## Counterparts and Actuality

March 4, 2008

I’ve been reading this paper by Delia Graff Fara for one of the discussion groups I’ve been going to. It’s basically a follow up to the Williamson and (Michael) Fara paper from a couple of years back, highlighting some problems counterpart theory would face if augmented by an actuality operator. I had some general methodological problems with these papers (for example, they would argue that CPT could not provide faithful interpretations of QML formulae – when CPT’s aims are to provide interpretations of English, and further, they claim, to do so more faithfully than QML). But that aside, there was one obvious response Lewis could make which neither paper seemed to consider. I mentioned it in the discussion group, but didn’t get a chance to think it through properly, so I though I might take this opportunity to expand it some more (so apologies in advance for any obvious errors!)

Consider a world of eternally recurring, qualitatively identical epochs (call it $w_e$.) Now Lewis wants to reconcile two things. He wants to deny a version of haecceitism that states that there can be qualitatively identical possible worlds which differ with respect to what de re possibilities they represent for some individual; while making sense of the intuitive claim that Bob, who, lets say, lives in the 17th epoch, might have lived in the 18th epoch (i.e. where is qualitatively identical twin, boB, lives.) To do this he allows the counterpart relation to hold between individuals that live in the same world. This amounts, as Graff Fara notes, to individuating possibilities more finely than possible worlds. For example, there is one possibility in which Bob lives in the 17th epoch, and one in which he lives in the 18th, yet there is only one possible world involved. In Lewis’s own words: “Possibilities are not always possible worlds. There are possible worlds, sure enough, and there are possibilities, and possible worlds are some of the possibilities.” (PoW, p230)

So why don’t we just interpret the actuality operator as being true in the actual possibility, rather than being true in the actual world? To fix ideas, let us think of a possibility as an ordered pair of a world, w, and a function from individuals in w to individuals in w. For example the first possibility we considered was $w_e$, with the identity mapping, taking Bob to himself, but when we considered the possibility that Bob might have lived in the 18th epoch we were considering the pair $w_e$ and the mapping that takes Bob to boB, Bob’s 18th epoch twin, (and which is the identity elsewhere.)

Does interpreting the actuality operator like this help? For example, do we get all the inferences we usually get from it? We can show that we do by simply interpreting QML+@ (quantified modal logic augmented with an actuality operator) in terms of possibilities and it should then be clear that it will validate exactly the same inferences as classical @ would.

We let the set of states be the set of possibilities, i.e. $S := \{ \langle w, \sigma \rangle \mid w \in W \wedge \sigma : Ind \rightarrow Ind(w), \sigma \subseteq C \}$. Let Ind be the set of individuals from any world, Ind(w) the individuals from w, and C the counterpart relation (I’ve relaxed the constraint that the function must go from and to individuals in the world.) I’ve idealised and assumed that $\sigma$ is a total function. We set one particular pair $s^* := \langle w^*, \sigma^* \rangle$ to be the actual possibility. The crucial truth clauses are as follows

$\langle w, \sigma, v \rangle \models Px_1, \ldots, x_n \Leftrightarrow \langle \sigma(v(x_1)), \ldots, \sigma(v(x_1)) \rangle \in [[P]]$
$\langle w, \sigma, v \rangle \models @\phi \Leftrightarrow \langle w^*, \sigma^*, v \rangle \models \phi$
$\langle w, \sigma, v \rangle \models \Diamond\phi \Leftrightarrow \langle w^\prime, \sigma^\prime, v \rangle \models \phi \mbox{ for some } \langle w^\prime, \sigma^\prime \rangle \in S$
$\langle w, \sigma, v \rangle \models \exists x\phi \Leftrightarrow \langle w, \sigma, \sigma \circ v^\prime \rangle \models \phi$
$\mbox{ for some } v^\prime \mbox{ which differs only from v in its assignment to x.}$

Since the clause for @, is exactly the same as in the standard semantics where we intepret S as the set of possible worlds, and the other truth clauses are sufficiently similar – we should get exactly the same inferences for @ as in the ordinary case.

Of course, counterpart theorists don’t like to use a language with primitive modal operators like QML+@, and will, if they can, phrase it all in first order logic. Standardly counterpart theorists will need the two primitive symbols: Iwx and Cxy. I is the relation of being a part of a world, C is the counterpart relation. We shall use one primitive, Rsxy, interpreted as $x = \sigma(y)$ where $s = \langle w, \sigma \rangle$. We can give a translation schema of for QML+@ as follows:

$(Px_1,\ldots,x_n)^s \mapsto$
$\exists y_1,\ldots,y_n(Rsy_1x_1 \wedge \ldots \wedge Rsy_nx_n \wedge Py_1, \ldots, y_n)$
$(\neg \phi)^s \mapsto \neg(\phi^s)$
$(\phi \wedge \psi)^s \mapsto (\phi^s \wedge \psi^s)$
$(\exists x \phi)^s \mapsto \exists x \exists y(Rsxy \wedge \phi^s)$
$(\Diamond \phi)^s \mapsto \exists s^\prime \exists y_1\ldots y_n \exists z_1\ldots z_n(Rsz_1x_1 \wedge \ldots \wedge Rsz_nx_n$
$\wedge Rs^\prime y_1z_1 \wedge \ldots \wedge Rs^\prime y_nz_n \wedge \phi^{s^\prime})$
$(@\phi)^s \mapsto \phi^{s^*}$

(Note in the $\Diamond$ clause, $x_1, \ldots, x_n$ are the free terms in $\phi$.)

1. Andrew,

Interesting. The stuff on true, necessarily false propositions in Lewis assumes that possible worlds exhaust the possibilities. There is presumably also the the sum of Bob and boB in @. Is that a possible object or an actual object? Taken on analogy with the sum of Bob and his counterparts in non-actual worlds, it probably wouldn’t count as either.

2. Hi Mike,

Thanks for your comments. I’m not sure I know the bit of Lewis you are referring to, but allowing multiple counterparts to exist in the same world was a change of mind for Lewis (his original counterpart theory had a postulate saying that only one counterpart could exist in a world.) So it’s likely that he didn’t always hold this view – but by this point, at least, he had a more liberal notion of possibility, in which parts of worlds could be possibilities, as well as de re possibilities.

As for the Bob boB fusion, I don’t think there’s anything precluding that from being actual. Possibilities on this view are entire worlds with extra information saying what the de re representation of each object is. What may be puzzling is that the Bob boB fusion may not be represented as the Bob boB fusion in this possibility.

3. I was referring to the literature (Parsons, Hudson, few others) on true propositions concerning objects that do not fully exist in any world. So, the fusion of you and your counterpart in the closest world in which you are left-handed (supposing there is such world and you’re not lefthanded). If that being is named AndrewL, then it’s true that there is a being named ‘AndrewL’ and necessarily false that there is a being named ‘AndrewL’. Similar problems arise for propositions that talk about the number of worlds. e.g., there are more than two possible worlds.
This “problem” (Lewis does not see it as a problem) arises for sums of course-grained possibilities (sums of counterparts in different worlds). This is why I was wondering what to say for sums of finer-grained possibilities (sums of possibilities within a world). But I suspect you’re right. The proposition that there is someone named ‘bob-boB’ is both true and actually true.

4. If I’m reading the details right, your $\sigma$ functions (proxy for the counterpart relation) are *functions*, right? How are you thinking about cases where Lewis wants a thing to have multiple counterparts in a world? Or are you just trying to save the fragment of Lewis w/o that feature…?

5. Hi Jason,

That’s right, they’re functions. I’m following Lewis in allowing multiple counterparts in the same *world*. But Lewis says this requires us to cut possibilities more finely than worlds. So in a *possibility*, there is exactly one counterpart. A possibility on this view is just a world equipped with information about how people are represented in a world.

6. Sorry, that should have been obvious from what you said at the beginning. I was just getting a bit lost in the nuts and bolts.

7. nice work, man

8. omg.. good work, guy

9. […] as are worlds supplemented with information about the de re representation of various objects (see here.) Since the fusion of two worlds is a possible object we can get our spatially disconnected epochs. […]