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A paradox about sets of properties

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Abstract

A paradox about sets of properties is presented. The paradox, which invokes an impredicatively defined property, is formalized in a free third-order logic with lambda-abstraction, through a classically proof-theoretically valid deduction of a contradiction from a single premise to the effect that every property has a unit set. Something like a model is offered to establish that the premise is, although classically inconsistent, nevertheless consistent, so that the paradox discredits the logic employed. A resolution through the ramified theory of types is considered. Finally, a general scheme that generates a family of analogous paradoxes and a generally applicable resolution are proposed.

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Notes

  1. Throughout I use ‘property’ in the sense of a (singulary) ‘attribute’, ‘feature’, or ‘trait’, in their ordinary senses. It is arguable that there are distinct but necessarily co-extensive properties in the relevant sense (e.g., triangularity and trilaterality, or being a valid formula of first-order logic and being a theorem of first-order logic), so that properties are not determined by their metaphysical intensions, i.e., by their associated functions from possible worlds to extensions.

  2. Here by ‘attribute’ I mean an n-ary relation-in-intension for n ≥ 1 (including a property regarded as a singulary relation-in-intension), an n-ary propositional function for n ≥ 0 (including a proposition regarded as a 0-ary propositional function), or any similarly intensional entity, such as a corresponding concept. (Impredicative definition of an extensional entity, such as a class or a truth value, does not pose the same difficulty.) This notion of impredicativity is a special case of, but stricter than, the broader notion, largely based on Henri Poincaré’s vicious-circle principle (1906), to wit, that of introducing (“defining”) a particular element of a class by quantifying over the elements of that class. Although it is not impredicatively defined in the stricter sense, in this broader sense the putative set involved in Russell’s paradox (the set of all and only those sets that are not elements of themselves) is said to be “impredicatively defined.” However, as F. P. Ramsey pointed out (1925, p. 204), so also is the idea of “fixing the reference” (Kripke) of a name by a superlative definite description, e.g., ‘the shortest spy’, ‘the first child to be born in the 22nd Century’, ‘the second shortest spy’, etc. (I thank C. Anthony Anderson for supplying this reference.) Compare Gödel 1944. See note 17 below. The stricter sense of ‘impredicative’, which is likely what is usually meant in the relevant literature, is uniformly adhered to throughout the present essay.

    The notion of definition by abstraction involved in the stricter notion of impredicativity is to be sharply distinguished from the distinct notion that goes by the same moniker (and which, as Frege showed, is in fact fictitious) of purportedly defining a function (e.g., the cardinality function) by defining what it is for arguments to the function to yield the same value. See my (2018).

  3. A logician objects that in order to establish that a particular rule of inference is logically fallacious one must show that it generates actual inconsistency. This objection confuses formal fallaciousness (e.g., satisfiability of the negation) and inconsistency (unsatisfiability). The sentence ‘∃x∃y(x ≠ y)’ has models and also has counter-models, and is thus both consistent and (taken as an axiom) fallacious, indeed classically invalid. Arguably, derivation of the classically valid ‘∀xFx ∴ ∃xFx’ and proofs of certain classical theorems (‘∃x(Fx ∨ ~Fx)’, ‘∃x(x = a)’, and the like), although consistent, also involve intuitively fallacious inference rules (classical UI, etc.).

  4. The paradox of property sets is not to be confused (as several readers have) with the unit-set variant of Russell’s paradox, to wit, the paradox of the set r = {{x}| {x} ∉ x}. Assuming that r exists, {r} is an element of r iff it is not. This variant of Russell’s paradox (which has been employed to refute Frege’s insufficient weakening of his Basic Law V in response to Russell’s original paradox) is a garden-variety set-theoretic paradox that turns on the inconsistency of naïve unrestricted set comprehension. By contrast, the paradox of property-sets explicitly invokes sets of properties (not sets of sets), and is independent of naïve unrestricted set comprehension. Significantly, the property-sets paradox invokes no comprehension principle not sanctioned by applied classical logic as based on the simple theory of types.

  5. The non-logical predicate ‘∈3’ is not the ‘∈’ of standard set theory, which is of altogether different type. It is not uncommon for philosophers to employ set-brace notation in combination with predicate letters to represent a set of properties, as in ‘{F, G, H}’. This notation implicitly employs ‘∈3’. The standard membership predicate ‘∈’ may be used instead of ‘∈3’ to formalize the property-sets paradox, by postulating that properties of individuals are special individuals governed by a property-comprehension schema sufficient to generate R—insofar as this can be achieved without also generating inconsistency. The present essay investigates the ramifications of the property-sets paradox for the more familiar higher-order intensional logic with λ-abstraction and is therefore formalized using that apparatus.

  6. The logic of ‘’ may be taken to be essentially that of Whitehead and Russell (1927), giving descriptions the narrowest possible scope, with the exception that definite descriptions are taken to be designators and a free logic is employed in connection with them. First-order free-logical UI (∀-Elim) licenses the inference from \({ \ulcorner }\forall \alpha \phi_{\alpha } { \urcorner }\) and the supplementary premise \({ \ulcorner }\exists \gamma (\gamma = \beta){ \urcorner }\) to ϕβ, where the variable γ does not occur free in the singular term β and ϕβ is the result of uniformly substituting free occurrences of β for the free occurrences of the variable α in ϕα. First-order free-logical EG (∃-Intro) licenses the inference from ϕβ and the same supplementary premise ⌜∃γ( γ = β)⌝ to ⌜∃αϕα⌝. (First-order free-logic involves similar modifications of ∀-Intro and ∃-Elim, but these are not relevant in connection with definite descriptions.) Stricter adherence to Whitehead and Russell would also serve the present purpose but introduces needless complexity.

  7. If the property R is identified with its corresponding propositional function, the lambda-abstract that ‘R’ abbreviates may itself be defined by means of the third-order definite description ‘Z∀y[ Zy = 2 ∃X(X∈3y & ~Xy)]’, where ‘ = 2’ is a dyadic logical predicate for identity between propositions (as well as between properties of individuals). See (Church 1974a, 1974b), pp. 29–30. Alternatively, the definite-description operator ‘’ is definable in terms of lambda-abstraction together with the higher-level function that assigns to any property the only object that has that property if such exists, and is undefined otherwise. Lambda-abstraction underlies all variable binding and is therefore more basic than definite-description formation.

  8. I thank C. Anthony Anderson, Saul Kripke, Romina Padro, and Teresa Robertson for discussion of the issues in this paragraph.

    There is an alternative way of looking at the matter. Like the unit set of a property, which bears a special relationship to its property element, any meaningful English adjective bears a special relationship to the property it expresses. Yet adjectives are to be treated as individuals rather than as entities that are of higher logical type than properties of individuals (whatever that would mean). The relation between an English adjective and the property it expresses is relevantly analogous to the relation between the unit set of a property and its element. The analogy is sufficient to support permitting the inclusion of sets of properties among the entities over which ‘x’, ‘y’, and ‘z’ range. In fact, sets of properties can simply be replaced by their canonical expressions, in combination with a suitably adjusted reinterpretation of set-theoretic notation (the predicate ‘∈’ for set-membership, set-theoretic braces, etc.)—so that for example ‘{R}’ is taken to designate itself. So interpreted there is no legitimate objection to letting ‘{R}’ designate something in the universe of individuals.

    It is worth noting also that it is critical to the proof of Frege’s theorem to treat classes of concepts X under which individuals fall as themselves individuals.

  9. Alternatively, Ex may be taken to be ‘∀X∃y(∀Z[Z∈3y ↔ X =2 Z])’. See note 6.

  10. In particular, ‘{F} = {F}’ fails if ‘{F}’ is an improper description. See note 6. More surprising, as we shall see the negation of Ex is a truth of classical third-order logic with free logic for definite descriptions.

    By Cantor’s theorem, if the universe of individuals is a set, then there are more sets of individuals than there are individuals. There are at least as many properties of individuals as sets of individuals, since for each set there is the unique property of being an element thereof. But Ex entails that there are at least as many individuals as there are properties of individuals, since according to Ex for each property of individuals a unique individual is the unit set thereof. Thus Ex has the unsurprising consequence that the universe of individuals is a proper class.

  11. Third-order free logic analogously modifies the classical logic of the quantifiers to take account of monadic predicates that do not designate any element of the universe over which the monadic-predicate variables range.

  12. See Church 1976; Russell 1908; and Whitehead and Russell 1927, *12, pp. 161–167. Church’s formulation of ramified type theory is followed here. The axioms of reducibility of Whitehead and Russell 1927 entail that every level n property for n ≥ 1 is co-extensive with a level 1 property. It is often said—following Chwistek (1921), Copi (1950), and Quine’s commentary on Russell 1908 (Quine 1967, p. 152)—that the axioms of reducibility defeat the purpose of ramified type theory by reinstating the paradoxes of impredicativity. The claim is incorrect, however, and Russell had explicitly noted as much in 1908 (last paragraph of section V). See also Church 1974b, p. 356; Church 1976, p. 758; and Myhill 1979.

  13. Ramified type theory does not provide the only possible resolution that employs stratification. An alternative resolution is to stratify just the set-membership relation. This theory posits a hierarchy of levels of sets: level 1 sets, whose elements that are properties are restricted to predicatively defined properties; level 2 sets, whose elements that are properties are restricted to ordinary properties and level 1 properties; level 3 sets, and so on. Presented with a choice between the two ramified resolutions, the present author believes that the ramified-type theoretic resolution is decidedly preferable on philosophical grounds. The rival resolution is part of a more piecemeal approach to paradoxes like that of (Russell 1903) and others of that ilk. Furthermore, the limitation it imposes on collecting properties into ordinary sets has little or no intuitive support.

  14. Thus un-superscripted ‘F’ could be taken as shorthand for ‘(λx[∃nFnx])’ and un-superscripted ‘p’ as shorthand for ‘∃npn’, except that the meanings of such expressions in ramified type theory with ‘λ’ involve vacuous quantification and are not what is intended here.

  15. The intermediate position has the further advantage that neither the liar sentence l = ‘∃p(l expresses p & ~ p)’ nor the truth-teller sentence t = ‘∃p(t expresses p & p)’ can be said to express any proposition, since neither sentence expresses any proposition of any particular level. The position also has the significant disadvantage that, although Ex can be maintained, it cannot be said that Ex expresses any proposition, since it too does not express any proposition of any particular level.

  16. This analysis reveals that the paradox need not be regarded as invoking the notion of logical product (conjunction). Cf. Salmón forthcoming and Robertson Ishii and Salmón 2019. The two “stone caster” paradoxes discussed there have an analogous analysis in terms of the encoding scheme.

  17. See note 12. The axioms of reducibility entail that in each case the property is co-extensive with a level 1 property.

    See note 2. The dyadic predicate ‘∈’ for set-membership is a primitive of the language of set theory. It is arguable, however, that set-abstraction (the set of individuals z such that) is conceptually prior to set-membership. If it is, then set-membership is properly analyzed as (λxy[∃Z(Zx & y = {w| Zw})]). This would have the result that (λy[y∉y]) is itself impredicatively defined (in the stricter sense used here).

  18. For a contrasting view of the matter see Martin (1977).

  19. Typically (not always), the third logical possibility, that whatever encodes ℜ also encodes some property not co-extensive with it, can be stipulated not to obtain.

  20. Meinongian theories face a problem, discovered by Romaine Clark, analogous to the property-sets paradox. See (Rapaport 1978). A resolution proposed by Alan McMichael and adopted by Edward Zalta, when adapted to the present paradox, bans the abstraction of any property that involves the membership relation ∈3 between a property and a set. See (Zalta 1983), p. 160. The purported resolution allows one to assert that the property of primality itself has specific properties and is an element of {primality}, while leaving no means for expressing (let alone inferring) that primality has the particular property of being an element of {primality}. This move is an evasion of the paradox rather than a genuine resolution.

  21. Although no such characterization has been attempted here, it would be useful to have an independent specification of exactly which subclass of impredicatively defined attributes must be rejected to avoid inconsistency. (Feferman 2005) provides a comprehensive survey of impredicativity.

  22. Russell says that either φ is impredicative or f does not exist, although “it may often be difficult to decide which of them to choose” (p. 35; p. 143).

  23. Priest 1994 and 1995 demonstrates, through judicious selections to fill the roles of φ and f—and in some cases with some finesse—that the range of paradoxes exemplifying generalizations of Russell’s scheme is remarkably broad. (I thank C. Anthony Anderson and Graham Priest for alerting me to this.).

  24. Since the identity function exists, Russell concludes that (λy[y ∉ y]) is impredicative. See notes 2, 17, and 22.

  25. An interesting exception is Berry’s paradox, presented in Russell 1906. It may be set out as follows. Say that something is concisely English-definable iff it is designated in unaltered English by a non-indexical definite description consisting of twelve words or less (e.g., as three is designated in English by ‘the third positive integer’). Since the English lexicon is finite, there are finitely many concisely English-definable positive integers. The paradox is that the smallest positive integer not concisely English-definable is evidently concisely English-definable by d = ‘the smallest positive integer indefinable in twelve English words or less’. See note 23. Priest 1994 (p. 29) shows how Berry’s paradox may be regarded as exemplifying Russell’s scheme, and asserts (p. 33) that Russell’s law is inapplicable. It should be noted, however, that a variant of Grelling’s paradox shows that there is a problem forming English descriptions that, like d, invoke designating in English. Cf. Church 1976, pp. 756–757. Russell’s law solves paradoxes of the relevant family by precluding ρ. By contrast, insofar as the English lexicon is finite, then (as may be expressed in a suitable metalanguage for English) there does exist a smallest positive integer not concisely English-definable. However, it cannot be correctly said of that integer in English that it is designated in English in the manner proposed.

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Acknowledgements

I am grateful to C. Anthony Anderson, Harry Deutsch, Saul Kripke, Romina Padro, and especially Teresa Robertson Ishii for discussion and comments. I am profoundly indebted to the late Alonzo Church for his superb tutelage on many of the topics of the present essay. The essay encountered an inordinate amount of intellectually improper resistance. I am grateful to Julien Murzi and the other Synthese editors for their academic integrity.

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    Nathan Salmón

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This article belongs to the topical collection “Non-Classical Approaches to Paradox”, edited by Julien Murzi, Lorenzo Rossi, and Brett Topey.

Appendix: Russell’s alternative scheme

Appendix: Russell’s alternative scheme

Russell (1907, p. 35; reprinting p. 142) provides the following scheme that is even broader in scope than the encoding scheme: (in addition to a particular kind K) (i) a putative property ϕ; (ii) its corresponding putative set w = df {y| ϕy}; and (iii) a function f purportedly from the power set P(w) to w and such that ∀z[z ⊆ w → f(z) ∉ z]. (As stipulated, f is surjective, i.e., onto w, since every element x of w is the value of f for w − {x}.) The paradox is that f(w) “both has and has not the property ϕ.” The resolution is that f(w) does not exist; hence either w does not exist, or else it does but f does not exist as stipulated (e.g., the stipulation that f is into w must be weakened so that ∀z[z ⊂ w → f(z) ∈ w]). Some cases go one way, some the other.Footnote 22

Russell asserts that “this generalization is important, because it covers all the contradictions that have hitherto emerged in this subject.”Footnote 23 Russell’s paradox emerges on this scheme as follows: Let K be the kind set. Let ϕ be the property (λy[y ∉ y]), so that w is the putative Russell set. Let f be the identity function. Then f(w) is also the putative Russell set.Footnote 24 Though Russell does not mention it, the Russell-Myhill paradox is obtained on this scheme as follows: Let K be the kind proposition; let ϕ be the putative property (λq[∃Φ(q = ∀p(Φp → p) & ~ Φq)]); and let f(x) = ∀p(p ∈ x → p). The putative proposition that ∀p(p ∈ w → p) both has ϕ and lacks ϕ. The property-sets paradox does not fit as neatly into Russell’s scheme, but a close relative emerges as follows: Let K be the kind set of properties. Let ϕ be the putative property R. Let f be the putative function that assigns to any set x of sets having R the particular unit set {(λy[y ∈3x])}. Then f(w) = {(λy[y ∈3w])}, whose sole element is an equivalent surrogate for R. Like {R}, assuming it exists {(λy[y ∈3w])} both has and lacks R.

Many paradoxes that exemplify Russell’s scheme, although not all, also exemplify the encoding scheme. This is generally accomplished through the following definitions:

Dencode*::

y encodes* X = df ∀z(Xz → ϕz) & y = f({z| Xz}).

Dℜ*::

ℜ* = df (λy[∃X(y encodes* X & ~ Xy)]).

Dρ*::

ρ* = df f(w).

Dencomp*::

x encompasses* y = df ∃Z(x encodes* Z & Zy).Footnote 25

By Dencode*, an entity of kind K encodes* every property co-extensive with any property it encodes*. Also by Dencode*, f(w) encodes* ϕ (assuming both exist). Furthermore, f(w) does not encode* any property not co-extensive with ϕ. For let F be a property such that ∀x(Fx → ϕx) but ~ ∀x(ϕx → Fx). By the stipulations on f, f({x| Fx}) ∈ w whereas f(w) ∉ w. (This remains true even if the stipulations on f are made consistent as long as ∀z[z ⊂ w → f(z) ∈ w] and ∀z[z ⊆ w → f(z) ∉ z].) It follows that f(w) ≠ f({x| Fx}), so that f(w) does not encode* F. Given the stipulations on f, ℜ* is co-extensive with (λy[∃z(y = f[z])]), which is co-extensive with ϕ (assuming all three properties exist). It follows that ρ* encodes* ℜ* and does not encode* any property not co-extensive with ℜ*.

On Russell’s scheme a variant of encoding is explained in terms of a variant of ℜ, whereas on the encoding scheme ℜ itself is directly defined in terms of encoding. For example, on Russell’s scheme a broad semantic or semantic-like relation of expressing* between heterological adjectives and properties is explained in terms of heterologicality, whereas on the encoding scheme heterologicality is defined in terms of semantic expressing. Unlike Russell’s scheme, the encoding scheme reflects how paradoxes of the relevant family arise.

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Salmón, N. A paradox about sets of properties. Synthese 199, 12777–12793 (2021). https://doi.org/10.1007/s11229-021-03353-8

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