Abstract
Individuals play a prominent role in many metaphysical theories. According to an individualistic metaphysics, reality is determined (at least in part) by the pattern of properties and relations that hold between individuals. A number of philosophers have recently brought to attention alternative views in which individuals do not play such a prominent role; in this paper I will investigate one of these alternatives.
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Notes
At least historically, such constraints have been motivated by crude theistic or verificationist concerns. Most who endorse such a constraint today, think that it is a defeasible constraint stemming from Occam’s-razor-style considerations (see, e.g., Pooley 2013).
There is also a distinct motivation from quantum mechanics—which I won’t touch on here—that has to do with the interpretation of states in quantum field theory. In a classical theory there are typically four possibilities for a two particle system where each particle can possess one of two properties F and G. In QFT there arise situations where we would only count three: both F, both G, and one of each. The strategies I’m considering in this paper take metaphysics involving individuals and outputs individual-free surrogates. If it turns out that QFT cannot be described in individualistic terms to begin with, our algorithm will have nothing to say about it.
Compare with Russell’s statement of ‘quantifier generalism’ (Russell 2017).
This analogy isn’t perfect. ‘Someone is German’ doesn’t contain a singular term, but expresses a proposition that is indirectly about Germany (it says, roughly, that someone is from Germany).
One important reason to seek a more coarse-grained theory of propositions is that the naïve version of the structured theory is actually inconsistent due to the Russell–Myhill paradox, and is thus unsuitable as a foundation of the metaphysics of propositions (see, for example, the discussion in Dorr 2016, Section 6). There are consistent coarser-grained theories of propositions that keep some elements of the structured theory: for example Dorr’s (2016) theory distinguishes propositions that have a different number of occurrences of a component, and in Goodman’s (2017) theory propositions can be different in virtue of the individuals that are their constituents, but both theories ignore other aspects of the structure.
Such as being closed under Boolean operations. We shall discuss further structural features later.
One might deny that there are worlds like this for all permutations of individuals: perhaps I could not have occupied the qualitative role of a boiled egg, so the permutation that switches me for a boiled egg cannot be applied to the actual world. Perhaps it is not metaphysically possible for me to be a hard-boiled egg, but the relevant sense of possibility at stake here might be a broader one (see Bacon 2017). More restricted versions of this idea can be also be developed: perhaps individuals can be partitioned into kinds, and we must restrict attention to permutations of individuals that preserves the kind they belong to.
The generalization of this definition to infinite sets of individuals is non-trivial. If F is a qualitative property, then \(Fa_1\wedge \cdots \wedge Fa_n\) is intuitively about \(a_1\ldots a_n\). But it is also fixed by any permutation that fixes all individuals except \(a_1\ldots a_n\), since permutating the conjuncts of a conjunction leaves it alone, making this conjunction count additionally as being about every individual except for \(a_1\ldots a_n\) (this is a general version of a counterexample mentioned by Fine (1977), although I got the generalization from Harvey Lederman). This is perhaps desirable, since this proposition can be equivalently expressed (in the constant domain setting) as ‘everyone except for \(a_{n+1}, a_{n+2},\ldots \) is F’. Be this as it may, on the assumption that there are infinitely many individuals apart from \(a_1\ldots a_n\), this puzzle falls outside the purview of our definition. Fine also raises an issue to do with infinite disjunctions, and discusses a number of different alternative definitions of aboutness; however, he proves that for propositions concerning only finitely many individuals the definitions are all equivalent.
One feature of my definition of aboutness is that if a proposition p is about distinct individuals \(\{a_1\ldots a_n\}\), then there is some qualitative relation R such that \(p=Ra_1\ldots a_n\). This qualitative relation is not in general unique, however there is always a unique strongest qualitative relation that p can be decomposed into.
Note that we must employ some sort of free logic in these contexts so that Nihilism has a chance of being true. Classical logic has the theorem that there is at least one thing: \(\exists x\, x=x\).
Fundamental Qualitativism is very close to a principle articulated in Russell (2017), that states that all determinate truths are qualitative. However it is not true in general that ‘fundamental’ and ‘determinate’ are synonymous: in particular, one can know non-fundamental truths whereas, insofar as I can make sense of the notion, indeterminate propositions are always unknown. For these reasons it is worth distinguishing these different theses; my theory of fundamentality is fleshed out below.
I am assuming here the principle that if P is fundamental, and is about some individual a, then a is also fundamental.
Both of these arguments rest on subtleties that were not present in the argument from Qualitativism to Nihilism. Assuming Booleanism, it’s not always true that propositions of the form Fa are haecceitistic, since \(\lambda x (Fx \vee \lnot Fx)a = \top \), and \(\top \) is surely qualitative. But plausibly contingent properties like being an electron, do not yield qualitative propositions when applied to individuals. Similarly, no principle I have stated so far guarantees that fundamental truths can’t be about non-fundamental individuals, although it seems plausible.
One might attempt to avoid the awkwardness we raised earlier by insisting that fundamentality is closed under definability, thus eliminating the distinction between fundamentality and fundamentality\(^*\). An objection to this idea is that it involves redundancy in what is fundamental. Adopting a metaphor from Sider (2011), when God is writing the ‘book of the world’, it seems like it would be redundant for him to include negation, disjunction and conjunction, when in fact the first two would have sufficed. However, this metaphor is prone to mislead if we imagine that God’s language is like English or standard presentations of first-order logic. One could imagine alternative languages in which there’s no difference between including negation and conjunction, negation and disjunction and having all three. Consider, for example, Ramsey’s notation for the propositional calculus, in which conjunction and disjunction are represented as \(\wedge \) and \(\vee \) as usual, and negation is represented by turning the formula upsidedown—it is simply not possible to include \(\vee \) and negation in this language without including \(\wedge \), and vice versa. A full defense of this conception of fundamentality, as closed under definability, would take us too far afield. We will thus continue to draw the distinction between what is fundamental and fundamental\(^*\), but leave it open whether they amount to the same thing.
A natural alternative is that metaphysical definability means definability from the combinators: operations in the type hierarchy that can be defined using only \(\lambda \)s and variables (although I’ve described them syntactically here, these operations can be given a non-syntactic characterization). However this conception is somewhat limited—for example, it doesn’t allow us to do things like definition by cases. The Permutation Criterion provides a much more comprehensive list of definitional devices by contrast.
Reason: if \(\pi _{\sigma \rightarrow \tau }(F)= F\) and \(\pi _{\sigma }(a) = a\), then \(Fa = \pi _{\sigma \rightarrow \tau }(F)\pi _{\sigma }(a) = \pi _\tau F \pi _\sigma ^{-1} \pi _\sigma a= \pi _\tau (Fa)\).
It follows by Booleanism that \(\vee (x)(y) = \lnot (\wedge (\lnot x)(\lnot y))\). It follows that \(\lambda y\lambda x\, \vee (x)(y) = \lambda y\lambda x\lnot (\wedge (\lnot x)(\lnot y))\) by the \(\xi \) rule, which says that if \(\vdash \alpha =\beta \) then \(\vdash \lambda x\,\alpha = \lambda x\, \beta \) (this step can also be made using the functionality principle discussed in Sect. 8, but this is stronger than we need). Finally \(\vee = \lambda y\lambda x\lnot (\wedge (\lnot x)(\lnot y))\) follows from two applications of the \(\eta \)-rule, which says that \(\lambda x\,\alpha x = \alpha \), to the left hand side.
In order to make this precise we assume that the model is one in which \(D^t\), the domain of propositions, just consists of all sets of possible worlds. We will also assume that the Boolean connectives (including the infinitary Boolean connectives) are qualitative and, for simplicity, that the domain of each world is the same, so that qualitative isomorphisms are just permutations. Then it can be shown that the qualitative isomorphisms of Sect. 1 are exactly the permutations defined here that fix the qualitative entities.
An example of an expression of type \(t\rightarrow e\) is the complementizer that. When you apply ‘that’ to a sentence, you get back something that behaves a bit like a singular term (e.g. ‘that snow is white’).
That is, we assume that precision is also closed under metaphysical definability. We also assume that there are vague propositions (as opposed to merely vague sentences), as defended in Bacon (2018).
One way in which this would be true would be if each truth was simply necessitated by a fundamental truth. However, my formulation is neutral on the question of whether there are any fundamental propositions. The idea that there are no fundamental propositions might flow naturally from a ‘no redundancy’ conception of fundamentality. For example, it would be redundant to include the proposition that Sparky is an electron as fundamental, if Sparky and electronhood are fundamental.
If the underlying algebra of propositions is complete and atomic, then for any world proposition w (possible or impossible), the conjunction of qualitative propositions w entails is also a consistent qualitative proposition that w entails. Since every world proposition entails a maximally consistent qualitative proposition it follows that the algebra is atomic.
Here is the argument in a little more detail. Suppose w is a metaphysical possibility where p is true and q false. Since the qualitative propositions are complete, the conjunction, r, of all qualitative propositions entailed by w is qualitative, and settles any other qualitative proposition by entailing either it or its negation. If \(w'\) were another metaphysical possibility where r is true, then w and \(w'\) would agree about all qualitative matters but disagree about some truth (e.g. whether w obtains or not). This contradicts supervenience, so w is the only metaphysical possibility where r is true. Every metaphysical possibility where \(q \vee r\) is true is either identical to w or a world where p is true. Thus any metaphysical possibility that makes \(q\vee r\) true is a metaphysical possibility that makes p true. So \(q\vee r\) is a qualitative proposition that necessitates p, contradicting our assumption that q was the weakest such proposition.
Given S5 this relation is reflexive, symmetric and transitive.
An anonymous referee has noted that the above appears to contradict something claimed in the introduction: that our theory recreates metaphysical possibilities corresponding to the swapping of qualitative roles in an individualistic metaphysics (although the recreated possibilities will be qualitatively different). One might be tempted to think this because each world in a cell ‘corresponds’, as it were, to the different ways of switching the roles of individuals in a individualistic theory. If only one world in a cell is metaphysically possible, then it looks like there can’t be two metaphysically possible worlds corresponding to a switching of qualitative roles. However, the above reasoning does not preclude there being two metaphysically possible, but switched, worlds in different cells. Indeed, in our reconstruction of these haecceitistic possibilities, there are primitive qualitative propositions that distinguish them, forcing them to belong to different cells. We shall treat this more thoroughly later; see Fig. 4 and the surrounding discussion.
It’s worth considering why this is so. Couldn’t we get by with two propositions, a single falsehood and a single truth which is the target of each false but seemingly true individualistic sentence? I take it that one of the reasons for pursuing a paraphrase strategy is to explain why it’s often useful to assert falsehoods. Suppose that there is both a bear and a rabbit behind you: it’s more urgent that I warn you that there’s a bear behind you, than that I warn you that there’s a rabbit behind you. If these warnings had the same true paraphrase we couldn’t explain this urgency. These considerations suggest that the nihilist needs there to be as many propositions as there would have been had the individualistic theory been true.
See Dasgupta (2009), Turner (2011). The basic idea was first articulated by Quine, and axiomatizations were later provided by Bacon (1985) and Kuhn (1983). The project is also closely related to the program of eliminating variables from theories, originating in Curry’s combinatory (Curry 1958). (See also variable free approaches in linguistics) (see Jacobson 1999).
Since I am presenting a version of functorese that is consistent with the simple theory of types outlined in Sect. 4, there are some minor differences between my presentation and standard presentations. For example, in the usual version of functorese \(\Delta \) is an operation that takes a predicate of any arity greater than 0, and returns another predicate of one less arity. As such this operation has no type, since the arity of the argument of a functor is uniquely determined by its type. Technically in the present setting we need an infinite collection of predicate functors \(\Delta _n\), one for each arity.
For every individual you can construct a permutation that moves it, but fixes all the standard predicate functors. Indeed for any permutation \(\pi \) of \(D_e\), there is a permutation of the entire structure generated by \(\pi \) on the individuals, and the identity permutation on the propositions, which fixes all the predicate functors.
The view in which individuals are sets of properties, as opposed to properties of properties, is not particularly plausible anyway since, on the assumption that an individual has a property if it contains it, it follows that individuals have all their properties essentially.
See for example Carpenter (1997, Chapter 2).
The operator \(=\top \) behaves like a very broad necessity operator—see Bacon (2017). In the present context it corresponds to be true in all possible and impossible worlds.
The feature of these sorts of models that is important for validating Modalized Functionality is that properties are uniquely determined by functions from worlds and individuals to truth values.
An anonymous referee has suggested to me a version of functorese nihilism in which predicates are not fundamentally analyzed in terms of things of type \(e\rightarrow t\) at all, but rather in terms of things whose types only involve ts. I am broadly sympathetic to this sort of response, and I pursue a similar line of thought in Sect. 9. But it is not clear to me that any such view can really be counted as a version of predicate functorese, since, as I have been using the term, a predicate simply is something of type \(e\rightarrow t\). (Of course, if the view described in Sect. 11 can be understood as a version of predicate functorese under a more liberal interpretation of ‘predicate’, then I have no objection to predicate functors; but this is just a verbal issue.)
Here we assume that the \(D_{e\rightarrow t}\) consists of all functions from \(D_e\) to \(D_t\). One can show this by considering the result of applying the Sparky predicate functor to the haecceity of Sparky: the property that maps Sparky to \(\top \) and every other element of \(D_e\) to \(\bot \).
This argument relies on the permutation criterion for definability. A weaker criterion that delivers many of the same results identifies definability with definability using combinators (i.e. things definable using only variables and \(\lambda \)). In the latter case it can be proven that no individuals can be defined from any set of predicates and predicate functors. This follows from the Curry-Howard isomorphism, which states that at least one expression of type \(\tau \) can be defined from things of type \(\sigma _1\ldots \sigma _n\) iff \(\tau \) is provable from \(\sigma _1\ldots \sigma _n\) in intuitionistic logic, treating the \(\rightarrow \) symbol as the conditional and e and t as propositional letters. It is easy to see that e cannot be proved from the types of predicates and predicate functors. Consider the following classical model (which is therefore also a model of intuitionistic logic): e is false and t is true. Every n-ary predicate type—t, \(e\rightarrow t\), \(e\rightarrow e \rightarrow t\), and so on—is easily verified to be true. A predicate functor is of the form \(\sigma _1 \rightarrow \sigma _2 \rightarrow \cdots \sigma _n \rightarrow \tau \), where \(\sigma _1,\ldots ,\sigma _n, \tau \) are all predicates. It follows that the type of any predicate functor is true in this model, since the type of any predicate is true. Since e is false one cannot prove e from the types of predicates and predicate functors.
More generally, every e-involving relational type is equivalent to a non-e-involving relational type. The relational types are defined as follows: t is a relational type, and if \(\sigma \) and \(\tau \) are relational types then so are \(\sigma \rightarrow \tau \) and \(e\rightarrow \tau \). First we define a translation from relational types to relational types over base types t and 1: \(t^* \mapsto t\), \((\sigma \rightarrow \tau )^* \mapsto \sigma ^* \rightarrow \tau ^*\), \((e\rightarrow \tau )^* \mapsto 1\). Then we define a translation from this type signature to the types only involving t as follows: \(t' \mapsto t\), \((1\rightarrow \sigma )' \mapsto \sigma '\), \((\sigma \rightarrow \tau )' \mapsto \sigma ' \rightarrow \tau '\). Finally, the result of composing these two translations results maps each e-involving type to a type only involving t. A simple induction shows that there is a bijection between \(D_\sigma \) and \(D_{\sigma ^{*\prime }}\) where we let \(D_e = \emptyset \), \(D_1 = \{*\}\), \(D_t\) may be any set, and we take the full space of functions when we define \(D_{\sigma \rightarrow \tau }\). (I suspect a more complicated argument would establish a similar result for arbitrary types.)
More formally, t is of hereditarily propositional type, and if \(\sigma \) and \(\tau \) are of hereditarily propositional type, so is \(\sigma \rightarrow \tau \).
This sort of redescription is similar to the sort of strategy outlined in broad strokes by Sider (2008) (for a slightly different purpose). Thanks to a referee for pointing this connection out to me.
This therefore includes states where two particles are assigned the same location. Such states are in fact needed in order to account for collisions between point particles, although this raises some subtle issues that we don’t need to get into.
See Field (1980). Field shows how to state derivatives using congruence and betweenness relations, which suffice for us to be able to state the two differential equations of the Hamiltonian formulation of classical mechanics.
Note, crucially, that this is not the same as the composite operator \([a]^*[b]^*\)—this is one reason why it’s preferable to take the particle operators and the obverse relation as primitive, rather than the obverses of particle operators as primitive.
This argument doesn’t hold when the range is empty or encompasses every possible value. In these two special cases our Fact holds vacuously.
If we wrote \(A^{p,l}\) for this proposition, then \(([a][b])^*A^{p,l}\) would correspond to the proposition that eithera has momentum p and b location lorb has momentum p and a location l.
For each variable \(x_i\) of type e we choose an operator variable \(X_i\) of type \(t\rightarrow t\), and for each atomic unary relation P we have a propositional letter \(A_P\) expressing the proposition corresponding to P:
\((a)^+ := [a]^*\)
\((x_i)^+ := X_i\)
\((Pt)^+ := (t)^+A_P\)
\((\lnot \phi )^+ := \lnot (\phi )^+\)
\((\phi \wedge \psi )^+ := (\phi )^+\wedge (\psi )^+\)
\((\exists x_i\phi )^+ = \exists X_i ({\mathcal {P}}X_i \wedge \phi ).\)
To illustrate the translation informally, the sentence ‘some particle is located in region r’, gets paraphrased as ‘for some particle operator X: X(some particle is located in region r)’.
In the following we assume that there is no contingency concerning what exists. Without this assumption the following conditions would need a slightly more careful formulation—these complications are mostly orthogonal to our concerns here.
The basic intuition is that given the cardinality constraint we can think of each world as a specification of the state of each object where the states are represented by \(S'\). Suppose that \(\rho {:}\;W \rightarrow D^S\) is a bijection. For each \(s'\in S'\) we define an object state as follows: \(s := \{(w,a) \mid \rho (w)(a) = s'\}\). It is then routine to show that the set of object states generated this way is a partition of \(W\times D\) and satisfies Combinatorialism.
We could also identify a monadic property with a relation whose first argument is redundant (the converse of our choice). A monadic property and the two dyadic relations it generates are all metaphysically interdefinable (given, e.g., the Permutation Criterion), so it actually doesn’t matter which we add to our basis.
It should be noted that theories that involve particles, space-time points and an asymmetric location relation do not straightforwardly satisfy Dyadic Combinatorialism: that would imply that the pair (a, b) and (b, a) could be assigned to a pair state that’s contained in the location relation, implying that a is located at b and b is located at a (thanks to Jeremy Goodman for spotting this fact). A space-time theory that does satisfy Dyadic Combinatorialism takes the symmetric relation \(Sxy :=\)x is located at y or y is located at x as primitive, along with \(Lx:= \)x is a location and \(Px :=\)x is a particle. In this theory the ordinary asymmetric location relation is a defined relation: \(Px \wedge Ly \wedge Sxy\).
This corresponds to a sort of reflection in state-space. If these states were removed there would be gaps in state-space and it’s not obvious how to formulate the laws in this setting (but see the theory of symplectic reduction which provides one framework for thinking about this issue; Butterfield 2006).
If they are indeed superfluous to science; see footnote 52.
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Acknowledgements
Thanks to: Jeremy Goodman, Cian Dorr, and to two anonymous referees for this journal for some very helpful feedback. Thanks also to the audience of Metaphysics on the Mountain 2017; thanks, in particular, to my commenter, Louis deRosset, who provided me with some great feedback and spotted several mistakes.
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Bacon, A. Is reality fundamentally qualitative?. Philos Stud 176, 259–295 (2019). https://doi.org/10.1007/s11098-017-1015-1
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DOI: https://doi.org/10.1007/s11098-017-1015-1