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\begin{document}
\title{Language use in a branching Universe}
\author{David Wallace \thanks{Balliol College, Oxford OX1 3BJ, U.\,K. Email: \texttt{david.wallace@balliol.ox.ac.uk}}}
\date{December 2005}
\maketitle

\begin{abstract}
I investigate the consequences for semantics, and in particular for the
semantics of tense, if time is assumed to have a branching structure not
out of metaphysical necessity (to solve some philosophical problem) but
just as a contingent physical fact, as is suggested by a 
currently-popular approach to the interpretation of quantum mechanics.
\end{abstract}

\section{Introduction}\label{introduction}

Aristotle thought that the future was open, in the sense that talk of the future need be neither
true nor false. Prior
(\citeyearNP{priormodality},\citeyearNP{priorpff}) modelled this openness by
giving time a branching structure and constructing a branching-time
semantics to match. \citeN{thomason1970}, McCall (\citeyearNP{mccall84,mccall}), 
Belnap (\citeyearNP{belnap,belnap02}),
and others concur. In each case their motivation is that certain aspects
of our everyday world, and our discourse about it, can be modelled only by 
branching time and branching-time semantics: examples
include causality, the objectivity of possibility and probability, free
will, and the intuition to say of things not yet determined that they
are neither true nor false.

But it is, I think, at least as widely believed that such a radical metaphysical strategy
is unnecessary, and that our everyday interactions with time can be understood perfectly well
on the conventional picture of time as non-branching. If so (and I am myself inclined to think that it \emph{is} so), 
philosophy gives us no reason to take seriously the proposal that time
is branching, or to adopt a semantics of tense which explicitly
accommodates branching.

However, it is a \emph{non sequitur} to suppose that because we are not
\emph{compelled} to accept a non-branching structure for time, it is not
philosophically \emph{possible} to accept one. This paper is concerned
with what semantics we should adopt for tensed discourse in a universe
in which branching is \emph{in fact} ubiquitous: its conclusion is that
if we live in a branching universe, we are required to adopt a branching
semantics.

Why take such a  supposition seriously in the first place? One
important reason is that `new developments in physics' do indeed give us
(some) grounds to suspect that time may be branching. In particular, the
Everett interpretation of quantum mechanics (often called the 
`Many-Worlds theory') makes precisely that claim: it supposes that
whenever we make a measurement of a quantum system\footnote{And at many
other times when measurement is \emph{not} occurring, too --- one of the
appealing features of the Everett interpretation is that `measurement'
plays no essential role in its formulation.} the Universe splits into
many copies, one\footnote{Rather more than one, actually; but counting
branches is a subtle matter. See \citeN{decprob} for more
discussion of this point.} for each possible outcome of the experiment.

It is not my purpose here to argue for the \emph{truth} of the Everett
interpretation (for such arguments, see Wallace (\citeyearNP{wallaceworlds,wallacestructure,decprob}) ). However, true
or not, it is now taken seriously by remarkably many physicists. As
such, philosophers have some obligation to ask, if not \emph{is it
true}, then at least \emph{could it be true}? That is, is it a possible
physical theory, albeit one which could turn out false? Or is it
metaphysically incoherent, something to be rejected out of hand without further ado?

One of the more powerful arguments for metaphysical
incoherence\footnote{Made with some force by \citeN{albertloewer},
amongst others.} is: how can it make sense to say, as we habitually do,
that we are \emph{uncertain} of the result of a quantum measurement when
in fact the Everett interpretation tells us \emph{with certainty} that the result is a
division of the universe into many copies? That is, how can we reconcile
the apparently indeterministic nature of quantum measurements with the
apparently deterministic, in-principle-perfectly-knowable nature of the
Everett interpretation?

To expand on the point: suppose an experimenter, asked what he believes
the result of the experiment will be, says `Either outcome A or outcome
B will occur, but not both, and I don't know which.' This statement (a
perfectly normal response for an experimenter who accepts quantum
mechanics) can be construed as the assertion of 
\begin{itemize}
\item `A will occur or B will occur', and
\item `Not both A and B will occur'
\end{itemize}
together with a refusal to assert either
\begin{itemize}
\item `A will occur' or
\item `B will occur'.
\end{itemize}
However, if quantum mechanics is supposed to mean that the world
branches in two, with A occurring in one branch and B in the other, it's
unclear how this can be a reasonable set of assertions to make. Do we
take A to occur if it occurs \emph{in some branch} (that is, somewhere
in the branching Universe)? Then it cannot be true that not both A and B
will occur, nor can it make sense to plead ignorance as to which will
occur (they both will). Or should we take A to occur if it occurs
period, not just occurs-in-some-branch? If so, neither A nor B occurs;
rather, some alien branching event occurs. Either way, the theory seems
to fit very badly with how experimental physicists actually behave.

Responses to this objection have generally taken one of two forms:
either bite the bullet and accept that we are just wrong when we say
that a quantum-mechanical experiment has an unknown outcome, but try to
argue that our behaviour should remain unchanged nonetheless
\cite{greaves}; or try to make sense of the apparent paradox by appeal
to considerations from the philosophy of personal identity (Saunders
\citeyearNP{saundersprobability}). I wish to suggest
an alternative possibility: that the `paradox' is a consequence of
mistakes in the philosophy of language. In particular, it is due
to the mistaken application of a semantic theory designed for a 
non-branching universe to the discourse of the inhabitants of a
branching universe.

In section \ref{review} I review the semantics of linear and branching time;
in section \ref{splitting} I apply each to a branching universe, and
show that linear, but not branching, semantics imply that if we live in
a branching universe then nearly all our beliefs are trivial, false or
meaningless. Section \ref{charity} makes the central claim of this paper: that the linguistic principle of
charity entails that branching semantics are correct for a branching
universe. Section \ref{uncertainty} addresses the frequently made objection that
talk of `uncertainty', which applies to a branching universe if
branching-time semantics are appropriate, is incoherent in situations
where all the objective facts are known. Section \ref{details}--\ref{possibility} 
develop the technical details of the proposal, and section \ref{conclusion} is the conclusion.


\section{Branching-time semantics: a review}\label{review}

Suppose that time is linear: that is, suppose that it can be represented
as some total ordering $<$ on some set \mc{T} of `times', to be read as ``earlier than''.\footnote{Recall that a total ordering of \mc{T} is an
antisymmetric, irreflexive transitive relation on \mc{T} such that for any $x,\,y \in
\mc{T}$, either $x=y$, $x<y$, or $x>y$.} Utterances concerned with time can then be divided
in to two classes: \emph{tenseless utterances}, such as ``$p$ occurs
(tenselessly) before $q$'' or ``$p$ (tenselessly) occurs at some time $t$'', 
the truth values of which are unaffected by when they are uttered; and
\emph{tensed utterances}, such as ``$p$ has happened'' or ``$q$ is (now)
happening'', which are true at one time, false at another. 
Put in terms
of sentences rather than utterances: tenseless sentences (ignoring ambiguity) are true or
false \emph{simpliciter}, tensed sentences have time-dependent truth
values.

There is a well-known relationship between the two sorts of utterance:
namely, a tensed utterance made at a particular time $t\in \mc{T}$ is
true iff a certain related tenseless utterance is true
\emph{simpliciter}. For instance, the tensed utterance ``$p$ will
happen'', uttered at time $t$, is true iff ``$p$ happens (tenselessly)
at some time $s>t$'' is true; ``$p$ is (now) happening'', uttered at $t$, is true
iff `$p$ (tenselessly) happens at $t$'' is true. In terms of sentences: each ordered pair
of a tensed sentence and a time determines a tenseless sentence; the
tensed sentence is true at that time iff the tenseless sentence is true
\emph{simpliciter}. 

Those, such as Prior \citeyear{priormodality,priorpff}, who take tense
as metaphysically fundamental will, at best, regard this relation between tensed
and tenseless sentences as in some sense definitional of the latter;
they might wish to go further and reject the framework above altogether
(or at least regard it as giving merely the abstract model theory for the logic of
tensed utterances). I regard this as untenable, for unoriginal
reasons\footnote{See \citeN[chapter 1]{sider} and \citeN{mellortimeII}
for recent criticisms of the notion of tense as fundamental.}, and for
the rest of this paper I follow Mellor (\citeyearNP{mellortime,mellortimeII}) \emph{et
al} in regarding the content of tensed utterances as exhausted by an explication of their 
time-dependent relation to tenseless truthmakers.

What if time is \emph{not} linear, but branching? Formally, this 
can be modelled by relaxing the
requirement that the relation $<$ on the set \mc{T} is a \emph{total}
ordering. We require instead only that it be a partial 
ordering\footnote{\iec, a transitive, antisymmetric relation.} 
which in addition satisfies:
\begin{enumerate}
\item \emph{Connectedness}: For any $x,\,y$, either $x<y$, or
$y<x$, or $x=y$, or for some $z$, $z<x$ and $z<y$.
\item \emph{No recombination}: If $z<x$ and $y<x$, then either $z<y$, or
$y<z$, or $y=z$.
\end{enumerate}

Branching structures for time were proposed by \citeN{priormodality}
with the intent of formalising the intuition that (some) future events
are not now either true or false. On the branching-time semantics that
Prior advocates, a future-tensed sentence (say, `There will be a sea battle', to borrow
Aristotle's example) uttered at $t$ is true iff there is a sea-battle in all
branches futurewards of $t$; false iff there is a sea-battle in no such
branch; neither true nor false otherwise.

More formally (and following \citeN{thomason1970}), define a
\emph{history} as a maximal totally ordered subset of \mc{T}. If we
restrict our attention just to a particular history, it looks exactly
like the `linear time' we described above. We can easily say what it is
for a tensed sentence uttered at $t$ to be true in some history $h$ (where $h$
contains $t$): the sentence is true at $t$ in $h$ if the associated
tenseless sentence is true in $h$. So in a branching-time picture, the
truth values of tensed sentences must be evaluated not just with
reference to a time but with additional reference to a history; the
truth values of tenseless sentences are not absolute, but require
reference to a history. (So `There (tenselessly) is a sea battle in 480
BC' is true in those histories in which there is a sea battle at that
date; false otherwise.)

Actual utterances are made at particular times, but in a branching-time scenario 
a time does not pick out a unique history. This gives Prior's 
non-bivalent semantics: a tenseless sentence $S$ uttered at $t$ is true iff
it is true in all histories $h$ with $t \in h$ and false iff it is true in no such histories. (Formally, this is a
supervaluational semantics; see \citeN{thomason1970} for technical
details). We will say, neutrally, that $S$ is \emph{not knowably true or
false} at $t$ if $S$ is true in some but not all histories containing
$t$.

Part of the motivation for branching time was to provide an objective
framework for future possibilities. In the linear-time framework then
possibility is an epistemic notion: given any two contradictory
statements, both may be \emph{possible} but one is true and one is
false, and both are possible only because we do not know which is which. 
If time is branching, however, it is natural to take ``it is
possible that $p$'' as true at $t$ whenever $p$ is true in some history
containing $t$. (This is actually entailed if we assume that (i) every
$p$ is either possible or impossible, and (ii) if something is
impossible it is also false). Formal semantics for `it is possible that'
(\eg \citeN{priorpff}) often take this condition as necessary \emph{and}
sufficient, but I shall not do so: it will be convenient to remain
neutral as to whether something can be possible but false, as is the
case in non-branching time. (I return to this issue in section
\ref{possibility}.)

For clarity, I should note two features of this presentation of
branching time:
\begin{enumerate}
\item Braching-time pictures are often (\egc, \citeN{mccall})
combined with essentially tensed views of time, so that as time passes
various `possible' futures succeed or fail to be actualised. This is
\emph{not} a feature of the branching time ontology I discuss in this paper: as with my presentation 
of linear time, the branching structure
is taken to have its structure tenselessly.
\item Even if one did not want to accept branching in the actual
\emph{structure} of time, it might still be possible to adopt a branching
\emph{semantics}. (One might, for instance, say that a future-tense
sentence was true at $t$ iff true in all physically possible worlds identical with
the actual world up to $t$, as is advocated by \citeN[pp.\,206--9]{lewisplurality}) 
By contrast, the picture of branching which
I wish to consider is explicitly metaphysical: the real structure of
time is branching.
\end{enumerate}

\section{Branching time and splitting worlds}\label{splitting}

Suppose, as does the Everett interpretation, that the universe in which 
we live is constantly splitting into
copies, so that I have not one unique successor but many.

A universe like this has a structure formally identical to that of a
branching time. It can be modelled via a set \mc{S} of
`instantaneous quasi-classical states', corresponding to the sorts of
instantaneous states that `the classical world' is generally taken to
have, together with a  relation $<$ on \mc{S} satisfying the same
axioms as we previously required for branching time and interpreted as causal dependence or some
dynamically defined equivalent. Elements of \mc{S}
describe such things as dead or alive cats, needles on detectors
pointing at various determinate values, and so forth; if $x\in \mc{S}$
is a state in which a cat is in a box, $y,z$ are similar states in which
the cat is respectively alive and dead, and $x<y$, $x<z$, $\sim y<z$
and $\sim z<y$, then we have a model of the ``Schr\"{o}dinger cat''
paradox.\footnote{In quantum mechanics, \mc{S} consists of time-indexed
projectors in some decoherence-defined basis; $\langle
\op{P}_1,t_1\rangle < \langle\op{P}_2, t_2\rangle$ iff $t_1<t_2$ and
$\tr (\op{P}_2\exp(-i/\hbar t \op{H})\op{P}_1\exp(-i/\hbar t \op{H}))\neq
0$. The splitting process is not part of the fundamental dynamics of the theory, but is emergent
from them; see \cite{wallacestructure} for details.}

Are such universes \emph{actually} examples of branching times? There
are two possibilities:
\begin{enumerate}
\item[Yes:] elements of \mc{S} should be identified with times; at each
time, the world has a roughly classical description.
\item[No:] despite the formal similarity, the branching structure of
worlds should not be regarded as a branching-time structure. Instead,
we should regard it as having a single, linear time, and having
instantaneous states which, far from being classical, consist of vast
numbers of causally isolated classical sub-worlds.
\end{enumerate}

In a sense, it is a semantic rather than a metaphysical question as to
which of these is correct. The actual structure of the world is not in
question (it is given by the set \mc{S} and its partial order). The practical
consequences of declaring that this system of branching worlds is a
branching-time or linear-time system is, rather, that we should use branching-time 
or linear-time semantics respectively to analyse language use within it.

To elaborate on these remarks: it is of course possible simply to talk
\emph{about} the branching structure described by the ordering on
\mc{S}, by uttering semantically unproblematic sentences (in which one might
say `some points with property $P$ are futurewards of some points with
property $Q$' for instance\footnote{In the context of the Everett
interpretation, statements like `the state of the Universe is
$\ket{\psi}$' are also of this form.} Let us call such utterances `branch-talk utterances' (or utterances
of `branch-talk sentences'). 

Furthermore, branch-talk sentences can unproblematically be uttered from within the universes that we are
considering, and in being made may incorporate some unproblematic
context dependence, For instance, someone aware of the branching might
say: `in the future, I will branch into two copies'. In this context,
the referent of `I' will be the person making the utterance, and the
statement is true if there are two distinct branches through the point
at which the utterance is made, each containing a copy of the person
making the utterance.\footnote{Certain issues of personal identity, passed over
lightly here, will be dealt with in more detail in section
\ref{details}.)}

What is in question, however, is not the semantics of branch-talk utterances, 
but the semantics of the utterances actually made
inside this world: everyday utterances like `it will rain tomorrow', for
instance. It is here that the two possibilities described above lead to
very different answers.

If branching-time
semantics are correct, the semantic theory for ordinary utterances is very familiar: each history
(that is, each maximal totally ordered subset of \mc{S}) is classical and unbranching, and 
provides a perfectly conventional (albeit history-dependent) semantics
for tenseless sentences, along the lines spelled out in section \ref{review}. 
A tenseless sentence like ``Athens [tenselessly] hosts the Olympics in 2500 AD'' uttered at $x\in\mc{S}$ is
true iff it is true in all histories containing $x$; 
`Possibly Athens hosts the Olympics in 2500 AD' is true at $x$ whenever
``Athens hosts the Olympics in 2500 AD' is true in some history
containing $x$ (and perhaps at some other points as well, since we are leaving it open the extent to which there
is a purely epistemic component to possibility even in a branching universe). In turn,
tensed sentences like `it will rain in Oxford tomorrow', uttered at a particular time, have the same truth value
as their associated tenseless sentence (`it [tenselessly] rains in Oxford on the
8th of December 2005', were I to utter it now.)

Things are more complicated if we apply linear-time semantics to
branching worlds. Recall that in this case, tenseless sentences are true
or false \emph{simpliciter}, not true or false relative to histories
(equivalently: there is only one \emph{history}, even though there are
many chains of classical sub-worlds). Tenseless sentences, in fact, should be
`God's-eye' descriptions of the entire branching structure: 
the ideal tenseless sentence would be actually be some `branch-structure sentence' like ``Athens hosts the Olympics
in some but not all 2500AD-sub-worlds futurewards of 2005-AD subworld $k$.''.

Ordinary people do not utter sentences like that, of course (or even
tensed variants of them), so what of `ordinary' tenseless sentences like
`Athens hosts the Olympics in 2500AD'? As we have noted in the case of branch-structure sentences, 
the content of such sentences
will have some contextual dependence even on linear-time semantics: `Athens'',
uttered at some subworld $k$, clearly refers to the Athens in that
subworld or its causally related future subworlds (much as terms like
``Me'', ``George'' or ``the Queen'' have denotations which depend on the context of
utterance). Nonetheless, if the basic assumption of linear-time
semantics is to hold then these sorts of utterances should have the same
truth conditions as some context-independent branch-structure sentence.

The trouble is that there are no very inspiring candidates. Some might include:
\begin{itemize}
\item ``Athens hosts the Olympics
in at least some 2500AD-subworlds futurewards of 2005-AD subworld $k$'' (where $k$ is the subworld
at which ``Athens hosts the Olympics in 2500AD'' was uttered), which is
almost certainly true at most plausible $k$.
\item ``Athens hosts the Olympics in all 2500AD-subworlds futurewards
of 2005-AD subworld $k$'',
which will no doubt be false at most plausible $k$.
\item ``The mereological sum of all copies of Athens in 2500AD-subworlds
futurewards of 2005 AD subworld $k$ hosts the Olympics'', which is false
since mereological sums scattered across causally isolated sub-worlds
cannot be said to host sporting events.
\end{itemize}
But none of these seem naturally equivalent in content to the original
utterance, though. None seem to have a great deal of
contingency; on the first and second proposal ``Athens hosts the
Olympics'' and ``Athens doesn't host the Olympics'' do not come out as
contradictory; on the third proposal we seem to be referring to an
object which has little to do with our ordinary understanding of
`Athens'.\footnote{cf Lewis's arguments \cite[pp.\,209--219]{lewisplurality} 
against the relevance of trans-world fusions to issues of trans-world identity.} 
Perhaps the best strategy to adopt is just to declare
``Athens hosts the Olympics in 2500 AD'' as meaningless according to the
linear-time semantics, and restrict our usage to more appropriate
tenseless sentences like `Athens hosts the Olympics in some but not all 2500 AD subworlds futurewards
of this subworld''.

On the linear semantics, it should be apparent, the hypothesis that our
world is branching entails that most of our ordinary beliefs about the future
are false, vacuously true, or just meaningless. It also entails that our ordinary description of quantum
theory's predictions is highly inaccurate: our ordinary reading of the theory is that the
correct attitude to (say) a quantum measurement  is uncertainty
as to which measurement result obtains, but under the Everett
interpretation it seems that all we can say is that \emph{all} `possible'
measurement results actually obtain, although only one obtains in each
branch.

As was noted in section \ref{introduction}, these radical conclusions have 
been used by skeptics about the
interpretation (such as \citeN{albertloewer} and \citeN{barrett}) to
argue that it cannot be taken as explanatory of the empirical data. They
have also been accepted by some advocates of
the Everett interpretation (such as \citeN{tappendenidentity}, and possibly
\citeN{greaves}), who make the point that such a radical change in our
ontology ought to have radical consequences for how we think about the
world in general and quantum experiments in particular.

However, it is important to note that such `radical consequences'
\emph{only} follow if the appropriate semantics for a branching world is
linear-time semantics. If instead we assume that branching-time
semantics are appropriate, virtually all of our existing beliefs remain
intact (the main exceptions are technical statements \emph{about} the topology of
time or the interpretation of quantum mechanics). Furthermore, the
semantics straightforwardly recovers the ordinary description of
measurement: if different measurement results occur on different
branches, then for each possible result $r$, ``$r$ will obtain'' is not
knowably true or false, and ``$r$ will possibly obtain'' is true.

So: the question of whether the Everett interpretation entails massive
error on the part of experimentalists and ordinary people comes down to
the question of what semantics are appropriate for a branching world.

It is not at all obvious that the burden of proof here lies with the
advocate of branching-time semantics. While there has been an extensive
debate on the topology of time within mainstream philosophy, it has been
almost exclusively concerned with whether branching time is
\emph{necessary} to recover some aspect of the everyday world (freedom
of will, objectivity of possibility, causation, etc --- see
\citeN{priorpff} and \citeN{belnap02} for samples of such arguments). I
am not aware of any attempt to argue for the \emph{a priori} necessity
of linear-time semantics. 

It is, however, possible to give a more constructive argument in favour
of branching-time semantics for a branching world. The next section will
present it.\footnote{Alternative arguments, from rather different
perspectives, have been proposed by \citeN{saundersprobability} and
\citeN{vaidman}; see \citeN{wallaceepist} for a critical evaluation.}

\section{The charity argument for branching-time
semantics}\label{charity}

Imagine a certain society, living in a Universe apparently not so different from
ours, in which philosophers and physicists have reached a consensus
about the correct metaphysics and semantics of time. Specifically: the
physicists are confident that their universe is a linear, non-branching
reality (whose instantaneous states are rather like `classical states'
of the world in our own universe); the philosophers are confident that
the universe is tenseless at a fundamental level and that the correct
semantics for tensed talk is linear-time semantics as described in
section \ref{review}. In their analysis, `possibility' is an entirely
epistemic notion: there is a unique actual future but other futures are
`possible' in a sense that is ultimately ``merely'' a matter of the ignorance of individuals about the actual future.

This notion of possibility can be expanded and quantified by the use of
probabilities: rational natives of the society, the philosophers argue, 
may be idealised as assigning a probability between 0 and 1 to the truth of each
tenseless sentence.\footnote{I ignore complications raised by the `problem of the
essential indexical', and although I follow \citeN{davidsonstructurecontenttruth} in
ascribing probabilities to the truth of sentences rather than (\egc) to propositions, this is purely for convenience
and does not affect the argument.} (This in turn may be derived from more primitive,
qualitative assumptions about which sentences they believe are more
likely to be true, using the methods of decision theory; see, \egc, \citeN{jeffrey}
and \citeN{savage} for two influential proposals as to how this is to be
cashed out; see \cite{kaplan,joyce} for
further details.).

The catch is that the natives' physicists are completely wrong. Their
universe is not, as they believed, described by the evolution of matter in an
approximately classical way: in reality it is constantly --- trillions
of times per second --- branching into quasi-classical sub-worlds, so
that their planet and everything on it is constantly being copied into
duplicates almost identical to one another.

Since none of the natives know this, it should be unsurprising that
certain aspects of their world-view are mistaken. The question is: which
aspects? There are two possibilities.

According to the \emph{Elite View}, despite the errors of the physicists in
describing the physical structure of the world, the philosophers'
analysis of semantics and rationality remains correct. That is:
tenseless sentences have history-independent truth conditions, modulo
some referential context-dependence of terms like `Athens'.
As we saw in the previous section, there are no candidate `branch-talk' sentences to
express such truth-conditions without making most of our `ordinary'  future-directed
sentences (like `it will rain tomorrow') trivial or
meaningless.
Since rationality requires agents to act in accordance with
probabilities over tenseless sentences (again ignoring referential
indexicality), it follows that `probability' remains an essentially
epistemic notion, measuring an agent's degree of belief in facts which
from a God's-eye view are fixed.

To adopt the Elite View is to continue to apply linear-time semantics to
this universe despite the physicists' radical discoveries about its structure.
This move reduces to falsehood,
triviality or nonsense almost every ordinary opinion held by natives of
this universe. 

The alternative \emph{Charitable View} accepts that the philosophers are wrong in some important
respects in their description of rationality and their linear-time semantics. Tenseless
sentences are true or false not \emph{simpliciter} but only relative to
a history; actual utterances are therefore to be classified as true, false or not-knowably-true-or-false
as dictated by branching-time semantics.

As such, rational agents should define probabilities to the truth of `tenseless
sentences' understood in this way. The price of this move is of course that it abandons the idea 
that (future-directed)
beliefs and desires should be interpreted in terms of ignorance about, and 
preferences between, possible tenseless states of the entire Universe, and 
that it abandons any attempt to provide determinate truth-values to future-tense 
sentences. On the other hand, by doing so, it saves the future-directed beliefs 
of agents from widespread falsehood. For instance, suppose someone says `in 2500 AD, 
either Athens will be hosting the Olympics, or some other city, but not both.' This can 
be analysed as having the tenseless (but history-dependent) truthmaker
\begin{quote} 
(Athens is hosting the Olympics in 2500 AD $\vee$ A city other than
Athens is hosting the Olympics in 2500 AD ) $\wedge \neg$ (Athens is hosting the Olympics in 2500 AD $\wedge$ A city other than
Athens is hosting the Olympics in 2500 AD )
\end{quote}
which is true at $x$ (according to the Charitable View) iff 
for every history containing $x$, in 2500 AD either the Olympics is being hosted by Athens or 
by some other city but simultaneously by Athens and another city --- which let us suppose is in fact the case. 
So (unlike under the Elite View) this banality does indeed come out true. 
The sentence `In 2500 AD Athens will host the Olympics', on the other hand, comes out as not knowably true or false, 
since in some histories through $x$ it is true and in others false. But this is perfectly in accord with everyday 
impressions of the openness of the future and with our native's 
perfectly sensible decision to give (as it happens) probability 0.4 to this outcome, 
rather than 1 (determinately true) or 0 (determinately false).



So, what is the correct move: to abandon the philosophers' original (linear-time) 
analysis of semantics, or to abandon the 
assumption that ordinary natives by and large have reasonable beliefs about the future? 
In one sense there is no definitive answer to give. For the natives' conceptual scheme, as 
a whole, fails to fit perfectly with reality, and so needs to be modified; yet if we 
accept (with Quine, Davidson, Lewis, et al) that the criterion of validity for a 
conceptual scheme are ultimately somewhat holistic, then there may be no absolute 
fact of the matter as to which part of a holistically wrong scheme is to be abandoned.

Nonetheless, as a practical matter there are enormously strong reasons why the 
`correct' move is to abandon the Elite View. For one thing, in doing so 
we have to accept as wrong only those parts of the natives' understanding of 
belief and desire talk which come from a fairly theoretical analysis --- 
furthermore, one motivated (\emph{ex hypothesi}) by a wildly wrong picture of what 
the universe as a whole is like. By contrast, in clinging to the Elite View we preserve the truth-value of the writings of native philosophers, 
but at the cost of abandoning the truth of most ordinary conversation about the future.

(As an analogy, consider a variant on Putnam's Twin-Earth example 
\cite{putnammeaning}. There is a general consensus that, if transported 
to Twin-Earth and given a glass of $XYZ$, I'm incorrect to say `that's 
water', and correct to say `water is $H_2 O$'. We interpret my language 
so that `water' refers to $H_2 O$ --- even at the cost of making most of 
my post-arrival statements about water come out false --- because this is 
the most explanatory interpretation of my linguistic community's overall language use.

But now suppose that I never left Earth at all, but that in fact the stuff 
in the rivers and seas has \emph{always} been $XYZ$. Only the tireless 
efforts of an International Conspiracy of Chemists has kept the public --- 
even quite educated members of the public, such as those with chemistry degrees --- 
duped into thinking that this stuff is $H_2 O$. When investigative 
journalists expose the Conspiracy, how should we react? Surely not 
by clinging to our technical, theoretical belief that water is $H_2 O$ --- 
thus preserving a few technical truths at the cost of making almost all the 
population's beliefs about water turn out false --- but rather by acknowledging 
that water was $XYZ$ all along and accepting that the technical connection 
between water and $H_2 O$ is simply false.)

Furthermore (though it is a related point), if we ask what justifies our semantics 
of the natives' language, it is hard to see how we could motivate 
the Elite View given the actual (i.e.\,branching) nature of 
the natives' universe.
For it has been powerfully argued (by \citeN{quinewordobject} and 
\citeN{lewislanguage}, for instance) 
that some sort of principle of charity must apply to translation 
and interpretation of a genuinely foreign language. 
That is, if we have a proposed  interpretation of such a language 
according to which nearly all uses of a large part 
of that language turn out not to be true (or possibly not even to 
be meaningful) then that seems simply to count against 
the accuracy of that proposed interpretation. Judged by this standard, 
the Elite View should be abandoned as being overwhelmingly less 
charitable than the Charitable View.

For these reasons, I claim, the correct semantics for the natives' 
language, and the correct analysis of their beliefs 
and desires, is that given by the Charitable View: that is, by applying branching-time semantics. 
The native 
philosophers, whose proposed semantics is motivated by 
an entirely reasonable but entirely false view of their world, are 
simply wrong in their understanding of their own language's semantics.
And of course, since \emph{their world} could well be our own world
except for some trivial simplifications, if our own world has the
branching structure that quantum mechanics suggest then the correct semantics of
our own languages are really branching.


\section{Probability and uncertainty}\label{uncertainty}

An objection remains to be responded to: I have argued that charity
considerations require us to adopt branching-time semantics and thus to
take an attitude of uncertainty towards the results of experiments which
have different results in different branches. But how can it even
\emph{make sense} to `take an attitude of uncertainty' towards the
future when we know all the relevant facts? 

This objection is frequently brought up in informal discussions of the
Everett interpretation; it has been made explicitly
by \citeN{greaves}, who writes:

\begin{quote}\label{greavesquote}
I can feel uncertain over $P$ only if I think that there is a fact of the matter
regarding $P$ of which I am ignorant. 
\end{quote}

This objection can be responded to on its own terms, by noting that it
is ambiguous between a truism of ordinary language and a technical
claim. As the former, it is indeed plausible to say that if a supposed concept of
`uncertainty' says that we can be uncertain about something
without there being something to be uncertain about, then it does not
deserve the name `uncertainty'. However, there \emph{is} something about
which I am uncertain if I am uncertain about whether Athens will host the
Olympics: trivially, that `something' is \emph{whether Athens will host
the Olympics}. Or if preferred, it is about whether ``Athens will host
the Olympics'' will be true.\footnote{`Will be true', not `is true', of course;
to find out whether it \emph{will be true}, wait and see. See
\citeN[pp.\,122-3]{priorpff} and \citeN[pp.\,15--34]{ryledilemma} for more on the mild linguistic tangles involved in
this wait-and-see notion of truth-conditions.}

The less trivial reading of Greaves' comment is that it asserts that
we cannot be uncertain about something unless there is a 
third-party fact --- a fact expressible via branch-talk sentences --- about which to be uncertain. But now it is
far from clear what motivates Greaves to accept this
technical claim. This doubt is strengthened when we note that
most philosophers do not accept the claim even in the absence of
branching: the `problem of the essential indexical'
(\citeN{perryindexical}, \citeN{lewisindexical}) seems amply to
demonstrate that there are other sorts of ignorance which are not
ignorance of 
third-party-accessible facts but are unavoidably indexical.

It would be to miss the point to respond to this objection by retreating
to the claim that we cannot be uncertain about something unless there is
\emph{either} a third-party-accessible fact about which to be uncertain,
or an indexical aspect to our ignorance. The point is that we didn't
realise that some ignorance was indexical until it occurred to authors such as Perry to
consider some such situation, to describe it in everyday language, and to note
that we intuitively classify such situations as ignorance even though no
relevant third-party-accessible facts are unknown. Essentially the same
has occurred in the case of Everettian branching: describe it in
everyday language (using the charity argument to tell us that `everyday
language' has a branching-time semantics) and it is intuitively clear that it is another
situation of ignorance.

(I should note in passing that on the `Lewisian semantics' to be described in the next
section, indexical ignorance is actually all that we need: the `essential
indexical' is \emph{my history}. This strategy will not work for the other semantic proposals made in that
section, though.)

There is also a more positive defence of the use of probability and
uncertainty talk in the face of branching. After all, even in the absence of branching
no \emph{reductive} analysis is given of what it means to assign 
probability 0.4 to the truth of some such proposition. Rather, the meaning of the 
statement is given implicitly, as part of a decision-theoretic analysis which treats 
`more likely than' and `preferable to' as theoretical terms whose engagement with empirical 
fact is via their role in the best explanation of my actual behavioural dispositions. 
The same applies to the case of branching-time semantics --- it's just that the set of 
propositions has a slightly different conceptual sense. (For more on
this issue see section 5 of \citeN{wallaceepist}).


\section{`Not-knowably-true-or-false': semantic details}\label{details}

So far, I have presented the formal structure of branching semantics and
provided an argument from the principle of charity that they are
applicable in a branching universe. However, I have so far been rather
unspecific as to how this semantics is to be understood --- and in
particular as to what meaning is to be ascribed to the 
`not-knowably-true-or-false' label which I applied to sentences which
are true in some but not all histories. In this section I will attempt
to remedy this, by providing three different strategies by which 
`not-knowably-true-or-false' may be understood.

The most straightforward such strategy might be called \emph{minimalist
semantics}, and it takes `not-knowably-true-or-false' at face value, as
a third truth-value: `indefinite' (with the semantics of that truth value understood
according to supervaluationist logic: that is, an utterance of a tenseless sentence $s$ 
is true (false) at $x$ iff $s$ is true (false) in all branches containing $x$). This has the advantage of
simplicity; it also accords well with the intuitions of an `open future'
which led Prior to tense logic in the first place. On the other hand, it
requires us to accept the failure of bivalence, with the consequent
logical difficulties (in particular, a rejection of Tarski's truth
biconditionals, which lead to contradiction in the absence of
bivalence\footnote{The Tarski biconditional for the sentence `it will
rain tomorrow' is `` `It will rain tomorrow' is true iff it will rain
tomorrow'', which has the form `` `It will rain tomorrow' is true iff $P$''; similarly, 
the biconditional for `It will not rain tomorrow' is `` `It will not rain tomorrow' is 
true iff it will not rain tomorrow'', which has the form `` `It will
rain tomorrow' is false iff $\sim P$''.
If it is neither true that it will rain tomorrow nor true that it won't, we can derive
$(P \& \sim P)$. See, \egc, \citeN[pp.\,187--197]{williamsonvagueness} for a more detailed presentation
of this argument.}).

An alternative strategy, which we might call \emph{ambiguity semantics}, 
was suggested to me by Simon Saunders (in
conversation). Suppose that just as a tensed sentence is associated
with a certain tenseless sentence, a tenseless sentence is
associated with a certain `branch-talk' sentence (e.g. `in branch
$x$, Y happens'). However, whereas in the tensed case it is usually
unambiguous which tenseless sentence is associated with an utterance, in
the branching case it is often highly ambiguous. This ambiguity often 
means that even a tenseless sentence cannot be given a truth value; however, if it
comes out true under all the disambiguations then it is true, period.
(Saunders has used the example of a proper name: if I use `David' as a
name for the 4-dimensional cradle-to-grave David, then it is unavoidably ambiguous
which of the vastly many Davids I am actually referring to. However, the
semantics of a branching universe cannot \emph{reduce} to the ambiguity
of proper names, because many future-directed statements we might wish
to make (\egc, `there will be a war next year') contain no names at all.)
Ambiguity semantics `demystifies' the `indefinite' truth value, making
it simply an unavoidable consequence of our inability to refer uniquely
to future branches; it has the technical advantage of maintaining
bivalence, at least at the theoretical level.

Motivation for yet a third strategy\footnote{Suggested in conversation by various people, notably
Simon Saunders.} comes from the philosophy of
personal identity: in particular, from considerations of `personal fission', the dividing of
an individual into two or more copies (by means of, for instance,
dividing the brain in two or using a Star Trek-style teletransporter),
which has been used by \citeN{parfit} to criticise the whole concept of
personal identity.
David Lewis, seeking (in
\citeNP{lewisfission}) to reconcile
common-sense views of identity with the case of fission, proposed
that when fission occurs in an person's life there are two persons present
\emph{all the time} --- it's just that they have shared person-stages
until the fission. Extending this to a branching universe, the natural
move is to regard a classical world not as an instantaneous thing (an element of \mc{S}, in our previous notation), but
as an entire history consisting of instantaneous `world-stages'.  
It would follow that there is no branching of worlds,
simply various periods when worlds cease to share world-stages. 

The strategy, which we could call \emph{Lewisian semantics}, might seem to suffer from extravagant multiplying of entities. 
If there are already
multiple identical-at-a-time worlds prior to branching then there are multiple utterances,
multiple agents, multiple labs, etc. On the other hand, if we subscribe
to an ontological framework where higher-level ontology is to be
understood as structures present in lower-level ontology (as defended in general by \citeN{realpatterns}
and \citeN{ladymanross}, and in the case of quantum mechanics by
\citeN{wallacestructure}), it does not
seem unreasonable to regard such a structure as a property of a history
rather than an instant --- especially as it is likely to be quite temporally
extended relative to the branching timescales (at least, this is true in
the case of quantum mechanics). 

Furthermore, the resultant semantics is
extremely straightforward, and has no need of truth-value gaps: ``The result of the
experiment will be Up'', uttered in world $w$ before branching, is true
iff in $w$ the result is indeed Up. The simplification is possible
because the physical event which serves as a world-stage for the
utterance in $w$ is also a world-stage for \emph{another} utterance in another
world $w'$. In one of $w$ and $w'$ the result is Up, in the other it is
Down. No rational agent prior to branching should commit to which result
he will see, but there is still a fact of the matter.

The difference between Lewisian and ambiguity semantics maps
reasonably well onto Sider's (\citeNP[pp.\,60--1]{sider}) distinction between the
`Worm
View' and the `Stage View' of how our ordinary talk about persisting
objects relates to the sequence of temporal parts making up those
objects' histories. The Worm View, identifying an object with its entire
history, leads (\emph{ceteris paribus}) to Lewisian semantics; the Stage
View, identifying an object with an instantaneous temporal part, leads
to ambiguity semantics, or possibly to minimalism.\footnote{A third possible view, not considered
by Sider but suggested to me in conversation by Simon Saunders, is that
an object at a time should be identified with its history \emph{up to
that time}; this view avoids the double-counting of the Worm View in the
presence of branching, and leads to the same semantics as for the Stage
View given that we are allowing only futurewards branching.}

Detailed considerations from philosophy of language are likely to bear
on which (if any!) of these three alternatives is preferable. For
instance, Lewisian semantics gives a very clear \emph{formula} for
reference: in that semantics, an utterance can be regarded as an
ordered pair of a microphysical event $x$ and a complete history $h$,
and a referential expression in the utterance refers to whatever thing
in $h$ it would have referred to on our conventional view of semantics
if we regarded $h$ as the only history. On the other hand, it is 
unclear what \emph{causal} story could be told as to how this reference
occurs (unless we wish to regard causation as itself emergent and branch-relative), 
since nothing intrinsic about the microphysical $x$ ties it to a
particular history.\footnote{For these sort of reasons, 
Lewis himself claimed \cite[pp.\,73--76]{lewisidentitypostscript} that
neither one of a pair of individuals who will be undergoing fission can
refer to himself and not his twin until the fission occurs; as such he
would in effect reject `Lewisian' semantics! (I am grateful to Oliver Pooley for
this observation.)}

By contrast, no unique formula for reference can be given at all in
ambiguity semantics: indeed, it is an explicit requirement of the
system that most referential terms fail to refer uniquely when used of
the future. This seems to put ambiguity semantics in direct conflict with what
Gareth Evans has called Russell's Principle \cite{evans}: the principle
that in order to refer successfully, I must know to what unique thing I
am referring.

However, I do not wish to attempt here to adjudicate between the various semantic
proposals that I have suggested: further discussion would involve too many controversies and 
would go beyond the scope of this
paper. I rather suspect that
there may not really even be any fact of the matter as to which is correct:
here perhaps, even if nowhere else, radical interpretation can lead to indeterminacy of translation, as
\citeN{quinewordobject} urged long ago.

\section{Propositions}\label{propositions}

In presenting branching semantics, and the arguments for it, I have
chosen to speak of tenseless sentences and their truth-values in the
hope of acheiving a presentation as neutral as possible between various
different semantic schemes. The conclusion of the charity argument ---
that one who knows his future involves branching should adopt an
attitude of uncertainty towards his post-branching state --- does not,
so far as I can see, depend on any specific theory of meaning.

One particular semantic proposal, however, seems worth discussing in
more detail, both because it is interestingly affected by the assumption
of a branching world and because it is currently rather popular in
many circles. This is the `possible worlds' semantics developed by
\citeN{kripke}, \citeN{stalnakercontent}, \citeN{lewislanguage}, 
\emph{et al}, according to which, in crude outline:
\begin{itemize}
\item Utterances express propositions;
\item The proposition expressed by an utterance is a function of the
context in which that utterance is made;
\item Propositions are sets of possible worlds, possible `ways the world might
be';
\item A proposition $P$ is true at a world $w$ if $w \in P$.
\end{itemize}

In this framework the difference between tensed and tenseless utterances
is that the proposition expressed by the former is dependent on the time
of utterance, the proposition expressed by the latter is
not.\footnote{Ignoring examples where the time of utterance has relevant `non-tense' 
implications: for instance, `the Queen' in `the Queen is [tenselessly]
called Elizabeth' denotes a different object when uttered in 1850 than in
2005.} 

In branching-time semantics, it is clear, a possible world is `a way a
particular history could be', not a way that the entire branching
structure could be. For the difference between the Elite and the Charitable
Views, then, is that on the Elite View a possible world is a way that
the entire branching structure could be; on the Charitable View it is a
way that a single history in the branching structure could be. This has
the consequence that in the Elite View most of our everyday utterances do not seem to
express propositions at all, whereas in the Charitable View it is
straightforward which proposition is expressed, but less straightforward
how the truth-value of that utterance is determined from the history-dependent truth-value
of the proposition expressed by it.

It is instructive to see how the various semantic proposals of section
\ref{details} look in the language of propositions. Lewisian semantics
is straightforward: to each utterance is assigned a unique history, and
the utterance is true iff the proposition expressed by the utterance is
true in the history in which the utterance was made. Minimalist
semantics, characteristically, offers simply a brute rule: an utterance
made at $x$ is true iff the proposition expressed by the utterance is
true in all histories containing $x$.

Ambiguity semantics is more interesting. The natural strategy would be
to regard it as ambiguous which proposition was expressed by a given
utterance, and this is particularly natural in the case of proper names
which Saunders considered. For suppose that my colleague Fred is to
make a quantum measurement with two possible results: Up and Down. Call
the two post-measurement Freds Fred$_1$ (who sees Up), and Fred$_2$ (who
sees Down). If, pre-measurement, I utter `Fred [tenselessly] sees Up',
perhaps it is ambiguous whether my utterance of `Fred' refers to Fred$_1$ or
to Fred$_2$, and hence whether the proposition which I express is the
proposition that Fred$_1$ sees Up (which is true) or the proposition
that Fred$_2$ sees Up (which is false). It is less clear where the
ambiguity in utterances like `there will be a war next year' lies, but
presumably it lies in the implicit `on this planet' which comes with such
utterances.

I should note that even if possible worlds are usually to be construed
as ways a history might be, there will be some contexts in which they
must be understood as the Elite View requires, as ways that the entire
branching reality might be: in particular, we may want to discuss
possible states of that branching reality; we may also wish to use
possible worlds in this second sense to ground the semantics of the
branch-talk sentences we use when we actually \emph{describe} this branching reality.
However, this does not seem particularly problematic either for the Charitable
View or for possible-worlds semantics in general.

\section{Possibilities}\label{possibility}

When I say that some future event is possible, sometimes I speak simply
of epistemic possibility, so that `possibly $P$' means just `I do not
know that $P$ is false'. As we have seen, the charity argument tells us:
\begin{itemize}
\item That if I know that it is true that $P$ on some
history containing my current location in the branching structure, then
I should assent to `possibly $P$'.
\item That if I know that it is true that $P$ on no such history then I should
reject `possibly $P$'.
\end{itemize}

Sometimes, however, we seem to use possibility in a non-epistemic sense (as
might be the case for physical, historical, logical possibility, etc.)
It is fairly clear that if $P$ is true of some history through $x$, then
`possibly $P$' must also be true at $x$: that which cannot in principle
be ruled out should not be impossible. However, nothing in the charity
argument entails that the converse holds: that for `possibly $P$' to be
true at $x$, $P$ must be true at some history through $x$.

This is not to say that some philosophically relevant notion of
possibility \emph{could} be defined via
\begin{quote}
`possibly $P$' is true at $x$ iff $P$ is true in some history containing
$x$;
\end{quote}
indeed, as far as I can see anyone tempted by the purely philosophical
arguments for branching time should embrace this notion of possibility,
for it seems to provide all that the advocates of such arguments desire
of `objective possibility'.

But this is not to say that it is \emph{required} that this particular
notion of probability has conceptual significance. As I mentioned in the
Introduction, I am myself sceptical that free will, causation \emph{et
al} require a branching-time picture. This raises the question of
how those analyses of possibility which traditionally have no notion
of branching time fare under the branching-world hypothesis.

In this context, recall that \citeN{lewisplurality} defined possibility
relative to some pre-defined class $X$ of `possible worlds' (such as the physically permitted
worlds, or the physically permitted worlds compatible with some data $x$, etc.): $P$ is
possible relative to this class iff it is true for some world $w$ in
the class. (See also \citeN{taylordennett} for a development of this position.) For future- directed possibility statements (that is, for
statements of the form `it is possible that Athens will host the
Olympics in 2500 AD', rather than of the form `it is possible that
Hitler won the Second World War') it is a requirement that the actual
world is in $X$.

Branching-time semantics makes only a slight alteration to this
framework. In such a semantics (ignoring Lewisian semantics) there is no
unique `actual history' but rather a set of such histories: all those through the point $x$
at which the possibility statement is considered. Any 
future-directed notion of possibility, defined via a set $X$ of possible
worlds (\iec, histories) and assessed at some point $x$, is coherent only if the 
entire set of histories through $x$ is contained within $X$. As far as I
can see, this alteration does not have profound effects for the
possible-worlds analyses of possibility. (Also note that on Lewisian
semantics, everything is almost exactly as in the absence of branching, with
exactly one world `actual' relative to a given utterance.)







\section{Conclusion}\label{conclusion}

If the only consistent ways to explain the semantics of tensed talk all
assume a certain structural property of time, then that would indeed be a reason to
assume that time does have that structure. This is
not the case for the question of the linearity or otherwise of time:
though I see no reason to doubt that nothing in our experience
\emph{rules out} linear time and linear semantics, I have argued here
that if time (or `the world' if you prefer) is actually branching, then the
correct semantics must also be branching.

As such, the semantics of tense are more naturalistic than meets the eye.
If the right
semantics for tense depends on the structure of time, we can be no more
certain of our semantic theory than of our theory of temporal structure.
(Perhaps this shouldn't be too surprising: if linguistic meaning
supervenes on linguistic behaviour and if the correct third-party
description of linguistic behaviour depends on facts about the world
about which we are ignorant, then our theory of linguistic meaning
should inherit that ignorance.)

In particular, if the Everett interpretation of quantum mechanics is
true, \emph{our universe} is branching. In such a situation, we are
required to assume that the semantics for our language is itself
branching. 
The apparent conceptual
paradoxes of the branching-universe are caused by insisting instead, without justification
and against the principle of charity,
on adopting linear-time semantics.

Once we recognise the semantic implications of the branching-universe
hypothesis, we realise that the statement `either A or B will occur, but not both, and I don't
know which', uttered at some point $x$ in the branching structure of the
Universe, becomes a perfectly rational thing for someone to say if A
occurs in some of the branches futurewards of $x$ and $B$ occurs in the
others. A rational agent at $x$ (and who knows the above facts) would
assert the truth of `either A or B will occur' (for in every branch one
or the other occurs) and the falsity of `both A and B will occur' (for
in no branch do both occur) but will assert neither the truth nor the
falsity of `A will occur' and `B will occur', for each occurs in some
but not all branches futurewards of $x$. As such, an interpretation of
quantum mechanics in which experiments cause the Universe to branch has
no difficulty in explaining why experimenters are entirely justified when, as they routinely do, they express
ignorance as to the outcome of their experiments. If so, one of the
major conceptual objections to the Everett interpretation of quantum
mechanics is unfounded.

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\end{thebibliography}

\end{document}
