatever language we use in constructing a semantic theory, we keep having to introduce additional ideology of higher and higher orders (cf. Linnebo 2006, 152f, n8). For a detailed examination of these issues, see (Rayo 2006).

  10. 10.

    If we also allow for a single language containing quantifiers of every finite order, and if we allow that for that language, too, we can construct an adequate semantics, we have to go beyond the hierarchy of finite orders of quantification. Since the desire to deal with absolute generality by itself does not seem to force us to allow for such a language, I set this (considerable) issue aside for the purposes of this paper. For discussion, see (Linnebo and Rayo 2012).

  11. 11.

    It is worth emphasizing that by speaking informally of a sentence’s truth-value I do not mean to commit myself to the claim that the best semantic treatment of sentences assigns to them a special sort of object, namely a truth-value. Indeed, the most natural implementations of the kind of semantic views I shall suggest to the higher-orderist do not do so, but take a shape closer to the Davidson–Tarski tradition in which we simply specify truth-conditions for sentences without assigning to them objects as their truth-values.

  12. 12.

    Unlike our above formulations, (SC1) implies that expressions make the same contributions to all sentences in which they occur. At least for the kinds of formal language we are concerned with, this further assumption appears unproblematic.

  13. 13.

    The analogous observation applies to the uniqueness condition in (SC1) and Linnebo’s Unique Existence.

  14. 14.

    My formulations lack the explicit implication that the contributions of monadic first-order predicates form a ‘distinctive kind’. If they do, this will presumably follow from the description of what they are like.

  15. 15.

    It may be worth emphasizing that the concession I make here for the sake of argument is substantive, and that it may also be possible to develop a tenable version of higher-orderism that endorses (P1+) and (N+) but blocks Linnebo’s objection at a later stage. I make the concession here to bring into clearer view the premise in Linnebo’s argument that I will argue the higher-orderist can reject.

  16. 16.

    This way of viewing the matter was suggested to me by Robbie Williams.

  17. 17.

    Note that (S-den n ) is a somewhat peculiar schema in that, in contrast to more familiar schemata such as ‘Fa’ or ‘\(p \rightarrow q\)’, its instances have different internal syntactic structures.

  18. 18.

    I shall henceforth suppress the parenthetical qualifications. We may assume the meta-language to extend the object language so that the relevant translation is simply the identity-mapping.

  19. 19.

    This of course holds only for extensional languages; for intensional languages, we should have to make use of intensional operators to construct ‘=’-like expressions for non-names.

  20. 20.

    We may also read ‘\(\hbox{den}_{2}^{i^2}\)’ as ‘\(\hbox{apples}^{i^2}\) to’. Unsurprisingly, there do not seem to be any independently familiar expressions of English corresponding to the ‘denotation’-functors for higher-order predicates.

  21. 21.

    It is worth mentioning that the higher-orderist may have independent reasons to use this kind of argument to explain the intuitive appeal of claims he rejects. In particular, it seems intuitively plausible that there is a general notion of existence corresponding to the hierarchy of existential quantifiers of higher and higher orders. Here as above, it seems to me that the natural strategy for the higher-orderist is to claim that what underlies the intuition is the analogy in inferential profiles between the quantifiers combined with implicit nominalization.

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Acknowledgments

I am grateful for many very helpful discussions with and comments from John Divers, Robbie Williams, Joseph Melia, Benjamin Schnieder, Alex Steinberg, Robert Schwartzkopff, Nick Haverkamp, Mirja Holst, and Øystein Linnebo, as well as two anonymous referees for this journal.

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Krämer, S. Semantic values in higher-order semantics. Philos Stud 168, 709–724 (2014). https://doi.org/10.1007/s11098-013-0157-z

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Keywords

  • Higher-order quantification
  • Semantics
  • Semantic values
  • Inexpressibility
  • Absolute generality