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\begin{document}

\section{Truth and Circumstance}
\label{section-truth-and-circumstance}
\label{sec-truth-circumstance}

Notoriously, `information' is ambiguous. Traditionally, the possession of
information that Smith is an anarchist, like the knowledge of the same, implies
that Smith {\em is} an anarchist.  With the advent of mechanisms for the
gathering, storing and retrieval of large amounts of information, the word has
taken on a more neutral meaning.  According to more modern usage, which will be
adopted here, the information that Smith is an anarchist may be stored on a
computer file in ASCII code, transmitted across the globe, converted to
speech and
heard over a loud speaker, without Smith's ever having had a subversive
political
thought in his life.

A problem with the modern usage is that a new word is
required to separate genuine items of information, in the traditional,
truth-implying sense, from mere infons.  Provisionally, we will use the
adjective `factual' for this purpose.  Thus, the infon
$\infon{{\sf anarchist};{\sf Smith}}$ is {\em factual} just in case Smith
is indeed an anarchist; otherwise it is non-factual, but information
nonetheless.

Our use of `factual' is provisional because the distinction on which it rests
is an important contention in situation theory.  No one would deny of the
statement that Smith is an anarchist, that it is true if and only if Smith is
an anarchist.  It is less clear that this biconditional may be used to determine
whether the corresponding infon is factual.  To see the difficulty, let us
suppose
that Smith had a rebellious youth, but has now settled down into conformist
middle age. An utterance of `Smith is an anarchist' during Smith's early period
would have made a true statement, but an utterance of the same sentence today
would not.  On the assumption that both utterances express the infon
$\infon{{\sf anarchist},{\sf Smith}}$, we arrive at an impasse: which
statement do we use to decide whether the infon is factual?

Of course, the problem can be resolved by denying the assumption that both
utterances express the same information.  We may say that the infons expressed
by the two utterances are distinguished by a temporal role which is filled by
the time of each utterance.  Instead of one infon, we have two:

\begin{quote}
$\infon{\mbox{\sf anarchist};\mbox{\sf subject: Smith},
 \mbox{\sf time: Friday 15th May,
1970}}$ and \\
$\infon{\mbox{\sf anarchist};\mbox{\sf subject: Smith}
     \mbox{\sf time: Monday 18th July,
1994}}$.
\end{quote}

The first infon is presumed to be factual, because of the truth of the statement
made in 1970, and the second non-factual, because of the falsity of the
statement made today.  On this account, which we will style the {\em hidden
parameter}\labelp{sec.hidden-parameter} account,
$\infon{{\sf anarchist};{\sf Smith}}$ is either excused from
considerations of factuality because it is unsaturated, or supposed not to exist
at all.

A quite different resolution of the difficulty is to take the truth or falsity
of a statement to depend on more than just the information expressed.
Following Austin (1961), two components of a statement are distinguished:
the situation, event or state of affairs which the statement is about, and
what is said about it.  On this account,  which we will  style the {\em
referential}\labelp{sec.referential} account, the difference between the two
statements arises not because of a difference in the information they
express, but because different situations are described: the utterance in 1970
correctly describes Smith's attitudes at that time, whereas today's
utterance falsely
describes his present state of bourgeois conformity.   The infon $\infon{{\sf
anarchist};{\sf  Smith}}$ is factual if taken to be about Smith's
beliefs and actions in 1970, but not if it is taken to be about his
current views.

Proponents of both accounts agree that the common information
expressed by the two utterances is neither factual nor straightforwardly
non-factual.  They differ on the reason for this failure.  According to the
hidden-parameter account,  the infon $\infon{{\sf anarchist};{\sf
Smith}}$, if it exists at all, is merely the unsaturated common part of the
distinct, saturated infons expressed by the two utterances.
On the referential account all claims to factuality, if intelligible at all, are
relative to the situation the information is about.   The infon
$\infon{{\sf anarchist};{\sf Smith}}$ is  factual if it is taken to be
information about the situation described by the first utterance, but not if
it is taken to be about the situation  described by the second.

On matters of detail, there is  room for  interplay between the
two positions.  For example, the referentialist may agree that
the two statements express different information by virtue of the indexical
nature
of tense, but still maintain that the factual status of the information
expressed is
determined by the described situation.  Moreover, the debate may be
conducted in different areas.  For example, instead of focusing on tense,
the reference theorist may claim an implicit relativity to a perspective
from which the assessment of Smith's political views is to be made---it may
be Smith's own, his fellow citizens',  `the authorities'', or even that of
the person
making the statement.   The proponent of the hidden-parameter account
may go on to insist either that the term `anarchist' is ambiguous, or that
the infon
expressed has an extra role  for the perspective from which the property of
being an
anarchist is judged.

In the next few sections we shall consider the hidden parameter account in
more detail, before returning to the referential account in
Section~\ref{sec.situation-structures}.

\subsection{Fact Structures}
\label{sec.fact-structures}

  From one perspective, the hidden parameter account is the more
straightforward of the two.  The information expressed by an unambiguous
statement is
modelled as a saturated infon, which  may or may not be factual. If it is
factual
then the statement is true; if not,  it is false.  Those roles of the infon
that are
not filled by arguments given explicitly by linguistic features of a statement
must be determined by other features of the context in which the statement
is made.
These additional  roles are called `hidden parameters'. For example, the
temporal
`parameter' of the statement that Smith is an anarchist is filled by the time
at which the statement is made.  If there are further hidden parameters, such as
the perspective from which this assessment is made, then the statement is
either
ambiguous or the additional role must be filled by some other aspect of the
context.

This account can be modelled using a `fact structures', defined as follows.

\begin{defn} \label{def.fact-structure} An infon structure
$\goth{F}=\struct{\goth{A}}{\Fact}$ is a {\em fact structure} if
for each $\si\in A$, if $\Fact(\si)$ then $\si$ is a saturated infon.
An infon $\si$ is a {\em fact} if $\Fact(\si)$.
$\goth{F}$ is {\em trivial} if every saturated infon is a fact.
% $\goth{F}$ is a {\em basic-fact structure} if it is a basic-infon structure.
\end{defn}




Here is our main example of a fact structure:
 If $\goth{M}=\tuple{M,R_1,\ldots,R_n}$ is a relational structure and
$\SInf{\goth{M}}$ 
information structure
constructed from it in Construction~\ref{con.standard},
then let
$F(\goth{M})$ be the fact structure extending $\SInf{\goth{M}}$  with
\begin{quote}
$\Fact(\tuple{3,R_i,\al})$ iff in $\goth{M}_0$,
     $R_i(\alpha(1),\ldots,\alpha(\nu(i)))$
\end{quote}
for each $i\leq n$ and each 
$\fun{\al}{\{1,\ldots,\nu_i\}}{M}$. 
Every infon in a standard structure is saturated, so
there is
nothing more to check.  We call $F(\goth{M})$ the {\em standard} fact structure
generated by
$\goth{M}$.\footnote{Unlike the construction of
 standard basic-infon structures, the construction of
standard basic-fact structure is one-one: every relational structure is
uniquely represented as a basic-fact structure by the above construction.
Conversely, by Theorem~\ref{thm-T0}, every fact structure satisfying axioms
($T_0$) is isomorphic to a standard fact structure; but in this direction the
representation is not unique, because of the arbitrary ordering of relations
in a relational structure.}


The main problem with the hidden parameter account is that there are many
statements
which would not usually be called ambiguous, but for which hidden
parameters remain
unfilled. These are statements which, when all contextual factors are
considered,
still only express unsaturated information. These are more common that one might
suspect.  Suppose that in the course of telling the story of Smith's
involvement in
a riot, we say that he  panicked and ran.  Superficially, the content of the
statement is the information that Smith panicked and ran, which might be
modelled by
the infon
\begin{quote}
$\infon{{\sf panic; Smith}}\ \wedge\ \infon{{\sf run; \sf Smith}}.$
\end{quote}
This infon is unsaturated because panicking and running are events, and so
occur at a particular time, which has not yet been specified.  Unlike in the
previous example, the tense of the statement (simple past) is not sufficient to
determine a saturated content.  Unless there are aspects of the circumstances in
which the statement is made which somehow provide a way of saturating the
infon, it
is difficult to  see how the truth value of the statement may be determined.

There are various ways in which the missing information may be provided.
The date
of Smith's alleged actions may be given explicitly at some earlier point in the
narrative (``On the 15th May, 1970, Smith was involved in a riot$\ldots$'') or
provided by some other means:  the date of the riot may be common
knowledge, or the
narrator may have special access to the facts, perhaps by being  witness to the
original event.

The hunt for contextual factors that determine the content of statements is
ultimately restricted by the lack of precision in ordinary language. A threat
to the hidden parameter view is that suggestion  that ordinary language is  {\em
indefinitely imprecise}: there are always ways of making a given statement more
precise, with successive approximations to the `missing' information, and no
definite level of precision beyond which such improvements cannot be made.

For example, it there is no way of finding the exact date of the riot from the
context---if for example, the context only specifies that it occurred in May,
1970---then we would be forced to conclude that the statement is ambiguous.
But even if the date is supplied by context, we may still be in trouble.
Supposing
the date to be determined to be 15 May, 1970, the content of the statement would
be the infon
\begin{quote}
$\infon{{\sf panic; subject: Smith},{\sf date:
5/15/70}}\ \wedge\\
\infon{{\sf run},{\sf subject: Smith},{\sf date:
5/15/70}}.$
\end{quote}
Unfortunately, this infon is still  unsaturated, as we can see by
supposing that
Smith was involved in two separate clashes with the police on the same day.
On the
first occasion,  he bravely stood his ground; on the second he panicked and
ran.
The mere possibility of two clashes is sufficient to show that
the infon is unsaturated and so, on the hidden parameter account, we are left
with ambiguity, even if there was only one clash.

In an effort to avoid the conclusion that there is widespread ambiguity in
our use of ordinary language, we may search for more elaborate ways in which the
context fills hidden parameters.  If the exact time is not supplied in the
linguistic context or by common knowledge, it may be determined by a causally
related chain of events starting with Smith's actions in the riot.

The danger in this move is that information expressed by a statement may be
unknown,
even to the person who makes the statement.  The narrator may well remember
the day
of the riot, and even whether it occurred in the morning or afternoon, but not the
exact time. If this time is to be a component of the infon expressed by the
statement then this is information which the narrator does not possess.

If, on the other hand, we are to resist the move to considering factors which go
beyond the knowledge of the narrator, there are other problems.  If we take the
narrator's beliefs to determine the temporal parameter, then falsehoods are
liable
to appear in surprising places.  Suppose, for example, that the narrator falsely
believed the incident took place at around noon on the 15th May, 1970; in
fact, it
occurred at about 11 a.m.  It is awkward to maintain that this false
belief, which
is otherwise irrelevant to the narrative, makes his statement that Smith
panicked
and ran false.

Finally, there is a problem in making the demand for ever-greater precision
compatible with our somewhat roughshod ontology of events and processes.
Although we think of the event of Smith panicking and running as having
duration, it is not the sort of thing which we can measure in seconds.

We may hope that for each kind of description there is a fixed tolerance
with which information about the values of hidden parameters must
be specified.  In this way, we could avoid the problems of requiring perfect
precision. In many cases, the truth of a statement in which a time is not
mentioned
explicitly is fairly robust: small variations do not alter the truth-value. But
there are circumstances in which small changes do matter---Smith's two
clashes with
the police may have happened within half-an-hour of each other---and so, the
precision required for the saturation of a given piece of information is
indefinite,
depending on particular facts about the circumstances of the original
event, and not
just about the later circumstances in which the statement is made.

A way out of this web of difficulty is to embrace the conclusion that many,
if not
all, of the statements we make are ambiguous; or, better, to cut the link
between
lack of a truth value and ambiguity. We may say that a statement is
unambiguous if
it expresses a determinate item of information, even if this information is
unsaturated and the statement neither true nor false.

A quite different approach is to claim that an unsaturated infon is factual
if there
is some saturated fact of which it is a part. By quantifying over the
possible ways
of saturating the infon, we arrive at the rather charitable position that a
statement is true just in case, in the circumstances, there is some way of
construing it as expressing a saturated fact.  This is reminiscent of
Davidson's
treatment (1967) of the same problem.  There are two main difficulties
with this
solution. Firstly, an unrestricted quantification over possible arguments
is clearly
too generous.  If in the course of discussing the riot in 1970 we state
that Smith
panicked and ran, then this statement cannot be made true by Smith's
panicking and
running last week. Thus, we must appeal to the context again, this time to
provide
suitable restrictions to the range of quantification.  Secondly, the
restrictions
on quantifiers are often interdependent.  For example, if in interpreting the
text `Smith panicked. He ran.' we may use of the quantificational strategy,
then
the restrictions on the possibly ways of saturating the two unsaturated infons
$\infon{\mbox{\sf panic; Smith}}$ and $\infon{\mbox{\sf run; Smith}}$ must be
related: if $\infon{\mbox{\sf panic; Smith, time: $t_1$,\ldots}}$ and
$\infon{\mbox{\sf run; Smith, time: $t_2$,\ldots}}$ are among the permissible
saturations then $t_1$ must be before $t_2$.  Such technical obstacles are not
insuperable 
(see the chapter on dynamics in this volume)
 but introduce complications
which we shall not go into here.


Before moving on, we should mention that many of the issues
in this chapter having to do with
context and how it changes have roots in
well-developed areas in the philosophy of language.
Some important references here are two papers of
Lewis,``General semantics'' and ``Scorekeeping in a language game,''
which appear in his 1983 volume. 
There is also much relevant  work by
Perry; see Perry (1979) and other papers in Perry (1993).


\subsection{Logic in Fact Structures}
\label{sec.logic-of-facts}

In Section~\ref{sec.complex-infons} we showed how to model compound infons using the
theory of structural relations.  There we were concerned only with the
structural
properties of those objects; now we shall examine their logical properties.
For example, any reasonable model of conjunctive facts should satisfy
the condition that
\begin{quote}
$\Fact(\si\wedge\ta)$ iff $\Fact(\si)$ and $\Fact(\ta).$
\end{quote}
Certain special cases of this condition may be determined by structural
properties
of conjunction. For example, in the infon structure of
Construction~\ref{con.idem-assoc-conjunction} conjunction is idempotent,
and so in any fact structure extending this, $\Fact(\si\wedge\si)$ iff
$\Fact(\si)$.
But structural properties alone will not usually suffice to determine every
instance of the above condition, and so we shall need to impose additional
axioms.  For example, a de Morgan infon structure 
(from Section~\ref{section-infon-structures})
$\goth{F}$ may be expected to
satisfy the following conditions:
\begin{enumerate}
\item $\Fact(\bigwedge\Si)$ iff $\Fact(\si)$ for each
$\si\in\Si$, and
\item $\Fact(\bigvee\Si)$ iff $\Fact(\si)$ for some $\si\in\Si$.
\item $\Fact(\neg\si)$ iff $\Inf(\si)$ and not $\Fact(\si)$.
\end{enumerate}
Let us say that a fact structure $\goth{F}$
 is {\em classical} if it does satisfy the above.

We could go on to define a host of different kinds of fact structure, catalogued
by their logical properties. However, there are several good reasons for
not doing
this. Firstly, it is a historical fact that logicians have failed to arrive
at a consensus on the logical properties of even the most fundamental compounds.
Indeed, there is much wisdom in treating logic as an applied subject, for which
the logical properties of compounds are determined by the application.  The
upshot
is that any list would be tediously long and necessarily incomplete. Secondly,
we wish the approach developed here to apply not just to the multitude of
existing
logics whose compounds have the familiar syntactic structure of (formal)
sentences, but to future logics acting on more intricate structures---circular
structures, infinite structures, and anything else that can be modelled
using the
theory of structural relations.

Instead of giving a list of logics on fact structures, we  give a general
method
for categorizing the logical properties of facts.

Our work is based on the notion of a consequence relation,
as it is studied in philosophical logic.  For a survey 
of this large field, see Makinson (1991). 

 \begin{defn}
A {\em consequence relation} $\vdash$ on an infon structure
is a binary relation between classes of infons such that
$\Ga\vdash\De$ iff $\Ga'\vdash\De'$ for every partition $\tuple{\Ga',\De'}$
of $\Inf^*$ such that $\Ga\subseteq\Ga'$ and 
$\De\subseteq\De'$.\footnote{$\tuple{X,Y}$ 
is a {\em partition} of $Z$ if $X\cup Y=Z$ and $X\cap
Y=\emptyset$.} Given a fact structure $\goth{F}$, a consequence relation
$\vdash$ on
$\goth{F}$ is {\em sound} if whenever $\Ga\vdash\De$ and every infon in
$\Ga$ is a fact then some infon in $\De$ is a fact.
\end{defn}

Consequence relations have the following familiar properties:
\begin{enumerate}
\item[] (identity) $\si\vdash\si$,
\item[] (weakening)  if $\Ga\vdash\De$ then $\Ga',\Ga\vdash\De,\De'$, and
\item[] (cut) if $\Ga\vdash\De,\si$ and $\si,\Ga\vdash\De$
then $\Ga\vdash\De$.\footnote{We use 
the usual abbreviations, e.g.,  $\De,\si$ stands for
$\Ga\cup\{\si\}$ when an argument of $\vdash$.}
\end{enumerate}
Moreover, if $\vdash$ is compact then these three conditions are
jointly sufficient to ensure that $\vdash$ is a consequence relation
(Barwise and Seligman, to appear).\footnote{$\vdash$
 is {\em compact} if whenever $\Ga\vdash\De$ there are finite
sets $\Ga_0\subseteq\Ga$ and $\De_0\subseteq\De$ such that $\Ga_0\vdash\De_0$.}
Also note that a consequence relation is entirely determined by the class of
partitions that lie in it.  It is easy to  see that a consequence relation
is sound
in a fact structure just in case the partition of $\Inf^*$ into facts and
non-facts
does not lie in the consequence relation.

In any de Morgan infon structure $\goth{F}$, if $\vdash$ is a
sound consequence relation on $\goth{F}$ and $\goth{F}$ is classical then
$\vdash$
must satisfy the following conditions:
\begin{enumerate}
\item $\bigwedge\Si,\Ga\vdash\De$ \quadif $\Si,\Ga\vdash\De$,
\item $\Ga\vdash\De,\bigwedge\Si$ \quadif  $\Ga\vdash\De,\si$ for each
$\si$ in $\Si$,
\item $\Ga\vdash\De,\bigvee\Si$ \quadif $\Ga\vdash\De,\Si$,
\item $\bigvee\Si,\Ga\vdash\De$ \quadif $\si,\Ga\vdash\De$ for each $\si$
in $\Si$,
\item $\Ga,\neg\si\vdash\De$ \quadif $\Ga\vdash\si\De$,
\item $\Ga\vdash\neg\si\De$ \quadif $\Ga,\si\vdash\De$.
\end{enumerate}
The conditions are analogues of the standard inference rules of
the sequent calculus for (infinitary) classical propositional logic; let us say
that $\vdash$ is  {\em classical}  if it satisfies all these conditions.
Then it is
easy  to check that if $\goth{F}$ is non-trivial and has a classical consequence
relation then $\goth{F}$ must be classical.  
 These observations amount to the following
characterization of de Morgan infon structures: a non-trivial de Morgan infon
structure is classical iff it has a sound classical consequence
relation.\footnote{Furthermore, there is a smallest classical consequence
relation
$\vdash_c$ on any infon structure $\goth{M}$,
 and so a non-trivial de Morgan infon structure $\goth{M}'$ 
extending $\goth{M}$  is classical iff
$\vdash_c$ is sound on $\goth{M}'$.}

In this way, we may study the logical properties of complex infons by studying
their consequence relations, without any explicit reference to fact structures.
A multitude of consequence relations have already been catalogued by
logicians, and
so we can import whichever logic we wish, defining the corresponding
class of fact structures on which the consequence relation is sound.



\subsection{Restriction}
\label{sec.restriction}

In Section~\ref{sec.application} we saw that properties defined by means of
abstraction inherit the appropriateness conditions of the infons over which
they are
defined. For example, the property
$\lambda x.\infon{\ob{fed}; \ob{Martha}, x}$ of being eaten by Martha is
appropriate
for just those objects $a$ for which $\infon{\ob{fed}; \ob{Martha}, a}$ is an
infon.

For semantics, it has proved useful to introduce a more flexible notion of
abstraction by which a much wider range of restrictions may be placed on
appropriate
assignments.  On some semantic analyses, the information expressed by the
sentence
`Martha fed him' is that the referent of `him'  has the property of being fed by
Martha, modelled by the abstract $\lambda x.\infon{\ob{fed};{\ob
Martha}, x}$. But the sentence also conveys the information that the person
fed by
Martha is male. One way of incorporating this information is to maintain
that the
abstract is restricted so that it may only be applied to males.
In other words, we want a property $p$ such that
$\infon{p;a}$ is
\begin{enumerate}
\item an infon iff $a$ is male and $\infon{\ob{fed}; \ob{Martha}, a}$ is an
infon
\item factual iff $a$ is male and $\infon{\ob{fed}; \ob{Martha}, a}$ is factual
\end{enumerate}


In some formulations, the restriction that $a$ be male attaches to the
abstracted variable (or `parameter') $x$, and a theory of {\em restricted
parameters}
is developed (Gawron and Peters 1990, Fernando (1991); in
others, the
situation-theoretic universe is extended to include {\em restricted
objects} quite
generally: such objects are identical to their unrestricted counterparts
when the
(often contingent) restricting condition is met, and are otherwise
``undefined'' or
of some other ontologically inferior status.  Plotkin (1990) adapts Curry's
``illative'' approach to logic, treating restriction by means of a connective
$\res{}{}$ whose formation rules require that the restriction be met:  if
$\si$ is
an infon and $\tau$ is a fact then $\res{\si}{\tau}$ is an infon; otherwise the
expression `$\res{\si}{\tau}$' is not even well-formed.

Barwise and Cooper (1993) introduce an elegant graphical notation for
restrictions and abstracts called Extended Kamp Notation, in honour of Kamp's
Discourse Representation Structures 
(see the chapter on DRS in this volume).   
They write situation theoretic
objects as boxes. A box  directly to the right of another box acts as a
restriction,
and a box directly above lists abstracted objects. For example, the restricted
property $p$ mentioned above would be written as

\begin{center}
\begin{tabular}{|c|c|} \hline
\multicolumn{2}{|l|}{$a$} \\ \hline  & \\
$\infon{\ob{fed}; \ob{Martha}, a}$ & $\infon{\ob{male}; a}$ \\  & \\ \hline
\end{tabular}
\end{center}


The approach adopted here is to extend the theory of abstraction and application
developed in Sections \ref{sec.abstraction} and \ref{sec.application}.

\begin{defn}
A  fact structure $\goth{F}$ has {\em restricted abstraction} if it
is also an abstraction structure and has an additional structural relation
$\Res^2$, such that if $\Res(\si,a)$ then $a$ is an abstract and for each
assignment $f$ for $a$, $f.\si$ is an infon. We say that $a$ is a {\em
restricted
abstract} and that $\si$ is a {\em restriction} of $a$.  For each object $a$,
set $B$ of objects in $\univ{\goth{F}}$, and set $\Si$ of infons
in$\univ{\goth{F}}$,  an object
$\Lambda B[\Si].a$ is the {\em abstraction of $B$ from $a$ restricted by
$\Si$} if
there is a function $\fun{\pi}{B}{\univ{\goth{A}}}$ such that
\begin{enumerate}
\item $\Abs(\Lambda B[\Si].a,\pi(b))$ for each $b$ in $B$,
\item for each $b$ in $B$, $\pi(b)$ is only of sort $\Abs^*$, and
\item $\Lambda B[\Si].a$ is of type
$\tuple{\Lambda B[\Si].a,T_{\pi.a}\cup
\set{\tuple{\Res,\pi.\si}}{\si\in\Si}}$.\footnote{This clause is 
slightly more complicated than the corresponding
clause
for unrestricted abstraction. It says that the restricted abstract $\Lambda
B[\Si].a$ has the  component structure of $\pi.a$, together with the additional
structural relationships $\Res(\pi.\si,\Lambda B[\Si].a)$ for each $\si$ in
$\Si$.}
\end{enumerate}
$\goth{F}$ is a {\em restricted-Lambda structure} if $\Lambda B[\Si].a$
exists for
each $a$, $B$, and $\Si$.
\begin{quote}
The key definition: an assignment
$f$ for $a$ is {\em appropriate for} $a$ if $f.a$ exists
and $f.\si$ is a fact.
\end{quote}
 As before, if $f$ is appropriate for $a$ then {\em the
application of $a$ to $f$}, written $a^*(f)$, is just $f.a$.  $\goth{F}$ has
{\em coherent appropriateness conditions} if for each infon
$\si=\infon{r;f}$, if $r$ is a  (restricted) infon
abstract then $f^*$ is an appropriate assignment for $r$.
$\goth{F}$ is {\em restricted-relation-closed} if
for each infon $\si$, each set $B$ of elements of $\goth{A}$, and each set
$\Si$ of
infons, the infon abstract $\Lambda B[\Si].\si$ exists and for each appropriate
assignment $f$ for $\Lambda B[\Si].\si$, the infon $\infon{\Lambda
B[\Si].\si;f}$ also exists.
\end{defn}

Technically there are no difficulties here.  Restricted Lambda structures
and restricted-relation-closed structures may be constructed using methods
similar to those used in
 Theorems~\ref{thm.Lambda-extension} and~\ref{thm.relation-closed-extension}.

In passing, we might note that the work of this section is 
a case where  linguistic applications has directly driven  
theoretical matters.
The work here may be seen as another link in a chain of results
going back at least to generalized quantifier theory
(see chapter 15).  

\subsection{Internal Definability}
\label{sec.internal-definability}

We have already seen how an internal function $f$ represents an
external function $f^*$ in a function structure, and how in an abstraction
structure an abstract $a$ represents a second-order function $a^*$ mapping
assignments to objects of the domain.  In fact structures, internal relations
represent external relations.  Suppose
$\goth{F}$ is a fact structure and $R$ is an $n$-ary relation on
$\univ{\goth{F}}$.  The basic idea is that  an internal relation
$r$ of $\goth{F}$ represents $R$ if for each $a_1,\ldots,a_n$ in
$\univ{\goth{F}}$,  there is a saturated infon
$\infon{r;a_1,\ldots,a_n}$ in $\goth{F}$ and
\begin{quote}
$R(a_1,\ldots,a_n)$ iff $\Fact(\infon{r;a_1,\ldots,a_n})$.
\end{quote}
However, there are several complications. Firstly, our use of the
functional notation for infons hides an assumed  correlation
between the roles of $r$ and the integers $1,\ldots,n$; this must be
made explicit.  Secondly, the condition requires that the infon
$\infon{r;a_1,\ldots,a_n}$ exists for every sequence $a_1,\ldots,a_n$
of elements of $\univ{\goth{F}}$. This will rarely be satisfied because most
internal relations have non-trivial appropriateness conditions---it fails
even in
standard fact structures.  We shall have to revise the definition so that
the domain of relation $R$ is restricted in some way.

\begin{defn}
Given a fact structure $\goth{F}$, an $n$-ary external relation
$R$ on $\univ{\goth{F}}$ is {\em represented by} elements
$r,i_1,\ldots,i_n$ of $\univ{\goth{F}}$ on the domain $A\subseteq
\univ{\goth{F}}$
if
\begin{enumerate}
\item $f$ is appropriate for $r$ iff $\fun{f^*}{\{i_1,\ldots,i_n\}}{A}$, and
\item if $f$ is appropriate for $r$ then $\Fact(\infon{r;f})$ iff
$R(f^*(i_1),\ldots,f^*(i_n))$.
\end{enumerate}
\end{defn}

In standard fact structures, every relation $r$ has a fixed finite arity,
$n$ say,
and  (by the generality principle) the infon $\infon{r;a_1,\ldots,a_n}$
exists for
each sequence $a_1,\ldots,a_n$  of ordinary objects. Thus every internal
relation
of a standard fact structure represents an external relation on the domain of
ordinary objects.

In the general case, however, internal relations may fail to represent for a
variety of reasons: they may have infinite or variable arity; they may generate
unsaturated infons; or they may have sortal---or even more
complex---appropriateness
conditions.  A simple generalization of the above definition handles the last of
these cases.

\begin{defn}
Given a fact structure $\goth{F}$, an $n$-ary external relation
$R$ on $\univ{\goth{F}}$ is {\em represented by} elements
$r,i_1,\ldots,i_n$ of $\univ{\goth{F}}$ relative to another
$n$-ary relation $D$ on $\univ{\goth{F}}$ if
\begin{enumerate}
\item $f$ is appropriate for $r$ iff $D(f^*(i_1),\ldots,f^*(i_n))$, and
\item if $f$ is appropriate for $r$ then $\Fact(\infon{r;f})$ iff
$R(f^*(i_1),\ldots,f^*(i_n))$.
\end{enumerate}
\end{defn}

Effectively, we regard $r$ as representing the relation $R$ only in the
context of $D$; outside of $D$ we do not care which sequences are in $R$ and
this is reflected by the failure of appropriateness for the corresponding
assignments for $r$.   In other words, there are three possible judgements
concerning a sequence $\vec{a}$: $\vec{a}$ in $D$ and $R$; $\vec{a}$ in $D$
but not
in $R$; and $\vec{a}$ not in $D$. These are represented by the corresponding
judgements for an assignment $f$: $\Fact(\infon{r;f})$; not
$\Fact(\infon{r;f})$;
and no infon of the form $\infon{r;f}$.

Another approach is to say that, in general, internal relations
represent {\em partial} relations on $\univ{\goth{F}}$.  A partial relation has
an extension and an anti-extension, which are disjoint but which need not
exhaust
the domain.   Facts with relation $r$  represent  sequences in
the extension of $R$, and infons that are not facts  represent  sequences in
the anti-extension of $R$.

The issue of infinite aries and variable arities can be handled with similar
generalizations of the our metatheoretic concept of `relation'.  Relations
that have
appropriate assignments generating unsaturated infons in an irregular way
are beyond
the scope of these methods. There is nothing about external relations that might
correspond to  this features of internal relations. The reason for this is that
internal relations may capture intensional differences of real relations
that are
lost in our rather crude characterisation of external relations as a
collections of
sequences.

Whichever characterization of representation is adopted,
it is natural to ask which external relations are represented in a given
fact structure.  For example, we may wonder if any of the structural relations
$\Arg$, $\Rel$, $\Inf$ and $\Fact$ are represented.

Plotkin (1990) has various negative results. Suppose that $\Fact$ is
represented by $r$ in a lambda structure with negation,
and let $p$ be the abstract $\lambda x.\neg\infon{r;x}$, which is sure to
exist so
long as there is any infon of the form $\neg\infon{r;\si}$.  A simple argument
shows that $p$ cannot have a fixed point: if $\si$ was such then
$\Fact(\si)$ iff $\Fact(\neg\infon{r;\si})$ iff $\neg\Fact(\infon{r;\si})$,
by the
properties of abstraction and negation; and  $\Fact{\infon{r;\si}}$ iff
$\Fact(\si)$
under the assumption that $r$ represents $\Fact$; so $\Fact(\si)$ iff
$\neg\Fact(\si)$, a contradiction.

In the structures used by Plotkin, every abstract has a fixed
point, and so $\Fact$ cannot be represented. Plotkin's uses Frege
structures (Aczel 1980), which are  constructed from models of the untyped
$\lambda$-calculus in which fixed point combinators such as $Y$ exist.
Similar problems arise in structures with internalized application, because they
also contain fixed point combinators.  Even without such combinators fixed
points
will exist in any structurally saturated fact structure.  For example, the
type  $\tuple{x,\neg\infon{r;x}}$ is instantiated in any such structure,
and  this
solution is a fixed point of $p$.

These considerations illustrate an important trade-off in the construction of
models for Situation Theory.  There are two ways in which we may measure
the power
of our structures as modelling tools. On the one hand, we may see which
operations
are  possible, desiring closure under such operations as abstraction,
infon-formation, and restricted abstraction.  On the other hand, we may see
which
external entities are internalized, which functions and relations are
represented, whether application can be internalized, and so on. We have seen in
Section~\ref{section-afa} that structures in which many types are
instantiated---in particular, structurally saturated structures---are very
useful
for ensuring closure properties under structural operations. However, the above
results show that if many types that are instantiated, then there are limits on
what can be internalized.  The boundary between the goals of structural
closure and
internalization are still far from clear.

\subsection{Situation Structures}
\label{sec.situation-structures}

Sections \ref{sec.fact-structures} to \ref{sec.internal-definability} have been
directed toward an articulation of what we have called the hidden-parameter
approach to the relationship between information and truth.   Now we turn to the
alternative, referential approach.

Consider again the example of telling a story about Smith's involvement in
a riot
during which he panicked and ran.  The straightforward approach is to say that
the statement that Smith panicked and ran conveys the information
\begin{quote}
$\infon{{\sf panic; Smith}}\wedge\infon{{\sf run; \sf Smith}}.$
\end{quote}
The puzzle was to determine the criterion for this (unsaturated) infon to
be factual.

On the hidden-parameter account,  for the statement to be true it must
express a
saturated infon whose additional roles are filled by various contextually
determined parameters.   Common usage dictates that the statement is true, or
at least {\em could} be true, and so that there are such hidden parameters
determining the fullly saturated information expressed by any unambiguous
statement.
The hidden-parameter approach to semantics therefore revolves around the
identifying and cataloguing of contextual parameters.\footnote{An
alternative is to
abandon the common sense idea that much of what is said is really true (or
false)
and try to explain semantic relationships in terms of information
conditions. This
is summed up nicely by Barwise's slogan `Information conditions not truth
conditions' and has become a core idea in Dynamic Semantics 
(see the chapter on dynamic semantics in this volume)}

The referential account is quite different, attributing the alleged gap between
information  and truth to the neglect of a fundamental component of descriptive
statements: the situation  being described.  In the above example, the
narrator is
describing a certain situation, a course of events twenty-six years ago,
in which (we may suppose) Smith panicked and ran. Consequently, the  description
is correct and the narrator's statement is true. However, the very same
sentence,
`Smith panicked and ran', may be used to describe a quite different situation in
which Smith was engaged in some activity incompatible with panicking and
running,
such as standing and fighting.  In this case, the statement---necessarily a
different
statement than the previous one---would be false.  A third possibility is that the
described situation is one  which encompasses many different events, making the
statement genuinely  ambiguous.

The range of outcomes shows why it is difficult for the reference theorist
to sort
infons into facts and non-facts.   Instead, a new predicate, `supports', is
introduced to describe the relationship between a situation and the information
expressed by a correct description of it. In other words, a statement
describing
a situation is true if the situation it describes supports the information
it expresses; otherwise it is false.  Consequently, a statement describing
an event
$e$ using the sentence `Smith panicked and ran' may be false, even if there is
another event $e'$ which would be correctly described by the same sentence.
Moreover, it is possible for two statements to be made using this sentence
but about the two different events $e$ and $e'$, and for the statements to
express the {\em same information}, even though one is true and the other false.
The difference in truth-value is explained as a difference in reference, not in
the information expressed.

Criticism of the referential account
(for example, in Seligman (to appear)) has centered on two related
problems. Firstly, it is difficult to account for just how people refer to
situations
when they make statements.  Various proposals  are akin to corresponding
proposal
for the reference of proper names: the intentions of the person making the
statement
may be involved; there may be some  kind of causal relationship between the
statement and described situation; and conventions within a linguistic
community may
be appealled to.  In addition,  the context of a statement has considerable
importance. A past tense sentence,  such as the one we have been discussing, may
well be referentially ambiguous when uttered out of context,  attaining a
definite
reference only in the context of  a discourse or narrative text
(see Seligman
and ter Meulen (1993)).  Nonetheless, no there is no general
account of how reference is achieved and this remains a serious lacuna.

The first problem is compounded by the second: that it is unclear what is
referred to when reference is achieved.  The word `situation' is intended
to be neutral between `event' and `state', so that it makes sense to speak both
of the situation in Bosnia, meaning the war (an event or course of events),
and the
situation on Capitol Hill, meaning (the state of) deadlock over the
 budget.\footnote{Indeed, the flexibility of 
the words `situation' and `situated'
has become
almost comical: any  traditional theory---of language, cognition, and even
sociology---gives birth  to a new `situated' theory, sometimes with little
more than
the liberal use of this adjective together with a strong, if vague,
commitment to
the idea that everything is relative to contextual factors.} The theoretical
uniformity of this terminology is bought at a price. Questions about the
identity of
situations appear much more pressing than the traditional (and equally
baffling)
metaphysical questions about the identity of material objects, persons and
members of a host of other ontological categories.

A short, bold answer to the question of the identity of situations is that
distinct
situations support different infons.  Contraposing we get the
\begin{quote}
{\em Principle of Extensionality:}  if for each infon $\si$, $s$ supports
$\si$ iff
$s'$ supports $\si$, then $s=s'$.
\end{quote}
If we accept this principle then we may think of situations and model them as
structured objects whose structure is determined by the infons they support.
This is the basis of a model of situations using the theory of structural
relations.
We introduce the structural relation $\Sit^1$ of being
a situation, and  $\HoldsIn^2$, which holds  between an infon  and a
situation just
in case the situation supports the infon.

\begin{defn}
An infon structure $\goth{S}$ with structural
relations $\Sit^1$ and $\HoldsIn^2$ is a {\em situation structure} if
\begin{enumerate}
\item if $x$ is an infon, a relation, or a role then not $\Sit(x)$, and
\item if $\HoldsIn(\si,s)$ then $\Inf(\si)$ and $\Sit(s)$.
\end{enumerate}
The elements of $\Sit^*$ are called {\em situations} and we
say that $s$ {\em supports} $\si$, written $s\supports\si$, if
$\HoldsIn(\si,s)$.
\end{defn}

Any fact structure $\goth{F}$ may be used to define a situation structure
$\goth{F}^\#$ by adding  a new element $s_{\Sigma}$ for each set $\Si$ of
facts, and
defining:
\begin{quote}
$\HoldsIn(\si,s_{\Si})$ iff $\si$ is in $\Si$.
\end{quote}
We say that a situation structure is {\em standard} if it is isomorphic to
$\goth{F}^\#$, for some standard fact structure $\goth{F}$.


\subsection{Parts of the World}

The relationship of smaller situations to larger 
situations is perhaps the most
controversial aspect of Situation Theory.  
To see why, consider the following two  situations:
Let $s_{{TW}}$ be the  situation in Taiwan,
or some relevant piece of that situation.
Let's suppose that  $s_{{TW}}$
 supports each infon of
form $\infon{\ob{ma};\ob{t}}$ that could be expressed by a sentence
of form `There was a motorscooter accident at time $t$', used to describe
correctly
events occurring in Taiwan. 
Let  $s_{{W}}$ be the situation on Earth as a whole.
We assume that $s_{{W}}$ supports each infon of
the form
$\infon{\ob{ma};\ob{t},\ob{z}}$ that could be expressed by a sentence
of form `There was a motorscooter accident at time $t$, time zone $z$', used to
describe correctly events occurring anywhere in the world. 
On an intuitive level, $s_{{TW}}$ is a part of $s_{{W}}$.

However, this assumption is not correctly modelled in 
out theory as it stands.  The reason is that the only order
we have around is the
`flat' order $\partof$ between situations obtained 
as an instance of our theory of partiality in Section~\ref{section-partiality}.
On this, $s_1 \partof s_2$ iff all infons holding 
   in $s_1$ situation also hold in $s_2$.
According to this definition, $s_{{TW}} \not\partof s_{{W}}$ since
the infons supported by $s_{{TW}}$ are {\em not} supported by $s_{{W}}$.
%For  $s_1$ to be a part of $s_2$ (in the intuitive sense)
% it is indeed sufficient that  $s_1\partof s_2$.  


%We are taking situations to be structurally determined by the infons they
% support.

 On closer inspection, it appears that there
are two reasonable ways of saying what it means
for  one situation to be contained in another. The
first is $\partof$ as above.
 The second way
 is that $s_1$ is  part of  $s_2$ if   all the
infons
holding  $s_1$ are {\em unsaturated versions} of the infons
holding in $s_2$.
For example, let $s_{{TW}^*}$ be the situation that supports each
infon of
the form  $\infon{\ob{ma};\ob{t},\ob{z}}$ 
that could be expressed by a sentence
of form `There was a motorscooter accident at time $t$, time zone $z$', used to
describe correctly events occurring in Taiwan.  It is in this second sense
that $s_{{TW}}$ is part of $s_{{TW}^*}$.  Also
$s_{{TW}^*} \partof s_{{W}}$.  Combining the two ways, we get the
following
definition:

\begin{defn}
$s_1$ is {\em a part of} $s_2$, written $s_1\sitpartof s_2$,  if:
\begin{quote}
$\forall\si$
if $s_1\supports\si$ then $\exists\tau$ $(s_2\supports\tau$ and
$\si\partof\tau$).
\end{quote}
  Two situations are {\em compatible} if they have an
upper bound in the $\sitpartof$-ordering.  A situation $s$ is {\em maximal}
if for each situation $s'$, if $s\sitpartof s'$ then $s=s'$.  A set of
situations may be {\em joined} if they have a least upper bound in the
$\sitpartof$-ordering, called the {\em join} of $S$.
\end{defn}


\begin{prop} Every standard situation structure satisfies the following
conditions:
\begin{enumerate}
\item[{\bf S1}] for every situation $s$, there is a maximal situation
of which it is a part,
\item[{\bf S2}] any two situations are compatible,
\item[{\bf S3}] there is a unique maximal situation of which every
situation is a
part,
\item[{\bf S4}] every pairwise compatible set of situations may be joined, and
\item[{\bf S5}] every set of situations may be merged.
\end{enumerate}
\end{prop}

These properties concern the  global properties of the
universe.  In standard situation structures, there
is a unique maximal situation, call it $w$, which is
given concretely by the set of facts from the underlying fact structure.
This gives is a simple relationship between facts and situations:
\begin{quote}
$\Fact(\si)$ \quad iff \quad  $\exists s. s\supports\si$ \quad iff \quad
$w\supports\si$
\end{quote}
As we move away from standard situation structures this relationship will
break down.  But first, we should consider what it would mean for  the
conditions listed above to fail.  How are we to make sense of incompatible
situations in the general case?   And how can there be two
distinct maximal situations?

Maximal situations are completely saturated, in the sense that information
about the filling of roles in infons is specified, and there is no compatible
way of adding further infons. This suggests the interpretation of maximal
situations as possible worlds.   If condition S1 is satisfied then every
situation
is part of some possible world, and so compatibility becomes
compossibility: two situations
are compossible iff they are part of the same possible world.  We
call this the {\em compossibility interpretation}.

Under this interpretation, one maximal situation $s_a$ must be
distinguished as the
{\em actual world}, 
and the other incompatible maximal situations are ways the
world might
have been.  There is a sharp metaphysical  distinction between those
situations  which are part of $s_a$ and those which are not; the former are {\em
actual situations}, the latter are merely {\em possible situations}.

The compossibility interpretation allows one to introduce many of the tools of
the theory of possible worlds. For example, standard analyses of
modality and conditionals  may be given.  However, such
analyses tend to run into difficulties due to  the partiality
of situations; we shall not go into the details here.\footnote{Our
construction of
standard situation structures may be modified in a fairly obvious way to
build standard `modal' situation structures from first-order Kripke
structures, by
generating situations from sets of compossible facts.}

On the {\em relativist interpretation}, incompatible situations are regarded as
embodiments of different perspectives on the same, actual world.  For example,
suppose we are facing each other across the dinner table, so that for you
the salt
is to the left of the pepper.  The situation $s_1$ concerning the arrangement of
objects on the table from your perspective supports the infon
$\infon{\ob{LeftOf}; \ob{salt}, \ob{pepper}}$,
 whereas the situation $s_2$ from my perspective supports $\infon{\ob{LeftOf};
\ob{pepper}, \ob{salt}}$.  On a relativist
conception, these situations may be incompatible because 
 there is  no situation
that supports both of these infons.

Even on the relativist conception, the two perspectives may be made
compatible by
appealing to a hidden parameter.  If there are infons
\begin{quote}
$\infon{\ob{LeftOf}; \ob{salt},\ob{pepper}, \ob{me}}$ and
$\infon{\ob{LeftOf}; \ob{salt},\ob{pepper}, \ob{you}}$,
\end{quote}
in which a role for an egocentric frame of reference is
filled by $\ob{me}$  and $\ob{you}$ respectively, then there could be a
situation
$s_3$ supporting both of these infons without conflict.  Moreover, other things
being equal, $s_1\sitpartof s_3$ and $s_2\sitpartof s_3$, and so $s_1$ and
$s_2$
are compatible after all.   In this way, we can account for the fact that there
are different perspectives in this example ($s_1$ and $s_2$ have no upper
bound in
the $\partof$-ordering), while showing how the two perspectives can be
joined using
hidden parameters.  On the relativist conception, situations are genuinely
incompatible, only if there is no way of adding a hidden parameter to
resolve the
conflict.\footnote{For more discussion of perspectives,
 see Barwise (1989b) and  Seligman (1991).}


Closely related to but strictly stronger than Extensionality, is the 
\begin{quote}
(\em Principle of Anti-symmetry) if $s_1\sitpartof s_2$ and $s_2\sitpartof
s_1$ then
$s_1=s_2$.
\end{quote}
In every saturated situation structure, Anti-symmetry is satisfied, making
$\sitpartof$ a partial order; but this is not so in general.  For example,
consider the infons
\begin{quote}
$\si_1=\infon{\ob{eat};\ob{Kara}}$ and
$\si_2=\infon{\ob{eat};\ob{Kara},\ob{trout}}$
\end{quote}
and suppose that $s_2$ is a (necessarily unique) situation which supports
only the
information $\si_2$ that Kara eats trout, and that $s_1$ is a situation
which also
supports the information $\si_1$ that Kara eats.  Then $s_1\sitpartof s_2$ and
$s_2\sitpartof s_1$ but $s_1\neq s_2$.

One might think that such examples should not arise.  This is a consequence
of the widely held
\begin{quote}
(\em Principle of Persistence) if $s_1\sitpartof s_2$ and $s_1\supports\si$ then
$s_2\supports\si$
\end{quote}
Extensionality ensures that Anti-symmetry follows from Persistence.
Indeed, Extensionality
ensures that situations are completely determined by the information they
support.
Persistence goes a step further and identifies the $\sitpartof$-order on
situations
with the $\partof$-order; this order  is always anti-symmetric.



We might also  want  axioms positing the existence
 of situations which support infons 
which are known to be supported
by other situations:
\factax{sitexist0}{If $\si\partof\tau$ and $t\supports\tau$ then $\exists
s\sitpartof t$ such that $s\supports\si$.}
\factax{sitexist1}{Let $t$ be such that
 for each $\si\in\Si$,  $t\supports\si$.  Then
$\exists s\sitpartof t$ such that
($s\supports\si$ iff $\si\in \Si$).}
\factax{sitexist2}{Let $t$ be such that 
    for each $\si\in \Si$,  there is a $\tau$ such that
$\si \partof  \tau$ and 
$t\supports\tau$. Then 
$\exists s\sitpartof t$ 
($s\supports\si$ iff $\si\in\Si$).}

\noindent Each of these
existence axioms is
satisfied by
every standard situation structure.

\medskip

\rem{THIS FINAL REMARK GOT DELETED. THE ORDER INVOLVED
IS NOT TRANSITIVE, AND IT DIDN'T SEEM IMPORTANT ENOUGH TO
FIX UP.

THE EXAMPLE ON TRANSITIVITY:

Suppose we have infinitely many roles, indexed by 
positive and negative integers.  Suppose  that
si_n fills roles m for  m \leq n.
So si_n \partof  si_m whenever n \leq m.
Let s_1 use  (or be) 
the even sigma's, and s_2 the odd ones.
Then s_1 and s_2 are different, but there are g
functions in both directions -- add 1 to the subscript on the 
infons.

THE TEXT:

Here is a final remark on order conditions and the relativist conception:
The identification of $\sitpartof$ with $\partof$, forced by Persistence, is
unfortunate for the relativist conception, since it makes the most trivial
difference of perspective result in incompatible situations.   However, the
alternative, when Anti-symmetry fails, poses  problems for the joining of
situations.  In saturated situation structures the least upper bound of a
set of
situations is unique if it exists, and so the join operation is well-defined.
When unsaturated infons are present, this is no longer the case, and we have
to find
a way of isolating a canonical representative of a potentially large class of
equivalent situations.
A solution to this problem is to define a refinement of the
$\sitpartof$ ordering.   In the principle  below,
we write $s_1\sitpartofstrict s_2$  to
mean $s_1\sitpartof s_2$ but $s_2\not\sitpartof s_1$.


\begin{quote}
$s_1\sitpartof^*s_2$ iff either $s_1\sitpartofstrict s_2$ 
or there is an injective function $g$ from the infon-set of $s_1$ to the
infon-set of $s_2$ such that for all $\si$ in the domain of
$g$, $\si\partof g(\si)$.
\end{quote}

Clearly, if $s_1\sitpartof^* s_2$ then $s_1\sitpartof s_2$, but in  
the opposite
direction, we have only that if $s_1\sitpartof s_2$ and $s_2\not\sitpartof s_1$
then $s_1\sitpartof^* s_2$.  The relation $\sitpartof^*$ is a partial order, and
{\tt proof required?}
so we may define a merge operation using least upper bounds, when they
exist.   The result agrees with intuition to the extent that the least upper
bounds in the $\sitpartof^*$-ordering are also least upper bounds in the
$\sitpartof$-ordering.

} 


\subsection{Logic in Situation Structures}

One can see the move from fact structures to situation structures as a
generalization in two directions.  First, the non-relational property $\Fact$
of being factual is replaced by the relational property $\HoldsIn$ of
holding in a
situation.  Second, this property may be had by infons which are unsaturated. In
other words, situations are {\em partial} in two respects: they need not
support all
of the facts, and the infons they do support may be only unsaturated parts of
facts.

More precisely, we may relate fact structures and situation structures as
follows.
Say that a situation $s$ is {\em saturated} if every infon it supports is
saturated.  For each situation $s$, let $\Fact_s$ be the property of holding in
$s$, i.e., $\Fact_s(\si)$ iff $s\supports\si$.  If
$\goth{S}=\struct{\goth{A},\Inf,\Sit,\HoldsIn}{\supports,\partof}$ is a
situation
structure, then for each saturated situation $s$ in $\goth{S}$, the structure
$\goth{F}_s=\struct{\goth{A},\Inf,}{\Fact_s,\partof}$ is a fact structure.

This construction suggests a way of extending our discussion of logic on fact
structures to the present setting.  We say that a situation structure
$\goth{S}$ is
{\em locally classical} if for each situation $s$ in $\goth{S}$, the fact
structure $\goth{F}_s$ is a classical fact structure.  By the previous
discussion,
we know that this is equivalent to the following: for each situation $s$ in
$\goth{S}$,

\factax{sitlog1}{$s\supports\bigwedge\Si$ iff $s\supports\si$ for each
$\si\in\Si$}
\factax{sitlog2}{$s\supports\bigvee\Si$ iff $s\supports\si$ for some
$\si\in\Si$}
\factax{sitlog3}{$s\supports\neg\si$ iff $s\nsupports\si$}

This is not the only way of understanding  entailment in situation
structures, and
may not be the best.   One potentially undesirable feature of locally classical
situation structures is that the set of infons supported by any given
situation is
bound to be infinite.  The set of facts in a fact structure is closed under
logical
consequence, and so will be infinite, given a few assumptions about the
structure
of compound infons.  The requirement that the set of infons supported by a
situation
form a fact structure therefore ensures that this set is infinite.

Whether or not this is acceptable depends on which theoretical role
situations are intended to play.  As objective truth-makers they may be
taken to support infinitely many infons without causing alarm; but if situations
are to represent the information state of a finite agent, some might argue that
supporting infinitely many infons is an unfortunate consequence of the theory.

Alternatives may be found by looking at different ways in which the notion
of a fact is to be related to that of a situation.  For example, given a
situation
structure, we may construct a fact structure by defining
\begin{quote}
$\Fact(\si)$ iff $\si$ is saturated and $\exists s$ $s\supports\si$.
\end{quote}
Say that a situation structure is {\em globally classical (intuitionistic)}
if the
fact structure so-defined is classical (intuitionistic).  Clearly every locally
classical structure is globally classical; indeed, it seems likely that
this is a
minimal requirement for a situation structure in which classical reasoning makes
any sense at all.

Intermediate positions may be taken by linking factuality to the $\partof$-order
between situations.  For example,  define the {\em eventual} fact structure
of each
situation $s$ by
\begin{quote}
$\Fact(\si)$ iff $\si$ is saturated and $\forall s'\cpartof s$ $\exists
s''\cpartof s'$ $s''\supports\si$.
\end{quote}
A situation structure is {\em eventually classical} if the eventual
fact structure of each situation is classical.  A situation structure
may be both locally intuitionistic and eventually
classical.\footnote{Indeed, this
is an attractive combination. Take an extensional Kripke-model for
intuitionistic
predicate logic, consisting of a partially ordered set of points, a domain
monotonically associated with each point, and an interpretation of each atomic
predicate in each domain (monotonically, again), with the property that for
points $p_1$ and $p_2$, if the domain and extensions of predicates in $p_1$ are
included in those of $p_2$ then $p_1\leq p_2$.    Now construct a situation
structure by defining basic infons to be tuples $\tuple{r,a_1,\ldots,a_n}$
for each
$n$-ary predicate $r$ and $a_1,\ldots,a_n$ elements of some domain. Compound
infons are terms built from the basic infons using $\bigwedge$, $\bigvee$,
$\implies$ and $\neg$.  Situations are points and the supports relation is
defined
inductively by
\begin{enumerate}
\item $\HoldsIn(\tuple{r,a_1,\ldots,a_n},p)$ iff $\tuple{a_1,\ldots,a_n}$
is in the
extension of $r$ in $p$
\item $\HoldsIn(\bigwedge\Si,p)$ iff $\HoldsIn(\si,p)$ for each $\si$ in $\Si$
\item $\HoldsIn(\bigvee\Si,p)$ iff $\HoldsIn(\si,p)$ for some $\si$ in $\Si$
\item $\HoldsIn(\neg\si,p)$ iff for no $p'\geq p$ $\HoldsIn(\si,p')$
\item $\HoldsIn(\si\implies\ta,p)$ iff for each $p'\geq p$ if $\HoldsIn(\si,p')$
then $\HoldsIn(\ta,p')$
\end{enumerate}
The resulting structure is locally intuitionistic, and may not be locally
classical, as there is no guarantee that either $\si$ or $\neg\si$ holds at a
given $p$. However, it is eventually classical because if $\si$ is not an
eventual
fact of $p$ ($\neg\forall p'\geq p$ $\exists p''\geq p'$
$p''\supports\si$) then there is a $p'\geq p$ such that $\forall p''\geq p'$
$p''\nsupports\si$, and so $p'\supports\neg\si$, and so $\neg\si$ is an eventual
fact of $p$.}


\subsection{Propositions}
\label{section-propositions}
\label{section-situation-semantics}


On the referential account  the informativeness of an infon
depends on which situation  it is taken to be about.
The information that Melanie has Jerry's car keys may be correct
information about some situations and incorrect information about
others.

It is natural to ask: what has happened to the concept of truth on
this account?  The short answer is that truth is a property of
{\em propositions}, which will be modelled later in this section.  The
longer answer is best given in the context of providing an analysis
of the use of language to make statements.


Suppose that a sentence $S$ is uttered.  The utterance
occurs as part of a situation $u$ involving the speaker, the hearer, and
perhaps many other things which play a part in determining what is
said; $u$ is called the {\em utterance situation}.  Among the many
facts supported by $u$ only some are linguistically relevant.  Let
$\si_u$ be some linguistically relevant information supported by
$u$. Typically, $\si_u$ might contain information about the
syntactic composition of $S$, the referents of indexicals and anaphors in $S$,
and so on.

Suppose further that by uttering  $S$ the speaker is making a
statement, rather than asking a question, giving a command, or any
of the many other things one can do with words.  For example,
suppose $S$ is the sentence `No one is wearing shorts', said by Alex
to Nicholas while looking out of the window at people walking along
the  street. In such a case,  the conventions of language use and
the particular circumstances of $u$, including perhaps the intentions of
the speaker, the discourse
context, the demonstrative actions of the speaker, and so on, all go
to determine what the speaker is talking about.  More specifically,
they determine which situation is being described.  In this
case, the {\em described situation} $d$ is the goings-on in the
street outside Alex's apartment.

Other conventions of language use, especially those fixing the
meanings of those words occurring in $S$ and their mode of syntactic
combination, go to determine what is being said about $d$.  In our
example, Alex said, about the situation $d$, that it is one in which
no one is wearing shorts.  The information (or misinformation)
$\tau_u$  that no one is wearing shorts is the {\em descriptive
content} of  Alex's utterance.

Now we are in a position to say what is true and what is not.  The
primary candidate for truth is Alex's statement; it is true iff the
described situation $d$ supports the descriptive content $\tau_u$.
But it is far from clear what sort of thing Alex's statement is and
how it is related to the various other components of our analysis:
the utterance situation, the described situation, the
linguistically relevant aspects of the utterance and the
descriptive content.  All these things are aspects of Alex's
statement, but its truth is determined just by $d$ and $\tau_u$.

Situation theorists and situation semanticists
have appropriated the word `proposition' to
refer to an abstract entity $\apr{s}{\si}$ consisting of the
combination of a situation $s$ and an infon $\si$.
On this modelling, propositions are
either true or false: $\apr{s}{\si}$ is true iff
$s\supports\si$.

The {\em propositional content} of Alex's statement is defined to be
the proposition $\apr{d}{\tau_u}$ consisting of the described
situation $d$ and the descriptive content $\tau_u$.  Thus her
statement is true iff it has a true propositional content.

We see that there are two different ``contents'' to a given
utterance $u$. First, there is the purely descriptive content
$\tau_u$. This is shared by a host of other statements. In this
case, assuming that there is no ambiguity involved, the descriptive
content of Alex's statement is the same as that of any other
utterance of the sentence $S$.\footnote{If Alex had used a sentence containing
indexicals or demonstratives then this would not be the case. For example,
Nicholas's utterance of the sentence `You are wearing shorts' does
not have the same descriptive content as an utterance of the same
sentence by Alex.} Second, there is the propositional content
$\apr{d}{\tau_u}$. Not all utterances of $S$ have the same
propositional content because they may be used to describe different
situations.  For example, if the shutters were closed and Alex was
whispering $S$ to Nicholas as their guests arrived at the
apartment, the situation $d'$ described would be quite different,
and so the proposition content $\apr{d'}{\tau_u}$ would also be
different.  Indeed, it could be that $\apr{d'}{\tau_u}$ is true,
but $\apr{d}{\tau_u}$ is false.


\subsection{Types and Proposition Structures}

A basic difference between infons and propositions is that
propositions are true or false absolutely, whereas infons may only be
determined as factual or not relative to a given situation.  Once
this perspective is adopted, it becomes tempting to consider a wider
class of propositions.  For example, whether or not a given object
$\si$ is an infon does not vary  from situation to situation, and so
we might also consider propositions determined by an object $\si$ and
a {\em type} of object, say the type $\Inf^*$ of infons, which is
true just in case $\si$ is an infon.  We write such a proposition as
$\pr{\si}{\Inf^*}$.

Extending this idea to other structural relations, we write $R^*$ to
denote the {\em type} corresponding to the structural relation $R$,
and $\pr{\al}{R^*}$ for the proposition that the objects in the
sequence $\al$ stand in the (structural) relation $R$.  This move
allows us to make do with one kind of proposition. Propositions of
the form $\apr{s}{\si}$ may be identified with those of the
form $\pr{s,\si}{\HoldsIn^*}$.

Moreover, it is now easy to see how to use our present framework to
study propositions and types.  We need structural relations $\Seq$,
$\Type$, $\Prop$, and a non-structural relation $\True$ with the
following interpretations:

\begin{tabular}{ll}
$\Seq(\al,p)$ & $\al$ is a sequence of objects in the basic
proposition $p$ \\
$\Type(T,p)$ & $T$ is the type in the basic
proposition $p$\\
$\Prop(p)$ & $p$ is a proposition\\
$\True(p)$ & $p$ is true.
\end{tabular}

The key is to observe that the axioms governing the structure of
propositions are exactly the same as those governing facts,
with the following relabelling of predicates:

$$\begin{array}{lcl}
\Ass & \rightarrow & \Seq \\
\Rel & \rightarrow & \Type \\
\Inf & \rightarrow & \Prop \\
\Fact & \rightarrow & \True \\
\end{array}$$

With this in mind, we make the following definition:

\begin{defn} A {\em proposition structure}
$\goth{P}=\relstr{\goth{A},\Fun,\Seq,\Type,\Prop;\True,\partof}$  is
a fact structure in which each structural relation in $\goth{A}$ is
represented.
\end{defn}

The requirement that each structural relation is represented is
exactly what is needed.  For example, if $\goth{A}$ is a
situation structure then it will have a binary structural relation
$\HoldsIn$, and so there must be a type $\HoldsIn^*A$, roles
$\HoldsIn^*_1$ and $\HoldsIn^*_2$, and for each situation $s$ and
infon $\si$, a function $f$ such that $\Fun(\HoldsIn^*_1,\si,f)$ and
 $\Fun(\HoldsIn^*_2,s,f)$, and a proposition $p$ such that
$\Seq(f,p)$ and $\Type(\HoldsIn^*,p)$, and
\begin{quote}
$\True(p)$ iff $\HoldsIn(\si,s)$
\end{quote}

A proposition  is {\em basic} if it has a type.  Compound
propositions, such as conjunctions, disjunctions and negations may
be modelled in the same way as compound infons. Typically, situation
theorists have proposed that the logic of propositions is classical.
This is compatible with any logic on infons, but a favourite
combination is to take a Kleene situation structure with a classical
proposition structure.  To see that this is a coherent combination,
note that the proposition $\apr{s}{\si}$ is either true or not,
and so in a classical proposition structure one of the propositions
$\apr{s}{\si}$ or $\neg\apr{s}{\si}$ is true.  If a situation $s$
fails to support $\si$ then $\neg\apr{s}{\si}$ is true, but this
does not imply that $s$ supports $\neg\si$ unless the underlying
situation structure is also (locally) classical.

Likewise, we may build models in which {\em abstract types} of the
form $\lambda x.p(x)$ exist (where $p(x)$ is a proposition for some
range of values of $x$), just as we did for abstract relations.


\medskip

The account of propositions developed by situation theorists, and
summarised here, leads to a two-story residence for logical
structure.

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On the ground floor there are infons.  Basic infons are
made up of assignments and relations; compound infons are structured
objects constructed from basic infons; and more complex relations
are formed from infons by abstraction.

On the upper floor there are propositions. Basic propositions are
made up of assignments and types; compound propositions are
structured objects constructed from basic propositions; and more
complex types are formed from propositions by abstraction.

The two floors may differ in matters of truth and factuality.
Typically, the ground floor is {\em situated}.  An infon needs a
situation to determine if it is factual or not; likewise, the
extension of a relation may vary from situation to situation. The upper
floor is {\em absolute}, the lower {\em relative}. 
 Propositions are either true or not, and
types have a fixed extensions.  The floors are linked by the
relationship between a structural relation $R$ on the ground floor
and the type $R^*$ which represents it on the upper floor.

Like all  metaphors involving a duality, the two-story picture
raises a question of redundancy: can we make do with just one of the
two, and if so, which one?
We do not have any final answers on this question,
but we believe that the conceptual and technical  work of
this chapter might help  sort out
the relationship between the absolute and the relative.

\subsection{Further Developments}

The aim of this chapter has been to give a general
framework for thinking several kinds of semantic issues.
The new feature is the development of the theory of
structural relations and its  use in building 
substitution structures, lambda structures, etc.
As we close, we want to mention some references to
papers that  either discuss issues which lead to our
work, or which  offer complentary points of  view.


For more on situation semantics, a good place to look
are the conference proceedings entitled
{\em Situation Theory and its Applications}, published
by CSLI press.  (These are the name of a biannual conference.
The name of the conference was changed in 1994
to ``Information-Theoretic Approaches to Logic, Language,
and Computation.)
We have listed in the references a number of papers from
these volumes.  One paper developing situation semantics
is Cooper (1991)
 Other source  for this are the books  Fenstad et al (1987) and
Gawron and Peters (1990). 
The latter   deals with anaphora, and the 
applications in that domain are among the key ones
for situation semantics.
 In addition,
 ter Meulen (1995)  studies the representation of
temporal information in
a related framework, as does Seligman and ter Meulen  (1993).  

Situation semantics has given rise to computational paradigms
as well.  For example, see
Tin and Akman (1993) and Tin, Akman, and Ersan (1995).
The general theory has also lead to work in
the social sciences; see Devlin and Rosenberg (1993) for
the first paper in a series of explorations 
intended to give a foundation for ethnomethodology.


In terms of the philosophical work on situation
theory, the key books are
are Barwise and Perry (1983),
Barwise (1989), 
Barwise and Etchemendy (1987),
and
Devlin (1991).
Perhaps the closest to our work here  is 
the Barwise and Etchemendy book; it was the first
work to consider the circular structures and  applied them
to build models of the Liar Paradox.  
Another work that applies circularity and the 
perspective of situations is Barwise (1989c);
this paper shows  what circularity can offer to the 
modeling of common knowledge. 



The concept of bisimulation is critical for our work.
It also plays a role in Chapter 10 of this handbook (dynamics),
where it is used to define equivalence of processes.
It  plays a simiilar role in Chapter 8 (feature logics),
but in that case the equivalence is on static structures.
The concept of bisimulation first appeared in modal logic and in work
in theoretical computer science on
the abstract notion of a process.  This latter development
lead directly to  Aczel (1988), 
the primary source on the set theoretic work of this book.
The general concepts behind the Antifoundation Axiom are
exposed in Barwise and Moss (1996), along with applications
to the theory of truth as well  as other fields.


Developments in logic motivated by situation theory include
the papers by  Aczel,  Aczel and Lunnon, Fernando, Plotkin,
Ruhrberg,
and Westerst{\aa}hl mentioned in the references.
The backdrop of these papers is that situation semantics
calls for theories which have to be formulated in new ways,
and also for models which do not come easily.  In short,
the mathematics related to the applications is still under 
development.  Some of the papers mentioned above bring in
methods from the lambda calculus, for example.
In this paper, we have gone in a different direction;
instead of basing our models on models of the lambda calculus
we have used set-theoretic structures with different  properties.


There are a number of works on the concept of 
information as we are using it in this chapter.
Among these are Dretske (1981), and Israel and Perry (1990) and (1991).
An ongoing development concerning the notion of information
has been the theory of {\em channels}.
The goal of this work is a framework for  understanding
how information flow results from lawlike regularities
in distributed systems, and to lay the logical-mathematical
foundation of a theory of information flow.
``Regularities'' in this sense admit exceptions, so the
work is also concerned with logical directions such as 
non-monotonic inference.
An early paper in this direction is Seligman (1991).
We have not exposed this line of research in this chapter,
though it would be a natural next step to take.
For more on this subject, see Barwise and Seligman (to appear).




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