Yet another formulation of my argument against a theistic multiverse

Here’s yet another way to formulate my omniscience argument against a theistic multiverse, a theory on which God creates infinitely concretely real worlds, and yet where we have a Lewisian analysis of modality in terms of truth at worlds.

1. Premise schema: For any first order sentence ϕ: Necessarily, ϕ if and only if God believes that ϕ.

2. Premise schema: For any sentence ϕ: Possibly ϕ if and only if ∃w(at w: ϕ).

3. Premise: Possibly there are unicorns.

4. Premise: Possible there are no unicorns.

5. Necessarily, there are unicorns if and only if God believes that there are unicorns. (Instance of 1)

6. Possibly, God believes that there are unicorns. (3 and 5)

7. Possibly God believes that there are unicorns if and only if ∃w(at w: God believes that there are unicorns). (Instance of 2)

8. ∃w(at w: God believes that there are unicorns). (6 and 7)

9. ∃w(at w: God believes that there are no unicorns). (from 1, 2, 4 in the same way 8 was derived from 1, 2, 3)

So, either there is a world at which it is the case that God both believes there are unicorns and believes that there are no unicorns, or what God believes varies between worlds. The former makes God contradict himself. The content of God’s beliefs varying across worlds is unproblematic if the worlds are abstract. But if they are concrete, then it implies a real disunity in the mind of God.

Premise schema (1) is restricted to first order sentences to avoid liar paradoxes.

Moving from world to world

If the A-theory of time is true, then it is (metaphysically) possible that the year 2010 have the objective property P of presentness and it is also possible that the year 2019 have P. For it is true that 2019 has P, and what is true is possible. But by the same token in 2010 it was true that 2010 has P, and so it was possible that 2010 have P. And what is metaphysically possible does not change. So even now it is possible that 2010 have P.

But a proposition is possible if and only if it is true at some possible world. Thus, if the A-theory of time true, there are possible world where 2010 has P and possible worlds where 2019 has P, and in 2010 we lived in one of the worlds where 2010 has P, while now in 2019 we instead live in a world where 2019 has P.

Consequently, given the A-theory of time, what world we inhabit continually changes.

This seems counterintuive. For now it looks like caring about what will happen is caring about what happens in some merely possible world.

World shuffling and quantifiers

Let ψ be a non-trivial one-to-one map from all worlds to all worlds. (By non-trivial, I mean that there is a w such that ψ(w)≠w.) We now have an alternate interpretation of all sentences. Namely, if I is our “standard” method of interpreting sentences of our favorite language, we have a reinterpretation I_ψ where a sentence s reinterpreted under I_ψ is true at a world w if and only if s interpreted under I is true at ψ(w). Basically, under I_ψ, s says that s correctly describes ψ(actual world).

Under the reinterpretation I_ψ all logical relations between sentences are preserved. So, we have here a familiar Putnam-style argument that the logical relations between sentences do not determine the meanings of the sentences. And if we suppose that ψ leaves fixed the actual world, as we surely can, the argument also shows that truth plus the logical relations between sentences do not determine meanings. Moreover, can suppose that ψ is a probability preserving map. If so, then all probabilistic relations between sentences will be preserved, and hence the meanings of sentences are not determined by truth and the probabilistic and logical relations between sentences. This is all familiar ground.

But here is the application that I want. Apply the above to English with its intended interpretation. This results in a language English^* that is syntactically and logically just like English but where the intended interpretation is weird. The homophones of the English existential and universal quantifiers in English^* behave logically in the same way, but they are not in fact the familiar quantifiers. Hence quantifiers are not defined by their logical relations. I’ve been looking for a simple argument to show this, and this is about as simple as can be.

Novels and worlds

As the length increases, the possibilities for good novels initially increase. It may not be possible to write a superb novel significantly shorter than One Day in the Life of Ivan Denisovich. But eventually the possibilities for good novels start to decrease, because the length itself becomes an aesthetic liability. While one could easily have a series of novels that total ten million words, a single novel of ten million words just wouldn't be such a good novel. Indeed, it seems plausible that there is no possible novel of ten million words (in a language like human languages) that's better than War and Peace or One Day or The Lord of the Rings.

If this is right, then there are possible English-language novels with the property that they could not be improved on. For there are only finitely many possible English-language novels of length below ten million, and any novel above that length will be outranked qua novel by some novel of modest length, say War and Peace or One Day.[note 1]

So, there are possible unimprovable English-language novels. Are there possible unimprovable worlds? Or is it the case that we can always improve any possible world, say by adding one more happy angelic mathematician? In the case of novels, we were stipulating a particular kind of artistic production: a novel. Within that artistic production, past a certain point length becomes a defect. But is an analogue true with worlds?

One aspect of the question is this: Is it the case that past a certain point the number of entities, say, becomes a defect? Maybe. Let's think a bit why super-long novels aren't likely to be that great. They either contain lots of different kinds of material or they are repetitive. In the latter case, they're not that great artistically. But if they contain lots of different kinds of material, then they lose the artistic unity that's important to a novel.

Could the same thing be true of worlds? Just adding more and more happy angels past a certain point will make a world repetitive, and hence not better. (Maybe not worse either.) But adding whole new kinds of beings might damage the artistic unity of the world.

How impossible can we get?

I've been thinking about a framework for really impossible worlds. The first framework I think of is this. A world w is a mapping (basically, a function, except that the propositions don't form a set) from propositions to truth values. Thus, if w₀ is the actual world, w₀(<the sky is blue>)=T and w₀(<2+2=5>)=F. But there will be a world w with all sorts of weird truth assignments, for instance where the conjunction is true but the conjuncts are false, or where p is false but its negation is also false.

But I then wondered if this captures the full range of alethic impossibilities. What about impossibilities like this: Worlds at which <2+2=4> has no truth value? Worlds at which every proposition is both true and false? To handle such options it's tempting to loosen the requirement that w is a mapping to the requirement that it be a relation. Thus, some propositions might not be w-related to any truth value and some propositions might be w-related to multiple truth values. But we can get weirder than that! What about worlds w at which the truth value of <2+2=4> is Sherlock Holmes? Nonsense, you say? But no more nonsense than something being both true and false. So perhaps w should be a relation not just between propositions and truth values, but propositions and any objects at all, possible or not. But even that doesn't exhaust the options of truth assignments. For what about a world where truth is assigned to every cat and to no proposition, or where instead of <2+2=4> having truth, truth has <2+2=4>? So perhaps worlds are just relations between objects, impossible or possible?

Of course, it feels like we've lost grip on meaningfulness somewhere in the last paragraph. But it's not clear where. My suggestion now is that none of the complications are needed. In fact, even the initial framework where a world is a truth assignment may be needlessly complicated. Let's take instead the simpler framework that a world is a collection of propositions.

Thus, p is true at w if and only if p is a member of w. And p is false at w if and only if ~p is a member of w.

But what about the bizarre options? On this framework, for any world w, either <2+2=4> is a member of w and hence true at w or it's not. What about the possibility that it is both true and non-true at w? I think the framework can handle all the bizarre possibilities provided that we understand them as world-internal. What is true at w is a question external to w, a question to be settled by the classical logic that is actually correct. Either p is true at w or it's not, and it can't be both true and non-true. But, nonetheless, although while it can't be that p is true at w and not true at w, it can be that p is true at w and p is false at w (just suppose both p and ~p are members of w). So that p is false at w does not imply the denial of the claim that p is true at w.

All the bizarreness, however, is to be found in world-internal claims. Let's say that p is (not) true in w provided that the proposition <p is (not) true> is true at w (in the external sense), i.e., <p is true> is a member of w. Likewise, say that p is (not) false in w provided that <p is (not) false> is true at w. And so on: in general, S(p) in w provided that <S(p)> is true at w. Then while truth-at w is relatively tame, truth- and falsity-in w can be utterly wild. We can have p true in w and p not true in w. We can have a world w in which <2+2=4> has the truth value Benjamin Franklin and is also false and true. There will be a world in which ~(2+2=4) but it is nonetheless true that 2+2=4. And so on. It's all a matter of getting the scope of the world-relativizing operator right.

A very impossible world?

In a criticism of the Pearce-Pruss account of omnipotence, Scott Hill considers an interesting impossible situation:

1. Every necessary truth is false.

While the criticism of the Pearce-Pruss account is interesting, I am more interested in a claim that Hill makes that illustrates an interesting fallacy in reasoning about impossible worlds. Hill takes it that a world at which (1) holds is a world very alien from ours, a world at which there are "infinitely many" "false necessary truths".
But that's a fallacious inference from:

2. (∀p(Lp→(p is false))) is true at w

(where Lp says that p is necessary) to

3. ∀p(Lp→(p is false at w)).

Indeed, there is an impossible world w with the property that (1) is true at w and there is no necessary truth p such that p is false at w. Think of a world as an assignment of truth values to propositions. A possible world is an assignment that can be jointly satisfied—i.e., it is possible that the truth values are as assigned. An impossible world is an assignment that cannot be jointly satisfied. Well, let w₀ be the actual world. Then for every proposition p other than (1), let w assign to p the same truth value as it has according to w₀. And then let w assign truth to (1).

Merging Lewisian worlds

According to Lewis, any pair (or, more generally, plurality) of concrete (he doesn't even restrict it this way) of objects has a mereological sum. Now, suppose that x and y are concrete objects in worlds w₁ and w₂ respectively. Let z be the mereological sum of x and y. According to Lewis, worlds are maximal spatiotemporally connected sums of objects. Now, here are some plausible principles:

1. Spatiotemporal connection is transitive and symmetric.
2. If a is spatiotemporally connected to a part of b, then a is spatiotemporally connected to b.

Consider any concrete objects a and b in w₁ and w₂, respectively. Then a is connected with x, since all objects in a world are connected. And y is connected with b. Moreover, by 2, a is connected with z since x is a part of z. And by 2, b is connected with z. Thus, by 1, a is connected with b. Thus, all objects in w₁ and w₂ are mutually connected, and so by Lewis's account of worlds, there is only one world. Which is absurd.

A cardinality argument against five-dimensional universalism

Five-dimensional universalism (hereby stipulated) holds that if f is a partially defined mapping f from worlds to regions such that (a) if f(w) is defined, then f(w) is a nonempty region of w's spacetime and (b) f(w) is nonempty for some w, there is an object O_f that exists in every world w for which f(w) is defined and occupies precisely f(w) at w. We will call a function f with the above properties a "modal profile", indeed the modal profile of O_f.

I think that to do justice to the vast flexibility of our language about artifacts, if we want to be realists about artifacts, we will need to be five-dimensional universalists. Mere four-dimensionalism mereological universalism is insufficient, because there can be always coincident artifacts with different modal properties.

But:

1. There is a set of all actual concrete objects.
2. There is no set of all modal profiles.
3. If there is no set of all modal profiles and five-dimensional universalism is true, there is no set of all actual concrete objects.
4. So, five-dimensional universalism is not true.

The argument for (2) is that there are way too many possible worlds with spacetimes to make up a set[note 1], and for each such world w there is a different modal profile[note 2], so there is no set of all modal profiles.

Eternalism and presentism

Here is an argument against eternalism:

1. If eternalism is true, times are like places.
2. Times are not like places.
3. So, eternalism is false.

There are a number of arguments for (2). Many, though not all, of them have something to do with the directionality of time, given that space lacks such directionality. Now consider this parallel argument against presentism, and hence for eternalism:

4. If presentism is true, times are like worlds.
5. Times are not like worlds.
6. So, presentism is false.

There are a number of arguments for (5). Here's a fun one. If I am happy now and miserable at all other times, I'm really unfortunate. If I am happy in the actual world and miserable at all other worlds, I'm really lucky one. In general, misery at other times matters in a way in which misery at other worlds does not.

So how to break this impasse? One way would be to opt for a theory other than eternalism and presentism, say growing block. Another way is to keep on adding disanalogies between times and places or between times and worlds until one of the disanalogies ends up being much stronger. Yet another way, and I think the most promising, is to embrace both (2) and (5), but explain the disanalogy in a way that is compatible with presentism or eternalism (whichever is one's preference).

One should also note that arguments from analogy tend not to be the strongest.

Psychological theories of personal identity and transworld identity

On psychological theories of personal identity, personal identity is constituted by diachronic psychological relations, such as memory or concern. As it stands, the theory is silent on what constitutes transworld identity: what makes person x in world w₁ be the same as person y in world w₂. But let us think about what could be send in the vein of psychological theories about transworld identity.

Perhaps we could say that x in w₁ is the same as y in w₂ provided that x and y have the same earliest psychological states. But now sameness of psychological states is either type-sameness or token-sameness. If it's type-sameness, then we get the absurd conclusion that had your earliest psychological states been just like mine, you would have been me. Moreover, it is surely possible to have a world that contains two people who have the same earliest types of psychological states. But those two people aren't identity.

On the other hand, if we are talking of token-sameness, then we seem to get circularity, since the token-sameness of mental states presupposes the identity of the bearers. But there is a way out of that difficulty for naturalists. The naturalist can say that the mental states are constituted by some underlying physical states of a brain or organism. And she can then say that token-sameness of mental states is defined in terms of the token-sameness of the underlying physical states. This leads to the not implausible conclusion that you couldn't have started your existence with a different brain or organism.

But I think any stories in terms of initial psychological states face the serious difficulty that it is surely possible for me to have been raised completely differently and to have had different initial psychological states. This is obvious if the initial psychological states that count are states of me after the development of the sorts of cognitive functions that many (but not me) take to be definitive of personhood: for such functions develop after about one year of age, and surely I could have had a different life at that point.

In fact this line of thought suggests that no psychological-type relation is necessary for transworld identity. But if no psychological-type relation is necessary for transworld identity, why think it's necessary for intraworld identity?

Two presentist ways of seeing worlds

If presentism is true, then right now, call it t₂, the proposition B that Bucephalus exists is false, but it once was true, namely at t₁. Now, at every time a token of the following sentence expresses a truth:

0. For all p, a proposition p is true if and only if it is true at the actual world.

Now, let's imagine ourselves at t₁. Then Bucephalus exists. Thus, B is true. Moreover, (0) expresses a truth, and so B is true at the actual world. So at t₁ the sentence

1. B is true at the actual world

expresses a truth. But now let's return to our time. B is false. But (0) expresses a truth, and so the sentence

2. B is not true at the actual world

does expresses a truth. Thus, (1) expresses a truth when said at t₁ but expresses a falsehood when said at t₂. This shows that either:

3. "The actual world" refers to different worlds at different times

or

4. The proposition that p is true at w can change in truth value, even if "p" and "w" refer rigidly to a proposition and a world, respectively.

Thus, the presentist has two ways of understanding possible worlds. Either possible worlds are tensed, so that at every time we inhabit a different possible world (that's option (3)) or else the "true at" relation is tensed, so that we inhabit the same world at different times, or when we say at t that p is true at w, we say something true if and only if p is true at t at w.

I think there is a problem for (4). Let p be the proposition that horses do or do not exist. Let t be the actual present time. Then p is true at every world, since it's a necessary truth. Now consider a world w where the time sequence does not include t. There are several options for this. Maybe in w, time comes to an end in 2011. Maybe time is discrete in w while in our world it is continuous, and so w either includes no times from our world or else w "skips over" t. Or maybe for some other reason the time sequence in w is radically different from our world's time sequence. Then p is true at w. But on (4), when we say that p is true at w, that is true if and only if p is true at t at w. But nothing is true at t at w, since t isn't a time at w.

Here's a slightly different way to see the point. When p is true at w, it is true either because there are or because there are not horses at w (this is an uncontroversial case of disjunctive grounding). Suppose it's true because there are not horses at w. But at which time are the horses not there at w? After all, w could have horses at some but not other times. Presumably, the relevant time is the present time. On proposal (3), every world comes along with its own present time, and this is fine. But on proposal (4), a world's relevant present time is our present time, and w doesn't have our world's present time.

One could try to solve this with counterpart theory for times. But one can suppose w won't have a counterpart to our time.

Here's a bolder move to defend (4) against our argument: The accessibility relation between worlds differs between times. The proposition p isn't true at all worlds, but only at all accessible worlds (this may or may not involve a denial of S5—S5 does not say that all worlds are accessible, but only that accessibility is an equivalence relation). And a world is only accessible if it includes the present time (or a counterpart to it?). This has the implausible consequence that what is metaphysically possible changes with time. For instance, if in w the time sequence comes to an end with 2011, then the proposition that w is actual was possible in 2011, but is no longer possible. But it's implausible that what is metaphysically possible changes with time.

If this is right, then the presentist should embrace (3). But is (3) plausible? Do we really live in different worlds at different times?

The presentist's other move is simply to abandon talking about worlds, and instead talk about, say, abstract times (in the Crisp sense).

A-theory and worlds

According to the best A-theories (i.e., those that accept an Aristotelian view of propositions as changing in truth value), there is an objective and non-relational fact as to what time it is, a fact that won't obtain tomorrow. Assume this. Here is one way to think about this. Let w be the actual world. The actual world holds all the actually true facts, including presumably the fact that it is January 18, since it is indeed Tuesday. Moreover, there will be a world w* at which everything happens just as at w except that at w* it is some time t on January 19. World w* will be a world that we will inhabit at t, tomorrow.

On this view, at every time, we are in a different world. We will then have an earlier-than relation between worlds defined as follows: w is earlier than w* if and only if at w* it is true that w was actual. Assume the earlier-than relation is transitive. Say that two worlds are directly temporally related if and only if either they are identical or one is earlier than the other. We then get:

1. The future is closed if and only if direct temporal relatedness is transitive.

Suppose the future is closed—the best A-theory will say that (or so I claim). Then direct temporal relatedness is an equivalence relation, and for any world w, we can form the equivalence class T(w) of all the worlds directly temporally related to w.

We need one more thing in the formalism. We need a way to compare times between worlds that aren't directly temporally related. Thus, there is a simultaneity relation between worlds. Worlds w and w* are simultaneous provided that at both worlds it is the same time. This relation is also an equivalence relation, and we can let S(w) be the equivalence class of all worlds simultaneous with w.

Each world w is then a member of two orthogonal equivalence classes: T(w) which contains all the directly past and future worlds, and S(w) which contains all the simultaneous worlds. This provides resources for the formation of new modal operators, using one or the other of the equivalence relations as an accessibility relation.

Enough formalism. Maybe in a future post I will try to criticize the view.

Why can't the past change?

If you're a B-theorist, it is no puzzle that the past can't change. It can't change because we are always in the same world, and so neither the past, nor the present nor the future can change. Today, let us suppose (I think correctly) that it is the case that on Wednesday it was raining. Could it tomorrow be the case that it wasn't raining on Wednesday? Not at all—for the very same world, the very same events, that make propositions true today is the one that we evaluate against tomorrow. The fact that the past can't change, thus, is a matter of mere logic—it just follows from the truth conditions for sentences.

But what if you're an A-theorist? So, you think that things will be objectively different tomorrow. Indeed, you already do think that some things about Wednesday will objectively change. For instance, while today (Saturday) Wednesday is objectively three days in the past, tomorrow it will objectively recede one more day into the past. So in fact we already have a change, but a change that the A-theorist doesn't mind. (Though she should.)

In any case, logic alone doesn't do the job. One way to see this is that some A-theorists actually think the future changes. Thus, today, it is false that either I am at Mass on November 8 or that I am absent from Mass on November 8. But come November 8, this disjunction will be true. But the clever tricks that open futurists use to make sense of an open future could be used, equally well, to make sense of an open past. (The parallel holds for B-theory. The B-theorist is committed to the claim that the future cannot change. This sounds fatalistic, but we must distinguish the ability to change the future from the ability to affect the future.)

In the setting of my earlier post on A-theory, the claim that the past cannot change corresponds fairly closely (and in fact exactly, if we assume a closed past) to the claim that the earlier-than relation is transitive. If today, a world where it rains on Wednesday is is past, tomorrow that world will also be past. So in the setting of that post, the explanatory challenge to the A-theorist is why the earlier-than relation E is transitive. The A-theorist who takes E to be fundamental can only say that it is a brute fact that it is necessarily transitive.

There may be A-theorists who can meet the challenge, however. Suppose that you think that there is a TimeShift operator which shifts tensed propositions time-wise. Thus, if p is the proposition that it is sunny, TimeShift(+1 day, p) is the proposition that in a day it'll be sunny. Suppose, further, we take worlds to be maximal consistent collections of propositions, or maximally specific consistent propositions. Then the TimeShift operator can also operate on worlds, and we can define E(w₁,w₂) to hold iff there is a t<0 such that w₁=TimeShift(t,w₂). Then it really is a matter of simple logic that E is transitive, and we have a perfectly good explanation of why the past cannot change.

Note, however, that an open-futurist cannot take this explanation. For her, the fixeity of the past remains a surd.

If this is right, then Tom Crisp is mistaken in taking the earlier-than relation between abstract times (which are just worlds in my terminology) to be primitive. An A-theorist should not say it's primitive—it needs explaining. Or at least I remember him taking it to be primitive, but my memory isn't so good.

Can timeless things change?

It seems that the answer has to be negative—isn't the idea utterly absurd? But suppose that an A-theory is true and Fred is a timeless being. Let W be the property of being (timelessly) in a world where a war is (presently) occurring. It seems that on A-theories this is a genuine property, and it was true in 1944 that Fred then had W and it is no longer true in 2008 that Fred has W. So it seems that Fred has changed in respect of W. The B-theorist is apt to deny the existence of such a property as W, and instead talk of the family of properties W_t of being (timelessly) in a world where a war is occurring at t. It was true Fred in 1944 had W₁₉₄₄ and it is true in 2008 that he does not have W₂₀₀₈, but that is not a change, since likewise it was true in 1944 that Fred had not-W₂₀₀₈ and it is true in 2008 that Fred has W₁₉₄₄.

So, if the A-theory is true (or at least if one of those A-theories is true that allow tensed properties like W), it follows that timeless beings change. Of course, the change is extrinsic. But even extrinsic change is puzzling in the case of a timeless being. Look at it from Fred's point of view. Does he or does not have W? It seems both, but that is absurd. In the case of a being in time, we would say that the question is ambiguous—does he have W at what time? But we cannot disambiguate this from Fred's point of view.

Here is something an A-theorist might say. She might say—in fact, I think that on independent grounds she should say it—that at every time, a different world is actual. (Right now, a world without a present world war is actual. In 1944, a world with a present world war was actual.) Then there is no contradiction in Fred's both having and not having W, since since in one world (the 1944 one) he has W and in the other (the 2008 one) he does not.

If we take this route, then the "objective change" that A-theorists are enamored of will be a movement (an orderly one) from one world to another. But Fred undergoes that movement just as much as you and I—in 2005 he was in the 2005 world, and in 2008 he is in the 2008 world—though there is a difference whose significance I am unable to evaluate at present (Fred exists timelessly in both worlds, while you and I exist presently in both worlds). It seems, then, that Fred undergoes objective change, while being outside of time. That seems absurd. Moreover, if we take this route then the following conceptual truth becomes really hard to account for: Nothing outside of time can undergo intrinsic change. But why can't Fred have one set of intrinsic properties in the 1944 world and another in the 2008 world? And if he did, then he would be changing in respect of intrinsic properties.

If the above is right, then it seems that what the A-theorist needs to do is to deny the possibility of timeless beings. This has some interesting consequences. If time began with the big bang, and if we are realists about mathematical entities, then the number 7 is about fifteen billion years old, give or take a couple of billion, and if time were to come to an end, then the number 7 would cease to exist. And once we've allowed abstracta to be in time, why should it be any more absurd to allow them in space? I do not know if these kinds of considerations form knock-down arguments against the view (and hence against the A-theory, if the A-theorist needs to go there), but they are worth thinking about.

Transworld identity without haecceities

Haecceities are individual essences, even of non-existent beings. Necessarily, entity exists iff its haecceity is instantiated. Some folks think we need haecceities to make sense of alien individuals—i.e., individuals that exist in other worlds but not in ours. We don't, as long as we are willing to be Leibnizian in denying the identity of indiscernibles. Here, then, is a simple theory of identity across worlds that entails the essentiality of origins. The theory applies both to substance-like and event-like individuals.

I will give the simplest version of the theory, for beings in an absolute time who do not engage in any time travel and without backwards causation. A general version of the theory requires the replacement of times by "causal (or maybe even explanatory) points"—points in the causal history of an entity. This is a bit tricky, and so I won't bother with it.

Let e be an entity and w a world. Say that H is a qualitative history of e up to t in w provided that e exists at t in w and H is a proposition giving an at least partial description of w such that:

1. H is purely qualitative except respect of e and t: i.e., the only particulars that are de re involved in H are e and t;
2. H states that e exists at t and gives a complete description of the intrinsic properties of e at t, subject to the restriction in (1);
3. For any state of affairs reported in H, any and all the causes in w of that state of affairs are also reported in H;
4. H reports that e exists at t;
5. H is a minimal proposition satisfying (1)-(4).

Now, let w₁ and w₂ be two worlds such that e₁ exists in w₁ and e₂ exists in w₂. Then e₁=e₂ if and only if there are times t₁ and t₂ and histories H₁ of e₁ up to t₁ in w₁ and H₂ of e₂ up to t₂ in w₂, such that H₁'s description of e₁ and t₁ coincides exactly with H₂'s description of e₂ and t₂.

To put it roughly, e₁ and e₂ are identical if and only if there are points in the existence of each one, such that their respective histories up to these points are the same.

This view entails essentiality of origins. It also implies that there cannot be two entities which, along with their causal histories, have been indiscernible up to some time. Thus, there cannot be completely identical twins. This consequence is counterintuitive, but may be but a small price to pay for avoiding haecceities.

Given this view, we can form something like a haecceity from the disjunction of all the histories of an entity in a world.

The view can be varied by relaxing or tightening the conditions (1)-(5) on histories. I do not yet know which is the optimal version.