Handbook of Philosophical Logic Second Edition 10

http://slidepdf.com/reader/full/handbook-of-philosophical-logic-second-edition-10 1/357
edited by D.M. Gabbay andF. Guenthner
Volume 1 – ISBN 0-7923-7018-X Volume 2 – ISBN 0-7923-7126-7 Volume 3 – ISBN 0-7923-7160-7 Volume 4 – ISBN 1-4020-0139-8
Volume 5 – ISBN 1-4020-0235-1 Volume 6 – ISBN 1-4020-0583-0 Volume 7 – ISBN 1-4020-0599-7 Volume 8 – ISBN 1-4020-0665-9 Volume 9 – ISBN 1-4020-0699-3
8/18/2019 Handbook of Philosophical Logic Second Edition 10
and
Ludwig-Maximilians-Universität München, Germany
A C.I.P. Catalogue record for this book is available from the Library of Congress.
Printed on acid-free paper
©
No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording
or otherwise, without written permission from the Publisher, with the exception
of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
ISBN 978-90-481-6431-8 ISBN 978-94-017-4524-6 (eBook)
DOI 10.1007/978-94-017-4524-6
2003 Springer Science+Business Media Dordrecht
J
ohn-J
Descriptions
Nathan
Salmon
Indexicals
since
the publication of the first edition and there have been great changes
in the landscape of philosophical logic since then.
The first edition has proved invaluable to generations of
students
and
weIl
as
to
Britannica
1999
has
are
confident
that
handbook
nity. It followed the
Logic, published in 1977, edited by the
late Jon
These areas were
which help and/or replace the human in his daily activity.
This
pressure
required
the
the computer
Philosophical Logic, which covered
these active communities, became
computer
forward,
stimulated
by
the
needs
of
of
departments
addition to
became active in
our times. Today we have a
handbook with
an
I t shows that the first edition is very close
to
topics were not included in the first edition, even though
D. Gabbay and F. Guenthner (eds.),
Handbook of Philosophical Logic, Volume 10, vii-ix.
©
viii
they were extensively discussed by all authors in a 3-day Handbook meeting.
These are:
• a chapter on combinatory logie and
A-calculus
We feIt at the time (1979) that non-monotonie logie was not ready for
a chapter yet and
moved.
1
major
cal logie, alongside default logies, labelIed deductive systems, fibring log
ies, multi-dimensional, multimodal and substructural logies. Intensive re
examinations of fragments of classieal logie have produced fresh insights,
including at time decision procedures and equivalence with non-classieal
systems.
in
with fallacy theory, informallogie and argumentation theory, attested to by
the Amsterdam Conference in Logie and Argumentation in 1995, and the
two Bonn Conferences in Practieal Reasoning in 1996 and 1997.
These subjects are becoming more and more useful in agent theory and
intelligent
and
reactive databases.
Finally, fifteen years after the start of the Handbook project, I would
like to take this opportunity to put forward my current views about logie
in computer science, computationallinguisties and artificial intelligence. In
the early 1980s the perception of the role of logie in computer science was
that of a specification
neat
data
structures and the use of logie was one of his options.
My own view at the time was
that
playa
and
to
this
rieh
and important application area and thus enhance its own evolution. The
relationship between logie and computer science was perceived as very much
like the relationship
its use as an essential tool, and so we
hoped for logie. Today my view has changed. As computer science and
artificial intelligence deal more and more
with
systems, processes, concurrency, agents, causes, transitions, communication
and control (to name a few), the researcher in this area is having more and
more in common with
1
I
am
the
omission
if
of
a
chapter,
developed in a more
the
are abstract
an similar to principles governing the cooperation of two large
organisation. A detailed rule based effective
but
and
manipulating
data.
the
same
as
formal philosopher
about
18.
The
that it
and
their
Thanks
our publication administrator Mrs J. Spurr for her usual dedication
and
excellence
Classical
databases
allows for
ing framework
Very important Becoming
of
1
INTRODUCTION
Knowledge
has
always been a topic central to philosophy (cf. e.g. [Glymour,
1992]). Since ancient times philosophers have
been
interested
in
the way
knowledge comes to us and in what way it relates to reality, the world in
which we live. As is
the
been
study these quest ons in a formal setting. These logics
are
'knowledge'. Mostly, also
the notion of 'belief', which is sometimes thought of as a weaker form of
knowledge
is debated among philosophers (cf. [Gettier, 1963;
Pollock, 1986; Voorbraak, 1993]) is considered, and we will also incorporate
this
logic(s)' for logics
approach to
adopting
style
of
or
two,
these
of
belief enabled one to
describe systems that are complicated both in physical and in logical respect:
for both so-called
ma
knowledge-based systems the concept
of knowledge is important, and to give these systems a sound basis it ap
peared
et al.,
Halpern
and
Moses,
1985; Halpern and Moses, 1990; Halpern and Moses, 1992; Moses, 1988;
Moses, 1992; Voorbraak, 1988].
we will give an overview of the development of epistemic
logic in recent years
Handbook of Philosophica/ Logic, Vo/ume
10. 1-38.
TO
The core (propositional) language
propositional atoms closed under
will treat
both knowledge and belief in the same model. Of course, one might leave
out one of the two, and then also simplify the models accordingly.)
Formally,
this
DEFINITION 1. Let
The
eore
t/J
for
-'<p
the
language with only B- (and N-) modal
ities will be denoted LB. Formulas without any occurrence of a modal oper
ator will be called objective, and for a set
q,
~ L
eore
0
(viz.
that
that
E
q,}. For
ease of notation, we will also use this convention for 0 = -'.
The
'it
is
believed
that',
while the dual operators M and N will denote possibility on
the basis of knowledge and belief, respectively.
This will be the core language
that
we will use for the moment. Later on,
we will extend this language to accommodate for further epistemic modali
ties.
truth
values.
is
3
•
per state;
the
is considered possible on the basis of the available
knowledge in state s. In some way, having a number of available alternatives
tl, t2, t3, ... for which R(s, ti) holds, expresses the knower's uncertainty about
the
the
relation
T
with respect to belief.
The interpretation of the language is now given by the following.
DEFINITION 3. Let M = (S,7l",R,T) be a given model and sES. Then:
• M, s
F
<pj
(S,7l",R,T), denoted
M F
<p, if
M ,
s
F
<p
for every sES. A formula<p is valid with respect to a set MOD
of models, denoted MOD
the
<po
2.
K<p
and F
B<p.
and
implication. Furthermore, validities are always known
and
though perhaps defensible as properties of idealized notions of knowledge
and
the
problem of logical omniscience, to which we will return in a later section.
In the
previous section
In this
with constraints
implies R(t,
and
and
T
is serial, transitive and Eudidean. We denote the dass of these models
by
S4/KD45.
model for S4-knowledge and KD45-belief such
that R
denoted
S5/KD45.
PROPOSITION
6.
3.
BB<p;
B-,B<p;
Items
are
'negative introspection'
properties. (The one for knowledge, item 6, is rather controversial, philo
sophically.)
Item
of
(K) K(<p t 'Ij;) t (K<p t K'Ij;);
(5) --,K<p-tK--,K<p
is
derivable from (K), (T) and (5) in the context of the
rules
(N
K
) and modus ponens, the system 85 can alternatively and more
concisely be defined as
KD45
(P) any axiomatisation
(D) --,BJ..;
together with the modus ponens rule and the rule
(NB) -ltp
We denote derivability of a theorem <p in a deductive systems 8 by 8
I-
<po
Let
84K +
KD45B
stand
ing
the
system
84
for
the
K-modality
KD45B.
of
Hughes
and
and
85K + KD45B I- <p
U ntil now there is no relation between the modalities of knowledge
and
belief whatsoever. To achieve a meaningful relation between these two
n(}
the
model for S4-knowledge (S5-knowledge)
s, t, U ES: R(s, t) & T(t, u) => T(s, u)
The
K
The
system
KL(S5jKD45)
(BK)
becomes derivable. Van der Hoek [1991a; 1993] has investi
gated an
edge by the positive introspection axiom (4) for knowledge,
and
adding
the
properties
other
solutions
to
and
belief
[Voorbraak, 1991; Voorbraak, 1993] argues that negative introspection for
knowledge is incompatible with the principle
that
the
S4
3 DESCRlBING KNOWLEDGE AND IGNORANCE BY MEANS OF
EPISTEMIC LOGIC
out what an
agent
knows and what it doesn't know. Sometimes describing the agent 's ignorance
is as
as giving a description of its knowledge. By using epistemie
logic one can express the an agent doesn't know a formula
<p,
standard
not
that
--,K
r,
q
and
r. In general, this would also be undesirable: it might be the case that
by asserting
an
agent is given, just as in ordinary propositional logic the assertion p 1\ q
might also be partial information of what is actually the case (e.g. p, q and
r hold ), so
In
can
Kp
to
doing this. In this chapter we will discuss two of these, viz.s(}-called
minimal
85,
whieh was in essence firstly proposed by Halpern and Moses [Halpern
and Moses, 1984b], and Levesque's 'All I know' logie [Levesque, 1990] which
has its roots in Moore's autoepistemie logic (AEL) [Moore, 1984; Moore,
1985a]. Abasie difference between these two approaches is that in the
minimal S5 approach the 'minimality of knowing' is obtained on
the
'meta
level' by considering a (non-monotonie) entailment relation, while in the
'All I know' logic of Levesque a special operator in the object language of
the
The
the
consideration
that
theory
wants
preference of 'models
contain less knowledge' over those in whieh more
is known. Considering such preferences on models falls in with a well
known stream of approaches within the area of nonmonotonie reasoning (see
Makinson's chapter in a later volume of this Handbook), where models are
ordered according to preference and mostly
the
Simple S5-models
We have seen that the S5-notion of knowledge corresponds to Kripke models
in which
equivalence relation.
by S5.
S5
M
2.
A
simple
formulas
the
set-indusion
ordering.
DEFINITION
Then MI -< M
this
DEFINITION 13. Let M = (S) be a simple S5-model. Then
the
knowledge
9
2
= (S2)
be
the
more
this
Minimal 85
N
ow
we are ready to define an entailment of the form cP r-H M Ij; expressing
that
Ij;
85.
REMARK 16. In Halpern and Moses' original approach [Halpern and Moses,
1984b]
it
preferred) model.
In this
the
entailment relation <p r-H M Ij; is defined. In the more
general approach above, which is also taken in e.g. [Lin and Shoham, 1992;
Schwarz and Truszczynski, 1994; Engelfriet, 1996],
we
also allow multiple
minimal models and consider as consequences the formulas that hold in
all of these. (In nonmonotonic reasoning this is called the skeptical ap
proach
(cf.
[Brewka, 1991].) The use of a preference relation on models
to
minimal
prominent way
ofthis
Handbook.
the
1.
4. pVq
that
this
important
take a slightly different approach to
the
concept
of
and
in
the
the
which
appears
to
the
DEFINITION
18.
B
• [w, Xl F
[w,
1jJ,
[w, Xl
E <p)
X
represents
the
entailment
However, F
'minimal' knowledge (or rather belief)
that
set. To
end we define the notion of an AE-extension (in the
literature
=
In
{B<p
=
'cB. Furthermore,
(classical) propositional closure
establishing a relation between AEL and 'standard' epistemic logic after
all.
DEFINITION
is a Kripke model
U =
{1fI1f
tE
S.
We
can
obvious way:
Let M
= (U, R) be the universal S5-model derived from the simple S5-
model M = (S). For s E U,
M,
s F B<p Hf M, t F <p for all t with
(s, t) E
interpretation
12 JOHN-JULES
eH. MEYER
One can now prove ([Vreeswijk, 1991; Meyer and van der Hoek, 1995bl)
that:
THEOREM
23.
M
be
the
of cP ijJ
=
of
U
to
AE
extensions.
simple S5-model
S, we
S*
= U\S. The dass of (enriched) universal S5-models is denoted by U.
We extend the language
I f
one calls
cP
has
an
AE-extension if it has a preferred belief model. This view renders
AEL a
even go a step further
and
we can
simply introduce
derived /rom M . Then, for 8
EU,
O<p
ijj, for alt t, t E S ~ M, t F= <p.
With
Theorem
and the
ijj
further
extend
a univer
sal S5-model as the necessity and possibility operators , respectively, on the
whole universe U of such models:
DEFINITION
29.
rived from M.
M,
•
for some
t EU.
3. D<p -+
<p
7.
1
formula
<p
9.
10.
<p
12.
<p
<p
13.
, B ~
<p
form
hte modalities 0 ,0 , B,
B* ,N
<p
16.
and
the
1994b]. •
REMARK
32.
The
logic
OK
(propositional frag
ment of
the) logic of 'All I know' as originally proposed by Levesque [1990].
A minor difIerence is that
instead of
But this
axiom fol
the
observation
(B<p
we
extra
inclusion in
the logic OK of the axiom 15, which essentially states that in a universal
model Rn
this
property
cannot be expressed by just using the modal language without
reference
to
There it
is shown
of
the duals <>1 and
one
obtains
a
complete logic with respect to Kripke models with two accessibility relations
that are each
to obtain
a complete
with
is reflexive
Assuming (2), we derive: Bip 1\ B*ip
*S5
1\
1\
D(ip
purposes, Boutilier's logic
can
thus be regarded as a kind of S4-variant of Levesque's logic
of
Halpern
for proposing
logics, they have turned out to be quite different. As
Schwarz and Truszczynski
name
a minimal simple S5-model,
whereas its analogue in AEL {Bp} has no AE-extension. The former de
scribes the knowledge of an agent containing
Kp
latter cannot really be the basis of the beliefs of
an
Bp
Minimal
Knowledge
and Truszczynski have proposed yet
another
knowledge
(as
to
be),
conservativity
with
tion of some formulas of the form q ++
ip,
=
{p V -,p}, where p is a propositional atom. Then <P r--H M
-,Kp.
set
<P'
has two minimal simple S5-models: one consisting of all truth assignment
conditions:
not
should preserve
An
agent regards its beliefs as knowledge, that is, it is unaware of
any
Schwan,;
and
(K)
structure
M
where S is a
set of possible worlds; 7f is a truth assignment function per possible world;
and
T
DEFINITION
E T,
while for the remaining worlds it is evaluated as usual
taking
the
belief
operator
can
be
=
it
operator
expresses in this logic something like 'considering it possible to
know'.
=
model
(S,7I",S).
that K<p
that
7I"N(t) =
F S T
= (S,7I")
be a reduced S5-model. We say that M is a preferred
ST-model of<p if
for every ST-model
•
states
satisfy
we
see
that
the
ST-approach is more in line with AEL, whereas in the first item above
it is more in line with minimal S5. So
we
observe
that
is
related
to
the
logic
84F,
in the following sense:
A reduced S5-model M is a preferred ST-model for <P
Hf the
(see [Schwarz
and
models gives us a number of properties, 'for free' so
to
the
the
axioms
(P),
(K)
and
the
'I,b)
L05
'I,b)
L06
may
view
this
the
eoneepts of knowledge and belief of an agent / knower, whieh might be
eonsidered eonvenient. On
situations, such as e.g.
partieular robot,
it seems unrealistie
sequenees of its beliefs
logical omniscience
in the
literature. Similar problems oeeur in other modal logies. For instanee, the
logieal omniscience problem is related
to
some
of
the
of the
we
K
ip -+
of
any
denying even
we
should
not
be
or
19
perhaps better to resort to cognitive science right away rather than trying
to
Instead
of them
which
we
want to deny while retaining the other ones, where it depends on the
applications which properties
In the
following subsection
nonstandard
models.
4.2
to
and
in
the
model
to
avoid
to
treating
and
posed in [Fagin
be
avoided,
can
state
to most radical form of using syntactical elements is called
'impossible world'
but we
the
the
• S* ~ S is the set of impossible worlds;
• a : (S\S* -+ (P -+ {tt, if}»
u
(S*
to the
s
S*.
Impossible world semantics resolves all logical omniscience as weH: also
in this semantics none of
LO
1-L07
into the
is
of
standard
where
•
DEFINITION 4l.
•
thus
this approach was first proposed by Fagin
and
T
as
the
and
involved. However,
it has
[1992] and
the authors
themselves [Fagin
et al.,
can
et al.,
above approaches are equivalent. In the literature several other approaches
have been proposed. However, these
appear
to
this
does
not
render
them
to
not
treat
all
of
~
(which is
where N :
5 --+
p(5) is a function that gives a set of worlds (the 'neighbour
hood' of its
B B'IjJ
So this approach is less general than the previous ones. Even further spe
cialisations
of
cluster semantics
by Fagin
semantics employing
the
usual
One can view a
*) as a Rantala model
F ep, if
al.
nonstandard
with
In
its
turn
this
logic
can
on
Fagin
et al.,
1995; van der Hoek and Meyer, 1989; Meyer and van der Hoek,
1995b; Blamey, 2001].
Epistemic logic has proven
describe properties
of distributed systems. Here we follow work by Halpern and co-workers
(e.g. [Fagin et
1995]), who developed 'dassical' epistemic logic further
for this purpose. One of the main contributions they made, is the enrich
ment of epistemic logic with a number of operators that express properties
of groups of 'knowers'. Moreover, by identifying these 'knowers' as the pro
cessors involved in a distributed system
they
trivial extension that is needed, is the introduction of mul
tiple K-operators: each knower (viz. processor) gets its own knowledge
operator. More interestingly, operators
knowledge.
The
will be explained later on.
In order to keep the treatment of these epistemic not ons as general as
possible (the notions of common and
distributed knowledge are useful in
any group of 'knowers'), we will in
the
of agents.
Let
Ag
be a given set of agents. For convenience, we will put Ag =
{I, . . . ,
core
dause
language
L
core
E LAg.
we
models. For this definition we use the following notation: if
R
DEFINITION 42. A Kripke model for LAg is a structure
M
RE,
Re,
RD),
where
• 8 is a non-empty set (the set of states or possible worlds;
atoms
per
state;
• R
I
accessibility relations for the agents;
• RE
= Ul <i<m Ri is the accessibility relation associated with shared
knowledge; -
• Re
with
tributed
knowledge
I
LAg by S5ECTJ
interpreted
by
means
DEFINITION 43.
PEP;
not
• M,
M,
(Kc) C('{J
(T
D
('{J
- 'L
Crp
25
This logic, and indeed already sublogics with only the D or C operator
added
on
K
i
pected problems when trying to establish completeness. For example, when
considering the subsystem consisting of those axioms and rules pertaining
to the operators K
Nc
(KD)
cf.
that
the
the
i
, so
that
more sophisticated methods are needed to obtain completeness (cf. [van der
Hoek and Meyer, 1992]). Adding all these extra operators yields
an
even
IJl
THEOREM 44. For
any'{J E
•
we can
through
(Sk,SkH)
We now have the following easy result.
PROPOSITION
46.
LetM
M ,
s F Cr.p for some S E 5 iff M F
Cr.p.
is
through' the states
epistemic logic, including questions such as whether common knowledge in
such systems is
aglobai
DEFINITION
47.
1. A run of a system is an infinite sequence of global states. A run r
over a set
y;
(r,
n)
Then we
interpreted system as folIows:
DOXASTIC
LOGIC
27
DEFINITION 48. An interpreted system I consists of a pair (R,
11"),
where
(i
=
state
s
E
point
in the system.
We associate with an interpreted system I = (R,11") a Kripke model
M I = (8,11", R
, • • • ,Rm) by taking:
• 8 is the set of points in I, i.e. in R .
• ~ r , n ) , (r',n')), mostly written as (r,n) ' i (r',n') and read as
"(r, n) and (r', n') are indistinguishable
to
agent i", holds iff r(n) ' i
r'(n'), where, for s = (se, SI, ... , Sm) and s' = ( S ~ , S ~ , ... , ~ ) , S
' i
s'
the
global
states
s
Note that a Kripke model MI = (8,11", R
1
the
this point one may wonder whether interpreted systems
are equivalent to Kripke models in the sense that also for every Kripke model
there is an associated interpreted system such that taking the associated
model of this system one ends up with a Kripke models that is isomorphie
to
the original one. This question has been addressed by Lomuscio and
Ryan [1997].
of Kripke models, viz. those models
(8,11",
R
1 ,
1. RD
2. 'VSI, ... ,
E 83s E 8 such that s ' i Si (i =
1,
. . . , m).
distributed knowledge
trivi
alizes, viz. that the formula <p t+ D<p is valid on the dass of these models,
whereas the second property states something
about
with
fuH
fuH
Cartesian
quences for
common knowledge,
the
next subsection.
Now we may define (I, r, n) 1= <p iff MI, (r, n)
1=
global
states
with
the
loeal
in all global
on the
Finally
we
define
the
notion
of
validity,
the
(I, r, n)
of synchronous
recall.
DEFINITION
51.
agent knows what time it is: there is no
doubt
in
the
the
time
information.)
• Let I = (R,7T) be an interpreted system, and let, for a point (r, n)
with respect to this system,
lssi(r, n) stand
states that agent
eonseeutive repetitions.
and
points
with perfect reeall every
vi; ,.
the
of
information.)
29
We see that these special classes of systems impose additional constraints
on
the
In
[Fagin et al., 1995] also other systems are introduced such as message
passing systems and message delivery systems. In the former send
and
receive actions (of messages) are considered as means of communication
between processes. A process' local
state
tion
about
its initial state, the messages that it has sent and received, and
the actions that it has performed internally. This enables them to analyse
the behaviour of so-called protocols between processes that are meant
to
transmit information (knowledge) from one process to another. In the com
puter science literature there have been proposed several protocols
to
trans
fer information between processes in various adverse circurnstances such as
unreliable channels between the processes. An important example is the
so-called alternating bit protocol [Halpern and Zuck, 1987]. These protocols
can now be investigated formally by means of episternic logic, and
it
the
sense
they are supposed to do.
In message delivery systems the emphasis is put on the aspect of delivery
of a message: here message delivery together with
what the
agents in
Common knowledge in distributed systems
The logic from section 5.1 can be applied straightforwardly in the setting
of distributed systems by extending the models by accessibility relations for
the
general setting.
So, formally we now consider Kripke models of the following form:
(8,7r,R
l
ofvectors ofthe
RE,
Re
and
the
general case in section 5.1.
Although this is all very straightforward, we can nevertheless obtain sorne
very fundamental results ab out the (common) 'knowledge' in a distributed
system. In particular, we will now
turn
to the issue whether common knowl
edge may increase during a computation, going from some global state to
another.
= (8,7r,R
l
this means
that
every global state s E g is encountered at sorne point (r, n)
in
the
system.
First of all, we have now the following basic easy corollary from Propo
sition 46.
7r,
R
l
7r,
R
l
Ci.p
iff
M F Ci.p.
Proof. Since a full model for the case m > 1 is strongly connected, the
result follows immediately from Proposition 46. •
This means
the
distributed
system
that
the
the last
proposition in
order to allow for an increase of common knowledge during computations.
But we can say
is
considered
to
be
to
impose
aB
simultaneous
tion step. Then, for all
i.p,
tation it is
et
et al.,
as
the
fact
that
of
the
system
(asynchronous
or
since it can
co-ordinated attack
(see [Fagin
Agents are entities that are supposed to act autonomously
and
rationally
(intelligently)
31
last decennium agents have been described by modallogics that consist of
epistemic / doxastic logic for representing their informational
attitudes
to
attitudes
[1991]
logic is
temporallogic
(see
in a rather complicated way. Possible worlds are themselves branching-time
temporal structures modelling
priori they
take the notions of Belief (B), Goal (G) and Intention (I) to be
all independent, given their own accessibility relations in the model. (Be
lief is considered in
the
possible
additional constraints on the models relating these accessibility relations in
some way. In their writings (in particular [Rao and Georgeff, 1991])
they
seem to favour the axiom that being a goal implies being a belief, which
they explain by stating that in order for something to be a goal it must be
achievable or realizable, Le., among the possibilities that you believe (possi
ble worlds
this is a reasonable explanation, it seems
to
possibility operator rather
Also in
of agents epistemic notions playa role, though somewhat smaller.
Here belief is taken as a primitive notion (in the KD45 sense) amongst such
notions as being a goal and notions pertaining
to
true
the
axiom
that
state
that
In [van der Hoek et
al., 1994]
epistemic logic is taken as a basis for describ
ing agents along with notions from dynamic logic pertaining to performing
actions.
In
this
so-called KARO framework epistemic logic is mixed with
dynamic logic (cf. Harel et al.'s chapter in Volume 2 of this Handbook) in a
way inspired by earlier work by Bob Moore [1985b], so that one can express
both the knowledge of change and the change of knowledge due
to
about
the planning
activities of agents. With this logic it is possible to express
the
portunities, viz. external possibilities, to perform actions and their results.
This
logic is extended with an ability operator to express also
the
internal possibilities or capabilities. The KARO framework has proven to
be a suitable basis for modelling
and
'informational' or 'belief-revising' ones such as observing, communication,
and reasoning by default, but also 'motivational' ones such as goals and
commitments (cf. van Linder et al., [1994a; 1994b; 1995; 1996;
1997]
and
Meyer
eH. MEYER
In order to construct agents one may use so-called agent (programming)
languages.
to
change
the
mental
state
of
the
agent,
by
(CON)GOLOG [Lesperance
et al.,
CAL
PROLOG).
However,
to the
the
mental
is satis: ied, mostly expressed by some (modal) operator. Thus,
we
true
application of epistemic logic here. It remains a research challenge for
the
future
to
investigate
the
and the
above mentioned logics for describing agents' atti tudes. Ideally one
would like to use these logics as speci: ications (and verification tools) for
programs written in such
appears
are
reasoning
of
describe how information from
'implementation' of default reasoning
by means of meta-level reasoning (e.g. [Tan and Treur, 1992)). Default
reasoning concerns reasoning
ab out what
hold
can fly;
therefore if Tweety
is a bird (and I do not know anything more about Tweety), I can conclude
(or rather: it is rational to assurne) that Tweety can
fly. I t appears
and
here briefly without plunging into
the
a level where
and
of
default
this
are
operator
r--
the
meta-level as weIl. This yields the possibility to apply default rules of
the
kind
this
by
the
formula
(Kc.p
1\
as
is
ward
this
information
der
al.,
1994]
we
have given a temporal semantics of this reasoning process where downward
reflection is modelled
paper
branching time tem
poral logic is employed to give a systematic treatment of all possibilities
of
to the
(and
ignorance)
to
beliefs,
so
that
the
form
2
X
can
be
treated
and
Treur,
1997]
via
lllinilllal 85
ral
epistemic
logic-an
default logic [Reiter, 1980].
we turned to
it
the
of knowledge?), for which several approaches have been proposed including
minimal S5
attention
in the literature. We have reviewed some of the solutions proposed
the
extremes
impossible
the
computer
science
and
the theory
and
intelligent
agents.
Of
course,
we
could
not
based
on
probability
der
Hoek
and
van
der
der Torre, Yao-Hua Tan, for discussions on topics related
to this
paper. Also
is gratefully acknowledged.
volurne 5. D. M.
Publishers,
Dordrecht,
2001.
and
pp.
33-85.
Modal
Approach,
[Brewka, 1991J G. Brewka, Nonmonotonie Reasoning: Logical Foundations 0/ Common
sense, Cambridge University Press,
[Chellas, 1980J B.F. Chellas, Modal Logic: An lntroduction, Cambridge University
Press, Cambridge / London, 1980.
42, 1990, pp. 213-261.
[Dunn and Restall, 2002J M. Dunn and G. Restall. Relevance logic. In Jlandbook
0/
Philosophical Logic, Second edition, Volume 6. D. M. Gabbay and F. Guenthener,
eds., pp. 1-128. Kluwer, 2002.
[Engelfriet,
1996J
Nonmonotonicity Monotonous,
in: J.J. Alferes, L.M. Pereira and E. Orlowska (eds.), Logics in Artificiallntelligence,
Proceedings JELIA '96, Springer, 1996, pp. 164-175.
Logic in Minimal Temporal Epistemic Logic, Technical Report, Vrije Universiteit,
Amsterdam, 1997.
[Fagin and Halpern, 1994] R. Fagin
and J.Y.
Halpern, Reasoning
[Fagin et al., 1995] R. Fagin, J.Y. Halpern, Y. Moses
and
Massachusetts, 1995.
[Fagin et al., 1984] R. Fagin, J.Y. Halpern and M.Y. Vardi, A Model-Theoretic Analysis
of Knowledge, in: Proeeedings
01 Computer
and
of Knowledge, J. 01 the ACM 38(2), 1991,
pp.
382-428.
[Fagin and Vardi, 1986] R. Fagin and M.Y. Vardi, Knowledge and Implicit Knowledge in
a
206.
ence on Theoretical Aspects 01 Knowledge, Morgan Kaufmann,
Los Altos, 1986, pp.
the Language
and Its
Applications, in: D.M. Gabbay and H.J. Ohlbach (eds.), Temporal Logie - Proc.
01
the
1st
[Get tier , 1963] E. Gettier, Is Justified True Belief Knowledge?
Analysis
to a Reflective Architecture: MILORD-II,
in: B. Bouchon-Meunier, R.R. Vager and L.A. Zadeh (eds.), Advanees in Intelligent
Computing
pp. 440-452.
[Glymour, 1992] C.N. Glymour, Thinking Things Through, an Introduetion to Philo
sophical Issues and Aehievements, The MIT Press,
Cambridge,
Massachusetts,
1992.
01
Reasoning about Knowledge,
Proeeedings 01 the 1986 Conlerence, Monterey, Morgan Kaufmann, Los Altos, 1986.
[Haipern, 1987] J.Y.
Ann. Rev. Comput. Sei.
2, 1987, pp. 37-68.
[Hai pern and Fagin, 1989] J.Y. Halpern and R. Fagin, Modelling Knowledge and Action
in
Distributed
Systems,
[Halpern
and
Moses, 1984b] J.Y. Halpern and Y.O. Moses, Towards a Theory of Knowl
edge and Ignorance, in:
AAAI, 1984, pp. 125-143.
[Halpern and Moses, 1984a] J.Y. Halpern and Y.O. Moses, Knowledge and Common
Knowledge in a
[Halpern and Moses, 1985] J.Y. Halpern
and
01
and
Common
Knowledge in a Distributed Environment, J. ACM 37(3), 1990, pp. 549-587.
[Halpern
and
Artificial Intelligence
54, 1992,
pp. 319-379.
pp.
917-962.
and
Way: Simple Knowledge-Based Derivations and
Correctness
puting,
1987,
pp.
269-280.
1999]
K. V. Hindriks, F. S. de Boer, W. van der Hoek and
J.-J. Ch.
Meyer, Agent
[Hintikka, 1962] J. Hilltikka, Knowledge and Belie/, Cornell University Press,
Ithaca
(N.Y.),
1962.
[Hughes
and
Modal logic, Methuen and Co. Ltd, London, 1968.
[Ilughes
and
Dogic,
Methuen,
Notre Dame
J. 0/
[Jaspars,
1991]
J.O.M.
0/
mentals 01 Ar·tificial Inte/ligence (WOCFAI'91), Paris, 1991, pp. 267-275.
[Jaspars, 1993J J.O.M. Jaspars, Logical Omniscience and Inconsistent Belief, in: M. de
Rijke (ed.),
J.
[Konolige, 1986] K. Konolige,
Pitman /
Morgan
Kauf
mann,
London
Time,
and Programming, Rennes, LNCS 226, Springer, Berlin, 1986.
[Kripke, 1963] S. Kripke, Semantic Analysis of Modal Logic, Zeitschrift Ir Mathematis
che Logik und Grundlagen der Mathematik 9, 1963,
pp.
67-96.
[Lesperance
Schert, Foundations to a Logical
Approach
to
Agent
Agents
II -
Agent
1037, Springer, 1996, pp. 331-346.
[Levesque, 1984] H.J. Levesque, A Logic of Implicit and Explicit Belief, in: Proceedings
0/ the National Conlerence
[Levesque, 1990] H.J. Levesque, All I Know: A Study in
Autoepistemic
and
Rationality
Maes
and
D.
[Marek and Truszczynski, 1993] V. W. Marek and M. Truszczynski,
Nonmonotonie
Logic, Context-Dependent Reasoning, Springer-Verlag, Berlin, 1993.
[Meyer and van der Hoek, 1995a] J .-J. Ch. Meyer and W. van der Hoek, ADefault Logic
Based
on
Epistemic
[Meyer
and
van
der
Cambridge
1998J
J.-J. Ch. Meyer and W. van der Hoek, 'V1odal Logics for
Representing
Handbook of Defeasible Reasoning and Uncertainty Management Systems, Vol. III,
1998.
Linder, A logical
approach to the dynamics of commitments. Artificial Intelligence 113, 1999, pp. 1-40.
[Moore, 1984] R.C. Moore, Possible-World
Semantics
Theory
Hobbs
and
R.C. Moore (eds.), Formal Theories of the Commonsense World, 1985, Ablex,
Norwood, New Jersey, pp. 319-358.
[Moses, 1988] Y. Moses, Resource-Bounded Knowledge, in: M. Vardi (ed.), Proceedings
of the
Altos, 1988, pp. 261-275.
[Moses, 1992] Y. Moses, Knowledge and Communication, in: Y. Moses (ed.), Proceed
ings
of
the
4th
Conference
on
Lon
Semantics
a
BDI-architecture,
in:
J.
Reasoning (KR'91),
[Reiter, 1978] R.
and J. Minker
(eds.), Logic and Data Bases, Plenum Press, New York, 1978, pp. 55-76.
[Reiter, 1980] R.
[Schwarz and Truszczynski, 1994] G. Schwarz and M. Truszczyllski, Minimal Knowledge
Problem: a New
Artificial Intelligence 67, 1994, pp. 113-141.
[Tan and Treur, 1992] Y.H. Tan and J. Treur, Constructive Default Logic in a Meta
Level Architecture, in:
tures IMSA '92,
[Thijsse, 1992] E. Thijsse, Partial Logic and Knowledge Representation, PhD Thesis,
Tilburg
on Agent Theories, Arrhitectures, and Languages, 1995, pp. 355-370.
[van der Hoek, 1991a] W. van der Hoek, Systems for Knowledge and Belief, in: J. van
Eijck (ed.), Logics in AI (Proc. JELIA '90), LNCS 478, Springer, 1991, pp. 267-281;
[van
der
and
van
Proceedings
of
'91,
van
2(1), 1992, pp. 81-123.
MEYER
[van
der
IIoek et al., 1994] W. van der Hoek, B. van Linder, and J .-J. Ch. Meyer, A
logic of capabilities, in: A. Nerode and Yu. V. Matiyasevich (eds.), Pmceedings of
the Third International Symposium
127-128, 1989, pp.177-194.
Logic, Logique et Analyse 133-134, 1991, pp. 251-270.
[van der Hoek and Meyer, 1992J W. van der Hoek and J.-J. Ch. Meyer, Making Some
Issues of Implicit Knowledge Explicit, Int. J. of Foundations of Computer Science
3(2), 1992, pp. 193-223.
J.-J.
Common
Knowl
[van
der
der
Formal
Semantics
of
Logic Pmgram Synthesis and Transformation - Meta-Pmgramming in Logic, 4th Int.
Workshops, LOPSTR'94
and META
352.
ester, 1994, pp. 331-335.
[van Linder et al., 1994b] R. van Linder, W. van der Hoek
and
and
Advances in Artificial Intelligence, LNCS 861, Springer-Verlag, 1994,
pp.
202-213.
[van Linder et al., 1995J B. van Linder, W. van der IIoek and J.-J. Ch. Meyer, Actions
that
Make
an
[van Linder
et al.,
ising Motivational Attitudes of Agents: On Preferences, Goals and Commitments, in:
M. Wooldridge, J.P.
[van Linder
et al.,
1997] B.
The
Dynamics of Default Reasoning, Data and Knowledge Engineering 21 (3), 1997, pp.
317-346.
[Voorbraak,
1988]
M .Y. Vardi (ed.), Proceedings of the 2nd Conf. on Theo1'l tical Aspects
01 Reasoning about Knowledge, Pacific Grove, Morgan-Kaufrnann Los Altos, 1988.
[Voorbraak, 1991] F. Voorbraak,
Rational
Belief,
in:
J.
van Eijck (ed.), Logics in AI (Proceedings 01 JELTA '90), LNCS 178, Springer,
1991, pp. 499-516.
Far as I
Complete
478, Springer,
1991, pp. 516-525.
[Wansing, 1990J H. Wansing, A General Possible Worlds Framework for Reasoning
about
[Weerasooriya et al., 1995J D. Weerasooriya, A. Rao and K.
Ramamohanarao,
and
N.R.
Jennings
(eds.),
Intelligent Agents - Pmc. of the 1994 Workshop on Agent Theor'ies, Architectures,
and Languages,
Language is primarily a medium of communication, particularly the com
munication of information. (Of course, language is more
than
for communication.) Declarative sentences are the primary vehicles for the
communication of information. Declarative sentences have various seman
tic attributes. They have truth value, either truth or falsehood. They have
semantic intensions, Le. correlated functions from possible worlds
to
truth
values,
and
l
semantic
and
prag
sentences encode information,
may be used not only to communicate
information to others, but also to record information for possible future use,
and perhaps even to register information with no anticipation of any future
1 A word or clarification is needed concerning my use of the semantic predicates 'en
code' and 'information'. The term 'information' is used here in a broad sense
to
in
Throughout
this
article we shall be concerned with discrete units of information
that
the
information
that
oJ
the
the
count noun phrase 'piece of informat ion', i.e.
as a general term whose extension is th class of pieces of information. Thus,
we
shall
write 'information that is such-and-such' to mean "pieces of information
that
information'
to
mean "different pieces of information", and so on. We shall use
the
verb
that
to
a given possible context c) a single piece of information, which is referred
to
(with
respect
to
hat' to the
A declarative
sentence mayencode (with respect to a given context) two or more pieces of information,
but
if it does so, it is ambiguous. Pieces of information encoded
by the
proper logical
consequences of
by
that
grass is green is different
information (a different piece of information) from the (piece of) information that snow
is white, though intuitively
latter included as part of the former. The sentence 'Snow
is white
the
the
information content
of a sentence is perhaps something like a class of pieces of information, closed under
logical consequence.
We shall not be concerned in this article with a notion of an amount of information
which arises in
The
informa
tion that snow is white and grass is green and Socrates is Socrates may be no more or
less information
than the
iJ
Nevertheless general considerations involving
Leibniz's Law strongly suggest
For instance,
Handbook
oJ
© 2009, Nathan Salmon.
philosophy of
language arise from these simple observations. To begin with, what is the
nature
that
pieces
of
informa
of
common with
both
of
these
are
directly
about
Socrates,
and
some
other
both
the
information
that
the
exhaust
the
an
expression
the
information
encoded by such sentences as 'Socrates is wise' and 'Socrates is snub-nosed',
and
the
the
'Socrates is wise'
and 'Plato is
which
it
is seeured which information is semantically encoded by which sentence is,
very roughly, that a sentence semantically encodes that piece of information
whose components are
manner
parallel
Bill',
where
of
the
rule
contents
complicated
difficulty arises in connection with the latter dause of the rule
and
with such quantificational locations as 'someone' in 'Sorneone is wise'.
Grammatically
to
'Socrates is wise', though logically and
semantically they are disanalogous. In 'Socrates is wise', the predicate 'is wise' attaches
to
the
singular
term
'Socrates'.
This
situation
the
information
the
A perhaps more
REFERENCE AND INFORMATION CONTENT 41
der to analyse the information eneoded by a sentenee into its eomponents,
one simply deeomposes
the
is no part of the information content
of
term'. Yet another important qualification concerns overlaid quantifiers. I t
is
necessary
to distinguish between the information contents of such constructions as:
(Al For everyone x there is someone y such
that
to
a singular propo
sition involving that individual. (Church himself applies the general method in such a
way as to invoke only Fregean functions from
pure
concepts
of
method
to the
attributes
(properties and relations) in place ofthese functions. The informat ion values ofthe quan
tifiers
'for
everyone'
and
'there
of
may
y
loves x, together with the information value of 'for everyone'. (Alternatively,
the property of
t,
where
t is the time of utterance, and the property of being loved by someone by
that
that
of
my
loves
y',
of
to
according to which the information value of a
compound
(ß)
of
sharply articulated
generate counterexamples to
Frege, according
made up
of
any
cornponent
of
'x there
information values
may suppose that the meaningful expres
sions that make up a language for example English, or some rich fragment
thereof
tives, certain
terms are
the
definite
article
'the'
'the
may be abstract
world, or a time) is
just
(First-order) n-place predicates (n >
polyadic
___
The
to
restricted,
able/sentence pairs)
which attach to (sequences of) singular terms, predicates,
or sentence, to form singular terms, predicates, or sentences, e.g. 'neces-
sarily', tense,
the
definite
description operator,
Le. the definite article 'the' in the sense of 'the one
and only', which attaches
form a singular term, and the 'that'
operator, which attaches to a sentence to form a singular term referring to
the information content of the sentence.
Sentences. We consider here only declamtive sentences, open or closed,
where an open sentence is an expression of the grammatical form of a sen
tence
but
for
the
presence of a free variable, e.g. 'x is wise'. Sentences are
the encoders of information, and when used in a particular context, are
to
true
now be stated as folIows:
What
expressions
that
is
the information
value of a singular term, such as 'Socrates'? What is the information value
of a predicate, such as 'is wise'? What is the information value of a sentential
connective? Similarly for
present article is
concerned primarily with the question of the information value of singular
terms, particularly proper names and singular definite descriptions.
THE NAIVE THEORY
One natural and elegantly simple theory of information value identifies the
information value of a singular term, as used in a particular context, with its
referent in
that
the 'Fido '-Fido
theory. Elements of this theory can be traced to ancient times. Likewise,
the
information value of a predicate, as used in a particular context, might
be identified with
to that
is, with the corresponding property
in the case of a monadic predicate, or the corresponding n-ary relation in
the case of an n-place polyadic predicate. Thus for example, the information
value
of
the predicate 'is tall' might be identified with the property of being
tall and
information value of the predicate 'is taller than' might be
identified with
(A
sophisticated
the
on a particular occasion, with a corresponding temporally indexed (and
spatially indexed, if necessary) attribute, for example
the
property of being
tall at t, where t is the time of utterance. This yields a more plausible notion
information encoded by a simple
atomic subject-predicate sentence such as 'Socrates is wise', as used in a
particular
By
sorts of entities (in a
certain
way).
On
this
the
be
the
the
the property
in
a
particular
context,
is
(I) (Declarative) sentences encode pieces
of
proposition encoded by a sentence, with respect to a given
context, is its
entity
constituents are semantically correlated systematically with expres
sions making up the sentence, typically
the
quotation
marks
and
spect
to
the
to
pp.
TI.
Oxford University Press, 1989.
REFERENCE AND INFORMATION CONTENT 45
(V) The information value, with respect to a given context, of a first-order
n-place predicate is an
n
= 1,
an
n-ary
relation
is
n
of the attached singular terms. Exceptions arise in connection with
quotation marks and similar devices.
(VI)
The
sentential connective
that
(VII)
The
quantifier or second-order predicate is a n-ary attribute, ordinarily of
the
predicates.
(VIII) The information value, with respect to a given context, of an operator
other than a predicate, a connective, or a quantifier is an appropri
ate
attribute (for sentence-forming operators), or operation (for other
types of operators), ordinarily an attribute of or operation on the sorts
of things that serve as referents for its appropriate operands.
(IX) The information value, with respect
to
is
is
theory.
It
the natural
liminary investigation into the nature and structure of information. Both
of
the
and Bertrand Russell, came
The
information
about
or
conceming
give
(or to know, etc.) the semantic content or 'meaning' of a sentence or state
ment is to give (know, etc.) its truth conditions. its notion of information
content is exemplary of the kind of notion of proposition
that
lutter
the sentence
'Socrates is wise', I assert something such that it is true if and only if the
individual, Socrates, has the property, wisdom. Moreover, what I assert is
such
that
with respect to an arbitrary possible world w if and
only if
obtains in w. It is not enough, for instance,
that
sembles or represents
w,
or
that
someone in w who fits a certain conceptual representation of the actual
Socrates be wise in w. I t must be the very individual Socrates. The naive
de re
the naive theory is its cogency
and
When lut ter
act
pick someone out, Socrates,
asserting that Socrates
to
Socrates
and
my
to) and
of
the
etc. is identified with its information value.
The
second
major
idea
uttered
in
encoded by such a sentence as
'the
definite description is a phrase which, like a sentence, has
parts
with
the
predicate 'wrote
the Republic'.
a sentence is made up of
the
parts,
the
in to
make up
is
like a sentence
at least in having information-valued parts, should also be something made
up of
INFORMATION
CONTENT
47
the information value of 'the individual who wrote the Republic' with its
referent, one should look instead for some complex entity made up partly
of something like
the Republic
than
t (which in turn is made up of the relation of having
written earlier than t, and the work the Republic), where t is the time of
utterance, and partly of whatever serves as the information value of the
definite-description operator 'the'.
author
singular proposition
but
place something involving the property of authorship of the Republic.
One extremely important wrinkle in the modified naive theory is
that
adefinite
description
sorts of singular terms, is seen
as having a two-tiered semantics. On the one hand, there is the descrip
tion's referent. This is
monadic predicate
applies, if there is only one such individual" and is
nothing otherwise. On the other hand, there is the description's informa
tion value. This is a complex made up, in
part,
of
the
other
simple singular term is
seen as having a one-tiered semantics: its information value (with respect to
a particular context) is just its referent with respect to that context). From
the
definite descriptions on the model of a proper name. Definite
descriptions are not single words but phrases, and therefore have a richer
semantic structure.
thus
of simple expressions, such as single-word connectives and quantifiers, may
be treated on the model of a proper name,
but
least in principle, capable of having a two-tiered
semantics. For example, though the information value of a sentence is its
information content, sentences might be regarded as referring
to
their
truth
values,
the
and
directions. An extremely compelling argument, due to Alonzo Church and
independently to
Gödel, however, seems to show that if a notion of
reference is to be extended to expressions other
than
to
4See Alonzo Church, 'Review of Carnap's Introduction to Semanties', The Philosoph
ical Review, 52,298-301, 1943, at pp 299-301;
and
The Philosophy
to its
to
referring expression,
then any
co-referential: (i)
<p
is exactly one such individual, (ii) trivially logically equivalent referring
expressions
are
co-referential;
and
marks,
the
a component singular
Since each
of
singular
term
reference, any attempt to extend the notion of reference to
other
sorts
propo
information content
a sentence is a complex, ordered
entity
in connection with quotation marks
and
erent with respect
an
n-place
>
quotation
marks
and
similar
devices.
128-129.
The
special
nouns in
n-place sentential connective is an attribute, ordinarily of the
sorts
of
things
that
n-place quantifier or second-order predicate is an
n-ary
attribute,
operand
operator other than a predicate, a connective,
or
of
entity
constituents
up
with
quota
tion
information value, with respect to a given context,
of
The
modified naive theory, with its theses (IV') and (V'), involves a
significant
departure
man'. This
is in
addition to
man' with
its referent,
and both
with
the
property
of being a man (at t). Instead, the modified naive the
ory
casts the dass of all men as the analogue in the case of 'is a man'
to
modified
naive theory attributes a two-tiered semantics even to single word predi
cates
as
theory
made
up
of
parts,
single-word components. The
information value of the definite description 'the tallest man' involves the
property
than any
the
the
oft
(induding
First
'The
Evening
Star
"given". According
of
precisely
the
same
components, and apparently in precisely the same way.5 Assurning a plau
sible principle of compositionality for pieces of information - according
to
the
same
the
the
point
the
first,
though
it
can
arise
5S
ee
Kote 2 above. I t has been argued, however, that the information
content
of a
sentence is a function not only of the information-values and the sequential order of the
information-valued
parts
structure
and
that
their
parts
are
Hilary
Putnam,
'Synonymity and the analysis of Belief Sentences' , Analysis, 14, L13-122,
1954, at pp.
and especially Note 8. For response, see Alonzo Church, 'Intensional
Isomorphism and Identity of Belief', loe. cit., at pp. 68-71 and my
Fr-ege
attitude contexts.
I f
but
the same heavenly body, he may sincerely
and reflectively assent to the sentence 'Hesperus appears in the evening' and
sincerely and reflectively dissent from
the
Le.
grasping their information contents. It seems, then, that Jones believes
the
Le.
that
Phosphorus appears in the
evening. This presents a serious problem for any semantic theory, since it
appears
to
the
Substitutivity
0/
Equality
or
substitution of an occurrence of any singular
term
any singular term a in any sentence, given r
a
Hesperus appears in the evening' and 'Hesperus is Phosphorus'
are both true, yet we seem prohibited from substituting 'Phosphorus' for
'Hesperus' in the first sentence. Of course, classical Substitutivity of Equal
ity is subject to certain well-known restrictions. Most notably, the inference
rule does not extend to contexts involving quotation marks and similar de
vices: the sentences
is the capital of France' are
both
the
sentence
'The
expres
sion 'the capital of France' is a five-Ietter word' is false. However, failure
of Substitutivity in propositional-attitude context poses an especially press
ing difficulty for
are unable to accommodate
apparent fact about J ones that he believes
the information encoded by the sentence 'Hesperus appears in the evening'
but does not believe the information encoded by the sentence 'Phospho
rus appears in the evening', since the theories ascribe precisely the same
information content to both sentences. Hence, the original naive theory
requires
the
contexts,
and
the modified naive theory requires the validity of a restricted,
but
the
same.
Failure of Substitutivity involving definite descriptions in temporal and
modal contexts presents a similar difficulty. For example, although the sen
tences 'In 1978 Ronald Reagan was a Republican'
and
US President
was a Republican' is false, since in 1978 Jimmy Carter was President
and
a Democrat. An analogous example involving modality, due to W. V. O.
Quine, effectively refutes
is
the sentence 'It
is
false,
rather
than
9.
Though the singular
terms '9' and 'the number ofplanets' refer to the same number, the sentence
truth,
whereas
the
sentence
'The
though true,
the
modified
contexts.
Thirdly,
there
term
sentence involving a singular term can be
true
only
term has a referent.
This appears to be a consequence of the naive theory in particular, since on
that theory, a sentence involving a singular term is
true if and
only if the
the
theory in connection with such negative existentials as 'Hamlet does
not
other
In
fact, on the (modified) naive theory the information content of any
sentence involving a (simple) singular
term
if any
sentence
'The
then,
that
term
can
have
that
then
that
not
bald,
yet
this
P or not-po
bald
probably conclude that he wears a wig".)
identity predicate,
of
that
the
sentence
about the
'Phosphorus' themselves, to the effect that they
are co-referential. I t is, of course, trivial that 'Hesperus' is co-referential
with itself, but
the
information
that
'Hesperus'
is
co-referential
and
informati veness.
an account of identity sentences in ordinary language,
and
the
true,
is
made
the
information
that
device, similar to quotation marks. This makes quantification impossible,
or at least highly problematic, as for example in 'For every
x and
x = y'. Moreover, the
account fails to solve the general problem of which Frege's puzzle is only a
special case: whereas
the
is the theory of
sentences as 'Hesperus is a planet
if
theory
outlined
above. Mill drew a distinction between what he called 'concrete general
names'
and
The
former
are
gener
'man'),
The
latter
and
attributes of
one, according to Mill, have the other as weIl. In
the
attributes.
All concrete general
names were held by Mill to have connotation. For example, 'man' was held
to connote the properties of "corporeity, animal life, rationality,
and
we
call
cording to Mill, proper names, such as 'Plato', (typically) have
denotation
but
as
'the
author
of
the name exdu
sively to that individual". This contrasts sharply with Mill's account of
proper
names.
As
. . .
is
"a
telling anything
about it".
we
doing, convey
As regards singular terms, Mill's theory resembles the modified naive
theory in
not
identify
the
informa
tion
its
formation encoded by such a sentence as 'Socrates is wise'
has at
least two
that
latter.
This
means
that
truth that
early
Frege,
though,
generalising
the
other three
puz
zles. (It is doubtful that this was part of Mill's motivation in propounding
his dual-component
of information value, though it doubtless would
have been seen as lending independent support to the theory.) By the same
token, the objections to Frege's early view apply a jortiori to the metalin
guistic component of Mill's theory of information value.
Even ignoring its metalinguistic component, Mill's
theory
names and definite descriptions -
diametrically opposed to Frege's later theory, the theory of direct
reference.
RUSSELL'S
THEORY
Inspired
by
promoted
to
theory
theory
with
or
function that assigns to
concerning
that
nmction
to
be
the
that
phrases') by supplementing
of so-called indefinite descriptions that is, restricted universal ür exis
tential quantifier phrases such as 'every woman', 'some logician',
'an
in
structor',
etc.
We
the
form
'IT(
is analysed as
(ß)[I/>ß
does
not
the
n
as 'every
in surface
and
embedded
within
any
sentential
operator
of
sentential context (as is done for example by a sentential
operator or
to
embed
the
analysis
of
'IT(
the
The other
57
for example, 'some treasurer is such that he or she must be found'. On
this reading the indefinite description 'some treasurer' is given wide scope,
and
for each additional context embedding. For example, the sentence 'Some
treasurer must not be poor' will have three readings: 'It must not be that
some treasure is poor' (narrow scope); 'It must be that: some
treasurer
is
such that he or she is not poor' (intermediate scope); 'Some treasurer is
such that he or she must not be poor' (wide scope).
Russell's General Theory of Descriptions is relatively uncontroversial
though its consequences for the theory of information value should render
the
model of a first-order predicate, the original
naive theory may take the information value of the indefinite description
'some man' to be
possessed by at least one man.
6
Using Russell's preferred notion of a propo
sitional function, the description 'some man' might be regarded as having
as its information value
the
proposition
instantiates F (where an individual x is said to in
stantiate
proposition obtained
by applying
F to
x is true). The proposition that some man is wise would
then be regarded as made up of the second-order propositional function f
and the first-order propositional function being wise.
7
an
second-order propositional function, however. Instead, he claimed
that
in
definite descriptions, and restricted quantificational constructions generally
("denoting phrases" ), "have no meaning in isolation". This claim may seem
puzzling, since for any such quantificational construction, Russell's theory
of higher-order propositional functions presents clearly identifiable entities
(such as
it
figures.
Russell was initially impressed with the fact that arestricted quantifica
tional construction, such as
of a sentence, e.g. 'A man is wise'.
He
assigned a "meaning in isolation" - for example the second-order propo-
6Cf. Richard Montague, 'The Proper Treatment
of
Thomason,
Press,
so
that
the
phrase
that
that
of
completion by a second first-order predicate. The General Theory of Descriptions also
conflicts with this extremely plausible
theory
concerning
the
logico-semantic
of
matical subject (see Note 2). In denying that restricted quantificational
constructions have meaning,
the
grammar,
however,
this
observation
information value.
quantifier phrases have information value.
On
the
general
Theory
the
x, x is
I t
something'
as
the
is wise' as a
man
the
tion
value
of
the
wise'.
There
is no place in this proposition for the second-oreler propositional func
tiOIl J
man and
wise - although the determiner 'sorne' may be regarded as contributing its
information value anel the information value of 'man' figures indirectly in the
construction
phrases such as 'every
determiner component of
rest of the description makes no contribution of its own
but
indefinite descriptions are justifiably regarded on the General Theory of
Descriptions as
Russell's Special Theory of Descriptions concerns singular definite de
scriptions, treating them as indefinite descriptions of a particular kind in
accordance with the General Theory ofDescriptions,
rather than
as semanti
surface
structure
consisting of an n-place
predicate and containing a complete definite description '1a4>0/ ' among its
n
occurrences of singular terms, is analysed into a conjunction of three
sentences:
(i)
4»;
(ii)
a = ß]
one 4»; and
.. , ß, .. . ]
, every
4>, ... ).
the
variables,
that
does
not
ß
rences of
1a4>0/,
...) ' has occur-
rences of
'1a4>0/ ' .
predicate and contains a further definite description '1"(tP,' - so
that
be analysed further by iteration of the method just given,
treat
:::)
and
the
procedure. A
definite description 'm4>O/ ' is said to be proper if conditions (i) and
(ii)
both obtain (Le. if there is exactly one individual that answers to it), and
is said
to
description '1a4>0/ ' is a necessary
condition for the truth of I l ( .. ,m4>O/, ... )'. The only other requirement
..
descriptions.
The
to
nating" definite descriptions, replacing an apparently subject-predicate sen
tence by a conjunction of quantificational generalisations.
complex generalisation:
(3ß)[(a:)(CPa == a:
= ß)&IT(. . . ,
'the author
the Republic' is regarded as semantically equivalent to the corresponding
uniqueness-restricted existential quantifier, 'some unique
author
of
cases where definite descriptions
embedded within sentential contexts,
one addit ional reading for each additional embedding. for example, the three
predicted readings of
must
not
'It
must
that
The
Special
Theory
1>'
(which
and the
'the
to
the
modified
supplement
the
similar
to
status
On both
theories, a
author
contributing the
As with
the
not
the
the
propositional TImction
(or property),
being both a unique author 0/ the Republic and wise, to the
effect
to
the
the
not
by combining his Special
or
indexicals
are
but
the
and
indexicals.
descriptive conditions associated
truncated
for
of Descriptions, analyses into (something logically equivalent to) 'There is
a unique ES'.
equivalent to) 'Something is both a unique ES
and
the
the
that Hesperus appears in
that
some
On the
the planet Venus) that it appears in the evening. Given
the
that
in the
evening. This
that
Phosphorus
appears
in
the
evening'.
the
the
evening',
about
information value of 'ES', while reading the
other
Since these are distinct pieces of information concerning different proposi
tional functions, it is
of Equality
not apply to
them. However, when
it is given that two propositional-function express ions (predicates) fjJ and
'IjJ
apply
to
fjJ
,-
some 'IjJ', will usually be interchangeable on other logical grounds. In the
special case where it is given that there is something
that
on
this
reason
propositional-attitude attributions. But
'lj;'
may still contribute differently to
the information contents of sentences in which they occur, they need not be
interchangeable when occurring within the scope
of operators,
analogous solutions to the problems of failure of substitutivity in
modal
and
temporal
manner,
not
true
of
France',
and
a
contradictory
wide-scope
reading,
'There
of
An unmodified sentence involving an improper definite description (con
cealed or not), such as 'the present king of France is
bald',
France. Its
not bald',
'there
bald',
wide-scope reading, 'Some unique present king of France is such
that
he
is
the more natural reading, but on this reading the
sentence is false, as is the original sentence. The Law of Excluded Middle
is preserved in the case of 'Either the present king of France is bald, or
the present king of France is
not bald',
of
France'
are
as
the
possibility
tion
semantically
in
accordance
with the
naive theory.
The dass
of possible
information values for genuine names was severely limited by Russell's Prin
ciple of Acquaintance: every proposition
that
posed entirely of
constituents with respect
sort
connectives,
and
be
information value of a genuine name is to
be its
to
refer
to
mental
her
con
'I'
used with introspective
deictic reference to oneself. Uses of genuine names for oneself or one's own
mental items, however, were held
to be
rare, since
singular proposition
encoded by a sentence involving such a name would be apprehended by
the
speaker of the sentence only very briefly, and never by anyone else. Even if
a speaker were
her
the
intended
be
circumvented
in this way, even when speaking abut oneself or one's own present expe
riences one might typically employ definite descriptions, disguised or not,
in lieu
are
referents seems sufficient
identity sentences involving genuine names for
an
item
of
not countenance genuine names lacking a referent; thus
the
fier,
'there
the
fact
that
the
the
the
cor
involving genuine names have a content, but are always trivially
true
and
Frege's later theory
fundamentally different. In his Begriffsschrift (1879),
Frege was generally sympathetic to something like the naive theory, propos
ing an ad hoc reinterpretation of the identity predicate to avoid the problem
of different informativeness among true identity sentences concerning
the
same individual. By the time he wrote his classic Über Sinn und Bedeutung
(1892), Frege no longer advocated any version of the naive theory. Posing
the problem of
the naive theory, Frege now proposed abandoning
that
the
modi
ther extended two-tiered semantics across
the
and even sentences.
an
to
what
is
the expression's information value. Frege's conception of sense is similar
to
that
asense.
Frege explained his notion of
the sense of an expression as something 'wherein the mode of presentation
is contained' and that 'is grasped by anybody who is sufficiently familiar
with the language or the total ity of designations to which it [the expression]
belongs;
but
with this the Bedeutung, in case it is available, is only one
sidedly illuminated."
something like a purely
conceptual representation, by mans of which a referent for the expression
is secured.
and the
referent of
the concept. Since the sense
of
a
expressions having the very same sense) must have the same
referent-
an
expression
as
the word 'concept' with a more

Documents

Handbook of Philosophical Logic Second Edition 10