1. List on Supervenience and Reduction

I start by giving a brief recapitulation of the apparatus used by List (Reference List2018) to analyze supervenience. For List, a system of ontological levels is a concrete posetal category wherein each function is surjective. That is, it consists of a class of sets ℒ equipped with surjective functions between them, such that (i) the class of all such functions is closed under composition, and for each set we include the identity function on that set (hence making this a concrete category), and (ii) for any sets A and B, there is a unique function σ mapping A to B (hence making the category posetal).

In order to relate this to issues of reduction, List introduces certain linguistic notions. Formally, he defines a language L to be a set (of “sentences”) equipped with a unary “negation” operator and a “consistency” property. Given a language L, List defines the ontology Ω_L of L to be the set of all maximal consistent subsets of L (where a consistent set A is maximal if and only if (iff) there is no consistent B such that $A\subset B$). Each element of this ontology he calls a world for L. For any $\varphi \in L$, ϕ is true at the world $\omega \in {\Omega }_L$ iff $\varphi \in \omega$; otherwise, it is false at ω. The propositional content [[ϕ]] of $\varphi \in L$ is the set of worlds at which ϕ is true: that is, $\left[\right[\varphi \left]\right]≔\{\omega \in {\Omega }_L:\varphi \in \omega \}$. Then, a system of descriptive levels is a system of ontological levels where each level is the ontology Ω_L of some language L.

The notion of reduction is defined as follows. Given some system of descriptive levels, suppose that $\sigma :{\Omega }_L\to {\Omega }_{L^{\prime }}$ is a supervenience mapping. List says that a sentence ${\varphi }^{\prime }\in L^{\prime }$ reduces to $\varphi \in L$ if $\left[\right[\varphi \left]\right]={\sigma }^{-1}\left(\right[\left[{\varphi }^{\prime }\right]\left]\right)$: that is, if ϕ is true at a world ω iff ϕ′ is true at σ(ω). And we say that the language L′ reduces to L if every ${\varphi }^{\prime }\in L$ reduces to some $\varphi \in L$: that is, if every higher-level sentence is reducible to some lower-level sentence.

Certainly, setting up this kind of association between the apparatus of possible worlds and languages is going to be necessary if we are going to talk about the relationship between supervenience (a map, on this treatment, between ontologies) and reduction (a relationship between languages). But it is not clear to me that this is quite the right way to set things up.

First, it seems a mistake to use maximal consistent sets of sentences to represent the ontology associated to a language. This is not because I have a problem with identifying worlds with sets of sentences—we are just doing mathematical modeling, after all, so “identification” is not really doing anything more than asserting a one-to-one correspondence. Rather, the problem is that there are intuitive reasons to think that, in general, there are more worlds (of a given language’s ontology) than there are maximal consistent sets of sentences of that language. Specifically, consider the class of models of a first-order language L. It seems plausible to suppose that there could be a world just like each model—one with the same number of individuals as there are elements in the model’s domain and with properties and relations distributed over those individuals just as the extensions of the predicates are distributed over the model. This suggests that we can take such models to represent worlds, with nonisomorphic models representing distinct worlds.Footnote ¹ But in first-order logic, there are nonisomorphic models that are elementarily equivalent, that is, that satisfy all the same sentences. So a maximal consistent set of sentences will, in general, be satisfied by several nonisomorphic models and so by several distinct worlds.

For this reason, I suggest that a friendly amendment to List’s proposal is that we take the ontology associated with a language to be the class of models for that language, rather than the set of maximal consistent sets of sentences of that language. This will require us to say a little more about the nature of the language: for the purposes of this essay, I will assume that we are working with finitary first-order languages, each with a particular nonlogical vocabulary. Note that we are including the possibility that the language is a propositional language (regarding propositional constants as nullary predicates).

That said, it does seem that we do not want to represent the ontology associated to a given level by all models of the associated language. After all, if the language contains predicates R for “red all over” and G for “green all over,” then there will be a model containing objects satisfying both R and G. That is, the worlds represented by the class of all models is intuitively the class of logically possible worlds, but that is typically not the species of possibility that we are interested in for the purposes of supervenience. Standardly, we want to consider supervenience relative to metaphysical possibility or relative to something more restrictive still (e.g., physical possibility). In List’s approach, one could code this up into the consistency property, so that only worlds corresponding to metaphysically or physically consistent sets of sentences are permitted (where “physical inconsistency” would mean, roughly, consistency with the physical laws).

In the friendly amendment I am suggesting here, a more natural way to do things would be to confine attention to the models of some theory, where that theory consists of all and only the sentences expressing things that are necessary—necessary, that is, relative to the standard of possibility we seek to capture. Thus, if we are thinking about metaphysical possibility, then we take the theory to consist of all the metaphysically necessary L-sentences (e.g.,$\forall x\neg (\mathrm{Rx}\wedge \mathrm{Gx})$); if we are thinking about physical possibility, then we take the theory to consist of all the L-sentences expressing physical laws; and so on and so forth. So every descriptive level is associated with a given theory, and the supervenience maps between levels are maps between the sets of models of the associated theories. We follow List in imposing the requirements that the sets of models, equipped with the supervenience maps, constitute a posetal category. Rather than requiring that these maps be surjective, however, it will be more helpful to reflect further on the relationship between models and the possible worlds they represent.

The relevant observation is that consistency with the laws at its own level is only a necessary condition for a model to represent a possible world, not a sufficient one: it is natural to think that the sense of possibility relevant for supervenience is something stronger. Just because a certain distribution of thermodynamical properties is possible according to the laws of thermodynamics, for instance, does not mean that that distribution is possible tout court; in particular, such a distribution may not be one that could actually be generated by any statistical-mechanically possible configuration of the microstructure. Thus, in the event that two descriptive levels, associated with theories T ₁ and T ₂, have a supervenience map $\sigma :\mathrm{Mod}\left(T_2\right)\to \mathrm{Mod}\left(T_1\right)$, then we should take the ontology of T ₁ (i.e., the possible worlds at the T ₁-level) to include only those models of T ₁ that lie in the image of the supervenience mapping: that is, that ${\Omega }_1\subseteq \sigma \left[\mathrm{Mod}\right(T_2\left)\right]$. And of course, for any other descriptive level on which the T ₁-level supervenes, the same requirement will hold.

By taking these to be the only such requirements, we may define the ontology associated with a given level as follows: suppose that the T-level supervenes on all and only the T_i-levels, for i in some index set I, with the supervenience mappings ${\sigma }_i:\mathrm{Mod}\left(T_i\right)\to \mathrm{Mod}\left(T\right)$. Then we let the ontology for the T-level be $\Omega ≔{\cap }_{i\in I}{\sigma }_i\left[\mathrm{Mod}\right(T_i\left)\right]$. It follows that if the T ₁-level supervenes on the T ₂-level, then the supervenience map is surjective on ontologies: that is, for every ${ℳ}_1\in {\Omega }_1$, there is some ${ℳ}_2\in {\Omega }_2$ such that $\sigma \left({ℳ}_2\right)={ℳ}_1$.

Finally, if the T ₁-level supervenes on the T ₂-level, with supervenience map σ, I will say that an L ₁-sentence ϕ₁ List-reduces to the L ₂-sentence ϕ₂ if $\left[\right[{\varphi }_2\left]\right]={\sigma }^{-1}\left(\right[\left[{\varphi }_1\right]\left]\right)$: that is, if for every $ℳ\in {\Omega }_2$, $ℳ\models {\varphi }_2$ iff $\sigma \left(ℳ\right)\models {\varphi }_1$. And T ₁ List-reduces to T ₂ if every L ₁-sentence ϕ₁ is reducible to some L ₂-sentence ϕ₂.

List observes that, within his framework, supervenience does not entail reducibility. In the version of that framework developed here, this is encoded by the following result.

List appeals to general combinatorial considerations for a proof. However, it will be illuminating (and helpful for the sequel) to construct an explicit counterexample to the hypothesis that supervenience entails reduction.Footnote ²

Proof. Let L ₁ be the propositional language whose only sentence-letter is F, and let L ₂ be the propositional language with sentence-letters {P ₀, P ₁, …}. Let $T_1=T_2=\varnothing$. Only two worlds are contained by Mod(T ₁): set ${ℳ}_F\left(F\right)=\top$ and ${ℳ}_{\neg F}\left(F\right)=\perp$. Let ℳ be the T ₂-model such that $ℳ\left(P_i\right)=\top$ for every P_i. Define $\sigma :\mathrm{Mod}\left(T_2\right)\to \mathrm{Mod}\left(T_1\right)$ as follows: for any $𝒜\in \mathrm{Mod}\left(T_2\right)$,

$\begin{array}{}\text{(1)}& \sigma \left(𝒜\right)≔\{\begin{array}{cc}{ℳ}_F& \text{if }𝒜=ℳ\\ {ℳ}_{\neg F}& \text{otherwise}\end{array}\mathrm{.}\end{array}$

Since the T ₂-level is the lowest, ${\Omega }_2=\mathrm{Mod}\left(T_2\right)$; hence, ${\Omega }_1=\sigma \left[{\Omega }_2\right]=\mathrm{Mod}\left(T_1\right)$.

Now observe that ${\sigma }^{-1}\left(\right[\left[F\right]\left]\right)=\left\{ℳ\right\}$. But ℳ is not a definable subset of Ω₂: that is, there is no sentence $\varphi \in L_2$ such that $\left[\right[\varphi \left]\right]=\left\{ℳ\right\}$.Footnote ³ So the sentence $F\in L_1$ is not reducible to any sentence in L ₂; thus, T ₁ is not reducible to T ₂. QED

Now, one natural (and I think appropriate) response to this argument is that it is unsurprising. After all, we have imposed almost no constraints on the supervenience map. A supervenience map between two ontological levels is just a map from the worlds of one level to the worlds of the other, with no constraints on how that map must relate to the structure present in those worlds. So for the rest of this essay, I would like to consider some ways in which we could impose such constraints on the maps and what consequences those constraints have for reducibility.

2. Definition and Reduction

Before going further, I want to pause to make contact with some more fully fledged ways of thinking about reduction. This will involve some of the apparatus of definability theory from logic.Footnote ⁴ An explicit definition of R in terms of Σ is a formula of the form

$\begin{array}{}\text{(2)}& {\delta }_R=\forall x_1\dots \forall x_n(R{x}_1\dots x_n\leftrightarrow {\tau }_R(x_1,\dots ,x_n\left)\right),\end{array}$

where τ_R is a Σ-formula. Given a Σ-theory T and Σ⁺-theory T ⁺, where $\Sigma \subset {\Sigma }^{+}$, T ⁺ is a definitional extension of T if T ⁺ is logically equivalent to the theory

$\begin{array}{}\text{(3)}& T\cup \{{\delta }_R:R\in {\Sigma }^{+}\setminus \Sigma \},\end{array}$

where for each R, δ_R is an explicit definition of R in terms of Σ. More generally, any Σ⁺-theory T ⁺ explicitly defines a symbol $R\in {\Sigma }^{+}\setminus \Sigma$ in terms of Σ if it entails an explicit definition δ_R of R in terms of Σ, that is, if $T^{+}\models {\delta }_R$ for some δ_R of the form (2).

Now, this may be compared to Nagel’s (Reference Nagel1979, 352–54) definition of reduction:

As it is often put: a theory T ₁ is reducible to a theory T ₂ just in case T ₁ can be derived from T ₂ together with so-called bridge laws. Now, as has been widely discussed (see, e.g., Dizadji-Bahmani, Frigg, and Hartmann Reference Dizadji-Bahmani, Frigg and Hartmann2010 and references therein), if we place no restrictions on the content of the bridge laws, then reduction becomes trivialized: any theory T ₁ may be reduced to any other theory T ₂, by taking as the bridge laws either T ₁ itself or any inconsistent set of statements. To avoid this problem, we put two restrictions on the bridge laws. The first is that the bridge laws do indeed serve to bridge the gap between T ₁ and T ₂: formally, we require that T ₂ plus the bridge laws explicitly defines every symbol $R\in {\Sigma }_1\setminus {\Sigma }_2$ (where Σ₁ and Σ₂ are, respectively, the vocabularies of T ₁ and T ₂). The second is that the bridge laws not sneak in “extra content” beyond this bridging function: formally, we require that T ₂ plus the bridge laws be a conservative extension of T ₂. (Recall that a Σ⁺-theory T ⁺ is a conservative extension of a given Σ-theory T, where $\Sigma \subseteq {\Sigma }^{+}$, if T ⁺ has no new Σ-consequences relative to T: i.e., if for every Σ-sentence ϕ, $T^{+}\models \varphi$ entails that $T\models \varphi$.)

But if these two conditions are satisfied, then T ₂ together with the bridge laws is a definitional extension of T ₂ (Hodges Reference Hodges1997, 53–54). Thus, to within logical equivalence, we may suppose that the bridge laws are explicit definitions of the novel terms (i.e., of the symbols in ${\Sigma }_1\setminus {\Sigma }_2$).Footnote ⁵ Note that this agrees with Schaffner’s requirement that bridge laws be “reduction functions” (statements that a certain term of T ₁ and a certain term of T ₂ are coextensional)—provided, that is, that we take the reduction functions to relate a simple term of T ₁ to a (possibly) complex term of T ₂.Footnote ⁶ Thus, we are led to posit the following relation: a theory T ₁ Nagel-reduces to T ₂ if there is some definitional extension $T_2^{+}$ of T ₂ such that $T_2^{+}\models T_1$.Footnote ⁷

We now want to think about how all this relates to the apparatus of supervenience mappings developed in the previous section. For these purposes, it is helpful to work with the notions of interpretation and translation rather than definition—although as we will see, these are closely intertwined (Barrett and Halvorson Reference Barrett and Halvorson2016a). So, define an interpretation from one language L ₁ to another language L ₂ to be a map $\tau :\mathrm{Form}\left(L_1\right)\to \mathrm{Form}\left(L_2\right)$, which commutes with the logical constants (where Form(L) is the set of formulas of L). And given theories T ₁ and T ₂, in languages L ₁ and L ₂, respectively, define a translation from T ₁ to T ₂ to be an interpretation $\tau :L_1\to L_2$ such that for any $\varphi \in \mathrm{Form}\left(L_1\right)$, if $T_1\models \varphi$, then $T_2\models \tau \left(\varphi \right)$. This lets us give a more compact characterization of Nagel-reduction, as follows.

Let the class of Σ-structures be denoted Str(Σ). Now, given an interpretation $\tau :{\Sigma }_1\to {\Sigma }_2$, the dual map of τ is a map ${\tau }^{*}:\mathrm{Str}\left({\Sigma }_2\right)\to \mathrm{Str}\left({\Sigma }_1\right)$, defined as follows: for any Σ₂-structure 𝒜, $\left|{\tau }^{*}\right(𝒜\left)\right|=\left|𝒜\right|$ (where |ℳ| denotes the domain of the model ℳ), and for any $R\in {\Sigma }_1$, its extension in τ*(𝒜) is the set of all and only those n-tuples that satisfy τ(R). It is straightforward to show that for any Σ₂-structure 𝒜 and Σ₁-sentence ϕ, ${\tau }^{*}\left(𝒜\right)\models \varphi$ iff $𝒜\models \tau \left(\varphi \right)$. As a corollary, if τ is a translation from T ₁ to T ₂, then ${\tau }^{*}:\mathrm{Mod}\left(T_2\right)\to \mathrm{Mod}\left(T_1\right)$ (i.e., if 𝒜 is a model of T ₂, then τ*(𝒜) is a model of T ₁).

Putting this all together, we get the following principle: if T ₁ Nagel-reduces to T ₂, then the reduction naturally induces a map from Mod(T ₂) to Mod(T ₁). Moreover, if the translations between a collection of theories constitute a posetal category, then their dual maps will as well: this follows from the fact that given translations ${\tau }_1:T_1\to T_2$ and ${\tau }_2:T_2\to T_3$, ${({\tau }_2°{\tau }_1)}^{*}={\tau }_1^{*}°{\tau }_2^{*}$. So such a collection of translations gives rise to a system of descriptive levels of the kind motivated in the previous section. Thus, Nagel-reduction entails supervenience. Moreover, it entails List-reduction: given any L ₁-sentence ϕ₁ and a translation $\tau :T_1\to T_2$, ϕ₁ List-reduces to τ(ϕ₂). It follows that the example in theorem 1 also shows that supervenience does not entail Nagel-reducibility.

Incidentally, the dual maps will not typically be surjective (as maps between classes of models), nor would it be appropriate to require that they were. To see this, observe that the following principle seems plausible: if T ₁ reduces to T ₂, and if T ₀ is a subtheory of T ₁ (i.e., if every sentence of T ₀ is a sentence of T ₁), then T ₀ reduces to T ₂. But it is an immediate consequence of this that not all dual maps can be surjective. Let T ₀ be a proper subtheory of T ₁, let τ be the translation from T ₁ to T ₂, and observe that it is thereby a translation from T ₀ to T ₂. If τ* is not surjective as a map from Mod(T ₂) to Mod(T ₁), then we are done. If it is, then since T ₀ is a proper subtheory of T ₁, there is at least one model ℳ of T ₀ that is not a model of T ₁. Since the image of Mod(T ₂) under τ* is Mod(T ₁), then ℳ is not in said image, and so τ* is not surjective as a map from Mod(T ₂) to Mod(T ₀).Footnote ⁸ As discussed previously, however, we can impose surjectivity of supervenience maps as maps on possible worlds by fiat.

With all this in hand, we now return to our main task: what kinds of constraints could be imposed on supervenience maps, in order to guarantee that they are associated with some form of reducibility? First, I will consider the idea that the supervenience maps must preserve similarity, as encoded in topological data about the models.

3. Supervenience and Continuity

Butterfield (Reference Butterfield2011b) asks us to consider the sequence of functions $g_n:ℝ\to ℝ$, for $n\in {\mathbb{N}}^{+}≔\mathbb{N}\setminus \left\{0\right\}$, where

$\begin{array}{}\text{(4)}& g_n\left(x\right)≔\{\begin{array}{cc}\mathrm{-1}& \text{if }x<-\frac{1}{n}\\ \mathrm{nx}& \text{if }-\frac{1}{n}\le x\le \frac{1}{n}\\ 1& \text{if }\frac{1}{n}<x\end{array}\mathrm{.}\end{array}$

The limit of this sequence (as determined by pointwise convergence) is the function g _∞, where

$\begin{array}{}\text{(5)}& g_{\infty }\left(x\right)≔\{\begin{array}{cc}\mathrm{-1}& \text{if }x<0\\ 0& \text{if }x=0\\ 1& \text{if }0<x\end{array}\mathrm{.}\end{array}$

As Butterfield notes, for each finite n the function g _n is continuous, yet the limit of the sequence of these individually continuous functions, g _∞, is discontinuous (at 0). Now consider the function $f_{•}:{\mathbb{N}}^{+}\cup \left\{\infty \right\}\to \{0,1\}$, defined by

$\begin{array}{}\text{(6)}& f_n≔\{\begin{array}{cc}0& \text{if }n\in {\mathbb{N}}^{+}\\ 1& \text{if }n=\infty \end{array}\mathrm{.}\end{array}$

We can now define a two-level ontological system, in the sense of List, by taking one ontology to be the set ${\left\{g_n\right\}}_{n\in {\mathbb{N}}^{+}}\cup \left\{g_{\infty }\right\}$, the other to be the set ${\left\{f_n\right\}}_{n\in {\mathbb{N}}^{+}\cup \left\{\infty \right\}}=\{0,1\}$, and the supervenience map to be $\sigma :g_n\mapsto f_n$. Alternatively, we could take the lower-level ontology to consist of all functions $ℝ\to ℝ$, the higher-level ontology to consist (again) of the set {0, 1}, and the supervenience function to be a function mapping continuous functions to 0 and discontinuous functions to 1.

This supervenience map fails to preserve limits. The limit of the sequence of lower-level worlds g ₁, g ₂, … is g _∞, and $\sigma \left(g_{\infty }\right)=1$. By contrast, the limit of the sequence σ(g ₁), σ(g ₂), … is the limit of the sequence 0, 0, …, which is of course 0. So it makes a difference whether we take the limit before or after applying the supervenience map; taking the limit does not commute with supervenience. Thinking of topological structure as giving a standard of similarity among the models (Fletcher Reference Fletcher2016), let us say that a supervenience map that fails to commute with limits in this fashion is not similarity preserving.

But now consider again the counterexample constructed in the proof of theorem 1. We can make this example resembles Butterfield’s example of the functions, by defining the models ℳ_n, for $n\in \mathbb{N}$, as follows: ℳ_n assigns ⊤ to P ₀ through P_n and ⊥ to all remaining sentence-letters. Then there is an intuitive sense in which the models in the sequence ℳ₀, ℳ₁, ℳ₂, … get “more and more similar” to the model ℳ, that is, a sense in which we can regard ℳ as the limit of the sequence ℳ_n as n goes to infinity. This intuition receives a precise expression in the concept of the Stone topology of a class of propositional models: in that topology, a sequence of models ℳ_n converges to a model ℳ iff the sequence of truth-values ${ℳ}_n\left(P\right)$ converges to the truth-value ℳ(P), for every sentence-letter P (Halvorson Reference Halvorson2012). However, the supervenience map does not play nice with this convergence structure: just as with Butterfield’s example, the supervenient image of the limit of the sequence (i.e., ℳ_F) is not the limit of the supervenient images of the sequence (i.e., ℳ_¬F). In other words, the supervenience map is not similarity preserving.

This suggests a conjecture: perhaps all such failures of reduction are associated with the failure of the supervenience map to be similarity preserving. In the propositional case, this turns out to be correct. More specifically, we have the following result.

4. The First-Order Case

Let us now turn our attention to the case of first-order theories. It will be helpful to split our inquiry into two parts. First, can we find conditions on a supervenience mapping sufficient for that mapping to be associated with a reduction? Second, can we think of those conditions as imposing a requirement of similarity preservation?

With regard to the first question, the following result—based on a result by van Benthem and Pearce (Reference van Benthem and Pearce1984)—provides an affirmative answer.

In order to explain this, I briefly review the relevant notion of an ultraproduct.Footnote ¹¹ First, an ultrafilter over a set X is a nonempty set U of subsets of X, such that

1. • $\varnothing \notin U$.

2. • U is closed under intersection: if $A\in U$ and $B\in U$, then $A\cap B\in U$.

3. • U is closed under supersets: if $A\in U$ and $A\subseteq B$ (for $B\subseteq X$), then $B\in U$.

4. • For every subset A of X, exactly one of A and $X\setminus A$ is in U.

Roughly speaking, one can think of an ultrafilter as equipping X with a notion of “almost all”-ness: relative to U, a subset $A\subseteq X$ contains “almost all” elements of X just in case $A\in U$.

Given an ultrafilter U on some set I (which we will call the set of indices) and an indexed family {${ℳ}_i:i\in I$} of Σ-structures, the ultraproduct Π_Uℳ_i of the family is a Σ-structure defined as follows. First, form the Cartesian product ${\Pi }_{i\in I}\left|{ℳ}_i\right|$ of the domains of the structures in the family: that is, each element of ${\Pi }_{i\in I}\left|{ℳ}_i\right|$ is an indexed family of elements $a=\{a^i\in |{ℳ}_i|:i\in I\}$, one from each ℳ_i. Then define the following equivalence relation: $a~b$ iff the set of indices i such that $a^i=b^i$ is in the ultrafilter. That is, a and b are equivalent if they are identical at “almost all” models in the family of structures. We then take the domain of the ultraproduct to be the quotient of ${\Pi }_{i\in I}\left|{ℳ}_i\right|$ by this equivalence relation (so its elements are sets of “almost-everywhere-identical” indexed families). Finally, we determine the extension of any Σ-symbol R as follows: for n-ary R, the tuple 〈a₁, …, a_n〉 is in the extension of R iff the set of indices i such that $〈a_1^i,\dots ,a_n^i〉\in R^{{ℳ}_i}$ is in U. Thus, for instance, an indexed family a satisfies a unary predicate iff it satisfies the predicate at “almost all” models in the family of structures. An ultraproduct all of whose factors are identical is called an ultrapower: an ultrapower of ℳ, relative to the ultrafilter U, will be denoted Π_Uℳ.

The key result for ultraproducts is Łoś’s theorem (see, e.g., Hodges Reference Hodges1997, theorem 8.5.3): given an ultraproduct Π_Uℳ_i and an n-ary Σ-formula ϕ, ϕ is satisfied by the n-tuple 〈a₁, …, a_n〉 in Π_Uℳ_i iff the set of indices i such that ϕ is satisfied by 〈$a_1^i$, …, $a_n^i$〉 in ℳ_i is in U. Thus, roughly speaking, an ultraproduct satisfies a formula iff “almost all” the models in the indexed family satisfy that formula. As a corollary, given some ultrapower Π_Uℳ, ϕ is satisfied in ℳ by the n-tuple 〈a ₁, …, a_n〉 (where every $a_p\in \left|ℳ\right|$) iff ϕ is satisfied by 〈a₁, …, a_n〉 in Π_Uℳ, where for every $i\in I$ and $1\le p\le n$, $a_p^i=a_p$.

With this in hand, we can turn to the second question raised at the start of this section: whether the conditions on theorem 4 can be thought of as imposing a requirement of similarity preservation. One way to show this would be to argue that an ultraproduct can be regarded as a kind of “limit” of a family of models. In support of this, we could observe that in the context of propositional logic (which, recall, is simply first-order logic where all predicates are nullary) the ultraproduct of a sequence of models is the limit of that sequence in the Stone topology (Halvorson and Tsementzis Reference Halvorson, Tsementzis and Landry2017). More generally, Łoś’s theorem provides a sense in which an indexed family of models “converges” on the ultraproduct. However, the sense of convergence here must remain rather loose. The standard axioms governing convergence require that if all members of a converging family are identical to some object, then the limit of the family is that object (see, e.g., Dudley Reference Dudley1964; Patten Reference Patten2014), yet, in general, the ultrapower of a given model is not identical to that model, nor is it even isomorphic to it.Footnote ¹² Nevertheless, I conclude that the above considerations do illuminate some aspects of the relationship between reduction (thought of as an essentially syntactic relation between theories) and supervenience (thought of as an essentially semantic relation between models of theories or the possible worlds they represent): we have learned that constraints on a supervenience map, statable in purely semantic terms, can enforce reducibility.