Posts on: Counterpart Theory
In around 2009, I got interested in counterpart-theoretic interpretations of modal predicate logic. Lewis's original semantics, from Lewis (1968), has some undesirable features, due to his choice of giving the box a "strong" reading (in the sense of Kripke (1971)), but it's not hard to define a better-behaved form of counterpart semantics that gives the box its more familiar "weak" reading.
Wondering if anyone had figured out the logic determined by this semantics, I found an answer in Kutz (2000) and Kracht and Kutz (2002). I also learned that counterpart semantics seems to overcome some formal limitation of the more standard "Kripke semantics". For example, while all logics between quantified S4.3 and S5 are incomplete in Kripke semantics (as shown in Ghilardi (1991)), many are apparently complete in the "functor semantics" of Ghilardi (1992), which I do not understand but which is said to have a counterpart-theoretic flavour. Skvortsov and Shehtman (1993) present a somewhat more accessible "metaframe semantics", inspired by Ghilardi's approach, and claim that the quantified version of all canonical extensions of S4 remain canonical (and hence complete) in metaframe semantics. Kracht and Kutz argue that their – much simpler – counterpart semantics inherits these properties of functor and metaframe semantics.
Informal talk about de re necessity is sometimes "weak" and sometimes "strong", in Kripke's terminology. When I say, 'Elizabeth II could not have failed to be the daughter of George VI', I mean – roughly – that Elizabeth is George's daughter at every world at which she exists. By contrast, when I say, 'Elizabeth II could not have failed to exist', I don't just mean that Elizabeth exists at every world at which she exists. My claim is that she exists at every world whatsoever. The former usage is "weak", the latter "strong".
When people give a semantics for the language of Quantified Modal Logic (QML), they typically treat the box as strong. '\( \Box Fx \)' is assumed to say that x is F at every accessible world, not just at every accessible world at which x exists.
Until recently, the Stanford Encyclopedia of Philosophy didn't have anything on
counterpart theory. The editors thought the topic isn't worth an entry of its
own, but at least it now has a section in the entry on "David Lewis's Metaphysics". This isn't ideal, since counterpart-theoretic approaches to
intensional constructions are best seen as metaphysically non-committal. But
it's better than nothing.
I also wrote an "appendix" to the entry with an overview over
counterpart-theoretic interpretations of quantified modal logic. It
explains some unusual features of counterpart-theoretic logics, how they arise,
and how they could be avoided.
Friends of singular thought typically assume that in order to have a singular attitude towards an object, one must either stand in a special acquaintance relation to the object, or have a special kind of mental representation for it. Both of these views face a challenge from our practice of attitude reports: we can seemingly attribute attitudes with singular content even if neither condition is satisfied.
In a well-known example from Sosa 1970, the army generals decide that the shortest man should go first. The Sergeant tells Shorty: 'they want you to go first'. Here the generals need not be acquainted with Shorty, and it is doubtful that they must have a "mental file" for him.
So far, we have looked at cases in which an agent has a descriptive belief (e.g., "the creature approaching through the woods is a bear"), which gets reported as a singular belief ("Mary beliefs Mark is a bear"). But sometimes we attribute singular beliefs even though the subject appears to have only a general (quantified) attitude about the relevant individual.
A murder has been committed. The detective has figured out that the culprit probably comes from a certain mountain village. She knows little about that village, but believes that all its inhabitants are poor peasants. You are one of the villagers. We might say:
Compare the following three sentences.
(1) I thought my husband was a bear.
(2) Mary thinks her husband is a bear.
(3) I think my husband is a bear.
(1) and (2) are ambiguous between a "de re" reading and a "de dicto" reading. But (3) only seems to have the "de dicto" reading. How come?
According to the semantics I have described in earlier parts of this series, an utterance of (3) is true on its de re reading iff (roughly) there is a suitable role R such that (i) in all the speaker's belief worlds, whatever plays R is a bear, and (ii) in the actual world, the speaker's husband plays R.
In the previous post, I have assumed that conversational context somehow determines a unique "suitable role" for each individual under discussion, relative to every epistemic subject. This is an unrealistic assumption.
For example, I believe that Canberra gets cold in winter. But Canberra is known to me as the occupant of many roles. Among other things, I know it as the capital of Australia, as the city in which I lived for most of 2012, and as the destination of my most recent international trip. When I say that I know (or believe) that Canberra gets cold, none of these roles may be particularly salient.
In this post, I'm going to present a first stab of a formal semantics for de re belief reports.
As I explained in the last post, I'm going to assume that for every epistemic subject at every time there is a set of doxastically accessible worlds, representing how the subject takes the world to be. I will sometimes refer to these worlds as the subject's 'belief worlds'.
On that background, we can make the guiding idea behind the Quine-Kaplan model more precise: 'S believes that x is F' is true iff there is a suitable role R such that (1) in all worlds doxastically accessible for S, whatever plays R is F, and (2) in the actual world, x plays R.
This is part 3 of a series on epistemic counterpart semantics (part 1, part 2).
Recall the guiding idea: A de re report 'S believes that x is F' is true iff there is a suitable role R such that (1) S believes that whatever plays R is F, and (2) in fact, x plays R.
I said that these truth-conditions naturally emerge if we treat 'believes' as a modal, quantifying over a set of accessible worlds. So I am going to assume that for any relevant subject in any relevant situation there is a set of "doxastically accessible" worlds which somehow characterise what the subject believes. I want to say a few words to clarify this assumption.
This is part 2 of a series on epistemic counterpart semantics. Part 1 is here.
I want to defend what I called the "Quine-Kaplan model" of de re belief ascriptions. According to this model, 'S believes that x is F' is true iff there is a suitable role R such that (1) S believes that whatever plays R is F, and (2) in fact, x plays R.
In this post, I mainly want to explain what I mean by a "suitable role". This will also bring to light some arguments in favour of the Quine-Kaplan model.
I have decided to write a series of posts on epistemic applications of counterpart semantics, mostly to organise my own thoughts.
Let's start with a motivating example, from Sæbø 2015.
On September 14 2006, Mary Beth Harshbarger shot her husband, whom she had mistaken for a bear. At the trial, she "steadily maintained that she thought her husband was a black bear", as you can read on Wikipedia.
Allen Hazen (1979, pp.328-330)
pointed out a problem for Lewis's counterpart-theoretic interpretation
of modal discourse: the fact that x is essentially R-related to y
should be compatible with the fact that both x and y have multiple
counterparts at some world, without all counterparts of x being
R-related to all counterparts of y. But the latter is what Lewis's
semantics requires for the truth of `necessarily xRy'.
Imagine a world with nothing but infinitely many duplicate dragons, aligned in one long row. Consider the second dragon in the row. Call it "Fred".
Fred could have failed to exist. There are many worlds where he does not exist. (The actual world is probably one of them). Some of these worlds where Fred does not exist contain no dragons at all, others contain some of the other dragons in Fred's row. In particular, there are worlds where all the other dragons exist, but not Fred. The dragons are after all distinct existences, and there are no necessary connections between those.
I'm somewhat stuck with the parts/counterparts paper. One of the problems is to find an acceptable semantics for time travel situations.
Part of the problem is that I'm often unsure what to say about these cases. I guess if time travel were more common, we would need some new linguistic conventions. Anyway, here are some sentences that seem true to me in the following scenario: Tina decides in 2025 to meet her younger self back in 2005. So at some time t in 2005, the younger Tina is in the living room and weighs 60 kg while the older Tina is in the kitchen and weighs 70 kg. Now, these all seem true to me:
I've written a little paper about the difference between theories on which ordinary things are fusions of parts located at various places, times, and worlds, and theories on which they instead have counterparts there. The dull conclusion is that there is no difference. I'm not sure I believe anything in there, and it's all quite rough, so comments are welcome: Parts and Counterparts (PDF).
Update 2005-03-08: Robbie Williams points out that my translation between Counterpart and Fusion Theory does not handle fission cases correctly. I should at least (following Lewis's translation scheme) say that names in the fusion language are indeterminate between all maximal eligible fusions of the corresponding counterparts from the counterpart language. But this is only a partial fix. I hope to come up with something better soon, though as I'm on the road for the next couple of days, that will probably have to wait until the weekend.
Ordinary objects - persons, planets, rivers and tables - are unextended atoms. They occupy only one point of space at only one time at only one world.
At first sight, this might sound absurd. Don't ordinary things
obviously exist at many different places, times and worlds? Isn't the
Yangtze clean in Geladandong and dirty in Shanghai? Wasn't it clean in Shanghai in 1500? And isn't it clean in Shanghai now at some other possible world?
Fortunately, Atomism need not deny any of this. For even though the
Yangtze is an unextended atom that strictly speaking only occupies a
single point, it has many counterparts at other points. And
these counterparts make all those statements true.
Well, I know what Counterpart Theory is not: it is not a theory according to which ordinary things do not really exist at other possible worlds.
There are two readings of "ordinary things do not exist at other worlds". The first is a neutral reading on which things exist at another world in the way they sleep at another world or win elections at another world: whatever possible worlds are, they somehow represent things as existing and sleeping and winning. In this sense, something exists at a world iff the world represents it as existing. Anyone who accepts possible worlds talk at all accepts that ordinary things exist at other worlds in this sense.
There is but one totality of worlds; it is not a world; it could not have been different. (Lewis, Plurality, p.80)
If the totality of worlds could not have been different, then presumably no possible world could have failed to exist.
Then in particular, the actual world, @, could not have failed to exist.
So there is an actually existing thing, namely @, that could not have failed to exist.
Even worse, arguably @ has some of its parts essentially. So there are some actually existing things besides @ that could not have failed to exist.
One might even say that all worlds have all their parts essentially, simply because worlds do not exist at other worlds. Then it follows that no actually existing thing could have failed to exist.
I've thought a bit more about the comments Michael Fara left last week, and I don't find my points very convincing any more. The following is partly a correction, but mostly just thinking out loud about a more general semantic question.
The general question is how to interpret sentences of the form
1) At i, A is F
2) At i, A is not F
where 'i' denotes something like a time or a place or a world. There are a dozen proposals for interpretations of (1) in the temporal case, invoking temporal parts or relations to times or whatever. Most of these proposals can be applied to other indices as well. But let's put that aside. Suppose we understand how to interpret instances of (1) in easy cases. The hard cases I have in mind are cases where A doesn't exist exactly once at i. The precise definition
of these cases depends on the question I've put aside, but I hope it
is reasonably clear what I mean anyway. Not existing exactly once at i
means either not existing at i at all, or multiply existing at
i. Plausible examples of the first kind: I do not exist in 1758; I do
not exist on Alpha Centauri; I do not exist at any world containing
only empty space-time. Controversial examples of the second kind: if I
get split into two persons tonight, I will doubly exist tomorrow; if
river R has two branches where is crosses the border to country C,
R doubly exists at the border to C; if at some world, two people
resemble me to exactly the same degree in all extrinsic and intrinsic
respects, I doubly exist at that world.
I need to tidy up this part of my belief space. Once I complained that literal trans-world identity (as opposed to trans-world identity based on similarity) is implausible because it entails that there can be no vagueness about a thing's essential properties (for determinate properties): either the thing has the property at all worlds or not. On the other hand, I also believe that there is no big difference between individuating things as worldbound and individuating them as trans-world fusions of worldbound counterparts. Unfortunately, these two views can't both be correct.
A few comments on Counterparts and Actuality by Michael Fara and Timothy Williamson (via Brian, of course).
Fara and Williamson argue that if Quantified Modal Logic is enriched by an "actually" operator, then given some further assumptions there is no correct translation scheme from QML to Counterpart Theory. Here, a correct translation scheme is one that translates theorems of QML into theorems of CT and non-theorems of QML into non-theorems of CT. (theorems of which QML? -- good question; read on.).
I just realized that I have inconsistent attitudes towards ontologically dependent entities, that is, entities x such that for some contingently existing entity y, i) necessarily, if x exists so does y, and ii) x and y are not parts or subsets or elements of each other. On the one hand, I don't believe that there are many such entities, except perhaps holes and borders. On the other hand, I also don't believe in general restrictions on the counterpart relation, or, perhaps equivalently, in restrictions on cross-world fusions of individuals. It follows that for any old property any world-bound thing has at our world, there is a thing which has this property essentially. For instance, there is somebody who leads exactly your life but who, unlike you, is essentially such that the cup in front of me is now empty. This somebody is a dependent entity: it can only exist if my cup does.
The other hand seems so obvious to me that I fear I must give up the one hand: there are lots of dependent entities. I can still say that they are not ordinary things, and that it is very hard or even impossible to refer to most of them (individually, of course -- I just managed to refer to them collectively). But still they exist. Hm.
Geoff at Too Much Text points out that the implausible hyper-essentialism implied by Kripke's account of rigidity can be avoided by adopting radical anti-essentialism, the view that there are no non-trivial (qualitative) essential properties at all. On this view, even though there is a precise boundary between a thing's essential and non-essential properties, the boundary is not very mysterious because it classifies virtually all properties as non-essential.
Yesterday, I said that it doesn't really matter whether we regard identity simpliciter as identity-at-our world -- individuationg referents extensionally -- or as identity-at-every-world -- individuating referents intensionally. Suppose we want to do the latter, so that the referent of "the amazon" determines a function from worlds to world-bound individuals, that is, an intension. So on the present account, we identify the amazon with something that completely determines the intension of "the amazon". The intension? What if, as two-dimensionalists argue, "the amazon" has two intensions? Which one is the one we want extensions to determine?
So there are several ways to make sense of restricted identities. Which is the right one? Maybe there is no fact of the matter.
The difference depends on which contexts are regarded as referentially transparent and which as opaque. And that in turn depends on how the referents are individuated. For instance, (de re) ascriptions of modal properties will be transparent iff the referents of singular terms are such that they determine the truth value of all such ascriptions, perhaps because they (the referents) are fusions of world-bound individuals with their counterparts, or because they are Carnapian individual concepts, or because they simply contain some hidden tag that determinately settles all their modal properties. At any rate, for de re modal contexts to be referentially transparent, the referents have to provide us with a function from worlds to world-bound individuals, as that's what we need to determine the the truth value of those ascriptions. Alternatively, if we hold that those contexts are referentially opaque, we decide that the referents do not contain that information. Instead, we put the information into another aspect of meaning, which we call the terms' intension. Is the difference really more than just a relabeling of semantic vocabulary?
Things are counterparts iff they are sufficiently similar to each other.
They needn't be similar intrinsically: For example, in "Individuation by
Acquaintance and by Stipulation" (§2), Lewis allows for counterparts that
are similar in standing in a particular relation of acquaintance to some
person. In fact, they needn't be similar at all: In On the Plurality of
Worlds (§4.4), Lewis accepts that, speaking unrestrictedly, everything
is an individual possibility for anything. However, in "Things qua
Truthmakers" (§5), he denies that things could be counterparts by living in
a world in which there are no unicorns. I wonder why. Lewis says that
such a respect of similarity would be too extrinsic and strike us as too
unimportant. But other eligible respects are extrinsic too, and what
strikes us as important certainly depends on the relevant context. I can
imagine theists who believe that there is a big difference between
living in a world where there is a God and living a duplicate life in a
Godless world. So in some special contexts, those of our counterparts who
live in Godless worlds might be excluded as being too different. Conversely, an atheist might exclude counterparts that live in worlds with Gods a being too different.
I'm currently writing a chapter on modal realism.
I don't like this topic because it always confuses me. Here is one such
confusion.
In some world w, pretty much resembling our world, there are two
individuals A and B. Let 'A-in-w' be an extremely rich descriptions of A
that implies every qualitative truth about w, similarly for 'B-in-w'
and B. Now the following two sentences might both be true:
1) If I were A-in-w, I would do X.
2) If I were B-in-w, I wouldn't do X.
Don't miss Brian
Weatherson's very insightful answer
to my posting on
rigidity (from which I've just stripped some irrelevant formalities). I
happily agree with everything he says, so I'll just add a footnote here.
Many advantages of the counterpart theory derive from its denial of the
equivalence between 'a=b', 'possibly a=b', and 'necessarily a=b'. For
example, this allows for a statue to be identical to a lump of gold even
though it might not have been. Since, as Weatherson argues, the rejected equivalence is
built into the customary ('strong') concept of rigidity, that concept must be weakened
to be useful for counterpart-theorists.
I wonder how rigidity can be characterized without begging the question
against a lot of good semantic theories.
Usually, a rigid expression is defined as an expression which has the same extension in all possible worlds (that is, as an expression with a constant intension, or C-intension).This characterization presupposes literal
trans-world-identity between extensions, which is bad, since it carries a
commitment to precise essences of individuals on the one hand and
(presumably abundant) universals as extensions of predicates on the other,
thereby ruling out counterpart theories and accounts on which tropes
or classes are the extensions of predicates.