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Sets and Plural Comprehension

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Abstract

The state of affairs of some things falling under a predicate is supposedly a single entity that collects these things as its constituents. But whether we think of a state of affairs as a fact, a proposition or a possibility, problems will arise if we adopt a plural logic. For plural logic says that any plurality include themselves, so whenever there are some things, the state of affairs of their plural self-inclusion should be a single thing that collects them all. This leads to paradoxes analogous to those that afflict naïve set theory. Here I suggest that they are the very same paradoxes, because sets can be reduced to states of affairs. However, to obtain a consistent theoretical reduction we must restrict the usual axiom scheme of Comprehension for plural logic to ‘stratified’ formulas, to avoid viciously circular definitions. I prove that with this modification to the background plural logic, the theory of states of affairs is consistent; moreover, it yields the axioms of the familiar set theory NFU.

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Notes

  1. We can treat some-of as primitive, or alternatively treat one-of as primitive and define some-of as follows: x are some of y if everything that is one of x is one of y.

  2. ‘I may mention that I was led to it [sc., the paradox] in the endeavor to reconcile Cantor’s proof that there can be no greatest cardinal number with the very plausible supposition that the class of all terms … has necessarily the greatest possible number of members.’ Russell [26], §100.

  3. Indeed, the same paradox will arise for any theory in which it would be natural to postulate a one for every many. For example, a theory of properties might want to say there is a property for every possible extension, (i.e., every plurality); a theory with primitive sets might want there to be a set for every plurality; a theory of Fregean sense might want there to be a Thought ‘about’ every plurality. All such theories will contain the species of paradox discussed here, and all of them could avail themselves of the solution I propose of restricting plural Comprehension. (This class of paradoxes are known as the ‘Russell-Myhill’ or ‘Principles of Mathematics Appendix B’ paradoxes. (Klement [12]). The presence of paradox in plural logic is discussed in McGee & Rayo [17]).

  4. Moore [20], pp. 144–8.

  5. Lewis [13].

  6. I adopt the following notational conventions. Since we are working in a plural logic, I use the variables ‘x’, ‘y’, ‘z’ etc. to range over things plurally: thus ‘∃x …x…’ is to be read ‘There exist one or more things such that … they …’. I reserve ‘u’, ‘v’ etc. for pluralities that comprise only one thing, i.e., for pluralities that are individuals, and I reserve ‘a’, ‘b’, etc. for individuals that are sets. I use ‘≤’ for the relation some-of, and ‘ε’ for the one-of relation, i.e., the relation in which a thing stands to some things, if it is one of them.

  7. The set theory NBG recognizes a so-called universal class, namely the proper class of all sets. But this is not a universal set in our sense, since it does not contain everything: for example, it does not contain itself.

  8. As can readily be verified, these axioms are exactly equivalent under a simple translation of notation to the standard axiomatisation of plural logic given in Linnebo [16]. (Cf. Rayo [25] and McKay[18], Ch 6.) The present axiomatisation allows the identity sign, treated as primitive, to be flanked by plural terms. Some authors prefer to regard primitive identity as a relation that can obtain only between individuals. On that approach plural identity is not primitive and instead is defined as plural coextensiveness, as follows: x = y = df x ≤ y ⋀ y ≤ x. It is then necessary to add an additional axiom scheme to secure that coextensive pluralities are indiscernible.

  9. Linnebo [15].

  10. Lewis [14], p63.

  11. In plural logic the formula ‘x = u’ is well-formed, and expresses the proposition that the (single-membered) plurality x is identical with the individual u. But in second-order logic, identity is a relation confined to items in the first-order domain, so that in any formula ‘v 1 = v 2’, the variables v 1 and v 1 must both be first-order: therefore we cannot simply reverse-translate ‘x = u’ as ‘X = u’, since the latter is not well-formed. (There are variants of second-order logic in which ‘X = u’ is well-formed but always false when X and u are of different orders, but this is no help, since on a plural interpretation ‘x = u’ is not only well-formed but sometimes true.) If plural identity is defined as coextensiveness, the plural formula ‘x = u’ will expand to ‘∀v (v ε x ↔ v = u)’ in primitive notation, and this is indeed reverse-translatable. However this treatment of identity requires us to go beyond the second-order language discussed by Boolos, and to adopt an extended language that has predicates of predicates. For an authoritative discussion of the exact similarities and differences between plural logic and second-order logic, see Linnebo [16].

  12. But see Hewitt [7] for a very interesting defence of a contrary view.

  13. That circular definitions and contradiction accompany one another is well known. If we thought that contradiction is unavoidable in human thought, because for example thought transcends the limits of thought (Priest [22]), then we might study contradictions in a tolerant spirit. If we thought that the semantic paradoxes reflect something circular and self-referential in the concept of truth, then we might with Gupta [5] treat certain circular definitions as acceptable. But here I take the opposite point of view, since I doubt that ‘the limits of thought’ is a coherent idea, and I think the paradoxes are better dispelled by holding to classical logic and taking care to avoid vicious circles.

  14. Poincaré [21], p 194. (I have amended Poincaré’s ‘A’ in the original to ‘E’, for consistency with his notation elsewhere.) It may be worth remarking here that the original ‘Vicious Circle Principle’ of Poincaré is not simply to be identified with a ban on impredicative definitions, i.e., definitions in which something is defined by quantification over a totality to which it itself belongs. Poincaré spoke not of ‘impredicative’ but of ‘non-predicative’ definitions:

    ‘Thus the definitions that must be regarded as non-predicative are those which contain a vicious circle. (Poincaré [21], p190, original emphasis.)

    Here Poincaré was following Russell’s terminology of 1905, who called a definition ‘non-predicative’ if it failed to define a class. (Russell [27], p141). It was only later on that Russell gave to ‘predicative’ a quite different sense which presupposes the theory of orders:

    ‘We will define a function of one variable as predicative when it is of the next order above that of its argument… If a function has several arguments, and the highest order of function among the arguments is the nth, we call the function predicative if it is of the n + 1th order…’ (Whitehead & Russell [28], p53).

    Thus by 1910 the ‘Vicious Circle Principle’ for Russell had ceased to be a general logical ban on circularity, and come instead to be understood as a ban on impredicativity, and subsequently to be associated with constructivism. But the purely logical principle that definitions should not be circular applies to any entities whatever, whether or not a constructivist account of them is correct. It is not undermined by doubts about impredicativity, a topic that has no direct bearing on the present discussion.

  15. The proof is that the axioms of plural logic have a model in the (singular) theory of sets of rank 1.

  16. List the countably many definitions of E in lexographical order. For each i, let r i be the number defined by the ith definition. Define N as follows: the integer part of N is 0; the ith decimal place of N is 0 if the ith decimal place of r i is 1, or 1 if the ith decimal place of r i is not 1. N has now been defined in a finite number of words, but cannot be one of E on pain of contradiction.

  17. Poincaré [21], p 190.

  18. For these and further examples see Quine [24], pp 36–37.

  19. One way to solve the problem is to impose a syntactic rule that forbids ‘cycles’ of variables in the formulas ϕ(u) permitted in Comprehension. We define a syntactic ‘path’ from u to v as a finite sequence of variables that starts with u and finishes with v, such that every pair of successive variables in the sequence figure in some atomic sub-formula of ϕ(u). A formula is acyclic if it contains no closed paths. Al-Johar, Holmes & Bowler [1] proves that a theory with (weak) extensionality and Acyclic Comprehension is equivalent to NFU.

  20. Mendelson [19] Chapter 4, §4.6.III.

  21. Its consistency is proved in Jensen [11].

  22. ‘Turing (1948) proposed ways of making type theory more acceptable to practicing mathematicians, but his work probably had an effect opposite to that intended.’ (Mendelson [19], p226)

  23. Two sets are ‘equinumerous’ if there is a one-to-one correspondence between them: a is ‘less numerous’ than b if there is a one-to-one correspondence between a and a subset of b, but not between b and any subset of a.

  24. We can however give essentially Cantor’s argument to prove there is no one-correspondence between the set of singleton subsets of a and P(a). The proof works as before by constructing a set b' = {u | ∃v (v ∈ a ⋀ u = {v} ⋀ ¬ u ∈ f(u))}, which this time does indeed exist, since the formula defining the plurality it collects is stratified. This proves that a set’s singleton subsets are strictly less numerous than the full power set. From a ‘naïve’ point of view it seems obvious that any set has exactly as many singleton subsets as it has elements, from which Cantor’s theorem would now follow. But we have no proof in our theory that the ‘natural’ one-to-one map u → {u} really does exist. A formula that would define the natural map is ∃v ∃w (u = (v, w) ⋀ w = {v}), but this is unstratified. Thus part (ii) of the proof of Cantor’s Theorem breaks down.

  25. Proof. Let f if possible be a singular-plural surjection from the individuals of z to the sub-pluralities of z. We show ∃u (1(u) ⋀ ¬ u ε f(u)). Assume the contrary: then for every x, 1(x) → x ε f(x). Now every individual is a unit plurality, and hence a plurality, so for any individual u, there is a v such that 1(v) ⋀ u = f(v), since f is a surjection. But by our assumption, v ε f(v) = u, so v ε u, so v = u, since u is an individual. So f(u) = u, so f maps every individual to itself; so any sub-plurality with more than one member is not the image f(u) of any individual: but such sub-pluralities exist, since z is not a unit plurality, so f is not a surjection, contrary to our assumption. So ∃u (1(u) ⋀ ¬ u ε f(u)). The formula ‘¬ (u ε f(u))’ is stratified, so by Stratified Comprehension, ∃y ∀u (u ε y ↔ ¬ u ε f(u))). Let x be this plurality. Then since f is a surjection, there is an individual u such that f(u) = x. So u ε x ↔ ¬ u ε f(u) ↔ ¬ (u ε x). Contradiction. See Florio [4], §3.3

  26. Until very recently it was not known whether Quine’s original system NF itself is consistent. However, Randall Holmes has announced that he is in possession of the outline of a proof of its consistency: see Holmes [9].

  27. An example of a full theory of states of affairs (as facts) may be found in Hossack [10], Chapter 2.

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Acknowledgements

I am grateful for discussion and comments to Stephen Barker, John Callanan, Dorothy Edgington, Simon Hewitt and Mark Textor, and for detailed written comments and discussion to Neil Barton, Salvatore Florio, Bob Hale and Øystein Linnebo. I wish particularly to acknowledge the invaluable help of an anonymous referee for The Journal of Philosophical Logic, who suggested many improvements to the original presentation of the material, saved me from several errors, and directed me to some enlightening papers.

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  1. Department of Philosophy, Birkbeck College, Malet Street, Bloomsbury, London, WC1E 7HX, UK

    Keith Hossack

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Correspondence to Keith Hossack.

Appendix—Adequacy and Consistency

Appendix—Adequacy and Consistency

1.1 Adequacy

Our reduction hypothesis says that the set whose elements are x is to be identified with the state of affairs that x are some of themselves: the empty set 0 can be identified with any arbitrarily chosen object 0* that is not a state of affairs. We prove that the reduction hypothesis and the plural theory of states of affairs yield the axioms of NFU. The only two assumptions about states of affairs that we need are (i) that states of affairs with the same constituents are identical and (ii) that for every plurality there exists the state of affairs that they are some of themselves.Footnote 27

Let L NFU be the first-order language whose only non-logical primitive symbol is ‘∈’. A formula ϕ(u) of L NFU is stratified if a natural number ‘type’ τ(v i ) can be assigned to each variable v i in ϕ(u) so that:

  1. (i)

    v i has the same type everywhere in ϕ(u);

  2. (ii)

    if ‘v i  = v j ’ occurs in ϕ(u), then τ(v i ) = τ(v j );

  3. (iii)

    if ‘v i ∈ v j ’ occurs in ϕ(u), then τ(v j ) = τ(v i ) + 1.

A formula of L NFU containing defined terms is stratified if it is stratified when written out in primitive notation.

Let L SOA be the plural language whose only non-logical function-symbol is ‘set’. A formula ϕ(u) of L SOA is stratified if and only if a natural number ‘type’ τ(v i ) can be assigned to each variable v i in ϕ(u) such that:

  1. (i)

    v i has the same type everywhere in ϕ(u);

  2. (ii)

    if ‘v i ε v j ’ or ‘v i  = v j ’ occurs in ϕ(u), then τ(v i ) = τ(v j );

  3. (iii)

    if ‘v j  = set(v i )’ occurs in ϕ(u), then τ(v j ) = τ(v i ) + 1.

A formula of L SOA containing defined terms is stratified if it is stratified when written out in primitive notation.

The reduction hypothesis leads to the following scheme for translating a formula A of L NFU into a formula A* of L SOA :

  1. i)

    ‘x = y’ of L NFU is translated as ‘x = y’ of L SOA .

  2. ii)

    ‘x ∈ y’ of L NFU is translated as ‘∃z (x ε z ⋀ y = set(z))’ of L SOA .

  3. iii)

    If A and B of L NFU are translated as A* and B* of L SOA , then ‘∀x A’, ‘¬ A’ and ‘A ⋀ B’ are translated as ‘∀x A*’, ‘¬ A*’ and ‘A* ⋀ B*’.

  • Theorem. Under the above translation scheme, the axioms of NFU are theorems of the plural theory of states of affairs.

  • Proof. The axioms of NFU are:

    • Extensionality. (∃u u ∈ a ⋀ ∀u (u ∈ a ↔ u ∈ b)) → a = b

    • Abstraction. If ϕ(u) is stratified and x is not free in ϕ(u), the following is an axiom: ∃a ∀u (u ∈ a ↔ ϕ(u)).

We obtain these as follows.

  1. (1)

    Extensionality. Let a and b be any sets. Assume ∃u u ∈ a and ∀u (u ∈ a ↔ u ∈ b). ∃u u ∈ a, so u ∈ b, so a ≠ 0 and b ≠ 0. So by the translation scheme ∃z a = set(z) and ∃z' b = set(z'), and u ε z ↔ u ∈ a ↔ u ∈ b ↔ u ε z'. So z = z', since pluralities with the same members are identical, so set(z) = set(z'), since if z = z' the state of affairs that z are some of themselves is identical with the state of affairs that z' are some of themselves. So a = b.

  2. (2)

    Abstraction. Let ϕ(u) be a stratified formula in L NFU in which x is not free, and let ψ(u)* be the formula of L SOA that comes from ϕ(u) by the above translation scheme. None of the translations disturb the type already assigned to any variable of ϕ(u), and where fresh variables are introduced they are bound by quantifiers, so their effect on the assignment of types is confined to the sub-formula in which they occur. So by relettering if necessary, we can find a stratified formula ϕ*(u) of L SOA logically equivalent to ψ(u)* in which x is not free. By the reduction hypothesis, ϕ(u) ↔ ψ(u)*, so ϕ(u) ↔ ϕ*(u). If ∃u ϕ*(u) then by Stratified Comprehension, for some x, u ε x ↔ ϕ*(u): so by Collections there exists a state of affairs a such that a = set(x). Then by the translation scheme, u ∈ a ↔ u ε x ↔ ϕ*(u) ↔ ϕ(u). So a is the set whose existence is required by Abstraction, since u ∈ a ↔ ϕ(u). Otherwise if ¬∃u ϕ*(u), then since 0* is not a state of affairs, ¬0* = set(x), so (u ε x ⋀ 0* = set(x)) ↔ ϕ*(u) ↔ ϕ(u). So by the translation scheme, u ∈ 0 ↔ ϕ(u) is a truth of L NFU , so 0 is the set whose existence is required by Abstraction.

1.2 Consistency

Let T be the theory of L SOA that adds Collections to the axioms of plural logic with Stratified Comprehension. The definitions and axioms of T are as follows:

  • Definition. 1(u) ↔ ∃x u ε x

  • P1. ∀x ∃u u ε x

  • P2. (∀u (u ε x ↔ u ε y)) ↔ x = y

  • P3. (1(u) ⋀ x ε u) → x = u

  • P4. If ϕ(u) is a stratified formula in which x is not free, the following is an axiom: ∃u (1(u) ⋀ ϕ(u)) → ∃x ∀u (1(u) → (u ε z ↔ ϕ(u)))

  • P5. ∀x ∃!a a = set(x)

Then T is consistent.

Proof. Reinterpret the formulas of L SOA into L NFU as follows: all variables are to be restricted to non-empty sets; ‘1(x)’ is to mean that x is a singleton; ‘x ε y’ is to mean that x is a singleton subset of y; ‘x = set(y)’ is to mean that x is the set of singleton subsets of y. Thus for each formula of L SOA we obtain its interpretation in L NFU by making the following substitutions, where in each case we choose a fresh variable for ‘z’:

  1. (i)

    ‘1(x)’ becomes ‘∃z x = {z}’, (in primitive notation, ‘∃z (y ∈ x → y = z)’)

  2. (ii)

    ‘x ε y’ becomes ‘∃z (x = {z} ⋀ x ⊆ y)’;

  3. (iii)

    ‘y = set(x)’ becomes ‘∀z (z ∈ y ↔ {z} ⊆ x)’.

  4. (iv)

    If A and B become A* and B*, then ‘∀x A’, ‘¬ A’ and ‘A ⋀ B’ become ‘∀x (∃z z ∈ x → A*)’ ‘¬ A*’ and ‘A* ⋀ B*’ respectively.

Under this scheme of interpretation, the axioms of T become the following propositions of L NFU , where to avoid clutter the ‘non-empty’ condition in formulas is omitted where it is redundant.

  • Definition* ∃z u = {z} ↔ (∃z u = {z} ⋀ ∃x u ⊆ x)

  • (A (non-empty) set is a singleton exactly if it is a singleton subset of some non-empty set.)

  • P1*. ∀x ((∃z z ∈ x) → ∃u ∃z (u = {z} ⋀ u ⊆ x))

  • (Every non-empty set has a singleton subset.)

  • P2*. (∀u (∃z u = {z} → (u ⊆ x ↔ u ⊆ y)) ↔ x = y.

  • (Non-empty sets are equal if they have the same singleton subsets.)

  • P3*. ∀x (∃z x = {z} → ((∃z y = {z}) ⋀ y ⊆ x)) → y = x))

  • (A singleton is identical with each of its singleton subsets.)

  • P4*. If ϕ*(u) is logically equivalent to the translation of a stratified formula ϕ(u) of L, then:

    $$ \begin{array}{*{20}c} {\left( {\exists u\;\exists z\;\left( {u=\left\{ z \right\}\wedge \phi *(u)} \right)} \right)\to } \hfill \\ {\;\;\;\left( {\exists x\left( {\left( {\exists z\;z\;\in\;x} \right) \wedge \forall u\;\left( {\exists z\;u=\left\{ z \right\}\to \left( {u\;\subseteq\;x\leftrightarrow \phi *(u)} \right)} \right)} \right)} \right)} \hfill \\ \end{array} $$

  • (If any singleton satisfies ϕ*(u), there is a non-empty set whose singleton subsets are exactly the singletons that satisfy ϕ*(u).)

  • P5* ∀x ((∃z z ∈ x) → ∃!a (∃z z ∈ a ⋀ (u ∈ a ↔ (∃v u = {v} ⋀ u ⊆ x))

  • (For any non-empty set, there exists the unique non-empty set of its singleton subsets.)

Each of these propositions is a theorem of NFU, as is obvious for Def1*, P1*, P2*, and P3*. For P4*, let ϕ(u) be any stratified formula of L SOA in which x is not free, and let ψ*(u) be the formula of L NFU that comes from ϕ(u) by the substitutions of the translation scheme. None of the substitutions need disturb the type of any of the original variables of ϕ(u), and where fresh variables are introduced they are bound by quantifiers, so their effect on the assignment of types is confined to the sub-formula in which they occur. So by relettering if necessary, we can find a formula ϕ*(u) which is logically equivalent to ψ*(u) and which is stratified in L NFU . Now let N = {u | (∃v u = {v}) ⋀ ϕ*(u)} be the set of singletons that satisfy ϕ*(u). N exists since its defining formula is stratified. If ∃u ∃v (u = {v} ⋀ ϕ*(u)), then for some v, {v} ∈ N, so v ∈ ∪N, so ∪N is non-empty. Now let {w} be any singleton subset of ∪N. Then {w} ⊆ ∪N, so w ∈ ∪N, so for some a, w ∈ a ∈ N, so a is a singleton, so a = {w}, so {w} ∈ N, so {w} is a singleton that satisfies ϕ*(u). Therefore ∪N is the required non-empty set whose singleton subsets satisfy ϕ*(u). For P5*, let a be any non-empty set, and let M be the set {u | (∃v u = {v}) ⋀ u ⊆ a}. Then M exists, since its defining formula is stratified: it is the required set of singleton subsets of a, it is non-empty since a is non-empty, and it is unique by Extensionality. Thus all the definitions and axioms of T can be reinterpreted as theorems of NFU. But Jensen [11] proves that NFU is consistent if arithmetic is consistent.

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Hossack, K. Sets and Plural Comprehension. J Philos Logic 43, 517–539 (2014). https://doi.org/10.1007/s10992-013-9278-2

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