Goodman and Quine and transitive closure

In the previous post, I showed that Goodman and Quine’s counting method fails for objects that have too much overlap. I think (though the technical parts here are more difficult) that the same is true for their definition of the ancestral or transitive closure of a relation.

GQ showed how to define ancestors in terms of offspring. We can try to extend this definition to the transitive closure of any relation R over any kind of entities:

1. x stands in the transitive closure of R to y iff for every object u that has y as a part and that has as a part anything that stands in R to a part of u, there is a z such that Rxz and both x and z are parts of R.

This works fine if no relatum of R overlaps any other relatum of R. But if there is overlap, it can fail. For instance, suppose we have three atoms a, b and c, and a relation R that holds between a + b and a + b + c and between a and a + b. Then any object u that has a + b + c as a part has c as a part, and so (1) would imply that c stands in the transitive closure of R to a + b + c, which is false.

Can we find some other definition of transitive closure using the same theoretical resources (namely, mereology) that works for overlapping objects? No. Nor even if we add the “bigger than” predicate of GQ’s attempt to define “more”. We can say that x and y are equinumerous provided that neither is bigger than the other.

Let’s work in models made of an infinite number of mereological atoms. Write u ∧ v for the fusion of the common parts of both u and v (assuming u and v overlap), u ∨ v for the fusion of objects that are parts of one or the other, and u − v for the fusion of all the parts of u that do not overlap v (assuming u is not a part of v). Write |x| for the number of atomic parts of x when x is finite. Now make these definitions:

2. x is finite iff an atom is related to x by the transitive closure (with respect to the kind object) of the relation that relates an object to that object plus one atom.

3. Axyw iff x and y are finite and whenever x′ is equinumerous with x and does not overlap y, then x′ ∨ y is equinumerous with w. (This says |x| + |y| = |w|.)

4. Say that D_yuv iff A(u−y,u−y,v−y) (i.e., |v−y| = 2|u−y|) and either v does not overlap y or and u ∧ y is an atom or v and y overlap and u ∧ y consists of v ∧ y plus one atom. (This treats u and v as basically ordered pairs (u−y,u∧y) and (v−y,v∧y), and it makes sure that from the first pair to the second, the first component is doubled in size and the second component is decreased by one.)

5. Say that Q⁰yx iff y is finite and for some atom z not overlapping y we have y ∧ z related to something not overlapping x by the transitive closure of D_y. (This takes the pair (z,y), and applies the double first component and decrease second component relation described in (4) until the second component goes to zero. Thus, it is guaranteed that |x| = 2^|y|.)

6. Say that Qyx iff y is finite and Q⁰yx′ for some non-overlapping x′ that does not overlap y and that is equinumerous with x.

If I got all the details right, then Qyx basically says that |x| = 2^|y|.

Thus, we can define use transitive closure to define binary powers of finite cardinalities. But the results about the expressive power of monadic second-order logic with cardinality comparison say that we can only define semi-linear relations between finite cardinalities, which doesn’t allow defining binary powers.

Remark: We don’t need equinumerosity to be defined in terms of a primitive “bigger”. We can define equinumerosity for non-overlapping finite sets by using transitive closure (and we only need it for finite sets). First let T_yuv iff v − y exists and consists of u − y minus one atom and v ∧ y exists and consists of v ∧ y minus one atom. Then finite x and y are equinumerous₀ iff they are non-overlapping and x ∨ y has exactly two atoms or is related to an object with exactly two atoms by the transitive closure of T_yuv. We now say that x and y are equinumerous provided that they are finite and either x = y (i.e., they have the same atoms) or both x − y and y − x are defined and equinumerous₀.

No fix for Goodman and Quine's counting

In yesterday’s post, I noted that Goodman and Quine’s nominalist mereological definition of what it is to say that there are more cats than dogs fails if there are cats that are conjoint twins. This raises the question whether there is some other way of using the same ontological resources to generate a definition of “more” that works for overlapping objects as well.

I think the answer is negative. First, note that GQ’s project is explicitly meant to be compatible with there being a finite number of individuals. In particular, thus, it needs to be compatible with the existence of mereological atoms, individuals with no proper parts, which every individual is a fusion of. (Otherwise, there would have to be no individuals or infinitely many. For every individual has an atom as a part, since otherwise it has an infinite regress of parts. Furthermore, every individual must be a fusion of the atoms it has as parts, otherwise the supplementation axiom will be violated.) Second, GQ’s avail themselves of one non-mereological tool: size comparisons (which I think must be something like volumes). And then it is surely a condition of adequacy on their theory that it be compatible with the logical possibility that there are finitely many individuals, every individual is a fusion of its atoms and the atoms are all the same size. I will call worlds like that “admissible”.

So, here are GQ’s theoretical resources for admissible worlds. There are individuals, made of atoms, and there is a size comparison. The size comparison between two individuals is equivalent to comparing the cardinalities of the sets of atoms the individuals are made of, since all the atoms are the same size. In terms of expressive power, their theory, in the case of admissible worlds, is essentially that of monadic second order logic with counting, MSO(#), restricted to finite models. (I am grateful to Allan Hazen for putting me on to the correspondence between GQ and MSO.) The atoms in GQ correspond to objects in MSO(#) and the individuals correspond to (extensions of) monadic predicates. The differences are that MSO(#) will have empty predicates and will distinguish objects from monadic predicates that have exactly one object in their extension, while in GQ the atoms are just a special (and definable) kind of individual.

Suppose now that GQ have some way of using their resources to define “more”, i.e., find a way of saying “There are more individuals satisfying F than those satisfying G.” This will be equivalent to MSO(#) defining a second-order counting predicate, one that essentially says “The set of sets of satisfiers of F is bigger than the set of sets of satisfiers of G”, for second-order predicates F and G.

But it is known that the definitional power of MSO(#) over finite models is precisely such as to define semi-linear sets of numbers. However, if we had a second-order counting predicate in MSO(#), it would be easy to define binary exponentiation. For the number of objects satisfying predicate F is equal to two raised to the power of the number of objects satisfying G just in case the number of singleton subsets of F is equal to the number of subsets of G. (Compare in the GQ context: the number of atoms of type F is equal to two the power of the number of atoms of type G provided that the number of atoms of type F is one plus the number of individuals made of the atoms of type G.) And of course equinumerosity can be defined (over finite models) in terms of “more”, while the set of pairs (n,2ⁿ) is clearly not semi-linear.

One now wants to ask a more general question. Could GQ define counting of individuals using some other predicates on individuals besides size comparison? I don’t know. My guess would be no, but my confidence level is not that high, because this deals in logic stuff I know little about.

Reducing de re to de dicto modality

In my previous post, I gave an initial defense of a theory of qualitative haecceities in terms of qualitative origins: qualitative haecceities encapsulate complete qualitative descriptions of an entity’s initial state and causal history. I noted that among the advantages of the theory is that it can allow for a reduction of de re modality to de dicto modality, without “the mystery of non-qualitative haecceities”.

I want to expand on this, and why qualitative-origin haecceities are superior to non-qualitative haecceities here. A haecceitistic account of de re modality proceeds in something like the following vein. First, introduce the predicate H(Q,x) which says that Q is a haecceity of x. Then we reduce de re claims as follows:

• x is essentially F ↔︎ ∀Q(H(Q,x)→□(∀y(Qy→Fx)))

• x is accidentally F ↔︎ ∃Q(H(Q,x)∧◊(∃y(Qy∧Fx))).

Granted, this involves de re modality for second-order variables like Q. But this de re modality is less problematic because we can suppose the Barcan and converse Barcan formulas to hold as axioms for the second-order quantifiers, and we can treat the second-order entities as necessary beings. De re modality is particularly difficult for contingent beings, so if we can reduce to a modal logic where only necessary beings are subject to de re modal claims, we have made genuine progress.

We will also need some axioms. Here are two that come to mind:

• ∀x∀Q(H(Q,x)→Qx) (things have their haecceities)

• ∀x∃Q(H(Q,x)) (everything has a haecceity).

Now, here is why I think that qualitative-origin haecceities are superior to non-qualitative haecceities. Given qualitative-origin haecceities, we can give an account of what H(Q,x) means without using de re modality. It just means that Qy attributes to y all of the actual qualitative causal origins of x, including x’s initial qualitative state. On the other hand, if we go for non-qualitative haecceities, we seem to have two options. We could take H(Q,x) to be primitive, which always should be a last resort, or we could try to define in some way like:

• H(Q,x) ↔︎ (□(Ex→Qx) ∧ □∀y(Qy→y=x))

where Ex says that x exists (it might be a primitive in a non-free logic, or it might just be an abbreviation for ∃y(y=x)). But this definition uses de re modality with respect to x, so it is not satisfactory in this context, and I can’t think of any way to do it without de re modality with respect to potentially contingent individuals like x.

Particularizers instead of haecceities

A haecceity of x is a property that, necessarily, x and only x has. For instance, it might be the property of being identical with x. If a particularly strong converse to the essentiality of origins holds, a good choice for a haecceity would be a complete history of the coming-into-existence of x. Haecceities are a useful tool. For instance, they let one replace de re modality with de dicto. For another, they help explain what God deliberates about when he deliberates which individuals to create.

There is a different tool that can do some of the same work: a particularizer. We can think of an x-particularizer as equivalent to the second order property of being instantiated by x. Thus, if A is an x-particularizer, then necessarily a property Q has A if and only if x has Q. I will occasionally read "Q has A" as "A particularizes Q". The main trick to using particularizers is to note that, necessarily, x exists if and only if x instantiates some property. Thus, if A is an x-particularizer, then, necessarily, x exists if and only if some property has A.

Suppose that any two distinct things differ in some property and that particularizers exist necessarily.

Then we can use particularizers for de re modals. Suppose A is an x-particularizer. Then, Q is an essential property of x if and only if necessarily: if any property has A, then Q has A. If we have an abundant account of properties, we can then account for more complex modals. And we can likewise account for God's creative deliberation about individuals: God deliberates about which particularizers should be instantiated.

A particularly neat thing about particularizers is that with some generalization they allow us to reduce quantification over particulars to quantification over properties. We need the primitive predicate P where P(A) if and only if A is a particularizer. If A is a property, I will use A(y) to abbreviate: y has A. Use E(A) to abbreviate ∃B(A(B)). If A is a particularizer of x, then E(A) holds if and only if x exists. Use A~B to abbreviate ∀C(A(C) iff B(C)). If A and B are particularizers, then A~B means that they are co-particularizers—i.e., there is an x such that they are both x-particularizers. Suppose now we want to say that there are exactly two dogs. Let D be the property of being a dog. We say:

• ∃A∃B(P(A)&P(B)&A(D)&B(D)&~(A~B)&∀C((P(C)&C(D))→(A~C or B~C)).

I.e., there are particularizers that (a) particularize doghood, (b) are not co-particularizers, and (c) any particularizer that particularizes doghood is a co-particularizer of one of them.

If we want to deal with relations, and not just unary properties, then we need to generalize the notion of particularizers. One way to do this would to be suppose a primitive "multiplication" operation that forms an n-ary particularizer A₁A₂...A_n out of a sequence A₁,A₂,...,A_n, where an n-ary relation B has A₁A₂...A_n if and only if x₁,x₂,...,x_n stand in B, where A_i is an x_i-particularizer.

Instead of names of particulars, we will then work with names of their particularizers. Note that if in a Fregean way we think of quantifiers as corresponding to second-order properties, then particularizers will correspond to quantifiers (and remember the Montague way of thinking of names as quantifiers—this all fits neatly together).

Abundant Platonists who think that for every predicate there is a corresponding property should not balk at the existence of particularizers. We can define a particularizer either in terms of an entity x, as the property of being instantiated by x, or in terms of a haecceity H, as the property of having an instantiator in common with H. Likewise, we can define a haecceity in terms of a particularizer. If A is a particularizer, then the property of having all the properties that are particularized by A will make a fine haecceity. Or we can take particularizers to be primitive, whether we have abundant or sparse Platonism.

The above shows that we could do without first-order quantification and without talking of particulars. Now I think that nobody should simplify their ontology by getting rid of objects. Yet the above shows that we can do so. How to resist this simplifying reduction? I think the best way is to say that it does not sit well with the fundamentality of claims such as "I exist" and "I am conscious." For on the above reduction, these claims end up being reducible to E(A) and A(consciousness), where A is a me-particularizer. But only someone with an ontology on which "I exist" or "I am conscious" can resist the reduction in this way.