Let P be the property of being a property that does not have itself. By classical reasoning and certain minimal syntactic assumptions, the rule,

(PD)

F(a) ⇔ a has the property of being F,


allows us to infer a contradiction (see Schnieder, 2017: 326 for a full derivation). The minimal assumptions are to the effect that the right-hand side of (PD) has a face-value, relational structure, ‘has’ acting as a two-place predicate and ‘the property of being F’ being a singular term.

The derivation of a contradiction from (PD) is different from the usual derivation of a contradiction in Russell’s paradox, where the contradiction is usually derived with the help of a comprehension scheme, like,

(CS)

For any meaningful predicate ‘F’, there is a set G such that G = {x: Fx} (the set of every x such that Fx).


However, given that every expression of the form, ‘the property of being F’ is a proper singular term, any meaningful instance of (PD) will, by existential generalization, have as a trivial consequence that there is a property associated with the predicate. The face-value analysis of property ascriptions is therefore more in line with the comprehension scheme than it may first appear.

Those who reject the face-value, relational analysis of property-ascriptions typically take the right-hand side of (PD) to be synonymous and identical in form to its left-hand side. This synonymy claim is an extreme form of deflationism about properties, akin to the redundancy theory of truth, which similarly imputes a strong synonymy claim about sentences and their corresponding truth-ascriptions. Both views were first advanced by Ramsey (1925).

Contemporary, self-professed deflationists about properties, like myself (Båve, 2015), and commentators like Schindler (2022), rather define deflationism about properties as a weaker equivalence claim about predications and their property-ascribing analogues. For instance, in Båve (2015), I held that they are “analytically equivalent”, which means that a necessary condition for being competent with ‘property’ is that one be disposed to infer between them (see 2015: 40, and, for my preferred understanding of such competence-grounding “dispositions”, see Båve, 2020). Analytic equivalence in this sense, however, does not entail synonymy.

A detailed discussion about related matters and a detailed defence of the face-value analysis of property-ascriptions can be found in a long series of papers by Benjamin Schnieder (see his 2005, 2006a, 2006b, 2010a, 2010b, 2017). A major reason for accepting the face-value analysis and rejecting the synonymy claim is that the latter is difficult to square with our inferential practice, e.g., our tendency to infer from, ‘a has the property of being F’ to, ‘a has some property’, and so on. Schnieder (2017: §6) compares the synonymy claim with Quine’s “virtual set theory”, on which ‘Fa’ and ‘a ∈ {x: Fx} are synonymous (cf. Quine, 1963: 15–21, 1970: 69–72). When so interpreted, it is not legitimate to infer from the latter sentence to, ‘∃y(a ∈ y)’. Still, Schnieder is open to the possibility that ‘property’ is ambiguous, so that in one sense, its meaning is given by the strong synonymy claim, and in the other, by the face-value analysis plus the weaker claim that his introduction and elimination rules for ‘property’ are “constitutive of our conception of properties” (more on this phrase below). But even if there is a sense in which the contradiction involving P cannot be inferred, we still face a paradox if there is another sense in which it can.

In his (2017), Schnieder offers an ingenious solution to the paradox, consisting in weakened inference rules governing ‘property’. Instead of (PD), Schnieder proposes the pair:

(PE)

a has the property of being F ⇒ F(a),

(PI)

F(a), ◇(the property of being F exists) ⇒

 

a has the property of being F


(2017: 328). This weakening of (PD) handles both the Russell variety and the Curry variety of the paradox, and has further attractive features:

(i)

it is simple,

(ii)

it does not rule out the pathological cases by explicitly mentioning ‘pathological cases’ or by mentioning particular pathological cases, like P,

(iii)

(PI), whose additional condition is what distinguishes these rules from the simpler (PD), has an independent motivation—and, hence, is not ad hoc.


The independent motivation is as follows (see Schnieder, 2017: 329). It is independently plausible that if an abstract object exists, it exists necessarily. Specifically, it holds necessarily that if a property exists, it exists necessarily. Formally,

(□EXIST)

□∀F(∃x(x = the property of being F) →

 

□∃x(x = the property of being F)).


But from this principle, the following comprehension principle follows (in S5):

(□COMPREHENSIONWEAK)

□∀F(◇∃x(x = the property of being F) → 

 

∃x(x = the property of being F)).


The latter principle, moreover, is consistent, in contrast to more standard comprehension principles, associated with the unrestricted rule (PD). And the weaker comprehension principle is the only one we can derive from Schnieder’s weaker rules. In this way, his weaker rules are motivated by a generally accepted principle about properties, formally expressed as (□EXIST).

Unfortunately, I will argue, the account suffers from a simple but insuperable defect in that it cannot account for reasoning with non-pathological properties. For consider how we might be justified in believing that roses have the property of being red. In order to infer this claim from the claim that roses are red, we need to be justified in believing also that this property can exist. This looks like something that should come easy (barring nominalist scruples): can’t we just infer from the claim that roses are red that they have the property of being red, from which it follows that that property can exist? Not with Schnieder’s rules.

A possible remedy is the assumption that we have an ungrounded justification in believing that the property of being red can exist. However, we would like to generalize this claim, for the same should then hold for the property of being large, being prime, etc. And now, the question is how to make this generalization without it entailing that we also have ungrounded justification in believing that P can exist. If we say that this justification holds only for non-pathological cases, or that it holds for all cases except for P and the Curry variety (and presumably further ones), then features (ii) and (iii) of Schnieder’s solution are lost (and, to some extent, also feature (i)).

We also cannot very well generalize by saying that for every property that can exist, we have an ungrounded justification for believing it can exist. Such selective justification, dependent on the truth of the proposition in question would be unheard of in epistemology. It would require significant defence, which I doubt Schnieder would want to be burdened with as part of his theory of properties. The upshot is that while we can decide, for P and other pathological cases, that they cannot exist, we cannot, with Schnieder’s rules, justify our beliefs about ordinary, non-pathological properties.

Schnieder’s main aim is to argue that our “ordinary conception of properties” harbours the restricted rule (PI) above, and is therefore consistent. (Note that he thus uses ‘ordinary conception’ in a non-standard way, since conceptions are usually thought of as sets of beliefs that are not necessarily analytic. Still, I will follow his terminology in what follows.) He thus opposes the “inconsistency theory of properties”, which is like Eklund’s (2002) inconsistency theory of truth, except concerning the notion of property instead of truth. On Eklund’s theory of truth, the meaning-constitutive principle for ‘true’ is the unrestricted truth equivalence schema, which Eklund takes to be inconsistent, thus making truth an inconsistent concept (other defences of such “inconsistency theories” include Azzouni, 2007; Båve, 2012, 2018; Patterson, 2009; Scharp, 2013: Ch. 2). The analogous view about ‘property’ is that (something like) (PD) is meaning-constitutive of ‘property’, making the ordinary notion of property an inconsistent notion.

Note that this disagreement does not (at least not directly) concern justification, but rather the non-normative question how to explain facts about what we actually accept/believe. Let us therefore set justification aside in what follows and consider the analogous argument concerning actual reasoning: how to explain how speakers arrive at the belief that roses have the property of being red, given that our conception of properties is governed by (PI)? The only possibility seems to be to assume that they make an ungrounded assumption that the property of being red can exist. But how can this be generalized without excluding pathological cases by fiat? It cannot, and, therefore, the theory fails as an account for our actual conception of properties. (Note that if this is right, then a new term, ‘property*’, governed by Schnieder’s rules, would also fail as a replacement, should our ordinary conception be inconsistent, since we cannot infer, for any x or F, that x has the property* of being F from the claim that x is F.)

If one appeals to an ungrounded disposition in speakers to believe that certain properties can exist, this ought to be considered part of our “ordinary conception”, since it would be arbitrary to think inference rules and unconditional acceptance rules (axioms) differ in this regard. When discussing why ordinary speakers are “pulled in” by the paradox (see Eklund, 2002: §§3–4), Schnieder says that they “accept the idea that for any F, the property of being F could exist” (p. 338). But taking this idea to be part of our ordinary conception commits one to taking that conception to be inconsistent. Taking it to fall outside of our ordinary conception, on the other hand, would not just be arbitrary; it would also violate the plausible claim that the acceptance facts constituting the meaning of an expression e should explain all acceptance facts about e, together with psychological laws that do not concern e (see Horwich, 2005: 37ff.). (Of course, I am here making several controversial assumptions about acceptance-dispositions being meaning-constitutive, etc., but these are assumptions shared by Schnieder, which are presupposed in any discussion about which principles constitute our “ordinary conception”, in Schnieder’s sense.)

Perhaps the most plausible response on Schnieder’s behalf is the claim that we have an ungrounded but defeasible disposition to accept every instance of ‘the property of being F can exist’. (For the discussion of justification, one could analogously propose that we are defeasibly justified in accepting every such instance.) This would be adequately general, it would explain speakers’ belief that non-pathological properties can exist, and it would be consistent with their denying that pathological ones cannot (assuming they have worked through the relevant paradox).

However, on this view, it is defeasibility, rather than the restriction in (PI) that does the real work. This view, which takes us to have a defeasible disposition to accept instances of ‘the property of being F can exist’, has the same results as, but is more complex than, a theory that simply says that we have a defeasible disposition to accept instances of (PD). It accommodates the same data, but needlessly posits a more complex inferential route to the relevant beliefs. And there seems to be no reason to insist that defeasibility is more plausibly appealed to in unconditional acceptance rules (axioms) than in conditional ones (inference rules).

If we appeal to defeasibility at all, then, we should do so directly, in the statement of the inference rules for ‘property’, and avoid the idle restriction relating to possible existence. The resulting theory would arguably yield a plausible account of ordinary property discourse, given its ability to accommodate, by simple means, data about actual reasoning both with pathological and non-pathological properties. The purpose of this note is not to support a positive theory, however, but to make the negative point that while the feature Schnieder proposes successfully handles the bug of paradox, it thereby also causes general breakdown in failing to handle ordinary, non-pathological cases.