Ordinal type theory

ABSTRACT

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorising about properties, relations, and states of affairs – or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). While STT, understood as a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities.

KEYWORDS:

• Properties
• relations
• states of affairs
• type theory
• higher-order metaphysics
• fundamentality

Acknowledgements

For valuable discussion of (or related to) material presented in this paper, I am grateful to Andrew Bacon, Kit Fine, Peter Fritz, Jeremy Goodman, Daniel Nolan, Robert Trueman, and Juhani Yli-Vakkuri. Special thanks to Francesco Orilia for detailed comments on an earlier draft. For financial support, I am grateful to the Swiss National Science Foundation (grants 100012_173040 and 100012_192200).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 I mainly have in mind Williamson (2013), Dorr (2016), Fritz and Goodman (2016), Goodman (2017), Bacon (2019; 2020), Dorr, Hawthorne, and Yli-Vakkuri (2021), and Bacon and Dorr (forthcoming). Major influences on this literature include Prior (1971) and Williamson (2003). Skiba (2021) offers a survey. Critical voices include Orilia and Landini (2019), Hofweber (2022), Whittle (2023, 1642–1645), Menzel (forthcoming), and Sider (MS). Also cf. Florio (forthcoming).

2 In the literature of higher-order metaphysics, the term ‘state of affairs’ is usually avoided in favour of ‘proposition’. Nonetheless, I shall here speak of states of affairs (or simply ‘states’), as this term goes more naturally together with ‘property’ and ‘relation’, whereas ‘proposition’ connotes something more fine-grained and sentence-like. (Cf., e.g., Dorr [2016, 54n.].)

3 A more detailed overview of LFO can be found in Shapiro and Kouri Kissel (2018).

4 This notation follows Williamson (2013, 221f.), whose notation is a variant of that introduced by Gallin (1975, 68).

5 Thiel (2002) gives a brief overview and helpful discussion of Behmann’s correspondence, relating to his proposal, with Gödel and Dubislav. Also cf. Feferman (1984). In a letter to Behmann, Ramsey wrote that ‘the hierarchy of functions and arguments is the one part [of the Theory of Types] which I feel can hardly be questioned’ (Mancosu 2020, 24). He motivates this assessment by pointing out that some functions are defined only on functions and some only on functions of functions (op. cit., p. 23). But the existence of such functions – or of properties that have instantiations only by properties – can be granted without having to embrace STT.

6 Cf. also Tarski (1935, §4), who proposes a form of STT based on considerations about the ‘semantical categories’ of natural-language expressions. In his postscript, however, he abandons his earlier standpoint and takes quite seriously the possibility that STT may need to be given up in the interest of expressive power.

7 A recent defence of STT that draws explicitly on Fregean ideas can be found in Button and Trueman (2022), which in turn invokes Trueman’s (2021) defence of ‘Fregean realism’. Trueman’s central thesis is that ‘it is nonsense to suppose that a property might be an object’ (2n.). Here an object is understood to be ‘anything which can be referred to with a singular term’, while a property is ‘anything which can be referred to with a predicate’ (1f.). Later in the book the relevant concept of reference undergoes a bifurcation when Trueman distinguishes between ‘term-reference’ and ‘predicate-reference’. At a crucial juncture, Trueman’s argument appeals to the requirement that ‘the notion of reference appropriate for predicates must allow us to disquote predicates’ (p. 52). But it is not very clear why this requirement should be accepted.

8 Cf. Jones (2018, §4.2). The rest of this paragraph is largely in line with §3.3 of the same paper.

9 In §3 of his ‘A Case for Higher-Order Metaphysics’ (forthcoming), Andrew Bacon argues for the virtues of STT over ‘property theory’ in part by maintaining that such questions as ‘Is wisdom located?’ or ‘Is wisdom concrete or abstract?’ do not have ‘straightforward higher-order analogues’. It is true that, in an STT-based setting, it is not obvious that these questions have higher-order analogues. But this just means that we are faced with the difficult question of whether they have such analogues; and here of course it is not enough to verify that they have no higher-order analogues in English. For instance, there may (assuming STT) be a fundamental entity of type $〈〈e〉,〈e〉〉$ that behaves just like a metaphysically primitive co-location relation among entities of type $〈e〉$. Although considerations of ontological parsimony give us a reason to think that there isn’t, this does not mean that the question does not arise.

A second way in which Bacon regards higher-order metaphysics as separate from ‘property theory’ rests on the idea, which goes back at least to Prior (1971, ch. 3), that quantification into predicate-position is a purely logical affair. For instance, the existential generalisation ‘Socrates somethings’ is supposed to follow unproblematically from ‘Socrates is wise’; and since one can accept ‘Socrates is wise’ without having to believe in properties, ‘Socrates somethings’ should be similarly free of ontological commitment to properties. (Cf. also Liggins [2021, 10028].) But notice that, if this argument works at all, it cuts both ways: one can accept ‘Socrates is wise’ without having to believe in STT or any of the ‘higher-order entities’ posited by higher-order metaphysicians. Hence, if ‘Socrates somethings’ does indeed follow logically from ‘Socrates is wise’, then that sentence should not carry any commitment to such entities, either.

A final point worth noting is that Bacon uses as his foil a rather specific form of ‘property theory’, whereas there are of course various possible ways to develop a theory of properties (and relations). As far as I can see, nothing hinders our regarding STT as one of these ways; though obviously this is not to say that STT, thus understood, will be unproblematic.

10 Thanks to an anonymous referee for raising this issue.

11 The subscript-less symbols ‘∀’ and ‘∃’ that are used throughout this section should be read as merely abbreviatory devices. Cf., e.g., Church (1940, 58).

12 On the intrinsic/extrinsic distinction, see, e.g., Plate (2018) and references therein.

13 Cf., e.g., Chierchia (1982), Menzel (1986, 3f.; 1993, 64f.), Bealer and Mönnich (2003, §10).

14 Similar remarks apply if one thinks of entities of type 〈〉 as sets of possible worlds. Indeed it might be wondered whether we should not, following Lewis (1986), conceive of intensional entities along these lines; but here I shall be taking for granted that the answer is ‘no’. (For relevant critical discussion, see, e.g., Schnieder [2004, 72f.].)

15 Prior (1961) and Bacon, Hawthorne, and Uzquiano (2016, 532) describe essentially the same conundrum. By similar reasoning (also within STT), already Chwistek reached the puzzling conclusion that ‘I cannot […] consider as interesting those and only those propositions which I wish so to consider’ (1921, 344f.). Related problems include the Russell–Myhill paradox and intensional analogues of, e.g., Grelling’s paradox and the ‘infinite liar’ paradox devised by Yablo (1993). A form of the Russell–Myhill paradox will be briefly discussed in footnote 39 below. For relevant discussion of Grelling’s paradox, see Church (1976).

16 One of the ‘special cases’ just alluded to is the case in which $Q^{〈〈〉〉}$ is the property of being identical with $s$, for some state $s$. (For example, let $s$ be the state that snow is white.) The question of whether $\mathrm{\exists }{x}^{〈〉}\left(Q^{〈〈〉〉}\left(x^{〈〉}\right)\wedge x^{〈〉}\right)$ obtains will then come down to whether $s$ obtains. Moreover, the stipulation that $Q^{〈〈〉〉}$ should itself be instantiated by $\mathrm{\exists }{x}^{〈〉}\left(Q^{〈〈〉〉}\left(x^{〈〉}\right)\wedge x^{〈〉}\right)$ ‘and by nothing else’ will be satisfied as long as states are individuated in a sufficiently coarse-grained manner. In particular, $s$ has to be identical with $\mathrm{\exists }{x}^{〈〉}\left(Q^{〈〈〉〉}\left(x^{〈〉}\right)\wedge x^{〈〉}\right)$ – i.e. with the state that some state identical with $s$ obtains. This requires a somewhat coarse-grained conception of states, but the required level of coarse-grainedness is arguably far from implausible. So here we have a case where, under not-implausible assumptions, the question of whether $\mathrm{\exists }{x}^{〈〉}\left(Q^{〈〈〉〉}\left(x^{〈〉}\right)\wedge x^{〈〉}\right)$ obtains is not left indeterminate (Thanks to Andrew Bacon and Jeremy Goodman for alerting me to this point). Even so, it is clearly a rather special case. If $Q^{〈〈〉〉}$ were instead, say, the property of being Quine’s favourite state, there would be no such easy path to determinacy.

17 The quotation is from Mortensen and Priest (1981, 385), who discuss a simpler truth-teller paradox.

18 In this connection, cases of ‘reciprocal negation’ – pairs of sentences, propositions, or states of which each ‘says’ of the other that it is false – also deserve consideration. (For discussion, see, e.g., Sorensen [2001, ch. 11; 2018, §9], Armour-Garb and Woodbridge [2006]; Greenough [2011]. As Sorensen notes, the problem goes back to Buridan.) Let $P^{〈〈〉〉}$ and $Q^{〈〈〉〉}$ be the properties of, respectively, being Prior’s favourite state and being Quine’s favourite state, let $p^{〈〉}$ and $q^{〈〉}$ be, respectively, the states $\mathrm{\forall }{x}^{〈〉}\left(P^{〈〈〉〉}\left(x^{〈〉}\right)\to \mathrm{\neg }{x}^{〈〉}\right)$ and $\mathrm{\forall }{x}^{〈〉}\left(Q^{〈〈〉〉}\left(x^{〈〉}\right)\to \mathrm{\neg }{x}^{〈〉}\right)$, and suppose that $p^{〈〉}$ is the only thing that instantiates $Q^{〈〈〉〉}$ while $q^{〈〉}$ is the only thing that instantiates $P^{〈〈〉〉}$. Question: Which of $p^{〈〉}$ and $q^{〈〉}$ obtains? The symmetry of the situation strongly suggests that they either both obtain or both fail to obtain, but either of these options leads to a contradiction.

19 For example, the extensionality axiom will be to the effect that no two sets have all their members in common.

20 For example, when below it is stipulated that ‘$\mathrm{\top }$’ abbreviates ‘$\mathrm{\&}\left(\right)$’, the quotation marks should be understood as serving to construct names of representations of $\mathcal{L}$-expressions. Using them instead to form names of $\mathcal{L}$-expressions, we would have to say that ‘$\mathrm{\top }$’ is identical with ‘$\mathrm{\&}\left(\right)$’. Quinean corner-quotes will be used in a similarly ambiguous way.

21 As for which letter is to be used to represent a given variable, any convention at all would do, and so I shall leave this open. That there are far more ordinals than letters won't matter in practice.

22 This formulation (‘everything there is’) might give rise to meta-ontological qualms, as it appears to presuppose an absolutely general (as opposed to ‘indefinitely extensible’) domain of entities. I acknowledge this presupposition, but will here not be trying to defend it.

23 Throughout this paper, unless otherwise specified, sequences may be of any set-sized length.

24 The conjunction of ‘no’ things is the empty conjunction, symbolised as ‘&()’ or ‘⊤’. It can be thought of as the state that obtains ‘no matter what’.

25 Cf., e.g., Goodman (forthcoming, §3.3).

26 It might be thought that talk of ‘individuals’ should be understood as referring to all and only those things that can be denoted by singular terms; but this is hardly convincing. Let x be some non-individual entity, for instance of type $〈e,〈〉〉$: on the face of it, nothing prevents us from introducing a singular term that denotes x. The STTler may point out that the relevant denotation relation would have to be of a different type than that which holds between singular terms and individuals. But this would be to presuppose, if not STT itself, then some similarly problematic division of entities into logical types. (Likewise if it is suggested that the individuals are just those things that are ranged over by ‘our first-order quantifiers’.) Alternatively it might be supposed that an individual is simply anything that is neither an attribute nor a state. (Cf. Whitehead and Russell [1910, 53], where individuals are understood to be ‘objects which are neither propositions nor functions’.) But someone who takes this route can of course no longer maintain that a commitment to the existence of attributes and states is a commitment to the existence of certain special kinds of ‘individuals’.

27 An objector might argue that there are, or may well be, multigrade relations, which have more than one adicity. (Cf. MacBride [2005, §2].) To adjudicate this issue, we would have to clarify what, exactly, a relation should be taken to be. For now it may suffice to say that (A1) recommends itself at least as a simplifying assumption, and that its main purpose within the present theory lies in settling certain questions that would otherwise be left as glaring lacunae. For an example that will become relevant in Section 5.4, consider the question of whether the tautologous formula ‘$x=x$’ analytically entails ‘$x=\mathrm{\lambda y}x\left(y\right)$’. With the help of (A1), this can be answered in the negative. For, under the semantics of $\mathcal{L}$, ‘$x=x$’ denotes a state relative to every variable-assignment that is defined on ‘x’, whereas ‘$x=\mathrm{\lambda y}x\left(y\right)$’ has a denotation relative to a given variable-assignment only if the latter maps ‘x’ to a property. Let now $g$ be some assignment that maps ‘x’ to the identity relation. By (S5), this relation is dyadic, and hence, given (A1), not monadic: it is not a property. Consequently, relative to $g$, ‘$x=x$’ denotes a state while ‘$x=\mathrm{\lambda y}x\left(y\right)$’ denotes nothing at all. So ‘$x=x$’ does not analytically entail ‘$x=\mathrm{\lambda y}x\left(y\right)$’.

28 Cf. Dorr (2016, 100). An earlier version can be found in Ramsey (1925/1931, 35). Anderson’s (1986, §1) criticism can be answered by pointing out that the domain of quantification contains all sets and hence contains proper-class many entities.

29 Note that, for any untyped variable $v$, this second conjunct is trivially satisfied. Hence, any untyped variable will simply denote_I,g whatever entity it is mapped to by $g$, provided that $g$ does map it to some entity. (Since there are proper-class many variables, no variable-assignment is defined on all of them.)

30 The concept of class serves here merely as an informal convenience; likewise the concept of a variable’s range. I am introducing the latter only for the purpose of formulating the basic idea behind the definition of ‘order of’, which is given below.

31 To see this, note that ‘$\mathrm{\top }$’ and ‘$\mathrm{\perp }$’ respectively abbreviate ‘$\mathrm{\&}\left(\right)$’ and ‘$\mathrm{\neg \&}\left(\right)$’. (Cf. Section 3.2.4 above.) So $\mathrm{\top }$ and $\mathrm{\perp }$ are respectively denoted, relative to any interpretation and variable-assignment, by ‘$\mathrm{\&}\left(\right)$’ and ‘$\mathrm{\neg \&}\left(\right)$’, which contain no occurrences of any atomic terms.

32 Let x be any particular or fundamental intensional entity, let $I$ be the empty set, let $t$ be any untyped variable, and let $g$ be any variable-assignment that maps $t$ to x. From clause (d3) of the above semantics, it then follows that $t$ denotes_I,g x; so condition (o1) on the right-hand side of Definition 16 is satisfied. Condition (o2) is also satisfied. And since $t$ contains no bound variable-occurrence, condition (o3) is satisfied for $\text{β}=0$.

33 Let $s$ be any state and $\alpha$ any ordinal. To prove the left-to-right direction, suppose that $s$ is of order $\alpha$. It is then clear that $\mathrm{\neg s}$ is of at most order $\alpha$, since, for any interpretation $I$, variable-assignment $g$, and term $t$: if $t$ denotes_I,g $s$, then $⌜\mathrm{\neg t}⌝$ denotes_I,g $\mathrm{\neg s}$. Now let $\text{β}$ be any ordinal such that $\mathrm{\neg s}$ is of order $\text{β}$. By what has just been said, we have that $\text{β}\le \alpha$. Further, by an argument analogous to the foregoing, it can be seen that $\mathrm{\neg \neg s}$ is of at most order $\text{β}$. But, by (S6), $\mathrm{\neg \neg s}$ is nothing other than $s$. This shows that $\alpha \le \text{β}$, and so we have that $\alpha =\text{β}$, as required. The right-to-left direction can be shown in a similar way.

34 Whether $N$ will really be classified as first-order depends, roughly put, on what other formulas it can be denoted by, and thus depends on how finely states are individuated. We will return to this topic in Section 5.5.3.

35 For an overview of the main approaches to avoiding semantic paradox, see, e.g., Horsten (2015, 687ff.) and Beall, Glanzberg, and Ripley (2018, ch. 5). For more detailed critical discussion of some recently popular approaches, see Murzi and Rossi (2020a; 2020b). Also cf. Hansen (2021), Sher (2023).

36 The reason for this lies ultimately in clause (d3) of the above semantics.

37 Incidentally, the property $P$, if it exists, will be nothing other than $\lambda x^1\mathrm{\neg }{x}^1\left(x^1\right)$. This can be seen from the semantics of lambda-expressions – given that $P=\mathrm{\lambda x}\mathrm{\neg R}\left(x,x\right)$ and $R=\lambda x^1,y{x}^1\left(y\right)$ – in combination with (A2). Regarding the question of whether $P$ does exist, cf. Section 5.1 below. Regarding the question of whether it is zeroth-order, see Section 5.5.3.

38 To verify this, one has to consult clauses (d4) and (d6) of the above semantics, bearing in mind that, as stated in Section 3.3.2, the negation of a state $s$ obtains iff $s$ itself doesn’t, and that the existential quantification of an attribute obtains iff that attribute has an obtaining instantiation.

39 On the reciprocal-negation paradox, cf. footnote 18 above. For a simple version of the Russell–Myhill paradox as it arises in the context of STT, consider the property $F$ of being Frege’s only favourite entity of type $〈〈〉〉$. Surely, for any distinct entities x and y of that type, the world will have to be at least a little bit different if $F\left(x\right)$ obtains than if $F\left(y\right)$ obtains; and so $F\left(x\right)$ has to be distinct from $F\left(y\right)$. (I am here omittingsuperscripts for the sake of readability.) But now let $M$ be a property of type $〈〈〉〉$, of being an entity x of type $〈〉$ that is identical with the state $F\left(y\right)$ for some entity y of type $〈〈〉〉$ such that $y\left(x\right)$ fails to obtain, or in symbols: $M=\lambda x^{〈〉}\mathrm{\exists }{y}^{〈〈〉〉}(\mathrm{\neg }{y}^{〈〈〉〉}\left(x^{〈〉}\right)\wedge (x^{〈〉}=F^{〈〈〈〉〉〉}\left(y^{〈〈〉〉}\right)))$. If we ask whether the state $F\left(M\right)$instantiates $M$, we are led to a contradiction. The upshot would seem to be that the STTler has to reject the existence of such properties as being Frege’s only favourite entity of type $〈〈〉〉$. But provided that one buys into STT and its type distinctions at all, this is certainly an awkward result.

To see how the OTTist avoids being saddled with an analogous result, let $G$ be the property of being Gödel’s only favourite entity. To construct an analogue of $M$, the best we can do is to let $N$ be the property of being an entity x that is identical with the state $G\left(y\right)$ for some zeroth-order entity y such that $y\left(x\right)$ fails to obtain, or in symbols: $N=\mathrm{\lambda x}\mathrm{\exists }{y}^1\left(\mathrm{\neg }{y}^1\right(x)\wedge (x=G(y^1\left)\right))$. From the type of ‘$y^1$’, it can be inferred that $N$ is at most first-order; yet nothing (on the face of it) commits us to holding that $N$ is zeroth-order. But if it isn’t, then it falls outside of the range of ‘$y^1$’, which is enough to block the paradox.

40 This ‘doctrine’ is loosely related to Russell’s vicious circle principle. It is also not too distantly related to Behmann’s (1931) proposal, already briefly mentioned in Section 2.2 above, that no predicate is predicable of a given object unless the predication can be rewritten in primitive vocabulary. Behmann has presented a more sophisticated version of his proposal in his (1959). However, neither this nor the original version offers an escape from Epimenidean paradoxes.

41 It is tempting to envision a broadly similar approach to the semantics of vague predicates, such as ‘is bald’ or ‘is a table’. The idea would be that, even though there are no properties of tablehood and baldness (i.e. no properties precisely corresponding to ‘is a table’ and ‘is bald’), there are properties that may be said to be picked out by possible precisifications of these predicates, and this is enough to keep them from being semantically defective – or at least enough to render them less semantically defective than, say, ‘is phlogiston’ or ‘orbits Vulcan’.

42 Let $R$ be the mentioned (hypothetical) relation, i.e. $\mathrm{\lambda x},y,z\mathrm{\exists w}\left(\left(w=y\right)\wedge \left(x=w\left(z\right)\right)\right)$. The reason why we are here considering this relatively complicated relation instead of the simpler $\mathrm{\lambda x},y,z\left(x=y\left(z\right)\right)$ lies in the fact that the latter relation would, if it were to exist, have an instantiation by entities x, y, and z, in this order, only if there actually existed an instantiation of y by z. It would thus not be a ‘generally applicable’ relation, whereas $R$, if it were to exist, would have an instantiation by x, y, and z, in this order, even if there were no instantiation of y by z.

To return to the paradox, let $Q$ be the property $\mathrm{\lambda x}\mathrm{\exists y}\left(R\left(y,x,x\right)\wedge \left(y\ne \mathrm{\top }\right)\right)$, and consider whether the state $Q\left(Q\right)$ obtains. We can first note that $Q\left(Q\right)$ is the state $\mathrm{\exists y}\left(R\left(y,Q,Q\right)\wedge \left(y\ne \mathrm{\top }\right)\right)$, which, given that $R=\mathrm{\lambda x},y,z\mathrm{\exists w}\left(\left(w=y\right)\wedge \left(x=w\left(z\right)\right)\right)$, and given (S6), is identical with (*) $\mathrm{\exists y},w\left(\left(w=Q\right)\wedge \left(y=w\left(Q\right)\right)\wedge \left(y\ne \mathrm{\top }\right)\right).$(*) From this it is easy to see that, if $Q\left(Q\right)$ obtains, then it is distinct from $\mathrm{\top }$. However, $R$ is denoted_∅,∅ by a term that contains no non-logical constants or free variables; and so the same goes for $Q$ and $Q\left(Q\right)$. Consequently, if $Q\left(Q\right)$ obtains, then it and $\mathrm{\top }$ necessitate each other, which by (S6) means that they are one and the same state. Since a state cannot be both distinct from and identical with $\mathrm{\top }$, we can conclude that $Q\left(Q\right)$ does not obtain. So it must be distinct from $\mathrm{\top }$ (since $\mathrm{\top }$ trivially obtains). But from this it can be inferred that (∗), i.e. $Q\left(Q\right)$ itself, does obtain: contradiction.

43 An objector might argue that I have put the cart in front of the horse: rather than to justify the hierarchy by relying on the assumption that my talk of instantiation is semantically non-defective, I should have established the hierarchy before indulging in talk of instantiation. Arguably, however, one can legitimately proceed in the opposite direction if there is independent reason to think that talk of instantiation is non-defective. Such a reason is given by the theoretical usefulness of the concept of instantiation in drawing up a theory of intensional entities. (Cf. also Section 6.1 below.)

44 To be sure, the existence of $R$ does not follow from the above ontology. But let us ignore this for the sake of the example.

45 Let x be any entity, and let $t$ be any term that denotes_∅,∅ x and contains no bound variable-occurrence at predicate- or sentence-position. By the semantics of $\mathcal{L}$, it can be seen that any atomic term that occurs free in $t$ is identical with ‘$\mathbf{I}$’ and hence denotes_∅,∅ the identity relation. By (F1) (together with Definition 16), it follows that x is zeroth-order.

46 Suppose for reductio that $P$ is zeroth-order, and let $Q$ be the property $\lambda y^1\left(\left(y^1=P\right)\wedge y^1\left(y^1\right)\right)$. Since $P=\mathrm{\lambda x}\mathrm{\neg \exists }{y}^1\left(\left(y^1=x\right)\wedge y^1\left(y^1\right)\right)$, the state $P\left(P\right)$ obtains iff $Q$ lacks an obtaining instantiation. Given that $P$ is zeroth-order, $Q$ has an instantiation by $P$, namely $\left(\left(P=P\right)\wedge P\left(P\right)\right)$. Suppose now that $P\left(P\right)$ obtains. It then follows that $Q$ lacks an obtaining instantiation, so that, in particular, $Q\left(P\right)$ does not obtain. But $Q\left(P\right)$ is the conjunction of $\left(P=P\right)$ and $P\left(P\right)$. Since $\left(P=P\right)$ trivially obtains, we thus have that $P\left(P\right)$ fails to obtain, contradicting the supposition. So we can conclude that $P\left(P\right)$ does not obtain. But clearly, $Q$ does not have an obtaining instantiation by any entity distinct from $P$. Hence $Q$ does not have an obtaining instantiation, which means that $P\left(P\right)$ obtains, after all: contradiction. This completes the reductio. We can thus infer that $P$ is not zeroth-order. So it must be first-order, since it is denoted_∅,∅ by ‘$\mathrm{\lambda x}\mathrm{\neg \exists }{y}^1\left(\left(y^1=x\right)\wedge y^1\left(y^1\right)\right)$’.

47 For related discussion, see, e.g., Sider (2011, 219), Bacon (2019, 1020; 2020, 569).

48 Bacon (2020, 566) has a roughly analogous principle of ‘Fundamental Completeness’.

49 The point of requiring the symmetry statements in question to be true_∅,g is to avoid weakening (F4) in such a way that, for any fundamental dyadic relation $R$ that is distinct from its non-trivial converse $\mathrm{\lambda x},yR\left(y,x\right)$, (F4) no longer entails, e.g., that the state $\left(R\ne \mathrm{\lambda x},yR\left(y,x\right)\right)$ does not necessitate $\mathrm{\forall x},yR\left(x,y\right)$. This would leave an unwelcome lacuna. (On the other hand, if $R$ were identical with $\mathrm{\lambda x},yR\left(y,x\right)$, then the state $\left(R\ne \mathrm{\lambda x},yR\left(y,x\right)\right)$ would necessitate $\mathrm{\forall x},yR\left(x,y\right)$, as it would be nothing other than the ‘impossible’ state $\left(R\ne R\right)$.) Similarly, the point of requiring the symmetry statements in question to be ontologically conservative relative to $t$ is to avoid weakening (F4) in such a way that it no longer entails, e.g., that for any particular or fundamental property x and any fundamental relation $R$ (other than identity), the state $\left(x=x\right)$ does not necessitate $\left(R=R\right)$.

50 See footnote 56 below for a version of this argument that relies on (F4) in its fully developed form.

51 More precisely: to ensure compatibility with (F2) in conjunction with the claim (which should arguably not be ruled out a priori) that there exists at least one fundamental relation $R$ with a distinct converse $R^{\mathrm{\prime }}$. To see the problem, let $u$ and $v$ be two variables that are under $g$ mapped to (respectively) $R$ and $R^{\mathrm{\prime }}$. Then the formulas $u=u$ and $v=v$ will respectively denote_∅,g the self-identity of $R$ and the self-identity of $R^{\mathrm{\prime }}$, which necessitate each other. Moreover, by (F2), $R^{\mathrm{\prime }}$ is fundamental, given that $R$ is. Yet $⌜u=u⌝$, even in conjunction with admissible symmetry statements, does not analytically entail $⌜v=v⌝$.

52 With some qualifications, (F4) can be considered a counterpart of Bacon’s (2020, 547) principle of ‘Quantified Logical Necessity’. An important difference (among several) lies in the fact that, where Bacon’s principle invokes a monadic notion of logical necessity, (F4) relies instead on the dyadic notion of necessitation.

53 I say ‘in principle’ because there are cases in which the procedure cannot be carried out: after all, given (F1), not every intensional entity is non-fundamental.

54 The proof of this assertion requires the use of (F3), which is needed to guarantee that there exists a term that denotes the respective set relative to a variable-assignment $g$ that satisfies clause (iii) of (F4). (For a similar reason, (F3) is also needed to prove the next assertion.)

55 By contrast, there may be fundamental instantiations of non-fundamental attributes. For example, suppose that there exists a fundamental state $s$. Then $s$ is zeroth-order, so that the non-fundamental property $\lambda x^1{x}^1$ has an instantiation by $s$. But that instantiation is nothing other than $s$ itself, and is therefore fundamental.

56 Let $s$ be any fundamental state (assuming that there is one), let $u$ and $v$ be two variables of type 1, let $g$ be the smallest variable-assignment that maps $u$ to $s$ and $v$ to $\mathrm{\neg s}$, and let $\tau$ be the formula $⌜\mathrm{\neg u}⌝$. The reductio succeeds because $⌜\mathrm{\&}\left(v\right)=\mathrm{\&}\left(v\right)⌝$ does not analytically entail $⌜\mathrm{\&}\left(v\right)=\mathrm{\neg u}⌝$.

57 Suppose for reductio that there are two fundamental states $s$ and $s^{\mathrm{\prime }}$ such that $s$ necessitates $s^{\mathrm{\prime }}$. By the definition of ‘necessitates’, there then exist an interpretation $I$, a variable-assignment $g$, and terms $t$ and $t^{\mathrm{\prime }}$ such that (i) $t$ and $t^{\mathrm{\prime }}$ respectively denote_I,g $s$ and $s^{\mathrm{\prime }}$ and (ii) $t$ analytically entails $t^{\mathrm{\prime }}$. Then $t$ also analytically entails $⌜t\wedge t^{\mathrm{\prime }}⌝$. But $⌜t\wedge t^{\mathrm{\prime }}⌝$ denotes_I,g $\left(s\wedge s^{\mathrm{\prime }}\right)$, which is therefore necessitated by $s$. Likewise, $s$ is necessitated by $\left(s\wedge s^{\mathrm{\prime }}\right)$. Hence, by (S6), $s$ is identical with $\left(s\wedge s^{\mathrm{\prime }}\right)$. So $s$ is a conjunction of two fundamental states. From the result stated in the text (namely, that the conjunction of any two fundamental states is not fundamental), it now follows that $s$ fails to be fundamental, contrary to hypothesis.

58 Let $P$ be this property, let $\rho$ be ‘$\lambda x^1\mathrm{\neg }{x}^1\left(x^1\right)$’, let $t$ be $⌜\rho =\rho ⌝$, and suppose for reductio that $P$ is zeroth-order. By Definition 16, there then exist an interpretation $I$ and a variable-assignment $g$ relative to which $P$ is denoted by a term $\tau$ that satisfies the following two conditions:

1. Any atomic term that is either identical with $\tau$ or has in $\tau$ a free occurrence at predicate- or sentence-position denotes_I,g either a particular or a fundamental intensional entity.

2. No variable has in $\tau$ a bound occurrence at predicate- or sentence-position.

   Without loss of generality, we may take $I$ to be the empty set and suppose that $g$ and $t$ jointly satisfy clauses (ii)–(v) of (F4). Given that $\tau$ denotes_∅,g the property $P$, which is purely logical and hence (as shown above) non-fundamental, it follows from (1) that $\tau$ is non-atomic. Now assume the following holds:

3. At least one atomic term has in $\tau$ a free occurrence at predicate-position.

Let $u$ be any such term, and let $t^{\mathrm{\prime }}$ be $⌜\rho =\tau ⌝$. Evidently $u$ does not occur free in $\rho$. Hence, unless $u$ is the constant ‘$\mathbf{I}$’, any conjunction of $t$ with zero or more admissible symmetry statements will not analytically entail $t^{\mathrm{\prime }}$; for there will exist an interpretation and variable-assignment relative to which $t$ has a denotation while $t^{\mathrm{\prime }}$ doesn’t. But, since $\rho$ and $\tau$ denote_∅,g the same entity (viz., $P$), we have that $t$ and $t^{\mathrm{\prime }}$ denote_∅,g one and the same state $s$. From (F4), it now follows that $t$, possibly together with one or more admissible symmetry statements, analytically entails $t^{\mathrm{\prime }}$. By what has just been said, we can thus infer that $u$ is the constant ‘$\mathbf{I}$’. But $u$ was any atomic term that has in $\tau$ a free occurrence at predicate-position. Hence, given (2), we have that no atomic term other than ‘$\mathbf{I}$’ occurs in $\tau$ at predicate-position. (At this point we can ‘discharge’ the above assumption (3).) Given (S5), it can now be seen, by induction over the complexity of $\tau$, that any property denoted_∅,g by $\tau$ must have an instantiation by any zeroth-order entity. So, since $\tau$ denotes_∅,g $P$, which by hypothesis is zeroth-order, $P$ has an instantiation by itself. But since $P$ is also denoted_∅,∅ by ‘$\lambda x^1\mathrm{\neg }{x}^1\left(x^1\right)$’, any instantiation of $P$ by itself obtains just in case it doesn’t. This completes the reductio.

59 Let x be any particular or fundamental attribute other than the identity relation, let $v$ be any variable of type 1, and let $g$ be the smallest variable-assignment that maps $v$ to x. Further, let $\alpha$ and $\text{β}$ be any two ordinals, and let $L$ and $M$ be any two lambda-expressions that denote_∅,∅ $\alpha$ and $\text{β}$, respectively. We have to consider three cases.

First, suppose that x is a particular. We can then show, with the help of (F4), that the zeroth-order fact $\left(x\ne \alpha \right)$ is distinct from the zeroth-order fact $\left(x\ne \text{β}\right)$. For let $\rho$ be $⌜v\ne L⌝$, let $\tau$ be $⌜v\ne M⌝$, and suppose for reductio that $\left(x\ne \alpha \right)$ is identical with $\left(x\ne \text{β}\right)$. We then have that $\rho$ and $\tau$ denote_∅,g the same state, and so, with $t=⌜\rho =\rho ⌝$ and $t^{\mathrm{\prime }}=⌜\rho =\tau ⌝$, we also have that $t$ and $t^{\mathrm{\prime }}$ denote_∅,g the same state. Moreover, clauses (ii)–(v) of (F4) are satisfied. Yet $t$, even in conjunction with one or more symmetry statements, does not analytically entail $t^{\mathrm{\prime }}$. (To see this, let $h$ be some variable-assignment that maps $v$ to $\alpha$. Then $\rho$ denotes_∅,h a non-obtaining state while $\tau$ denotes_∅,h an obtaining state. As a result, $t$ denotes_∅,h an obtaining state while $t^{\mathrm{\prime }}$ doesn’t.) This completes the reductio, and we can conclude that $\left(x\ne \alpha \right)$ is distinct from $\left(x\ne \text{β}\right)$. But $\alpha$ and $\text{β}$ were any two ordinals. Hence there are as many zeroth-order facts as there are ordinals.

Next, suppose that x is a fundamental property. The argument proceeds exactly as before, only that, for some untyped variable $u$, we let $\rho$ be the formula $⌜\mathrm{\lambda u}v\left(u\right)\ne L⌝$. Finally, suppose that x is a fundamental $\gamma$-adic relation, for some $\gamma >1$. The argument proceeds again largely as before, only that now we are considering not $\left(x\ne \alpha \right)$ and $\left(x\ne \text{β}\right)$ but rather the somewhat different zeroth-order facts $\left(x\ne R_{\alpha }\right)$ and $\left(x\ne R_{\text{β}}\right)$, where, for any ordinal $\delta$, $R_{\delta }$ is the $\gamma$-adic relation $\lambda x_0,x_1,\dots \left(\delta \left(x_0\right)\wedge \underset{1\le i<\gamma }{\bigwedge }{x}_i=x_i\right)$. (The symbol ‘$\bigwedge$’ should here be read as an abbreviatory device for conjunctions. Basically, $R_{\delta }$ is $\delta$ with one or more ‘dummy’ argument-places tacked on.) Correspondingly, for some $\gamma$-sequence of pairwise distinct untyped variables $u_0,u_1,\dots$, we let $\rho$ and $\tau$ be (respectively) the formulas ${\phi }_L$ and ${\phi }_M$, where, for any term $t$, ${\phi }_t$ is the formula $\lambda u_0,u_1,\dots v\left(u_0,u_1,\dots \right)\ne \lambda u_0,u_1,\dots (\left(t\right)\left(u_0\right)\wedge \underset{1\le i<\gamma }{\bigwedge }{u}_i=u_i).$Suppose for reductio that $\left(x\ne R_{\alpha }\right)$ is identical with $\left(x\ne R_{\text{β}}\right)$. Then $\rho$ and $\tau$ will denote_∅,g the same state, and so, with $t=⌜\rho =\rho ⌝$ and $t^{\mathrm{\prime }}=⌜\rho =\tau ⌝$, we also have that $t$ and $t^{\mathrm{\prime }}$ denote_∅,g the same state. Yet $t$, even conjoined with one or more symmetry statements, does not analytically entail $t^{\mathrm{\prime }}$. (To see this, consider a variable-assignment $h$ that maps $v$ to $R_{\alpha }$.) This completes the reductio, and we can conclude that $\left(x\ne R_{\alpha }\right)$ is distinct from $\left(x\ne R_{\text{β}}\right)$. As before, it follows that there are as many zeroth-order facts as there are ordinals.

60 Cf., e.g., Quine (1960, 257f.) on ‘ordered pair’ and Lewis (1986, 55f.) on ‘property’.

61 See, e.g., Leśniewski (1992, 209ff.), Black (1971), Bealer (1982, ch. 5), Lewis (1991, §2.1). For relevant historical discussion, see Kanamori (2003).

62 It might be suggested that the mystery could have been avoided by thinking of those ‘ways of grouping’ not as sets of sets but rather as pluralities of pluralities. However, it is somewhat hard to see why pluralities should be any less mysterious than sets. The metaphor of the ‘invisible plastic bag’, used by Bealer to lampoon the notion of set, is no less applicable here. (Cf. also Florio and Linnebo [2021, ch. 8], where this becomes clearer.) Moreover, pluralities lead to trouble within an ontology that also admits non-fundamental intensional entities, such as instantiations of fundamental attributes. For example, if we think that there is a plurality of the flowers in the field as well as the state of John’s loving Mary, then presumably we will also want there to be such a thing as the state of John’s loving the flowers in the field. But as innocent as this kind of commitment may seem, it tends to lead to a form of the Russell–Myhill paradox. (Cf. McGee and Rayo [2000], Pruss and Rasmussen [2015].) We thus seem to be faced with a choice between renouncing pluralities on the one hand and renouncing non-fundamental intensional entities on the other. Given the enormous theoretical utility of the latter, it is arguably pluralities that have to go. (A third option, which is to make pluralities behave more like sets, has recently been advanced by Florio and Linnebo op. cit.)

63 This talk of what we ‘need’ should be understood in the sense of what has to be put in place (at least on one possible approach) in order to construct a philosophically satisfying theory that accommodates sets, in particular in accordance with the proposed conception of sets as complex properties.

64 This definition is, like the previous one, essentially borrowed from Plate (2016, 30f.). I have added the qualifier ‘logically’ in order to leave terminological room for different notions of analysability.

65 Suppose there exists some abyssal entity x. By (F3) (and regardless of whether x is abyssal), there exists a set $X$ such that $x\prec X$ and every member of $X$ is either a particular or a fundamental intensional entity. But since x is abyssal, $X$ contains an entity y that is analysable in terms of some non-logical entity z that fails to be analysable in terms of y. Again by (F3), there exists a set $Z$ such that $z\prec Z$ and every member of $Z$ is either a particular or a fundamental intensional entity. Since z is non-logical, $Z$ has to contain some entity $w$ distinct from $\mathbf{I}$. Further, since z fails to be analysable in terms of y, so must $w$. Consequently, $w$ is distinct not only from $\mathbf{I}$ but also from y, and is also not a converse of y. Moreover, since $y\in X$ and $w\in Z$, we have that each of y and $w$ is either a particular or a fundamental intensional entity. So, by (F4), y is independent of $w$. (Cf. Section 5.5.1.) But this cannot be, since y is, via z, analysable in terms of $w$.

66 Let x be any abyssal entity, and let $X$ be any set with $x\prec X$. Suppose for reductio that $X$ contains no abyssal entity. For each $w\in X$, there then exists a set $S_w$ such that $w\prec S_w$ and $S_w$ does not contain any intensional entity y that is analysable in terms of some non-logical entity that fails to be analysable in terms of y. The union of the $S_w$, i.e. $\bigcup _{w\in X}{S}_w$, will then also not contain any such entity. At the same time it can be seen that $x\prec \bigcup _{w\in X}{S}_w$. So x fails to be abyssal, contrary to hypothesis.

67 Plate provides an account of logical complexity in terms of ‘reductions’. The details of that account need here not interest us, but it is worth noting that, while Plate’s main account is concerned with the concept of logical simplicity as applied to attributes, he provides an analogous account applicable to states on pp. 34f. of the same paper. With minor changes, the result in question can be stated as follows (cf. op. cit., 32):

• (R) If an attribute $A$ is analysable in terms of some non-logical entity that is not fully analysable in terms of $\left\{A,\mathbf{I}\right\}$, then $A$ is logically complex (i.e. not logically simple).

This result is obtained on the basis of assumptions about the individuation of intensional entities that, in all relevant respects, also hold in the present system. An analogous result could be obtained for states. Meanwhile, from Definition 17 above it can – with the help of the transitivity of analysability – be inferred that any abyssal entity x is analysable in terms of some non-logical entity that is not fully analysable in terms of $\left\{x,\mathbf{I}\right\}$. By (R) and its analogue for states, it now quickly follows (given that every abyssal entity is an attribute or state) that every abyssal entity is logically complex.

68 For example, Russell (1918/1919, 180) seems to have taken seriously an analogous possibility in the case of facts. Decades later, theories of non-well-founded sets have been studied by, e.g., Aczel (1988). Around the same time, Barwise (1989, 192ff.) has defended the existence of ‘situations’ that have themselves as constituents. His examples include announcements and mental acts that are in some sense ‘about’ themselves. It seems tolerably clear, however, that such cases need not be interpreted as evidence for the existence of logically non-well-founded entities.

69 For discussion and references, see Cotnoir and Varzi (2021, §5.2.1).

70 Cf. Plate (MS). Also see, e.g., Nolan (2012).

71 See, e.g., Prior, Pargetter, and Jackson (1982), Prior (1985, ch. 7), Campbell and Pargetter (1986), Pargetter (1988), Lewis (1997).

72 For context, see, e.g., Bongard (1970).

Additional information

Funding

This work was supported by Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung [grant number 100012_173040, 100012_192200].