Gallois on occasional identity
In the (Northern) summer, I wrote a short survey article on contingent identity. The word limit did not allow me to go into many details. In particular, I ended up with only a brief paragraph on Andre Gallois's theory of occasional identity, although I would have liked to say a lot more. So here are some further thoughts and comments on Gallois's account.
In his 1998 monograph Occasions of Identity, Gallois defends the view that things can be identical at some times and worlds and non-identical at others. For simplicity, I'll focus only on the temporal dimension here. Gallois begins with a long list of scenarios where it is intuitive to say that things are identical at one time but not at others. For example, when an amoeba A fissions into two amoebae B and C, it is tempting to say that B and C were identical prior to the fission and non-identical afterwards.
Usually, when philosophers want to account for contingent or occasional identity, they offer a semantic story according to which, roughly speaking, operators like 'at t2' can affect the referent of embedded singular terms. Thus according to counterpart theory, 'B=C' and 'at t2, not(B=C)' can both be true because in the scope of 'at t2', the terms 'B' and 'C' denote counterparts of their original referents, and a single object can have two counterparts at other times.
Gallois offers no such explanation, and in fact rejects all proposals along such lines. His basic idea is much simpler. It is that predications must always be relativised to a time. Things aren't just short or tall, they are short at t1 or tall at t2. Predications of identity are no exception. Thus 'B=C' does not express a complete proposition. As Gallois says: "When told that an identity holds always ask at what time" (p.76). Gallois argues that if we follow this rule and always make the temporal argument explicit, then there is simply no reason to think that 'at t1, B=C' would have to entail 'at t2, B=C'.
For example, consider the objection from the transitivity of identity. The objection says that since the initial amoeba A is supposed to be identical to both of the resulting amoebae B and C, B and C must be identical to each other. Gallois replies that A is only identical to B and C at t1, and at this time B and C are indeed identical to each other. At t2, on the other hand, A is not identical to B and C, so it does not follow that B would have to be identical to C at that time.
Along similar lines, Gallois responds to the objection from Leibniz's Law. Here we also learn something interesting. Suppose at t2, B is in a pond while C is in a slide. Then it looks like there is something that distinguishes B and C already at t1, namely their whereabouts at t2. Not so, says Gallois. It is true that at t1, B has the property of being in a pond at t2:
(1) At t1, at t2, B is in a pond.
Moreover, at this stage, B is identical to C:
(2) At t1, B=C.
It follows by Leibniz's Law (with the temporal relativisations made explicit) that at t1, C has the property of being in a pond at t2:
(3) At t1, at t2, C is in a pond.
We also know that C is not in a pond at t2:
(4) At t2, C is not in a pond.
So what we learn is that iterations of 'at ti' do not collapse: 'at t1, at t2, p' does not entail 'at t2, p'. In other words, things can change their time-indexed properties. Gallois tries to explain this "surprising metaphysical implication" (p.90) by the following principle governing iterated temporal modifiers (p.84):
(E) at t1, at t2, A(x) <-> (Ey)(at t1, x=y & at t2, A(y)).
In the case of the amoeba, (3) is true because there is a y, namely B, such that at t1, C=y and at t2, y is in a pond.
It is interesting to compare Gallois's account with the results of more traditional counterpart-theoretic or intensional accounts. On these views, we have to distinguish a "meta-language" and an "object language" (see Hazen 1979). In the meta-language, one can truly say things like 'Hubert Humphrey is a very short-lived stage that only exists at the present time'. In the object language, by contrast, it might be true that Hubert Humphrey is 50 years old and exists (or existed) at many times. Counterpart theory offers translation rules from the object language of ordinary thought and talk into the counterpart theoretic meta-language. What would a counterpart theorist say if she were to stick to the object language? She would say things very similar to the things Gallois says.
For example, she would vehemently deny the hypothesis that ordinary objects are short-lived stages, and insist that Humphrey (he himself exists at other times and worlds. So she would reject the meta-language statements of counterpart theory as highly revisionary and counterintuitive. As does Gallois.
What would the counterpart theorist say about a case of fission? Suppose it is now t2, after the fission. At this point, B is not identical to C. On the other hand, at t1, before the fission, B was identical to C, because B and C have the same counterpart at t1, namely A. So we have
(2) At t1, B=C.
Since the pre-fission object A has two present counterparts, B and C, it is not obvious what we shall say about statements like
(5) At t1, now, A is in a pond.
But it is not too implausible, I think, that (5) is true. Then so is (6):
(6) At t1, now, C is in a pond.
On the other hand, we know that C is not in a pond at t2:
(7) Now, C is not in a pond.
So the double modifier 'at t1, now' does not coincide with 'now'. It looks like things can change their time-indexed properties!
Finally, let's talk about the world from an atemporal, God's eye perspective. Most counterpart theorists are B-theorists about time, so that should be possible. What can we say in the object language about A, B and C from this perspective? Without any time to fix the context of utterance, the three names are indeterminate between a large number of stages. Presumably, the counterpart theorist can truly say that A and B and C all exist, since that is true on any resolution of the indeterminacy. But she may not want to answer the question whether B and C are identical, which is true on some resolutions and false on others. She can say (even from the timeless perspective) that at t1, B and C are identical. So our counterpart theorist can only say whether B and C are identical if a time is provided at which the identity is supposed to hold.
Given these observations, it is noteworthy that in section 5.5 of his Four-Dimensionalism, Ted Sider argues at length against Gallois's account, without realising how closely this account is related to his own temporal counterpart theory. In particular, Sider complains that Gallois's temporal logic is non-standard and implausible, due to phenomena like non-collapsing iterated modalities. The complaint is repeated in Sider's review of Gallois's book, and also takes center stage in Achille Varzi's review. I don't think the complaint is fair, especially coming from Sider, as his own account plausibly yields very similar results.
But there are other things to complain about. Some of Gallois's claims would not be vindicated by the assumption that he speaks a counterpart theoretic object language. And most of those claims I find problematic.
First, there is Gallois's insistence on speaking from God's point of view. I don't think there is anything incomplete about a bare identity statement like 'B=C'. We do not have to say at what time the identity is supposed to hold, or at what world. This would be problematic if the utterance took place outside of time and space, and outside the logical space of worlds. But there are no such utterances, nor could there be any. By adding a time and world coordinate, we turn a statement that expresses a contingent condition on possible situations into a non-contingent and eternal statement -- a statement that is either true at all possible situations or at none. But what's the point of uttering such a sentences, if it rules nothing in or out?
Second, there is something odd about Gallois's use of quantifiers. Given the close connection between identity and quantification, one might have thought that if identity is time-relative, then so is quantification. But Gallois's quantifiers are explicitly not relativised to a time. Have another look at (E):
(E) at t1, at t2, A(x) <-> (Ey)(at t1, x=y & at t2, A(y)).
Here the right-hand side does not say that there is an object y at t1 such that x=y at t1 and A(y) at t2. If the quantifier were time-relative in this manner, ranging over objects at t1, then we could not distinguish B and C, which are after all one and the same object at t1. We would have to ask whether this one object satisfies A(y) at t2. For Gallois, by contrast, the quantifier ranges timelessly over timelessly individuated objects. From this perspective, B and C are non-identical, and it is B, not C, as value of y that renders (3) true. Similarly, Gallois holds that B satisfies `at all times, x=B', while C does not.
This suggests that Gallois's time-relative "identity" is in fact a kind of time-relative coincidence relation that can obtain between absolutely non-identical objects. In the scenario of the amoeba, there are two individuals in the quantifier domain both of which are found at the same place at t1.
Gallois even gives a little argument in defense of timeless quantification. The argument is hidden on p.264f., in a discussion of an argument of Quine's in favour of four-dimensionalism. Here Gallois accepts Quine's observation that we use quantifiers to range both over things with temporal characteristics and over things like numbers without temporal characteristics. But the same is true for identity: maths is full of timeless identity predications, just as it is full of timeless quantifiers. I don't see any advantage of endorsing timeless quantification but only time-relative identity.
The worry that Gallois's "identity" is really a time-relative coincidence relation is reinforced by his use of semantic notions. For example, what object is picked out by 'B', or 'B at t1' in the fission scenario? One might expect that this has only a time-relative answer: at t1, `B' picks out the single object A=B=C, while at t2, it picks out B and not C. But this is not what Gallois thinks, as we learn in chapter 4. Here Gallois generally takes on the timeless perspective from which the episode of fission appears to involve exactly two amoebae, B and C, which are determinately and timelessly picked out by 'B' and 'C', respectively.
Consider Gallois's comments on the original amoeba A. Suppose we have stipulated that 'A' denotes the sole amoeba at t1. According to Gallois, this is not sufficient to bestow a determinate meaning on the name, because it doesn't settle which of the two amoebae is meant to be picked out: the stipulation "underdescribes the use we are making of that expression. [...] We need to say more before we can use ['A'] to refer to the amoeba existing at [t1]" (p.104). That's odd. I would have thought that according to the hypothesis of occasional identity, there aren't two amoebae at t1; there's only one. So how can our stipulation leave open which of two amoebae is named?
(As Gallois admits, it is usually impossible to make a name fully determinate, because we don't know in advance which objects will undergo fission. So we are stuck with referentially underdetermined names like 'A'. How do we evaluate such names in the scope of intensional operators for which it matters how the indeterminacy is resolved? What should we say about 'at t2, A=B' and 'at t2, A=C'? Gallois quickly rules out the option that they are both true, since that would conflict with the tensed transitivity of identity. His own answer is to treat the names like Russellian descriptions, and thus to read 'at t2, A=B' like 'at t2, (the amoeba at t1)=B', which comes out false (pp.106-109).)
Summing up. Unlike other accounts of occasional or contingent identity, Gallois assumes that identity is at bottom a time-relative relation; facts about identity at a time cannot be reduced to facts involving classical, untensed, two-place identity. Many aspects of his view, especially some of its more surprising logical implications, resemble implications one might get if one takes a counterpart-theoretic account but never leaves the object language. However, other aspects of Gallois's view do not fit this picture, and rather suggest that his time-relative "identity" is not identity at all but some form of coincidence. A full-blooded theory of occasionally identity should vindicate that there may be a single object A such that the pair (A,A) satisfies 'x=y and at some time, not(x=y)'. Gallois's account does not deliver this result. From the timeless perspective from which he always considers his scenarios, a case of fission involves exactly two objects -- two values of timeless quantification. In second-order logic, we could use his timeless quantifiers to define a timeless identity predicate, which would make explicit that for Gallois, the fission products B and C are simply and timelessly non-identical, although they occasionally stand in a certain time-relative relation that Gallois (somewhat misleadingly) calls 'identity'.