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.