Suppose a ship (b) leaves port and two ships (a and c) dock, but (for the usual reasons having to do with the replacement of bits) we want to say it is indeterminate whether a=c, and indeterminate whether b=c, yet determinately not the case that a=c. Pinillos argues this cannot be right, and a key premise of his argument is that the set of ships which left port - {b} - is distinct from the set of ships which docked - {a,c} - because these two sets have different cardinalities. (This leads him to conclude that there must be something in one set which is not in the other - so that there is some pair consisting of one member from each set such that it is (determinately) the case that the two things in that pair are not identical - contradicting the assumption that all the relevant identity claims are indeterminate.)

I won't go into the details of Dominic's response to the Pinillos argument, since (as far as I know) it's not yet publicly available. But my own reaction (different from Dominic's) was to wonder whether the defender of this sort of vague identity should accept that the two sets are distinct. Maybe she should say that it's indeterminate whether the set of ships which left port is identical to the set of ships which docked.

Sure, there is a strong intuition that something is wrong with '{b} = {a,c}', but perhaps this could be explained as an intuition to the effect that, because of the difference in cardinality between the sets, this identity claim is

*necessarily not true*, which does not (for people - like Dominic - who are into three-valued logics) entail that it is false. (By analogy, a claim Dominic argued for in one of his other talks here is that although we have a strong intution that claims like 'Patch p is red and non-red' are not true, we should not therefore regard them as false when p is a borderline case.)

In fact, is the defender of vague identity even compelled to accept the claim that the sets have distinct cardinalities? If it really is indeterminate whether a=b and indeterminate whether b=c, then the set of ships which left port contains b, and it is indeterminate whether it contains a and indeterminate whether it contains c. And the set of ships which docked contains a and c, and it is indeterminate whether it contains b. Maybe, therefore, we should say that the cardinality of each set is indeterminate, in such a way that it is not determinately the case that the two sets have distinct cardinalities.