What plurals are there?

Plural quantification is meant to be a logical way of avoiding some technical and/or conceptual difficulties with sets and second-order quantification. Instead of quantifying over one thing, one quantifies over pluralities. Thus, a theist might say: For all xs, God thinks of the xs in their interrelationship.

What plurals are there? Intuitively, for any finite list of objects, there is a plurality of precisely those objects. After all, we can easily have a sentence about any finite plurality of things we have names for: Alice, Bob and Carl like each other. But what furthe pluralities are there?

An expansive proposal is plural comprehension: the axiom schema that says that for any formula F with free variables that include y, for any values of the free variables other than y, there are xs such that y is one of the xs iff F. Unlike the comprehension schema in naive set theory, there does not seem to be any direct Russell-type paradox for plural comprehension, because the xs are not in general an object, but multiple objects.

But plural comprehension on its own does not seem to quite settle what plurals there are. Suppose we have a plurality of nonempty disjoint sets. We can for instance ask: Is there a plurality of objects that includes exactly one object from each of these sets? If (a) there is a set of these disjoint sets, and (b) the Axiom of Choice holds for sets, then the answer is affirmative by plural comprehension. But of course whether the Axiom of Choice holds for sets is itself not philosophically settled, and further not every plurality of sets is such that there is a set of the sets in the plurality.

Observations of this sort show that plural quantification is not as metaphysically innocent as it may seem. You might have hoped that there is no further metaphysical commitment in allowing for plural quantification than in singular quantification. But we can now have substantive questions about what pluralities there are even after we have fixed what singular objects there are, even if we assume plural comprehension. For instance, suppose we think that the objects are the physical objects of the world plus the elements of a model of ZF set theory with ur-elements and with the negation of the Axiom of Choice. We can know what all the objects are, and it still not be decided what pluralities there are. For in the case of a set of disjoint nonempty sets that lacks a choice set, as far as I can tell, there still might be a "choice plurality" (a plurality that has exactly one object from each of the disjoint sets) or there might not be one. (And if you say, well, the Axiom of Choice is obviously true, I may try to come back with a similar issue regarding Choice for proper classes.)

Or I might make a similar point about the Continuum Hypothesis (CH). The following story seems quite coherent. Every uncountable subset of the real numbers is in a bijection with the set of reals (i.e., CH is true), but there is an uncountable plurality of real numbers not in bijection with the plurality of reals. (It's easy to define bijections of pluralities in terms of pluralities of pairs.) But it's also coherent that CH is true, but there is no such uncountable plurality of reals--i.e., that CH is true for sets but its analogue for pluralities is false.

We might try to get out of this by insisting that, necessarily, the right set theory has to have a stronger version of the Schema of Separation that allows for formulas free plural variables and for the plural-membership relation. But that's conceding that the theory of pluralities is metaphysically non-innocent, because now what pluralities there are will constrain what objects there are!

So the question of what restrictions we put on plurals is a really substantive question.

Next note that following point. There seem to be two particularly simple and non-arbitrary answers to the Special Composition Question which asks which pluralities compose a whole: nihilism (there are no non-trivial cases of composition) and universalism (every plurality composes a whole). But once we have realized that it is a substantive question what pluralities there are, it seems that what objects there are and affirming universalism, even with mereological essential thrown in, doesn't settle the question of what wholes there are. There is substantial metaphysics to be done to figure out what pluralities there are!

I say the above with a caution: there are various technicalities I am glossing over, and I wouldn't be surprised if some of them turned out to be really important.

Ordinary language and "exists"

In Material Beings, Peter van Inwagen argues that his view that there are no complex artifacts does not contradict (nearly?) universal human belief. The argument is based on his view that the propositions expressed by ordinary statements like “There are three valuable chairs in this room” do not entail the negation of the Radical Claim that there are no artifacts, for such a proposition does not entail that there exist chairs.

I think van Inwagen is right that such ordinary propositions do not entail the negation of the Radical Claim. But he is wrong in thinking that the Radical Claim does not contradict nearly universal human belief. Van Inwagen makes much of the analogy between his view and the Copernican view that the sun does not move. When ordinary people say things like “The sun moved behind the elms”, they don’t contradict Copernicus. Again, I think he is right about the ordinary claims, but nonetheless Copernicus contradicted nearly universal human belief. That was why Copernicus’ view was so surprising, so counterintuitive (cf. some remarks by Merricks on van Inwagen). One can both say that when people prior to Copernicus said “The sun moved behind the elms” they didn’t contradict Copernicanism and that they believed things that entailed that Copernicus is wrong.

People do not assert everything they believe. They typically assert what is salient. What is normally salient is not that the sun actually moved, but that there was a relative motion between the rays pointing to the elms and to the sun. Nonetheless, if ordinary pre-Copernicans said “The sun doesn’t stand still”, they might well have been contradicting the Copernican hypothesis. But rarely in ordinary life is there occasion to say “The sun doesn’t stand still.” Because of the way pragmatics affects semantics (something that van Inwagen apparently agrees on), we simply cannot assume that the proposition expressed by the English sentence “The sun moved behind the elms” entails the proposition expressed by the English sentence “The sun doesn’t stand still.”

Something similar, I suspect, is true for existential language. When an ordinary person says “There are three chairs in the room”, the proposition they express does not contradict the Radical Thesis. But if an ordinary person says things like “Chairs exist” or “Artifacts exist”, they likely would contradict the Radical Thesis, and moreover, these are statements that the ordinary person would be happy to make in denial of the Radical Thesis. But in the ordinary course of life, there is rarely an occasion for such statements.

This is all largely a function of pragmatics than the precise choice of words. Thus, one can say: “Drive slower. Speed limits exist.” The second sentence does not carry ontological commitment to speed limits.

So, how can we check whether an ordinary person believes that tables and chairs exist? I think the best way may be by ostension. We can bid the ordinary person to consider:

1. People, dogs, trees and electrons.

2. Holes, shadows and trends.

We remind the ordinary person that we say “There are three holes in this road” or “The shadow is growing”, but of course there are no holes or shadows, while there are people (we might remind them of the Cogito), dogs, trees and (as far as we can tell) electrons. I think any intelligent person will understand what we mean when we say there are no holes or shadows. And then we ask: “So, are tables and chairs in category 2 or in category 1? Do they exist like people, dogs, trees and electrons, or fail to exist like holes, shadows and trends?” This should work even if like Ray Sorensen they disagree that there are no shadows; they will still understand what we meant when we said that there are no shadows, and that’s enough for picking out what we meant by “exist”. To put in van Inwagen’s terms, this brief ostensive discussion will bring intelligent people into the “ontology room”.

And I suspect, though this is an empirical question and I could be wrong, once inducted into the discussion, most people will say that tables and chairs exist (and that they have believed this all along). But, van Inwagen should say, this nearly universal belief is mistaken.

This story neatly goes between van Inwagen’s view that ordinary people don’t believe things patently incompatible with the Radical Theory and Merricks’ view that ordinary poeple contradict the Radical Theory all the time. Ordinary people do believe things patently incompatible with the Radical Theory, but they rarely express these beliefs. Most ordinary “there exist” statements—whether concerning artifacts or people or particles—do not carry ontological commitment, and those of us who accept the Radical Theory normally aren’t lying when we say “There are three chairs in the room”. But the Radical Theory really is radical.

A method for blocking deflation of ontological debates

Consider Hirsch-type deflationary views on which many differences in ontology are simply verbal differences. A standard case is nihilism and universalism about composition: the nihilist says that multiple things can never compose a whole and the universalist says that every plurality must compose a whole. The deflationist sees the two views as notational variants. The universalist’s sentences describe the same facts as the nihilist’s. We can maybe even translate with little if any loss between the two idioms, replacing the nihilist’s quantifiers with quantifiers restricted to simples on the universalist’s side, and replacing the universalist’s quantifiers with plural quantification, or quantification over sets, or some other device acceptable to the nihilist.

Note, first, that in this particular case there is a bit of a problem. The universalist might allow for composed objects that have no simple parts—“gunk”. The claim that possibly there is gunk is one that cannot be translated into any statement in the nihilist’s language that has a hope of being true. The nihilist’s usual way of translating a universalist’s statement is to use plural quantification. So the statement that possibly there is gunk is going to get translated into something like the statement that possibly there is a plurality of things none of which is a simple. But that’s obviously false given nihilism, since the nihilist’s quantifiers can only quantify over simples, and so the statement basically says that there are simples none of which is a simple. Thus, we have a genuine, non-verbal disagreement.

So the only way we can take a nihilist-universalist disagreement to be merely verbal is if both theorists deny the possibility of gunk. I think they should deny the possibility of gunk.

Here is a second case, where disagreement on composition cannot be deflated. Consider a brutal composition view like Markosian’s. On this view, there will be possible worlds with the same simple objects standing in the same non-mereological relations but differing as to composition facts. For instance, in one world there might be three rocks that make up a whole and in the other world the very same three rocks do not make up a whole, even though they are arranged in exactly the same way. Any nihilist or universalist description of the two worlds will be unable to distinguish such worlds, but on a brutal composition view, there can be such pairs. Here we have a real disagreement, one that cannot be taken to be merely verbal. The brutal composition theorist has more possibilities than the nihilist and universalist. And the brutal composition theorist’s statement that the two worlds differ in composition facts but not in non-mereological facts either has no translation into either nihilist or universalist language or translates into something that is clearly false on the given theory.

The brutal compositionalist has an additional “degree of freedom”, as the scientist would say, on her theory than the nihilist or universalist does. The case here is similar to those dualists who believe in the possibility of zombies. While the disagreement between a dualist who thinks the mental supervenes on the physical and the pure physicalist could seem to be merely verbal to some (though I think it’s a mistake to see it that way), the disagreement between a dualist who thinks that the mental does not supervene on the physical and the pure physicalist is certainly not merely verbal.

In general, thus, the modal ramifications of theories can block deflationary moves. One theory may allow for a possibility that simply rules out the other theory (e.g., gunk ruling out nihilism), or one theory may posit contingent facts that do not supervene on reality as describable in the other theory (the brutal composition or zombie cases).

This leaves the possibility that there will be some ontological debates that are merely verbal. Perhaps the debate between the nihilist and the anti-gunk universalist is merely verbal. But that some pairs of ontological theories disagree merely verbally is not a very interesting deflationary thesis.

Moreover, I think that once we see that there are nearby debates that are clearly not merely verbal, the plausibility of the deflationary move in the cases that looks more verbal goes down. Once we realize that among the views under discussion there is a brutal composition view on which there is a possible world just like ours but where nihilism contingently holds and a possible world just like ours but where universalism contingently holds, it becomes pretty clear that holding nihilism to hold necessarily will also differ from holding universalism to hold necessarily. (That said, there may be particular variants on universalism that just are notational variants on nihilism. Say, ones where the quantifiers are stipulated in terms of plural quantification over simples.)

Grounding graphs, new take

In a previous post, I looked at the idea of grounding graphs as global entities. But I think there is a more natural way of looking at them. There are two main views about grounding. On the truthmaker view, true propositions are grounded in entities that make them true. On the propositional view, true propositions are grounded in other true propositions. But I think a more natural approach is to say that propositions are grounded in graphs.

A candidate grounding graph for a proposition p is a directed graph G satisfying the following properties:

1. all the vertices of G are true propositions
2. p is a vertex of G
3. all the vertices of G other than p are ancestors of p
4. p is not the only vertex of G.

The grounding relation is then a relation between a proposition p and a candidate grounding graph for p. For instance, the proposition <The sky is blue or (roses are red and violets are blue)> is grounded in a graph with two vertices, one of which is <The sky is blue> and the other being the target proposition, with one arrow from the former to the latter. But it is also grounded in a more complex graph with four vertices: <Roses are red>, <Violets are blue>, <Roses are red and violets are blue>, and the target propositions, with arrows from the first two propositions to the third, and an arrow from the third to the target.

Define a proposition as fundamental provided that it is true but has no grounding graph. Say that a grounding graph for p is a candidate grounding graph for p that in fact grounds p. A vertex is initial provided that it has no ancestors and is final provided it has no children. A candidate grounding graph has exactly one final vertex. Say that G* extends G provided that (a) G* has the same final vertex as G and (b) every vertex of G that has a parent in G has exactly the same parents in G* as in G. The following are important properties of grounding graphs: Say that a graph where every initial vertex is fundamental is a fundamental graph.

• Acyclicity: Every grounding graph is acyclic.
• Extensibility: If G is a grounding graph for p, then there is a fundamental extension of G that is also a grounding graph for p.
• Adjoining: If G₁ is a grounding graph for p, and G₂ is a grounding graph for some initial vertex q of G₁ such that G₂ has a fundamental extension whose only vertex in common with G₁ is q, then the graph whose vertex collection is the union of the vertex collections of G₁ and G₂ and whose arrow collection is the union of the arrow collections of G₁ and G₂ is also a grounding graph for p.
• Truncation: If G is a grounding graph for p, then any subgraph of G that is a candidate grounding graph and that has the property that if it contains any one of G's arrows to q then it contains all of G's arrows to q is a grounding graph.

The following is very controversial but very helpful:

• Well-foundedness: No grounding graph contains an infinite chain of arrows.

This is compatible with some grounding graphs being infinite. For instance, we could have a fundamental grounding graph for an infinite conjunction. There, the infinite conjunction will have infinitely many parents. Moreover, there may be arbitrarily long chains in the graph—the first parent might be fundamental, the second might have a chain of length two to a fundamental ancestor, and so on.

I think that if we reject well-foundedness, we should reject acyclicity. For the most plausible putative counterexamples will be infinitely nested propositions like p₁&(p₂&(p₃&...)). But if we accept such propositions, we will also accept p&(p&(p&...)), and these will be cyclically grounded if the former will be non-well-foundedly grounded. But we shouldn't reject acyclicity, so we should accept well-foundedness, and if there are such infinitely nested propositions, we should ground them all at once in the symmetric conjunction p₁&p₂&... which then is grounded in each of its conjuncts.

Finally, we want to say something about how this interacts with logic. Say that p is free of q provided that is a fundamental grounding graph for p that does not contain q.

• Disjunction introduction: If p is free of (p or q), then the following is a grounding graph: p→(p or q).

Two metaphysical convictions

I have two basic metaphysical convictions that drive my metaphysics.  One is that there is no vagueness at the fundamental level.  So anything there is vagueness about is non-fundamental.

The other is that humans are among the fundamental substances.  We are not logical constructions out of other entities; we are not reducible to other entities; we are fundamental substances.

Together, these two convictions lead to various controversial things, especially since some people will think there is a tension between the two.