Abstract
This paper argues that we should solve paradoxes for propositions (such as the Russell–Myhill paradox) in essentially the same way that we solve Russellian paradoxes for sets. That is, the standard, iterative approach to sets is extended to include properties, and then the resulting hierarchy of sets and properties is used to construct propositions. Propositions on this account are structured in the sense of mirroring the sentences that express them, and they would seem to serve the needs of philosophers of language and metaphysicians who rely on such propositions. The resulting account has limitations, comparable to those faced by the iterative approach to sets, but it is argued to be in important ways preferable to others that have been proposed.
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Notes
See Rosen (2010) for an account of grounding (i.e. the in-virtue-of relation) in terms of structured propositions.
Two notable antecedents are Charles Parsons’ approach to the liar paradox (1974), and a suggestion of George Bealer’s about how we should solve paradoxes for properties (1982). Parsons’ solution to the Liar is in terms of propositions, and he advocates seeing these as stratified in a way analogous to sets under ZFC. He does not however give a worked out account of propositions along these lines. (I believe that the account proposed here is the natural way of implementing his basic idea, even if the resulting picture seems to differ in certain respects from that which he paints; I will not discuss such differences because the paper is already long enough as it is). Bealer, on the other hand, suggests a solution to paradoxes for properties that is based on ZFC. The basic scheme is thus similar to that of the account proposed here. The details, however, seem to be rather different: see note 19. In addition, in Whittle (2017), I proposed an account of properties and propositions that is in certain respects similar to that of this paper. However, the account proposed here is different in many points of detail, and seems more straightforward (at least, it fits more comfortably with standard classical logic).
I use ‘collection’ rather than ‘set’ or ‘class’ to emphasize that in this informal presentation we are not relying on a technical notion to be understood in terms of some specific theory. Also, if S is a sentence, then by \(\langle S\rangle\) I mean the proposition that S expresses.
This principle should be understood as making a universal claim about any collections of propositions B and C.
The idea behind Fritz’s argument can be illustrated (as in his forthcoming) using arbitrary conjunction. (The official argument uses only standard connectives and quantifiers, but it is more intricate). Thus, for any plurality pp of propositions, we assume that there is a proposition \(\langle \bigwedge \textit{pp}\rangle\) that is the conjunction of these. The key assumption (which is a version of a standard principle about grounding) is that for any plurality ff of facts (i.e. true propositions), \(\langle \bigwedge \textit{ff}\rangle\) is immediately grounded precisely by the members of ff. It follows that for distinct pluralities of facts ff and gg, \(\langle \bigwedge \textit{ff}\rangle \ne \langle \bigwedge \textit{gg}\rangle\). We can then give a Russellian paradox of a familiar shape: by considering the plurality rr of facts f such that, for some facts gg, \(f = \langle \bigwedge \textit{gg}\rangle\), but f is not one of the gg. We get that \(\langle \bigwedge \textit{rr}\rangle\) is in rr iff it isn’t. For the official version of the argument, see Fritz (forthcoming) [and for further discussion, (2021)].
If A is a set, then \(\mathcal {P}A\) is its powerset: the set of its subsets.
So ‘rank’ is used in two ways: for these increasing sets \(U_0\), \(U_1\), etc., and also (when we talk about the rank of a set) for the indices of these sets.
The union of a set is the set of its members’ members. Indeed, I would argue that the only axiom of ZFC that cannot be read off the picture in this way is that of replacement: for discussion of this axiom, see Potter (2004: 296-98). Boolos (1971) claims that Choice is also an exception. I disagree, but will not try to make that case here.
Here < is the relation of (full) ground.
It will perhaps be helpful to have the definition of n-tuples before our minds in what follows, so I give this now. Thus, the ordered pair of x and y is defined:
$$\begin{aligned} \langle x,y\rangle =_\text{df} \{\{x\}, \{x,y\}\}. \end{aligned}$$Then, for \(n \ge 2\),
$$\begin{aligned} \langle x_1,\dots ,x_{n+1}\rangle =_\text{df} \langle \langle x_1,\dots ,x_n\rangle ,x_{n+1}\rangle . \end{aligned}$$(The ordered single of x, \(\langle x\rangle\), is simply x itself).
We are, then, using the same notation for propositions as for ordered tuples: something that, given how we will construct propositions, is natural, even if it is strictly speaking an abuse.
A natural question at this point: now that we have all these properties, do we still need the sets? One advantage of having sets in addition to properties is that, given that the former (but apparently not the latter) are extensional, we are freed from having to make certain arbitrary choices. For example, if we want to consider a model whose domain consists of Socrates, the Eiffel Tower and Sicily, we can simply take this to be the set of these things; we do not have to choose from among the (presumably) many properties that apply to them. Further, since so much work in both mathematics and philosophy has already been developed in terms of sets, it is convenient to be able to avail ourselves of this, without having to reformulate it. Having said that, however, it would be perfectly possible to give a version of the account of propositions to follow purely in terms of properties (and urelements), if one so desired. Thus, when our theory of properties is an addition to a pre-existing one of sets, our axioms for properties can essentially piggyback on those for sets, asserting (roughly) that for every set there is a corresponding property, and vice versa (see Sect. B.2). In the absence of sets this strategy would of course no longer be available. One would instead give axioms directly for properties that are akin to those of ZFC. One would then substitute properties for sets in standard applications: e.g. one would define ordinals as properties, and one would then use those (rather than the set-theoretic ordinals) to describe the hierarchy of properties.
Bealer (1982) suggests an approach to paradoxes for properties inspired by ZFC. The implementation of this basic idea is quite different from the one in this section, however: rather than using ZFC as a guide to which properties there are (as in this section), Bealer uses it as a guide to how the application relation behaves. Thus, Bealer does not solve Russell’s paradox for properties by denying that there is a general property of not self-applying; he solves it by rejecting the intuitive account of which things this property applies to, i.e. he denies that it applies to exactly the properties that do not apply to themselves. An advantage of the account proposed here, I believe, is that our universe of properties (and sets and urelements) has a simpler and more straightforward shape.
However, I will restrict attention to first-order languages that do not contain function symbols (other than individual constants). The proposal could certainly be extended to allow for functions as constituents (i.e. in analogy with function symbols). But this would require us to address an issue that I think would be a distraction here [see Kaplan (1977: 496) and Whittle (2017: 5032-33)].
If Q is \(\forall\) or \(\exists\), then the scope of an occurrence of Qx in a formula is Qx together with the formula that immediately follows it. An occurrence of a variable x is free if it is not within the scope of an occurrence of \(\forall x\) or \(\exists x\).
So, to be clear, propositional functions (as I use the term) are not in fact functions in the usual sense. This use of the term fits with at least some of the things that Russell says in (1908).
On the approach proposed here, both propositional functions and propositions will contain variables. Since structured propositions are intended to mirror the sentences that express them, and the latter contain variables, this seems quite natural to me. Nevertheless, I explain how to give a version of the proposal that eliminates variables in an appendix.
I discuss alternative possible choices for propositional connectives and quantifiers in note 47.
I insist that these domains are disjoint from \(\mathsf {Var}\) to avoid the difficulty of distinguishing propositional functions with free variables from propositions about variables. And the domains must be disjoint from \(\mathsf {Value}\) to allow us to distinguish, for example, atomic propositions about propositions from negations.
Well, there is actually one possible exception: if the domain D is a singleton, then (for all that we have assumed) \(\forall _D = \exists _D\), meaning that the universal functions are identical to the existential ones. This exception is natural: since if there is only one thing, to be true of everything comes to the same thing as merely being true of something. However, one could, if desired, insist that in this case \(\forall _D\) and \(\exists _D\) are coextensive but distinct properties, eliminating even this exception.
These are defined as in the linguistic case. Thus, if \(\langle \langle \forall _D,x\rangle ,p\rangle\) occurs in a propositional function, then the scope of that occurrence of \(\langle \forall _D,x\rangle\) is this whole pair \(\langle \langle \forall _D,x\rangle ,p\rangle\). (And similarly in the case of \(\exists _D\)). An occurrence of a variable is free if it is not within the scope of an occurrence of \(\langle \forall _D,x\rangle\) or \(\langle \exists _D,x\rangle\).
Note that since propositions are ordered pairs, i.e. sets, we have already defined their rank.
If p is a quantified proposition, in virtue of containing either \(\forall _D\) or \(\exists _D\), then D is the unique (and so in particular the least) domain of p.
In the interest of full disclosure, I should confess that there is something that this talk of propositions of ‘form (\(\dagger\))’ is passing over: specifically, that on the proposed account there is no completely general membership relation, and similarly no completely general property of truth. (Just as, in ZFC, there is no set of all ordered pairs \(\langle x,y\rangle\) with \(x \in y\)). Rather, for any given rank \(\alpha\), there will be a membership relation \(\in _\alpha\) that applies to x and y of rank less than \(\alpha\) such that x belongs to y. And similarly there is a property true\(_\alpha\) that applies to propositions of rank less than \(\alpha\) that are true. Thus, a more careful account of the solution of the Russell–Myhill would have to incorporate such indices, but its essential shape would be as described.
A reader might at this point have the concern: if there is no overarching membership relation, how are we to make sense of the claims of set theory itself, for these are made using the single (unsubscripted) predicate symbol \(\in\)? The answer is that these can be understood not as quantifying over absolutely all sets, or as being about an overarching membership relation, but as being about some specific \(U_\alpha\) and \(\in _\alpha\): because we can choose \(\alpha\) so that talking about \(U_\alpha\) is for many purposes interchangeable with talking about the entire universe of sets (see Sect. 6).
To be more precise: in saying this I am thinking of the version of Fritz’s argument that involves properties (see 2021). The version of (forthcoming) instead involves pluralities (as in note 10), and is not yet solved on the proposed account simply because I haven’t said anything about these. I would suggest that the natural thing to say is that talk of pluralities should, under the proposal, be explicated as talk of sets. This would mean that there is no plurality of all sets, or all propositions, for example. However, there is on the face of it something rather self-defeating, and inelegant, about adding propositions about such pluralities (e.g. about all propositions) ‘on top’ of those we have already introduced. For we will not then have succeeded in adding propositions about all propositions, but only all of those we had previously introduced. A similar point can be made about the possibility of admitting propositions about all sets (even if the issues in that case are a little more delicate). Of course, I do not imagine that these brief remarks will convert a devotee of plural quantification (understood along the lines of Boolos (1984), where there is, for example, a plurality of all sets), but it should at least convey the basic thought that a fuller treatment would develop.
I should note however that when such writers specify the exact structure of propositions, they do not tend to allow these to contain variables. But in appendix A I explain how to give a version of the proposal that does away with variables. Thus, even if one wants to stick to that aspect of the literature, a version of the account is available.
See Sect. 6 for a discussion of how theorizing about sets can proceed even in the absence of a general membership relation or quantification over all sets.
There is an additional point that is worth making here. Probably the most criticized, even mocked, element of Russell’s theory is the axiom of reducibility. This states, very roughly, that any propositional function is coextensive with one that is quantifier-free. This means, in effect, that the restrictions on quantification (in that theory) can in many cases be ignored: since even if a propositional function is at a high level of the hierarchy, in virtue of its quantifiers, there will be an equivalent at a lower level. This axiom was needed to enable various forms of mathematical reasoning, hampered by the restrictions on quantification, but it is seen by many as being ad hoc. It is thus worth noting that the analogue of the axiom of reducibility in our framework in fact holds, apparently as a result of entirely natural, and intrinsically motivated choices. The relevant principle says essentially that every propositional function is coextensive with a property. That is, since we can regard propositional functions as ‘structured’ properties, the principle says that every structured property is coextensive with an unstructured one. This follows (from standard principles about sets together with) our assumption that every set of n-tuples corresponds to an n-ary property (see Sect. 3). From this perspective, then, the axiom of reducibility can be seen as asserting simply that unstructured properties are abundant. I would suggest that one could mount a similar defence of the original axiom, in the context of Russell’s theory, but that would require a much more in-depth discussion.
In fact, I am simplifying somewhat, since ‘true’ and ‘false’ must also be understood schematically here. A more careful schematic formulation of (1) would rather use a schematic variable \(\alpha\) over ranks. Thus,
every\(_\alpha\) proposition\(_\alpha\) is true\(_\alpha\) or false\(_\alpha\).
Here ‘every\(_\alpha\)’ is a universal quantifier ranging over everything of rank less than \(\alpha\), while the other subscripted words denote the rank-restricted properties we have already met.
Although (3) and (4) are universal-existential, and thus might seem to be similarly out of reach of this tactic, in virtue of their existential component, in these cases a purely universal surrogate would seem to be available (i.e. one whose only schematic variable is understood universally):
-
(3S)
for every\(_\alpha\) property\(_\alpha\) x there is some\(_\alpha\) y such that y is the extension\(_\alpha\) of x
-
(4S)
for every\(_\alpha\) set\(_\alpha\) x there is some\(_{\alpha + 1}\) y such that y is the powerset\(_{\alpha + 1}\) of x.
Still, this sort of trick won’t work for every universal-existential, or of course more generally, by any means.
-
(3S)
For higher-order logic in general, see Gamut (1991), Carpenter (1997) or Williamson (2003). For discussions of the Russell–Myhill paradox within this framework, see Dorr (2016), Goodman (2017) or Fritz (2021). The focus in this section is on the higher-order approach to properties: specifically, the way that these are stratified. In contrast, propositions are not usually stratified on this approach, with the consequence that they must be more coarse-grained than on structured accounts (see, e.g., Goodman 2017).
See, for example, Prior (1971: ch. 3).
See Fritz (2021).
I have discussed non-cumulative versions of higher-order logic, but what about cumulative versions (as in Linnebo and Rayo (2012), for example)? The relation of such a theory to the framework proposed here would, to a great extent, depend on whether it is extensional. Suppose first that the cumulative theory is extensional (as in Linnebo and Rayo 2012). In this case, when it comes to giving an account of structured propositions, the theory would seem to be limited in essentially the same way that set theory is (see Sect. 3). For, in metaphysical terms, extensional higher-order entities would seem to be just as different from properties as sets are. On the other hand, if the higher-order cumulative theory is not extensional, then its entities could more plausibly be counted as genuine properties. In that case, the theory would be similar to the version of the present proposal with properties but no sets (see note 18). In particular, the absence of sets (or, at least, the absence of sets of higher-order entities) would lead to the same sort of inconveniences as in that case (note 18).
There is also a general ‘reflection’ theorem to the effect that for any sentence S of the language of set theory, there is \(\alpha\) such that S is true when interpreted as quantifying over absolutely all sets iff it is true when interpreted as quantifying over \(U_\alpha\). See theorem II.5.3 of Kunen (2011) for the result about set theory without urelements, but it also holds once urelements are permitted.
There are of course more approaches to the Russell–Myhill paradox than I can discuss in any depth here (see note 3). For example, in addition to the accounts considered in the text, there are proposals within the framework of Alonzo Church’s Logic of Sense and Denotation, such as those of C. Anthony Anderson (1987) and Sean Walsh (2016). On Anderson’s approach, all talk of propositions must be relativized to a language: there is no general notion of a proposition, only that of a proposition\(_\mathcal {L}\) (for a given language \(\mathcal {L}\)). However, at least from certain perspectives (e.g. those of metaphysical applications), one might hope for an account of propositions that is not tied to language in this way. Walsh’s proposal is predicative in the sense that a formula defining a certain sort of higher-order entity cannot quantify over entities of that sort. The general project, which motivates this restriction, is a treatment of higher-order quantifiers that encompass only those that ‘fall within our referential ken’ (2016: 296). Again, though, for at least some applications (especially in metaphysics) one might want an account of propositions that is not tied to our cognitive abilities. A very different approach, closer in spirit to that of the present work, has been put forward by Giorgio Sbardolini (2021). This is based on the modal set theory of Linnebo (2013), and readers sympathetic to that may find the analogous treatment of propositions congenial. On the other hand, if one believes that the theoretical benefits of adding modal operators to our set theory can be attained more straightforwardly using only the resources of ZFC, then one might prefer the approach of this paper.
I am ignoring Schönfinkeling (i.e. the practice of turning n-ary properties, for \(n > 1\), which on this approach are thought of as functions, into unary functions).
Uzquiano (2015) makes a related point, but about pluralities rather than properties.
For help with this paper, I am grateful to Peter Fritz, Stephan Krämer, Stephan Leuenberger, Bryan Pickel, Joshua Schechter, and two referees for this journal. This work was supported by the Arts and Humanities Research Council [Grant No. AH/M009610/1].
I should note that sometimes writers make different choices about propositional connectives and quantifiers which, on the sort of iterative approach pursued here, would not work. For example, sometimes propositional connectives are thought of not as properties of truth values, but as properties of propositions. On an iterative approach, that would have the consequence that negation, for example, could not apply to propositions that contain it (since no property can apply to propositions that contain that property). Similarly, propositional quantifiers are sometimes thought of as properties, not of subsets of the domain, but of proposition-valued functions (such as g above). But this (on an iterative approach) would rule out propositions in which a single quantifier occurs twice, one occurrence within the scope of the other (since a property cannot apply to functions that have within their range propositions that contain that property). I do not believe that this is a great cost of our approach, however, since the choices of propositional connectives and quantifiers that are available to us seem perfectly natural. Of course, this treatment of the standard propositional connectives cannot to be extended to non-truth-functional ones, e.g. connectives for notions of grounding. On the proposed approach, these would be treated a properties of propositions, and so could not apply to propositions that themselves contain the notion of grounding in question. This certainly is a limitation, but as we have already seen, essentially, there is much to be said by way of mitigation (Sects 6 and 7).
In place of separate predicate symbols Prop\(_n\) for each n, we might use a binary predicate that applies to an n-ary property and the natural number n. This would be more expressive but less straightforward, it seems to me.
Thus, in the main text, I used ZFC for the version of this theory that allows for urelements, but in fact the standard version restricts attention to pure sets. For the (standard) axioms of ZFC, see, e.g., Kunen (2011: 16–17).
Here \(W_0^n\) is the set of ordered n-tuples of members of \(W_0\). And if A and B are sets, then \(A \times B\) is the set of ordered pairs \(\langle a,b\rangle\) with \(a \in A\) and \(b \in B\).
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Appendices
A Eliminating variables
On the account proposed above, propositions contain variables. This seemed natural: we want propositions to mirror the structures of sentences. The latter contain variables. So shouldn’t the former? I note however that when philosophers of language have expressed views on the nature of quantified propositions, they have conspicuously refrained from pursuing this option. Instead, they have done things in terms of proposition-valued functions, as follows.Footnote 46
The basic idea is that \(\langle \forall xFx\rangle\) is the pair of \(\forall _D\) and the function that sends an object a to the proposition \(\langle Fa\rangle\). It is not clear to me that there are good reasons for wanting to avoid variables in this fashion. Nevertheless, in this appendix I explain how to eliminate variables from the proposed account, if desired.
We define a mapping from our propositional functions to variable-free propositions, relative to an assignment, as follows. Thus, by an assignment on a propositional domain D I mean a function from \(\mathsf {Var}\) into D. Given such an assignment \(\sigma\) on D, we essentially extend \(\sigma\) to a mapping from propositional functions (on D) into variable-free propositions, in the following way. (Here, if \(a \in D\), then \(\sigma a\) is simply a itself; and \(\sigma (x/a)\) is the assignment that sends x to a and is otherwise just like \(\sigma\)).
-
\(\langle R, \langle a_1,\dots ,a_n\rangle \rangle ^\sigma = \langle R, \langle \sigma a_1, \dots , \sigma a_n\rangle \rangle\)
-
\(\langle {{\boldsymbol{\lnot }}},p\rangle ^\sigma = \langle {{\boldsymbol{\lnot }}},p^\sigma \rangle\)
-
\(\langle {\varvec{\wedge }},\langle p,q\rangle \rangle ^\sigma = \langle {\varvec{\wedge }},\langle p^\sigma ,q^\sigma \rangle \rangle\) (similarly for \({\varvec{\vee }}\), \({\varvec{\rightarrow }}\) and \({\varvec{\leftrightarrow }}\))
-
\(\langle \langle \forall _D,x\rangle ,p\rangle ^\sigma = \langle \forall _D, g\rangle\), where g is the function that sends \(a \in D\) to \(p^{\sigma (x/a)}\)
-
\(\langle \langle \exists _D,x\rangle ,p\rangle ^\sigma = \langle \exists _D, g\rangle\) (same g).
Of course, if p is a proposition, then \(p^\sigma = p^\tau\) for any assignments \(\sigma\) and \(\tau\). Eliminating variables in this way leaves the main features of the account, for example, its solutions to paradoxes for propositions, essentially unaffected.Footnote 47
B Axioms and consistency
In this appendix I extend ZFC to incorporate properties, as in the account of Sect. 3, and then sketch an argument for the relative consistency of the resulting theory.
1.1 B.1 Language
We work in a first-order language with equality, whose non-logical predicate symbols are:
Unary: Set, Ind, Prop\(_n\)
Binary: \(\in\)
\((n + 1)\)-ary: \(\eta _n\).
The intended interpretation of Ind is to apply to urelements. As mentioned in Sect. 3, I abbreviate \(\eta _nQa_1\dots a_n\) as \(Qa_1\dots a_n\). Further I will use s to range over sets. Thus, \(\forall sFs\), for example, is an abbreviation of
(for some variable x). Similarly, I use \(P_n\) to range over over n-place properties.Footnote 48
1.2 B.2 Axioms
Basics
-
(i)
\(x \in y \rightarrow \text{Set}\ y\)
-
(ii)
\(\eta _nxy_1\dots y_n \rightarrow \text {Prop}_n x\)
-
(iii)
\(\lnot (Fx \wedge Gx)\), where F and G are distinct unary predicate symbols
Sets Our axioms for these are simply those of ZFC. Or, more carefully, they are the standard axioms of ZFC, modified in the usual way to allow for urelements.Footnote 49 So, for example, the axiom of extensionality does not say that any objects with the same members are identical, but simply that any sets with the same members are.
Properties Our axioms for properties assert a correspondence between these and sets. That is, every set (of the relevant sort) corresponds to a property, and vice versa. For ease of reading, I use abbreviations rather than writing these out in gory detail.
-
(i)
x is a set of n-tuples \(\rightarrow \exists P_n \forall y_1\dots y_n (P_ny_1\dots y_n \leftrightarrow \langle y_1,\dots ,y_n\rangle \in x)\)
-
(ii)
\(\,\,\exists x[x\) is a set of n-tuples \(\wedge \forall y_1\dots y_n (P_ny_1\dots y_n \leftrightarrow \langle y_1,\dots ,y_n\rangle \in x)]\)
1.3 B.3 Consistency
I now sketch an argument for the claim that if ZFC with urelements is consistent, then so is our extended theory. The natural way to do this is to work within ZFC with urelements, and construct a proper class model W of our theory. It is not that we are really assuming the existence of proper classes, however. Rather, talk of proper classes is, in the manner familiar from texts on set theory, officially to be understood as talk about formulas that correspond to these classes. Further, for simplicity I assume that there are infinitely many urelements (this assumption isn’t essential). Thus, let \(u_1\), \(u_2\),...be a countable infinity of urelements. The domain of our model is the union of the following sets, which follow the notation of Sect. 3.Footnote 50
Our logical vocabulary is then interpreted in the obvious way. Thus, \(\text{Ind}^W = W_0\); \(\text{Set}^W\) is the union of the sets \(S_\alpha\) for an ordinal \(\alpha\); and Prop\(^W_n\) is the union of the sets \(P^n_\alpha\). The membership symbol \(\in\) holds of an object x and a member A of Set\(^W\) iff \(x \in A\). Finally, \(\eta _n\) holds of a member Q of Prop\(_n^W\) and \(a_1\), ..., \(a_n\) iff \(\langle a_1,\dots , a_n\rangle\) belongs to the first member of Q. It is then relatively routine to verify that each of the axioms of our theory holds in this model.
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Whittle, B. The iterative solution to paradoxes for propositions. Philos Stud 180, 1623–1650 (2023). https://doi.org/10.1007/s11098-022-01901-7
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DOI: https://doi.org/10.1007/s11098-022-01901-7