- God and Jesus differ.
- Things which differ are two (i.e. are not numerically identical)
- Therefore, God and Jesus are two (not numerically identical). (1, 2)
- For any x and y, x and y are the same god only if x and y are not two (i.e. are numerically identical).
- Therefore, God and Jesus are not the same god. (3,4)
- There is only one god.
- Therefore, either God is not a god, or Jesus is not a god. (5, 6)
- God is a god.
- Therefore, Jesus is not a god. (7,8)
James Anderson and his commenters note that the analogous argument fails miserably for certain understandings of composition cases, noting that analogues to (4) in composition cases require assuming that every kind of numerical sameness is numerical identity. But an additional, although less obvious, problem with this argument is with (2), the claim that difference guarantees that things are not numerically identical, which raises questions of interpretation. Tuggy tries to conflate (2) with the indiscernibility of identicals. Superficially this is plausible. The indiscernibility of identicals, x=y → ∀F(Fx ↔ Fy), tells us that the operation of identifying two particular things requires positing that everything predicated of one is predicated of the other; this does mean that a difference is inconsistent with the operation of logical identity. But what counts as 'everything' here? It's not obvious that we should be using unrestricted rather than domain-relative quantification with identity. It is not required by the notation that we be thinking of every possible predicate; we can also just be thinking of a narrower range of predicates, and the logical operation of identity still would make perfect sense.
For instance, it's plausible to say that I am one and the same person as I was yesterday. The person, Brandon, who existed yesterday, is identical to the person, Brandon, who exists today. But the indiscernibility of identicals would require that yesterday-Brandon and today-Brandon share all predicates. They do not do so if we are considering all predicates. Thus in justifying (2), Tuggy has to make an ad hoc exception for times, which he explicitly does. But if (2) doesn't work across times, which certainly involves differences affecting predicates, why would we assume that it necessarily and self-evidently works across every other kind of difference? There are always analogies between times and other modalities, for instance; we would expect to find that there are at least some other modalities that are at least similar enough to temporal ones to work the same way in these cases.
It's clear yesterday-Brandon and today-Brandon only share all predicates if we are talking only about what makes a person a person. So within the domain relevant to determining what is a person, the operation of logically identifying yesterday-Brandon and today-Brandon still makes complete sense, and we can with perfect reason say that yesterday-Brandon and today-Brandon share all the predicates within that domain. Likewise, if we are considering George H. W. Bush at age 40 and George W. Bush at age 40, and we want to prove that they are not the same person, what we would do is identify one difference they have within the domain of things relevant to being a person; it's even more obvious that we don't need unrestricted quantification here, because we just need to look at what's relevant to the topic at hand, sameness of personhood in particular.
This requires no elaborate theories of relative identity; it just requires recognizing the obvious formal fact that the logical operation of identity can make perfect sense even if we are quantifying over a restricted domain rather than over everything whatsoever -- just like any other logical operation. Likewise, although there are lots of reasons to think the tendency of philosophers to quantify over everything is a bad practice in general, and that they should fall in with the common practice of mathematicians and computer scientists in being more restricted and specific about what they are quantifying over in every particular case, it's not necessary to insist upon it: the point above just requires the obvious logical fact that you can quantify over specific, restricted domains. And to be sure, this means that the logical operation of identity might work for given objects in some domains and not in others -- i.e., it might make sense to hold that yesterday-Brandon and today-Brandon are identical if we are only considering persons, but not if we are considering phases of a person's life -- but nothing whatsoever about the formal operation itself rules this out.
Some people have tried to block this by claiming that there is a special identity relation, what might be called proper numerical identity, which only exists when we are quantifying over all predicates. This sort of move is highly artificial and ad hoc; there are reasons to doubt that we ever use this super-special identity concept outside of very limited formal contexts in which we just posit that it applies. If you say that something, a, is identical to something else, b, are you ever in real life taking into account every possible thing that could possibly be said about them? I'm highly skeptical of the idea that in most cases you really are taking every predicate into account. When we talk about these things in ordinary English, in ordinary contexts, without trying deliberately for a peculiarly philosophical consistency, we are surely not so thorough. And even if we were deliberately holding ourselves to a very high degree of thoroughness, as in philosophical contexts we often are, we still can't run through every predicate, so we would have to be doing some kind of higher-order logic dealing with predicates that apply to predicates, in order to sort them and make sure we covered them all. And the problem becomes more acute when we look at modalities -- like the temporal modality distinguishing me from my younger self -- the sort of thing we do all the time. This super-special concept is not what we usually mean by saying that things are the same at all. My younger self and I do not share all the same predicates; I've changed, even if it has been but a day. But I am exactly the same person as I was yesterday! Everybody knows what I mean when I say something like that. If I think about my younger self, and think about myself today, and ask how many people I have thought about, the obvious answer is that I have thought about one, and only one, person, exactly one and the same person.
In order to get something like what Tuggy needs for his argument, we need to identify numerical sameness with numerical identity with unrestricted logical identity. (That is to say, the attributes 'not two [or more]', 'numerically identical', and 'represented by the logical operation of identity' have to be the same.) But if we're really to make sense of sameness and identity, it's difficult to see how we can do this without breaking at least one of these identifications: either numerical sameness and numerical identity are not the same, or numerical identity and logical identity are not the same. The view that numerical identity (the identity by which we count things as precisely one and the same, hence the name) and logical identity (as we find it in, say, the indiscernibility of identicals) are the same is required by Tuggy's (2); the assumption that (2) is equivalent to the indiscernibility of identicals requires that numerical identity actually be the logical operation of identity. The view that numerical sameness (the sense in which we say that things are the same so as to be one) at least requires numerical identity is required by Tuggy's (4): talk about the same god or divine being is talk of numerical sameness, and (4) says it requires numerical identity. But Tuggy's view is even stronger than this, since his justification of (4) assumes that they're really the same thing. So [numerical sameness] = [numerical identity] = [ unrestricted logical identity]. We will keep running into puzzles if we attempt to do this, however.
Numerical sameness doesn't actually seem to work like logical identity when the latter is not restricted in domain. For one thing, the natural way to talk about numerical sameness is to say that things are the same F (whatever F may be); but this kind of talk doesn't show up at all when talking about logical identity -- in the latter we don't talk about things being the same F but just the same. We can equate these expressions if we are not quantifying over everything -- if we just quantify over the domain relevant to F, then 'the same' will just mean 'the same F'. But if we're not quantifying over every predicate, then we aren't using the super-special concept of identity; we're just using a logical operation in a particular domain. Thus things that differ might still be one in some other way, if we look at some other domain. If we're talking about my younger self and my present self, they differ if we're only talking about phases of life; but they are still in a very real sense one thing if we are talking not about phases of life but about persons. But if things are in some real sense one thing, then in that sense they are numerically the same. I am the same person yesterday and today. But since I differ between yesterday and today, the sameness can be the kind of identity we use in logic only if we are only considering predicates that are not affected by this difference of time. But if this is how we're handling things, then knowing that two things differ does not tell us that they are not in some other way the same. And quite clearly this would make Tuggy's argument useless.
We get a broadly analogous set of problems if we look at composition, which is what Anderson was looking at (Anderson accepts the identity of numerical identity and logical identity found in (2)). [Tuggy's initial response to this does not seem particularly appropriate, because he very oddly treats Anderson's response as being about material composition rather than as being about ways in which Tuggy's original argument could fail to be sound, since Tuggy's challenge was explicitly to give reasons for denying or withholding assent to the premises of the argument. Anderson notes this later. And, for some reason, Tuggy only considers denial and not withholding of assent in his response, despite the fact that the challenge was explicitly about denial or withholding of assent. Identifying a puzzle that arises on some metaphysical views certainly gives at least some reason for being cautious in assenting. In addition, it is an error to focus solely on material composition, since material composition is just being put forward as an example of why you might reject an argument of this general kind; due to analogies among arguments, you would have to also look at the question of whether similar reasons to reject this kind of argument might arise for (say) temporal composition, or transworld identity, to take just two examples of things that often have broad structural analogies to material composition. Likewise, it doesn't matter at all whether divinity is very like matter; if the analogue of (4) were to fail for material composition, or any other kind of case, we would need a reason to think (4) itself is so obvious and undeniable in the case of divinity.]
Numerical identity likewise doesn't seem to work like unrestricted logical identity. Numerical identity here means the identity used to indicate that things are such as to count as the one and very same thing. There are a number of possible assumptions under which numerical identity could turn out not to be unrestricted logical identity; for instance, if there is relative identity, if there is temporary identity, if there is contingent identity, if there is vague identity, if there is partial identity, and so forth, all of which have been proposed for reasons having nothing to do with the Trinity, and any of which would problematize Tuggy's argument. But set all of these controversies aside for the moment. Do we really require the full logical power of unrestricted logical identity in order to count things as exactly one thing? I am counting chicks running around, One, Two, Three, Four, and you say, "No, you just counted the original one again." And I ask, "Are you sure that it's the very same? It's in a different place and it's not cheeping like the one I counted as One." And you say, "Yes, it very definitely is one and the same." Are you actually taking into every predicate? I just told you that One and Four differed in at least two predicates, and you still insisted that One and Four were numerically identical. Why aren't you taking numerical identity to require the indiscernibility of identicals? Of course, you could still be using the indiscernibility of identicals, if you're not actually quantifying over every predicate -- if, for instance, you are only allowing predicates indexed to times, or are only allowing predicates relevant to being a chick. But that, again, is not what Tuggy needs for his argument.
The point of all of this long (and yet all too brief for the complexity of the topic) discussion is that there is at least some reason to doubt that we can get all these samenesses to be the same sameness, [numerical sameness] = [numerical identity] = [logical identity when we are not restricting quantification]; but if we are restricting quantification, something may differ in one sense and yet be the same in another, which makes Tuggy's argument useless -- (1)-(5) could give us no more than the claim that God and Jesus are not the same divine thing if we are looking at some predicates, which is not at all controversial, and which cannot get us (7) -- and if any of these identities of sameness break, the argument equivocates.
None of this is unknown; Tuggy himself recognizes for (4) that philosophers have noted lots of problems with this particular identification. His response is that "to save their various theories, sometimes philosophers deny what is obviously true." And, indeed, yes, to save his theory Tuggy does seem to be denying what is obviously true, namely, that a number of problems have been identified with the kind of claim in (4), if we are actually trying to be precise about it; accepting it as true lands us in a number of puzzles, any one of which could be pointed to as a reason for being cautious when considering whether to accept (4), or even for withholding assent.