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Inexpressible properties and Grelling’s antinomy

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An Erratum to this article was published on 31 July 2010

Abstract

The paper discusses whether there are strictly inexpressible properties. Three main points are argued for: (i) Two different senses of ‘predicate t expresses property p’ should be distinguished. (ii) The property of being a predicate that does not apply to itself is inexpressible in one of the senses of ‘express’, but not in the other. (iii) Since the said property is related to Grelling’s Antinomy, it is further argued that the antinomy does not imply the non-existence of that property.

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Notes

  1. The view is denied, e.g., in Schiffer (2003, p. 71) and Hofweber (2006).

  2. As has been pointed out, e.g., by Strawson (1953/54, p. 256f) and Peterson (1986, p. 298).

  3. For a more detailed account of the semantics of nominalized property designators see Schnieder (2005; 2006).

  4. See, for instance, Hofweber (2006). He argues at length that there are no inexpressible properties, simply presupposing an understanding of what it is for a predicate to express a property instead of explicating the relevant notion of expressing.

  5. Cp. Hofweber (2006, pp. 172–73).

  6. However, the distinction plays a role for the question of whether there are properties which are—even if not strictly inexpressible—inexpressible in English. A standard argument for there being such properties is the Cardinality Argument (see, e.g., Tye 1982, p. 53): there are only countably many English expressions, but uncountably many properties (for any real number x, there is the property of being greater than x). So, there are more properties than English expressions, and hence some properties are inexpressible in English. As Hofweber (2006, pp. 172–78) points out, this argument relies on the first of the two senses of ‘expressible’ distinguished above and it cannot show that there are properties inexpressible in English in the second sense of ‘inexpressible’. Moreover, it is hard to see whether a modified version of the argument could achieve this goal.

  7. For similar proposals see, e.g., Cartwright (1966, p. 685) and Soames (1998, p. 330).

  8. See, e.g., Montague (1970), p. 193).

  9. ‘Circumstances of evaluation’ is used in Kaplan (1977, p. 494) sense.

  10. As defended, e.g., by Jackson (1998, p. 126f.). A modal individuation of properties is compatible with several views on their nature; they might, e.g., be functions (Montague 1974, p. 152), sets of possible individuals (Lewis 1986, p. 55), or entities sui generis.

  11. Just notice that terminological care is highly important here: as I use ‘apply’ this predicate signifies a relation holding between linguistic entities and entities of any kind, while ‘exemplify’ signifies a relation holding between entities of any kind and properties. Hence, Grellingness is a word-property while the Russellian property would be a property of properties—if it existed.

  12. Cp. Hofweber (2006, pp. 186–87). While most philosophers who accept (C) deny the existence of both properties, Hofweber endorses the existence of both.

  13. I cannot discuss the more radical solutions to the paradoxes which, e.g., tamper with classical logic (see e.g. Field 2004, or Priest 2007) or reject unrestricted quantification (Fine 2006). Even if one adopts one of them, it is still interesting to see what the direct consequences of the paradoxes would be on a more conservative view.

  14. This is, e.g., suggested by Goldstein (2003). However, as most philosophers do, Goldstein construes Grelling’s Antinomy with the artificial term ‘heterological’. He argues that it cannot receive a meaning from the stipulation ‘let “heterological” apply to all and only those terms that do not apply to themselves’. This may well be correct; but Grelling’s Antinomy is not essentially concerned with defined predicates. As Quine (1961, p. 6) emphasized, the problem already arises with the natural language predicate ‘does not apply to itself’, which need not be defined by stipulation.

  15. See, e.g., von Wright and Henrik (1960, pp. 7–8) (he argues that because Grellingness cannot be expressed by any predicate, it does not exist; however, his argument suffers from a confusion of naming and expressing a property).

  16. Many of the different formulations of Grelling’s paradox mention not only predicates but also non-linguistic entities corresponding to them, such as concepts (e.g., Grelling and Nelson 1908), properties (e.g., Grelling 1936; Reach 1936, p. 106), or functions (e.g., Copi 1950). For some such versions it is a correct reaction to deny the existence of the relevant non-linguistic entity; see, e.g., Newhard (2005) for a demonstration that Grelling’s 1936 version of the antinomy requires us to deny the existence of a certain word-property. But, as the above version shows (used, e.g., in Martin 1968 and Chihara 1976), a version of Grelling’s paradox can be stated without recourse to any non-linguistic entities. For such a stripped-down version of the paradox which reveals its basic core it is of no help to deny the existence of some non-linguistic entity.

  17. See Thomson (1962, p. 104).

  18. Cp. Fine (1994, p. 13) and Sorensen (1996, p. 204). Yet another conception of words is what Kaplan called the common currency view; it is unclear whether, on that view, words could have had other meanings than their actual ones (cp. Kaplan 1990, p. 117f.).

  19. See, e.g., Tye (1981, p. 23) and Yagisaw (1997, p. 46).

  20. There are nominalizations of different forms (‘coyness’, ‘convergence’), but since it is debatable whether they are coreferential with the gerundive nominalizations (see the controversy in Moltmann 2004 and Schnieder 2007), I rely only on the latter in what follows.

  21. For a similar account see, e.g., Sosa (1993, p. 191).

  22. I am grateful to Kit Fine, Jussi Haukioja, Miguel Hoeltje, Wolfgang Künne, Øystein Linnebo, Panu Raatikainen, Moritz Schulz, Wolfgang Schwarz, Alex Steinberg, and to audiences in Berlin, Hamburg, Konstanz, München, and Paris, for comments and discussion.

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Authors and Affiliations

  1. Institut für Philosophie, Humboldt-Universität zu Berlin, Unter den Linden 6, Berlin, D-10099, Germany

    Benjamin Schnieder

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Correspondence to Benjamin Schnieder.

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An erratum to this article can be found at http://dx.doi.org/10.1007/s11098-010-9592-2

Appendix

Appendix

1.1 Proof of the inexpressibility of grellingness

For a rigorous presentation of the argument given in Sect. 2.2, notice first that the validity of DN implies the validity of its semantically descended counterpart:

DN* :

The property of being F exists → ∀x (x has the property of being F ↔ x is F).

DN* can be reached from DN via the following two valid schemata:

  1. (i)

    ‘the property of being F’ is a non-empty designator ↔ ‘the property of being F’ denotes the property of being F.

  2. (ii)

    ‘the property of being F’ denotes the property of being F ↔ the property of being F exists.

Abbreviating ‘the property of being a non-self-applying expression’ as ‘P*’, we can run the following derivation in first-order predicate logic:

1

(1)

P* exists

A

2

(2)

t expresses P*

A

2, Ex

(3)

∀x (t applies to x ↔ x has P*)

2, Ex

1, DN

(4)

∀x (x has P* ↔ x applies to x)

1, DN*

1, 2, DN, Ex

(5)

∀x (t applies to x ↔ x applies to x)

3, 4 FOPL

1, 2, DN, Ex

(6)

t applies to t ↔ t applies to t

4 ∀E

1, DN, Ex

(7)

t expresses P*

6,2 FOPL (RAA)

1, DN, Ex

(8)

∀y y expresses P*

6, ∀I

Remarks

The proof (some elementary steps are omitted) relies on three assumptions: on the validity of DN and E x, and on the existence of P*. Since first-order predicate logic is not a free logic, the existence of P* is already implicitly presupposed when the term is used. However, to make the assumptions of the argument vivid and to be able to directly apply DN*, it is explicitly assumed in line (1). DN* is needed for deriving line (4) from line (1); since DN* is an implication of DN, (4) depends on the latter.

According to line (8), there is no phrase that expresses P*. The rejected assumption (2) was that an arbitrarily chosen term t expresses P*, and it could be chosen from any possible language. Hence, P* is strictly inexpressible.

1.2 Proof of Thomson’s Theorem

Assume (i) a is a member of S, which (ii) bears R to all and only those members of S which do not bear R to themselves, where (iii) R is defined over S. Then we have:

1

(1)

a is a member of S

A

2

(2)

∀y (y is a member of S → (a bears R to y ↔ y bears R to y))

A

3

(3)

∀x (x is a member of S → (x bears R to x ∨ x bears R to x))

A

1,2

(4)

a bears R to a ↔ a bears R to a

2∀E; 1,2 MPP

1,3

(5)

a bears R to a ∨ a bears R to a

3∀E; 1,3 MPP

6

(6)

a bears R to a

A

1,2,6

(7)

a bears R to a

6,4 ↔E

1,2,6

(8)

a bears R to a & a bears R to a

7,8 &I

9

(9)

a bears R to a

A

1,2,9

(10)

a bears R to a

9,4 ↔E

1,2,9

(11)

a bears R to a & a bears R to a

9,10 &I

1,2,3

(12)

a bears R to a & a bears R to a

5,6,8,9,11 ∨E

So, from (i) to (iii) (line (3) is equivalent to (iii)), a contradiction is derived. This proves the Theorem.

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Schnieder, B. Inexpressible properties and Grelling’s antinomy. Philos Stud 148, 369–385 (2010). https://doi.org/10.1007/s11098-008-9329-7

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