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Hierarchical Propositions

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Abstract

The notion of a proposition is central to philosophy. But it is subject to paradoxes. A natural response is a hierarchical account and, ever since Russell proposed his theory of types in 1908, this has been the strategy of choice. But in this paper I raise a problem for such accounts. While this does not seem to have been recognized before, it would seem to render existing such accounts inadequate. The main purpose of the paper, however, is to provide a new hierarchical account that solves the problem.

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Notes

  1. I have mentioned sets, but one could make similar points with reference to other areas where there are paradoxes and both hierarchical and non-hierarchical responses (e.g. truth). For non-hierarchical set theories see, e.g., Forster [4] and Maddy [12]; for a non-hierarchical approach to propositions see, e.g., Prior [13]. The reader should consult these works to get a sense of the extent to which going non-hierarchical requires sacrificing simplicity and naturalness.

  2. Note that this ‘Russellian’ approach is to be distinguished from Russell’s ramified theory of types: the former is a general approach to propositions, while the latter is a much more specific theory, concerning the hierarchical structure of the space of propositions.

  3. For the Fregean approach to propositions, see, e.g., Frege [5]. For the Russellian one, see, e.g., Russell [15, 42–52] and Kaplan [8].

  4. This paradox is similar to Russell’s ‘appendix B’ paradox (see his [15, 527–28]). However, I focus on the paradox in the text, rather than on that one, because the appendix B paradox involves ‘classes’ of propositions which seem to introduce unnecessary complications.

  5. The main difference between the response sketched and the ramified theory of types is that the latter is ‘stricter’ in various ways (e.g. there can be no properties that apply both to objects and to level 1 properties). There is one point of interpretation that I should mention, however. In taking the ramified theory of types to be a response of the same sort as that sketched I am taking the former to be an account of properties and structured propositions. This clearly seems to be the correct interpretation of Russell [16]: see, e.g., Goldfarb [6]. But there is some debate about how exactly Whitehead and Russell [18] is to be interpreted: see, e.g., Goldfarb [6], Linsky [11] and Klement [10]. However, since I am in this paper concerned with structured propositions, I will ignore understandings of the ramified theory on which it is not an account of these (e.g. on which it is an account of linguistic entities). For more on the ramified theory of types see, e.g., Ramsey [14], Gödel [7], Chihara [2], Church [3], Anderson [1] and Klement [9].

  6. Thus, the paradoxes are blocked in virtue of the fact that on the sort of hierarchical account sketched properties can only apply to things at lower levels of the hierarchy. However, the sort of account sketched will also have the feature that properties and propositions can only quantify over things at lower levels. And since both of the paradoxes that I have given involve quantification (in characterizing the property R I said ‘for some property F and object a’; and the version of the Liar paradox clearly involved quantification over propositions), one might wonder if one could get by simply with these restrictions on quantification. However, although it is possible that a hierarchical account that only had such restrictions on quantification would block the paradoxes in the text (it would depend on the details), there are alternative versions of these paradoxes that do not involve quantification. The most obvious example would be a version of Russell’s paradox simply for properties: i.e. consider a property Q that applies to a property F iff F does not apply to itself. Another example would be a version of the Liar paradox that does not involve quantification: e.g. involving a proposition p of the form ¬T(p), or a similar proposition constructed using a ‘diagonal’ function for propositions. Thus, restrictions on quantification will not be sufficient for an adequate account: one will also need the restrictions on which things a property can apply to.

  7. To spell this out: R can be defined as follows. For any proposition p, R(p) iff there is some property F and object a such that p = F(a) and ¬F(p).

  8. Further, quantifiers give rise to a similar problem. For we surely want there to be propositions both of the form ∀x∃y H(x,y) and of the form ∃y∀x H(x,y), without having to say that ‘ ∀x’ and ‘ ∃y’ mean different things in the two cases. But, again, it is hard to see how this could be legitimately permissible on a hierarchical account.

  9. As I said above, there are ways in which Russell’s theory differs from the simpler sort of hierarchical account sketched in Section 1. However, it is easy to see that none of these differences help at all with the problem that I have raised. For on Russell’s theory, just as on the simpler account, there can be no propositions of the form F(F(a)). But then how can it be acceptable for there to be propositions of the form ¬¬p (for example)? Since no justification for this differential treatment is given, it is hard not to see the resulting theory as ad hoc.

  10. Thus, there is an extensive body of literature discussing the foundations of Russell’s theory: see, e.g., the work cited in note 5. Further, large sections of this work are devoted to the question of how Russell’s hierarchical restrictions might be justified (see, especially, Gödel [7], Chihara [2], Goldfarb [6] and Linsky [11]). But in none of this work does there seem to be an awareness of the fact that this question is made considerably harder by the fact that these restrictions are allowed to be flouted in certain central cases: for, once one allows exceptions in the cases of standard logical operators, it is hard to see what could justify insisting on the restrictions in every other case.

  11. Indeed, with the exception of unstructured accounts in terms of possible worlds (which, as I have noted, face serious problems), no non-hierarchical account of propositions seems to have been worked out in any detail. (Prior [13] proposes such an account but does not so develop it.) It would seem therefore that this problem with hierarchical accounts means that we are not in possession of any adequate response to the paradoxes that has been worked out in any detail.

  12. I should forestall a possible confusion at this point. For Russell’s ramified theory of types gives a hierarchy not only of properties and propositions but also of (what he calls) ‘propositional functions’. One might thus wonder if what I am suggesting is not something that Russell has already proposed. That is not so, however. Firstly, Russell’s propositional functions are not constituents of propositions (they seem rather to be the result of replacing certain constituents of propositions with variables). Secondly, if these propositional functions are literally functions at all, they are not functions to truth values (and thus nothing like the tallness function just mentioned), and they will not help with the problem raised above in anything like the way in which functions to truth values will, as will become clear once I explain that way in which functions to truth values help with the problem.

  13. I use angle brackets to distinguish propositions constructed out of functions from the values of functions. That is, I will use g〈f〉 for the proposition just mentioned, to distinguish it from the value of g at f (i.e. what g outputs when given f as input), which I will denote g(f).

  14. Or, strictly, everything else of lower level than ¬ to f. (I will sometimes omit such qualifications below.)

  15. An alternative, perhaps ‘purer’, implementation of the basic idea might try to do without these variables (i.e. making do with nothing more than objects and functions). However, it simplifies things to use variables, and so, at least for the purposes of this paper, this is what I will do.

  16. That is, f is an n-place unstructured function from objects to objects, g is an m-place such function, and a 1, …, a n , b 1, …, b m are each either objects or level 0 variables.

  17. A ‘purer’ implementation of the basic idea, doing without this additional constituent, is perhaps ultimately to be desired. However, for reasons of space I will not attempt such an implementation here.

  18. Thus, + is a 2-place unstructured function from objects to objects such that, for any natural numbers n and m, +(n,m) is the sum of n and m (and any pair of objects at least one of which is not a natural number will be sent to some object that is itself not a natural number). Similarly for ×.

  19. A reader may at this point have the following thought: why, in giving an account of functions, have I not availed myself of standard existing notation for these, specifically, Church’s λ-calculus? The reason is essentially as follows. In the λ-calculus, for any variables x and y and formula A, if λ x.λ y.A is wellformed then so is λ y.λ x.A. However, if one is giving a hierarchical account of propositions, then it is no less ad hoc to introduce operators λ x and λ y that can ‘permute’ like this than it would have been to insist from the start that standard connectives and quantifiers can iterate, permute, etc. Thus, the λ-calculus is not something one can help oneself to in giving a hierarchical account of propositions.

  20. That is, standard quantificational logic, the basic terms of which are: names of objects, function symbols for functions from objects to objects, predicates of objects, quantifiers and variables over objects, and truth functional connectives.

  21. Thus, ¬ would send an n-place level 1 concept of the form \(g\langle x_{1}, \dots , x_{n}\rangle \) to its ‘complement’, i.e. a concept \(h\langle x_{1}, \dots , x_{n}\rangle \) that sends objects a 1, …, a n to t iff \(g\langle x_{1}, \dots , x_{n}\rangle \) sends them to f (and similarly in the cases of level 1 concepts of other forms; ¬ would send anything that is not a level 1 concept to f). Other connectives could be similarly reconceived.

  22. That is, R〈¬,Z〉 works just like the version of negation described in the previous note: sending level 1 concepts to their ‘complements’ (and sending anything that is not a level 1 concept to f).

  23. Specifically, there will be a distinct raising function R n,m for each pair n, m with n≥1 and m≥2. Thus, R n,m will in effect turn the n-place logical operators of level less than m into level m operators, so as to allow them to compose with other level m operators (such as quantifiers over level m−1).

References

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Acknowledgments

For comments and discussion, I am grateful to George Bealer, Susanne Bobzien, Justin Khoo, Zoltán Gendler Szabó, audiences at Philosophy Today (Mexico City) and the Society of Exact Philosophy (Ohio State University), and two referees for this journal.

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  1. Department of Philosophy, Yale University, PO Box 208306, New Haven, CT 06520, USA

    Bruno Whittle

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Appendix

Appendix

In this appendix I construct a model of the proposed account in standard set theory. In particular, in ZFU. I write \(\mathbb {N}\) for the set of natural numbers \(\{0, 1, 2, \dots \}\), and \(\mathbb {N}^{+}\) for \(\mathbb {N}-\{0\}\). For simplicity, I assume that there exist the following infinite, and pairwise disjoint, sets of urelements: LEV 0, VAR 0, VAR 1, … (i.e. a set VAR n for each \(n \in \mathbb {N}\)). I assume also that there is some urelement ∘ not in any of these sets, and that the truth values t and f are in LEV 0. LEV 0 is the model of level 0 of the hierarchy.

Let X and Y be non-empty sets. By an n-place set-theoretic function from X to Y (\(n \in \mathbb {N}^{+}\)) I mean a set Z of ordered n+1-tuples of members of \(X\cup Y\) such that: the first n members of each member of Z are in X, and the last member is in Y; and each n-tuple of members of X is the initial segment of exactly one member of Z.

UF 1={f:f is an n-place set-theoretic function from LEV 0 to LEV 0 for some \(n \in \mathbb {N}^{+}\}\). UF 1 is the model of the class of unstructured functions of level 1.

The model of the class of structured functions of level 1 is as follows. I use \([{\kern -2.3pt}[a_{1}, \dots , a_{n}]{\kern -2.3pt}]\) for the n-tuple of a 1, …, a n in that order. SF 1 is defined recursively as follows:

if f is an n-place member of UF 1 (for some \(n \in \mathbb {N}^{+}\)), and each a i is either a member of LEV\(_{0}\cup \text {VAR}_{0}\) or \([{\kern -2.3pt}[\circ , g]{\kern -2.3pt}]\) for some g∈SF1, then \([{\kern -2.3pt}[ f, a_{1}, \dots , a_{n}]{\kern -2.3pt}] \in \text {SF}_{1}\).

LEV\(_{1} = \text {LEV}_{0}\cup \text {UF}_{1}\cup \text {SF}_{1}\). LEV 1 is of course the model of level 1 of the hierarchy.

The model of level 2 is as follows. UF\(_{2} = \text {UF}_{1}\cup \{f:\) f is an n-place set-theoretic function from LEV 1 to LEV 1 for some \(n \in \mathbb {N}^{+}\}\). SF 2 is then defined:

  • (i)  SF\(_{1} \subseteq \text {SF}_{2}\); and

  • (ii)  if f is an n-place member of UF 2−UF1 (for some \(n \in \mathbb {N}^{+}\)), and each a i is either a member of LEV\(_{1}\cup \text {VAR}_{1}\) or \([{\kern -2.3pt}[\circ , g]{\kern -2.3pt}]\) for some g∈SF2, then \([{\kern -2.3pt}[ f, a_{1}, \dots , a_{n}]{\kern -2.3pt}] \in \text {SF}_{2}\).

LEV\(_{2} = \text {LEV}_{0}\cup \text {UF}_{2}\cup \text {SF}_{2}\) is the model of level 2 of the hierarchy.

The models of subsequent levels are defined similarly. Thus, the model of level m+1 for \(m \in \mathbb {N}\) with m≥2 is as follows. UF\(_{m+1} = \text {UF}_{m}\cup \{f:\) f is an n-place set-theoretic function from LEV m to LEV m for some \(n \in \mathbb {N}^{+}\}\). SF m+1 is defined as follows:

  • (i)  SF\(_{m} \subseteq \text {SF}_{m+1}\); and

  • (ii)  if f is an n-place member of UF m+1−UF m (for some \(n \in \mathbb {N}^{+}\)), and each a i is either a member of LEV\(_{m}\cup \text {VAR}_{m}\) or \([{\kern -2.3pt}[\circ , g]{\kern -2.3pt}]\) for some g∈SF m+1, then \([{\kern -2.3pt}[ f, a_{1}, \dots , a_{n}]{\kern -2.3pt}] \in \text {SF}_{m+1}\).

LEV\(_{m+1} = \text {LEV}_{0}\cup \text {UF}_{m+1}\cup \text {SF}_{m+1}\).

For \(n \in \mathbb {N}^{+}\), I write VAR <n for \(\bigcup _{m < n} \text {VAR}_{m}\). I define the notion of a member of VAR <n being ‘free’ in a member of SF n as follows. Let x∈VAR<n and f∈SF n . For some g∈UF n and a 1, …, a m (\(m \in \mathbb {N}^{+}\)), \(f = [{\kern -2.3pt}[ g, a_{1}, \dots , a_{m}]{\kern -2.3pt}]\). x is free in f if: for some i (1≤i≤m), a i is x; or a i is \([{\kern -2.3pt}[\circ , h]{\kern -2.3pt}]\) and x is free in h. f∈SF n is r -place if exactly r members of VAR <n are free in f.

The ‘values’ of members of SF n are defined as follows. I write VAR\(_{\mathbb {N}}\) for \(\bigcup _{n \in \mathbb {N}} \text {VAR}_{n}\) and LEV\(_{\mathbb {N}}\) for \(\bigcup _{n \in \mathbb {N}} \text {LEV}_{n}\). An assignment is a 1-place set-theoretic function A from VAR\(_{\mathbb {N}}\) into LEV\(_{\mathbb {N}}\) such that for any \(x \in \text {VAR}_{\mathbb {N}}\) and \(n \in \mathbb {N}\), if x∈VAR n then A(x)∈LEV n . Let f∈SF n (\(n \in \mathbb {N}^{+}\)) and let A be an assignment. The value of f at A is defined as follows. For some g∈UF n and a 1, …, a m (\(m \in \mathbb {N}^{+}\)), \(f = [{\kern -2.3pt}[ g, a_{1}, \dots , a_{m}]{\kern -2.3pt}]\). For each i with 1≤i≤m, define b i as follows: if \(a_{i} \in \text {LEV}_{\mathbb {N}}\), then b i = a i ; if \(a_{i} \in \text {VAR}_{\mathbb {N}}\), then b i = A(a i ); if a i is \([{\kern -2.3pt}[\circ , h]{\kern -2.3pt}]\) for some h∈SF n , then b i is the value of h at A. The value of f at A is then \(g(b_{1}, \dots , b_{m})\).

It is easy to show that all the assumptions and claims made in Section 4 hold in the model, when interpreted in the obvious way. These fall essentially into two groups: existence and distinctness claims. An example of an existence claim is as follows. In Section 4.2 I claimed that if f is an n-place unstructured function of level 1, and each of a 1, …, a n is either an object, a level 0 variable, or ∘S, where S is a structured function of level 1, then \(f\langle a_{1}, \dots , a_{n}\rangle \) is a structured function of level 1. This holds in our model because for any n-place f ′∈UF1, and \(a_{1}^{\prime }\), …, \(a_{n}^{\prime } \in \text {LEV}_{0}\cup \text {VAR}_{0}\cup \{[{\kern -2.3pt}[ \circ , g]{\kern -2.3pt}]: g \in \text {SF}_{1}\}\), \([{\kern -2.3pt}[ f', a_{1}^{\prime }, \dots , a_{n}^{\prime }]{\kern -2.3pt}] \in \text {SF}_{1}\) (by the definition of SF 1). Similarly, the existence of unstructured functions such as ¬, ∃ x (for x a level 0 variable), and the raising functions of Section 4.6 follows easily from the axioms of ZFU. An example of a distinctness claim is the assumption made in Section 4.2 that no structured function is an unstructured function. This holds because for any \(n \in \mathbb {N}^{+}\) and f∈SF n , f is an ordered m-tuple (for some \(m \in \mathbb {N}^{+}\)). It follows that f is finite. In contrast, for any \(r \in \mathbb {N}^{+}\) and g∈UF r , g is infinite (because g is a set-theoretic function from LEV r−1 into LEV r−1, and LEV r−1 is infinite). Similarly for the other claims made in Section 4. It follows that the account of Section 4 is consistent.

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Whittle, B. Hierarchical Propositions. J Philos Logic 46, 215–231 (2017). https://doi.org/10.1007/s10992-016-9399-5

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