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General Semantics
Author(s): David Lewis
Source: Synthese, Vol. 22, No. 1/2, Semantics of Natural Language, II (Dec., 1970), pp. 18-67
Published by: Springer
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DAVID
GENERAL
I.
LEWIS
SEMANTICS
INTRODUCTION
that all natural or artificial languages
On the hypothesis
can be given transformational
of a certain
grammars
to
it
becomes
sort,
possible
give very simple general
of interest to us
not-very-special
answers
to the
questions:
(1)
What
(2)
What
sort of thing is a meaning?
is the form of the semantic
are built up from
compounds
tuent parts?
rules whereby meanings
of
the meanings
of their consti
It is not my plan to make any strong empirical claim about language.
I want to propose a convenient
the contrary:
format for semantics
a
to
for
of
work
general enough
great variety
logically possible languages.
This paper therefore belongs not to empirical linguistic theory but to the
To
thereof.
philosophy
to
will not conform
regarding the nature of meanings
of those linguists who conceive of semantic interpreta
to sentences and their constituents
tion as the assignment
of compounds
of 'semantic markers' or the like. (Katz and Postal,
1964, for instance.)
My proposals
the expectations
Semantic
markers
are symbols:
call Semantic
in the vocabulary
of an artificial
Markerese.
Semantic
by
interpretation
to a translation algorithm from the object
items
language we may
means of them amounts merely
language Markerese.
language to the auxiliary
But we
can know
the
Markerese
translation of an English sentence without knowing
the first
thing about the meaning of the English sentence :namely, the conditions
under which
it would
be true. Semantics
is not semantics. Translation
conditions
with
no
into Markerese
treatment
of
truth
is at best a substi
tute for real semantics, relying either on our tacit competence
(at some
or on our ability to do real seman
future date) as speakers of Markerese
tics at least for the one language Markerese.
Translation
into Latin might
Synthese 22 (1970) 18-67. All Rights Reserved
Copyright
?
1970 by D.
Reidel
Publishing
Company,
Dordrecht-Holland
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GENERAL
serve as well, except insofar as
into it useful features on symbolic logic - that might
than for Latin. (See
Markerese
to build
The Markerese
SEMANTICS
19
the designers of Markerese may choose
freedom from ambiguity, grammar based
it easier to do real semantics for
make
Vermazen,
1967, for similar criticisms).
in part just because it deals with
is attractive
finite combinations
of entities of a familiar sort out
method
nothing but symbols:
of a finite set of elements
of finitely many
by finitely many applications
no
risk of alarming the ontologically
rules. There is
But
parsimonious.
it is just this pleasing finitude that prevents Markerese
semantics from
dealing with the relations between symbols and the world of non-symbols
- that
we should be
semantic relations. Accordingly,
is, with genuinely
a
more
to
find
in
that
adequate method, meanings may turn out
prepared
to be complicated,
various
infinite entities
built up out of elements
belonging
to
categories.
ontological
will
also not conform
proposals
to the expectations
of those who,
My
to the psychology
turn immediately
in analyzing meaning,
and sociology
of language users : to intentions,
and mental
ideas, or
sense-experience,
to social rules, conventions,
I distinguish
two topics:
and regularities.
as abstract
of possible
first, the description
languages or grammars
are
associated
with
semantic systems whereby
symbols
aspects of the
of
the
the
and
and
world;
second,
description
psychological
sociological
facts whereby a particular one of these abstract semantic systems is the
one used by a person or population.
comes of mixing
Only confusion
these two topics. This paper deals almost entirely with the first. (I discuss
the second elsewhere: Lewis, 1968b and 1969, Chapter V.)
are in the tradition of referential, or model-theoretic,
My proposals
semantics
descended
recent work
of Kripke
from Frege, Tarski, Carnap (in his later works), and
and others on semantic foundations
of intensional
1947 and 1963, ? 9;
1892; Tarski,
1936; Carnap,
(See Frege,
and
1971 ;Scott, 1970.)
1963
;
1964; Montague,
1960,1968,
Kaplan,
Kripke,
The project of transplanting referential semantics from artificial to natural
logic.
in various ways, by several phi
has recently been undertaken,
and
1967;
Parsons,
1968; Montague,
losophers
linguists
(Davidson,
no
I
and
have
1970b; Keenan,
1969, 1970a,
1969.)
quarrel with these
languages
indeed, I have here adapted features from several of them. I hope,
that the system set forth in this paper offers a simpler way to
however,
do essentially
the same thing. But simplicity is a matter of taste, and
efforts;
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DAVID
20
LEWIS
simplicity at one place trades off against simplicity
these trade-offs that my approach differs most from
II.
A
CATEGOR?ALLY
grammar
1964, Part
categorial
Bar-Hillel,
the following
BASED
It is in
elsewhere.
the others.
GRAMMARS
in the sense of Ajdukiewicz
II) is a context-free
phrase
1935;
(Ajdukiewicz,
structure grammar of
sort.
First, we have a small number of basic categories. One of these is the
category sentence (S). Others might be, for instance, the categories name
(N) and common noun (C). Perhaps we can get by with these three and
no more;
went so far as to do without
the category
indeed, Ajdukiewicz
common noun. Or perhaps we might
do better to use different basic
categories ;we will consider dispensing with the category name in favor
noun
verb phrase
of an alternative
basic category
(VP), or perhaps
phrase
(NP).
c, cx, ...,
Second, we have infinitely many derived categories. Whenever
or
we
a
are
have
derived
either basic
any categories,
derived,
cn(n^\)
category which we will write (c/c1... cn). (However, we will usually omit
the outermost
parentheses.)
Third, we have context-free
c-+(c/c1...cn)
corresponding
c,cl9...9cn,
+
phrase-structure
c1+?+cH
to each derived
the
result
of
rules of the form
category.
concatenating
That
any
is to say: for any categories
expression
of
category
of category cl9 then..., and finally any
then any expression
(c/c^.Xj),
an
we
expression of category c. Accordingly,
expression of category cn is
...
and a cn and makes a c. The
takes a ct and
will say that a (c/ci...cn)
rules are implicit in the system of derived categories.
phrase-structure
a lexicon wherein finitely many expressions - words
we
have
Finally,
- are
or word-like morphemes
assigned to categories. The categories of
these lexical expressions may be either basic or derived; unless some
lexical expressions belong to derived categories, no non-lexical compound
can be generated. Notice
that although
there are infinitely
expressions
and infinitely many
derived
rules,
many
phrase-structure
categories
nevertheless
with
any given
lexicon
all but finitely many
categories
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and
GENERAL
rules will
be unemployed.
This
SEMANTICS
is true even
21
though many
l?xica will
generate infinitely many compound
expressions.
To specify a categorial grammar, we need only specify its lexicon. The
rest is common
to all categorial grammars. Consider
this lexicon:
<a
(S/(S/N))/C><pig
(believes
(every
(grunts
(is
(loves
(Petunia
(S/N)/S>
(S/(S/N))/C>
S/N>
(S/N)/N>
(S/N)/N>
N>
C>
(piggishly
(Porky
(something
(the
(which
(yellow
(S/N)/(S/N)>
N>
S/(S/N)>
(S/(S/N))/C>
(C/C)/(S/N)>
C/C>
It gives us a categorial grammar which is simply a notational
this rather commonplace
context-free
grammar:
NP+VP
VP+Npr
Adv+VP
VP
NP
Nco
Adj
Vt+Npr
Vs + S
Art+Nco
Adj + Nco
Rel + VP
Npr
NP'
J
of
Porky
[Petunia
something
Nco
pig
VP
vt
variant
grunts
J
loves
[is
Vs
Art
believes
a
every
the
Adj
yellow
Adv
piggishly
Rel
which
There are three peculiarities
about the grammar. First, proper nouns are
noun
from
distinguished
phrases. Proper nouns or noun phrases may be
word order) but only proper nouns may
with
different
subjects (though
be objects. Second, there is nothing to prevent inappropriate
iteration of
Third, the word order is sometimes odd. We will see later how
these peculiarities may be overcome.
The employed rules in this example are the eight phrase-structure
rules
to the eight employed derived categories.
corresponding
modifiers.
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DAVID
22
LEWIS
In this example, I have used only derived categories of the form (c/c^
that take a single argument. I shall adopt this restriction for the most part
in practice, but not in principle.
It is apparent that categorial grammars of this sort are not reasonable
grammars for natural language. For that matter, they are not reasonable
- the
grammars for most artificial languages either
exception being sym
bolic logic in Polish notation. Hence,
their
despite
elegance, categorial
have largely been ignored since the early 1950's. Since then,
we
have become interested in the plan of using a simple phrase
however,
structure grammar as a base for a transformational
grammar. The time
grammars
therefore
mars,
seems ripe to explore categorially based transformational gram
by taking an Ajdukiewicz
categorial grammar as base and
a transformational
far as I know, this proposal
So
component.
obtained
adding
has been made
only once before
(Lyons,
1966), but
it seems an obvious
one.
to the
that by adding a transformational
component
our
we
of
could
the
word
order
and
categorial grammar
example,
rectify
out
we
filter
iterations of modifiers.
Less obviously,
could
inappropriate
com
provide for noun phrase objects by means of a transformational
lexical items - items that need never
ponent together with a few additional
It is obvious
appear in the final generated
If reasonable
categorially
sentences.
can be
transformational
grammars
to
can
all
of
if
this
interest
and
be
under
the
done
us,
given for
languages
are to be determined entirely by base structure,
constraint that meanings
so that the transformational
is irrelevant to semantics, then
component
:
it becomes extremely easy to give general answer to the questions What
based
is a meaning? What
is the form of a semantic
how this can be done.
III. INTENSIONS
projection
rule? Let us see
FOR BASIC CATEGORIES
In order to say what a meaning
is, we may first ask what a meaning
and then find something that does that.
A meaning
under which
for a sentence
the sentence
of the sentence
various places,
in various
for various
is something that determines
is true or false. It determines
possible
speakers,
does,
the conditions
the truth-value
states of affairs, at various times, at
and so on. (Imean this to apply even
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GENERAL
SEMANTICS
23
to non-declarative
Similarly, a meaning
but postpone
of them.)
consideration
sentences,
for a name is something that determines what thing,
states of affairs, at various
the name names in various possible
if any,
times, and so on. Among
'things' we include things that do not actually
exist, but might exist in states of affairs different from the actual state of
affairs. Similarly,
determines which
noun
a meaning
(possible
to in various
applies
for a common
noun
is something
that
common
if
that
any,
actual)
things,
states of affairs, at various times,
possible
or
so on.
and
We call the truth-value of a sentence the extension of that sentence;
we call the thing named by a name the extension of that name; we call
the set of things to which a common noun applies the extension of that
common
noun.
The
extension
in one of these three cate
of something
and, in general, on other things as well:
on its meaning
the world, on the time of utterance, on the place of utter
ance, on the speaker, on the surrounding discourse, etc. It is the meaning
of
which determines how the extension depends upon the combination
gories depends
on facts about
sort of things determine how something
factors. What
on something else? Functions, of course; functions
in the most
in
which
set-theoretic
domain
of
the
sense,
arguments and the
general
sort
of
of
of
values
consist
entities
may
any
range
whatever, and in which
it is not required that the function be specifiable by any simple rule. We
have now found something to do at least part of what a meaning
for a
other
relevant
depends
sentence, name, or common noun does :a function
an appropriate
extension when given as input a
factors on which the extension may depend. We
package of relevant factors an index; and we will
indices to appropriate
an
extensions
for a sentence,
yields as output
package of the various
will call such an input
which
call any function from
name, or common noun
intension.
Thus an appropriate intension for a sentence is any function from indices
an appropriate intension for a name is any function from
to truth-values;
to
intension for a common noun is any
indices
things ; an appropriate
function from indices to sets. The plan to construe intensions as extension
originated with Carnap. (Carnap, 1947, ? 40, and
us call such functions Carnapian
let
intensions. But
1963.) Accordingly,
as
whereas Carnap's extension-determining
functions take
their arguments
models or state-descriptions
representing possible worlds, I will adopt the
determining
functions
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24
DAVID
LEWIS
be
1968; Scott, 1970) of letting the arguments
(Montague,
suggestion
of miscellaneous
factors relevant to determining
extensions.
packages
We may take indices as ?-tuples (finite sequences) of the various items
that may enter into determining
other than meaning
extensions. We call
these various
of the index, and we shall assume
some arbitrary fixed order.
items coordinates
that the
are given
we
must
sentences
coordinate. Contingent
have a possible-world
First,
on
so
are true
for
their
truth
value
facts
about
the
and
world,
depend
coordinates
at some possible worlds and false at others. A possible world corresponds
to a possible totality of facts, determinate
in all respects. Common nouns
at different possible worlds;
and so do
also have different extensions
some names, at least if we adopt the position
in
Lewis, 1968a)
(defended
that things
strong
are related
to their counterparts
rather than identity.
in other worlds
by ties of
similarity
to
Second, we must have several contextual coordinates corresponding
familiar sorts of dependence on features of context. (The world coordinate
itself might be regarded as a feature of context, since different possible
utterances
of a sentence are located in different possible worlds.) We
must have a time coordinate,
in view of tensed sentences and such sen
tences as 'Today is Tuesday'; aplace coordinate, in view of such sentences
as 'Here there are tigers' ; a speaker coordinate in view of such sentences
as T am Porky' ; an audience coordinate in view of such sentences as
'You are Porky' ; an indicated-objects
coordinate in view of such sen
tences as 'That pig is Porky' or 'Those men are Communists';
and a
previous discourse coordinate in view of such sentences as 'The afore
mentioned
pig is Porky'.
to have an assignment
it
coordinate: an infinite
is convenient
Third,
as
of
sequence
giving the values of any variables that
things, regarded
occur free in such expressions as 'x is tall' or 'son of y9. Each variable
employed in the language will accordingly be a name having as its inten
sion, for some number n, the nth variable intension: that function whose
value, at any index /, is that thing which is the nth. term of the assignment
may
coordinate
thing is the extension, or value, of the variable at /.
there is more than one possible
thing, the variable
are distinct: nothing
is both the ?xth and the n2th variable
of /. That
that because
(Note
intensions
intension
nt and n2.) The extensions of 'x is tall'
on the assignment and world coordinates
of indices
for two different numbers
of 'son of y9 depend
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GENERAL
SEMANTICS
25
just as the extensions of T am tall' of'son of mine' depend on the speaker
cannot naturally
coordinate
and world coordinates. Yet the assignment
among features of context. One might claim that variables
do not appear in sentences of natural languages; but even if this is so,
it may be useful to employ variables in a categorial base. In any case, I
be included
seek
sufficient
generality
to accommodate
languages
that do
employ
variables.
But
Perhaps other coordinates would be useful. (See the Appendix.)
let us stop here, even though the penalty for introducing a superfluous
is mere clutter, while the penalty for omitting a needed one
coordinate
is inadequacy. Thus an index is tentatively any octuple of which the first
the second coordinate
coordinate
is a possible world,
of
is a moment
a
a
is
the
fourth
the
third
coordinate
coordinate
is
person
time,
place,
is a
(or other creature capable of being a speaker), the fifth coordinate
set of persons (or other creatures capable of being an audience), the sixth
coordinate
is a set (possibly empty) of concrete things capable of being
is a segment of discourse,
and the
at, the seventh coordinate
an
is
infinite
of
coordinate
sequence
eight
things.
our functions from indices to extensions, are designed to
Intensions,
pointed
do part of what meanings
do. Yet they are not meanings;
for there are
in meaning
in
differences
differences
It
intension.
unaccompanied
by
would be absurd to say that all tautologies have the same meaning,
but
they have the same intension ; the constant function having at every index
are part of the way to meanings,
however, and
they are of interest in their own right. We shall consider later what must
be added to an intension to obtain something
that can do all of what a
the value
truth. Intensions
does.
meaning
We may permit Carnapian
from
intensions to be partial functions
some
name
indices. A
indices, undefined at
may not denote anything at
a given possible world.
'Pegasus', for instance, denotes nothing at our
so
be taken as undefined at any index having
its
intension
may
world,
our world as its world coordinate. A sentence that suffers from failure
of presupposition
is often
to lack a truth-value
(for instance in
Strawson,
1950; Keenan,
1969; McCawley,
1969). If we adopt this treat
ment
of presupposition,
sentences
to lack of truth-value
susceptible
should have intensions that are undefined at some indices. They might
even have intensions that are undefined at all indices; a sentence with
thought
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DAVID
26
LEWIS
inconsistent
function,
Hitherto
tensions
should
presuppositions
defined at no index.
have
as
its intension
the empty
of 'things'. Things are name ex
spoken uncritically
name
sets of things are common-noun
of
and values
intensions;
I have
and values
of common-noun
sequences of things
set of
the
Change
underlying
sets
of
and
the
extensions,
indices,
change
Carnapian
intensions. What,
then, are things? Of course I want to say, once and for
all: everything is a thing. But I must not say that. Not all sets of things
extensions
are assignment
things and we
coordinates
of
intensions;
indices.
can be
things; else the set of things would be larger than itself. No
at certain
intension can be a thing (unless it is undefined
Carnapian
... a member
a
of
of
must
We
it
would
be
member
itself.
else
indices);
of extensions,
the above definitions
indices, and Carnapian
of compositional
definitions
intensions,
(and the coming
a
as
to
set
of
relativized
chosen
and
l?xica)
things. Can
tacitly
meanings,
we choose the set of things once and for all? Not quite; no matter what
set we choose as the set of things, the system of intensions defined over
intensions for certain terms - 'intension', for
that set will not provide
to that choice.
instance
of the semantic metalanguage
corresponding
understand
intensions
Consider
extension
language
must move
the language of this paper (minus this paragraph) with the
semantic meta
of 'thing' somehow fixed; it is an adequate
for some languages but not for itself. To do semantics for it, we
language in which 'thing' is taken more inclusive
ly; to do semantics for that language we must move to a third language
in which 'thing' is taken more inclusively still; and so on. Any language
can be treated in a metalanguage
in which
'thing' is taken inclusively
to a second
limited by the
enough; but the generality of semantics is fundamentally
fact that no language can be its own semantic metalanguage
(Cf. Tarski,
no
can
But
universal
semantic
be
hence
there
and
metalanguage.
1936)
we can approach generality as closely as we like by taking 'thing' in
enough. For the remainder of this paper, let us proceed on the
that the set of things has been chosen, almost once and for
assumption
all, as some very inclusive set: at least as the universe of some intended
clusively
model
possible,
set theory with all the non-sets we want, actual or
as individuals. Let us ignore the sequence of semantic
that still escape treatment.
of standard
included
metalanguages
In that case there is overlap
between
things,
sets of things, and truth
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SEMANTICS
GENERAL
27
(Not all sets of things can be things, but some should be.) More
if we adopt the
sets and truth-values
over, there is overlap between
common conventions
of identifying the truth-values truth and falsity with
values.
1 and 0 respectively, and of identifying each natural number
extensions and in
Thus the appropriate
the set of its predecessors.
the numbers
with
for
function
that is the intension
of the name
sentences,
names,
of all contradictions
we want
however, is harmless. Whenever
a
of
intensions by ordered pairs
category
for that category.
FOR
INTENSIONS
The
overlap.
is also
same
the intension
noun
'zero' and of the common
IV.
nouns
common
and
tensions
'round square'. Such overlap,
to get rid of it, we can replace
and an intension
DERIVED
appropriate
CATEGORIES
it is best to foresake extensions and Carna
Turning to derived categories,
for instance, a
in the interest of generality.
Sometimes,
pian intensions
- that
- has an extension like that of a common noun :
an
is,
C/C
adjective
a set of things to which (at a given index) it applies. Probably
'married'
is such an extensional adjective. But most adjectives do not have exten
sions. What
is the set of things to which
'alleged' applies? An alleged
is not something which is, on the one hand, an alleged thing
Communist
and, on the other hand, a Communist.
In
general,
an
takes
adjective
common
and
the
noun
a common
to make
a new,
the new common
common noun in a manner
of
intension
com
noun
noun;
deter
depends on the intension of the original
of the adjective. A meaning
for an adjective,
mined by the meaning
intension
therefore, is something that determines how one common-noun
pound
depends on another. Looking for an entity that does what a meaning does,
we are led to say that an appropriate
intension for an adjective is any
intensions. In
function from common-noun
intensions to common-noun
more
from
detail
indices
: it is a function whose
to sets. Thus
domain
the intension
and range consist of functions
'alleged' is a function that,
of
'windshield', or,
given as argument the intension of 'Communist',
'chipmunk' yields as value the intension of the compound common noun
or 'alleged chipmunk' respec
'alleged Communist',
'alleged windshield',
when
tively. Note
common-noun
that
it would
extensions
not work
to use
(sets) to common-noun
instead
a function
extensions
from
;for at certain
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28
DAVID
LEWIS
and 'Maoist' have the same extension but 'alleged
and 'alleged Maoist' do not - or, at other indices, vice versa.
generally, let us say that an appropriate intension for a (c/c1...cn),
'Communist'
indices
Communist'
More
where
function
from
cn are
and
...,
c, cu
any
basic
categories,
..., and
^-intensions,
or
is any
derived,
?-place
to c-intensions.
?-?-intensions
That
is,
it is any function (again in the most general set-theoretic
sense) having as
as
its range of values a set of ?-intensions,
its
domain
of first
having
as
set
of
the
and
its
domain
of nth
...,
arguments
having
q-intensions,
set
arguments
the
and makes
a c by concatenation;
takes
of
a q-intension
and
as function
?-intension
A
??-intensions.
... and
(c/c1...cn)
correspondingly,
a ??-intension
value. We
will
a ct
takes
as
call these
... and
and
a cn
a (?/q.-.O-intension
and
arguments
intensions
a
makes
for derived
some of my
intensions. (Intensions
resembling
compositional
in Kaplan,
intensions are discussed
1964; in Scott, 1970;
compositional
- in
and - as appropriate
intensions for adjectives and other modifiers
1970a. The latter discussion
is due in part
1968 and Montague,
Parsons,
to J. A. W. Kamp.) The general form of the semantic projection rules for
categories
an interpreted categorial grammar is implicit in the nature of composi
tional intensions, just as the general form of the phrase-structure
rules
is implicit in the nomenclature
for derived categories.
The result of
a
concatenating
with
(c/ci...cn)
intension
a
<j>09 ct with
intension
(?>u ...,
and a cn with intension <j>nis a c with intension 0o(^i---0?)
We have considered
already the derived category adjective C/C.
another example, take the derived category verb phrase, S/N.
For
to make a sentence. (We rely on the trans
to
component
change the word order if necessary.) An ap
- an
- is therefore
a
for
verb
intension
phrase
propriate
S/N-intension
a function from name intensions to sentence intensions. That is, it is a
A verb phrase
takes a name
formational
function
from functions
to truth values.
The
from
to things to functions from indices
'grunts', for instance, is that function
any function (?)lfrom indices to things,
indices
intension
of
(j)whose value, given as argument
is that function <?>2
from indices to truth values
truth if
,
... _
*2W"
world
such that, for any index i,
grunts
</>!(/) is something which
and time given by the appropriate
dinatesofi
falsity
otherwise.
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at
the
coor
GENERAL
SEMANTICS
29
rule, we find that the sentence 'Porky grunts' is
i such that the thing named by 'Porky' at i grunts
that is the world coordinate of i at the time which
the projection
Applying
true at just those indices
at the possible world
is the time coordinate
it comes
is spurious;
intension of a word
of /. (The appearance of circularity
of the fact that I am using English
of English.)
in this account
to specify
the
example, take the derived category adverb (of one sort),
An
adverb of this sort takes a verb phrase to make a verb
(S/N) (S/N).
so
an
intension for such an adverb - an (S/N)/(S/N)
appropriate
phrase;
intension - is a function from verb-phrase
in
intensions to verb-phrase
For another
from functions
tensions; or, in more detail, a function from functions
to func
from indices to things to functions from indices to truth-values
tions from functions from indices to things to functions from indices to
truth-values.
I deliver functions
I promised
from functions
from func
simplicity;
to functions from functions
to functions. And worse
tions to functions
is in store ifwe consider the sort of adverb that modifies ordinary adverbs :
the category ((S/N)/(S/N))/((S/N)/(S/N)). Yet I think no apology is called
are complicated
but the principles
of their
constructs,
are extremely simple. The situation is common : look at any
account of the set-theoretic construction
of real numbers, yet recall that
the real numbers rather well.
children often understand
to find simpler intensions, but at
In some cases, it would be possible
for.
Intensions
construction
an exorbitant
cost: we would
have to give up the uniform function-and
form for semantic projection
rules. We have noted already
arguments
that some adjectives are extensional,
though most are not. The exten
sional adjectives could be given sets as extensions
and functions from
indices to sets as Carnapian
intensions. Similarly for verb phrases: we
iff there is a function (/>from indices
may call a verb phrase extensional
to sets such that if (f)l is the (compositional)
intension of the verb phrase,
name
is
/ is any index, then
is
and
intension, 03
(?)2 any
(?)i(4>2)9
,
._
is a member
{truth if 02(0
otherwise.
{falsity
of </>(/)
If there is any such function 4>, there is exactly one; we can call it the
intension of the verb phrase and we can call its value at any
Carnapian
index / the extension of the verb phrase at /. 'Grunts', for instance, is an
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DAVID
30
LEWIS
verb phrase; its extension at an index i is the set of things that
and the
grunt at the world and the time given by the world coordinate
time coordinate of the index /. Verb phrases, unlike adjectives, are ordi
but Barbara Partee has pointed
out that the verb
narily extensional;
seems
to
in
'The
of
milk
is
be non-extensional.
price
phrase
rising'
extensional
There is no harm in noting that extensional adjectives and verb phrases
intensions as well as compositional
have Carnapian
intensions. However,
it is the compositional
intension that should be used to determine
the
of
intension
an
or
extensional-adjective-plus-common-noun
extensional
If we used the Carnapian
combination.
intensions,
verb-phrase-plus-name
we would have a miscellany
of semantic projection
rules rather than the
uniform
function-and-arguments
rules using Carnapian
projection
for reconstructing
compositional
with
the
rule. (Indeed, the best way to formulate
intensions might be to combine a rule
intensions
rule
function-and-arguments
we would sacrifice generality:
Moreover,
verb phrases
would
have
or not at all. This
ones,
of adjectives;
but not
to be treated
from Carnapian
intensions
for
compositional
intensions.)
non-extensional
adjectives and
from
the
extensional
separately
loss of generality would be serious in the case
in the case of verb phrases since there are few,
verb phrases.
if any, non-extensional
For the sake of generality, we might wish to take account of selection
restrictions
intension to be undefined
for
by allowing a compositional
some arguments of appropriate type. If we thought that 'green idea' should
lack an intension, for instance, we might conclude that the intension
intensions
'green' ought to be a partial function from common-noun
of
to
common-noun
intensions, undefined for such arguments as the intension
of 'idea'. It proves more convenient,
however, never to let the intension
be undefined but rather to let it take on a value called the null intension
(for the appropriate category). The null intension for the basic categories
will be the empty function;
the null intension for any derived category
that
will
be
whose value for any combina
(c/cl...cn)
(?/?^..??)-intension
tion of appropriate arguments
is the null intension for ?. Thus the inten
sion of 'green', given as argument the intension of 'idea', yields as value
the null intension for the category C. The intension of the adverb 'furious
ly', given as argument the intension of 'sleeps', yields as value the null
intension for the category (S/N, and that in turn, given as value any name
intension, yields as value the null intension for the category S. (I dislike
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GENERAL
this treatment
of selection
SEMANTICS
restrictions,
31
the option
but provide
for those
who want
it.)
It is worth mentioning
that my
of
and
the
categories,
corresponding
pendent of my account of
and N-intensions
S-intensions
or ideas in someone's
mind
of
intensions
for derived
form for projection
rules, is inde
for basic categories. Whatever
intensions
- even
of Markerese
may be
expressions
- it still is
possible
from N-intensions
functions
account
to S-intensions
of 'Porky grunts' by applying the intension
intension of 'Porky' as argument.
V.
to take S/N-intensions
as
and to obtain the intension
of 'grunts' as function
to the
MEANINGS
have already observed that intensions for sentences cannot be iden
in meaning - for instance, between
tified with meanings
since differences
in intension. The
may not carry with them any difference
tautologies
same goes for other categories, basic or derived. Differences
in intension,
We
in meaning. For fine differences in
say, give us coarse differences
we
to
must look
the analysis of a compound
into constituents
meaning
For instance 'Snow is
and to the intensions of the several constituents.
we may
or it isn't' differs finely in meaning
from
of the difference
in intension between
white
because
'Grass is green or it isn't'
sentences
the embedded
and 'Grass is green'. For still finer differences in meaning
look in turn to the intensions of constituents of constituents,
and
'Snow is white'
we must
on.
so
Only
when
we take sameness
Carnap,
'analytic
we
come
of intension
1947, ? 14, on
to
lexical
non-compound,
constituents
can
as a sufficient condition
'intensional
isomorphism';
of synonymy. (See
C. I. Lewis, 1944, on
meaning'.)
It is natural,
with semantically
inter
therefore, to identify meanings
their terminal nodes: finite ordered trees
preted phrase markers minus
intension. If we asso
having at each note a category and an appropriate
ciate a meaning of this sort with an expression, we are given the category
of the expression;
and if the expression
is compound, we
are given also the categories and intensions of its constituent parts, their
constituent
parts, and so on down.
and
intension
Perhaps we would thereby cut meanings
will be unable to agree with someone who
too finely. For instance, we
says that a double negation
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DAVID
32
LEWIS
has the same meaning as the corresponding
affirmative. But this difficulty
does not worry me :we will have both intensions and what I call meanings,
and
sometimes
one and
as an
be preferable
some
about meanings.
Perhaps
can also be found, but I doubt that there
the other will
sometimes
of our
explication
ordinary
entities of intermediate fineness
discourse
is any uniquely natural way to do so.
It may be disturbing that in our explication of meanings we have made
- for
in an index.
instance, of the order of coordinates
arbitrary choices
- how can we choose to construct them in one
are
Meanings
meanings
to set
is a general objection
way rather than another? The objection
so
not
I
will
theoretic constructions
Benacerraf,
(see
1965),
reply to it
are
here. But if it troubles you, you may prefer to say that real meanings
I call 'meanings' do duty for
sui generis entities and that the constructs
a
because there is
real meanings
natural one-to-one
be
correspondence
tween
them and the real meanings.
also be disturbing
that I have
spoken of categories without
a matter of arbitrary choice;
are.
what
This
is
they
saying
again
we might, for instance, take them as sets of expressions
in some language,
or as sets of intensions, or even as arbitrarily chosen code-numbers.
It
It might
hitherto
turns out
to be most convenient,
to identify cate
if slightly unnatural,
own
names:
in
with
their
the proper way
expressions
gories
composed
out of the letters 'S', 'N', 'C (and whatever others we may introduce later
in considering
revisions of the system) together with parentheses
and
our
slashes. This does not prevent
from being
category-names
diagonal
names of categories:
involving
they name themselves. All definitions
categories
are to be understood
categories
and
in accordance
with
the identification
of
category-names.
Some might even wish to know what a tree is. Very well : it is a function
that assigns to each member of the set of nodes of the tree an object said
to occupy or be at that node. The nodes themselves are finite sequences
of positive numbers. A set of such sequences is the set of nodes of some
tree iff, first, it is a finite set, and second, whenever
it contains a sequence
<[b1.. .bk} then it also contains every sequence that is an initial segment
of <?>i..A> and every sequence <ei..A-i?>fc>
with 0k<0k- We regard
zero
as
of
the
the
sequence
topmost node; (b{) as the Z^th
< >,
length,
node from the left immediately beneath < >; (bt b2} as the b2th node
from
the left immediately
beneath
<[b{); and so on. We
can easily define
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GENERAL
SEMANTICS
33
notions
of tree theory in terms of this construction.
identified meanings with semantically
interpreted phrase
it becomes natural to reconstrue
the phrase-structure
rules of
all
the requisite
Once we have
markers,
grammar, together with the corresponding
projection rules, as
of well-formedness
for meanings.
1968.) Ac
(Cf. McCawley,
cordingly, we now define a meaning as a tree such that, first, each node
is occupied by an ordered pair <? 0> of a category and an appropriate
intension for that category; and second, immediately beneath any non
categorial
conditions
terminal node occupied by such a pair <? <?>are two or more nodes, and
these are occupied by pairs <?0 0O>, <?x^X
..., <?? 0?> (in that order)
such that ?o is (c/c1...cn) and (?>is ^oO^i---^?)
A meaning may be a tree with a single node; call such meanings simple
and other meanings
are, as it were,
compound. Compound
meanings
in
built up from simple meanings
which several meanings
by steps
(simple
or compound)
are combined as sub-trees under a new node, analogously
are built up by concatenating
in which expressions
shorter
a
a
a
m'
constituent
call
We
may
meaning
of meaning m iff
expressions.
m' is a subtree of m. We may say that a meaning m is generated by a set
of m belongs to that set.
iff every simple constituent
of simple meanings
to the way
generally, m is generated by a set of meanings
(simple or compound)
iff every simple constituent of m is a constituent of some constituent of m,
possibly itself, which belongs to that set.
as though they were
We shall in many ways speak of meanings
sym
More
expressions generated by an interpreted categorial grammar, even
is the
though they are nothing of the sort. The category of a meaning
as
of
its
intension
found
the
first
node.
The
topmost
category
component
bolic
of its top
is the intension found as the second component
of a meaning
name
most node. The extension at an index i of a sentence meaning,
or
common-noun
is the value of the intension of the
meaning
meaning,
i. A sentence meaning
is true or false at i
for the argument
or
a
as
names
name
at
i
truth
its
is
extension
falsity,
according
meaning
at i that thing, if any, which is its extension at /; and a common-noun
meaning
meaning
we have
derived
things belong to its extension at /. As
applies at i to whatever
in
seen, extensions might also be provided for certain meanings
a
non
as
or
cannot
such
but
be
done
in
this
S/N,
categories
C/C
artificial, general way.
Given as fundamental
the definition
of truth of a sentence meaning
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at
DAVID
34
LEWIS
an index, we can define many derivative
truth relations. Coordinates
of
the index may be made explicit, or may be determined
by a context of
over all coordinates,
utterance, or may be generalized over. Generalizing
we can say that a sentence meaning
is analytic (in one sense) iff it is true
over the world and assignment
at every index. Generalizing
coordinates
be determined by context, we can
and letting the remaining coordinates
is analytic (in another sense) on a given
say that a sentence meaning
occasion
iff it is true at every index / having as its time, place,
and previous-discourse
coordinates
indicated-objects
speaker,
audience,
ively the time, the place, the speaker,
to, and the previous discourse
pointed
over
the time and assignment
cluding world) be determined
respect
the audience,
the set of objects
on that occasion. Generalizing
coordinates
and letting the others (in
we
define eternal truth of a
by context,
over the assignment
on an occasion;
co
sentence meaning
generalizing
ordinate and letting all the rest be determined
by context, we define
and so on.
simply truth on an occasion;
We also can define truth relations even stronger than truth at every
index. Let us call a meaning m' a semantic variant of a meaning m iff m
and m' have exactly the same nodes, with the same category but not
a common
the same intension at each node, and, whenever
necessarily
in m, a common
intension also
us
an
m'
call
appears
s-fixed semantic
iffm and m' are semantic
variant ofm, where s is a set of simple meanings,
of s which
is a constituent
of m is also a
variants and every member
intension
constituent,
appears
at those
at
the
at two terminal nodes
two nodes
in m'.
same
ofm'.
place,
Let
Then
we
can
call
a sentence
meaning
s-true
iff every ?'-fixed semantic variant of it (including
itself) is true at
s
we would
of
If
the
set
whose
is
bearers
index.
every
simple meanings
we
as
s-true
call
sentence
then
may
logical vocabulary,
classify
meanings
logically true; if s is the set of simple meanings whose bearers we would
classify as mathematical
(including logical) vocabulary, we may call s-true
we can define a
true. Analogously,
sentence meanings
mathematically
and
relation of s-fixed semantic variance between sequences of meanings;
we can say that a sentence meaning m0 is an s-consequence
(for instance,
a logical consequence or mathematical
ml9 ... iff, for every ?-fixed semantic
(m0 mx ...> and every
true at i. (The premises
consequence) of sentence meanings
of the sequence
variant <jnQm\...y
index / such that all of m'l9 ... are true at i9m0 is
mi9 ... may be infinite in number. Their order is
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GENERAL
SEMANTICS
35
in terms of
insignificant.) These definitions are adapted from definitions
truth in all logically or mathematically
standard interpretations of a given
language. However, we have been able to avoid introducing
of alternative
of a language, since so far we
interpretations
with
entirely
meanings.
the notion
are dealing
VI. GRAMMARS RECONSTRUCTED
of meanings may serve, in effect, as a universal base for
transformational
based
grammars. There is no need to repeat
categorially
as a base com
rules of categorial well-formedness
the phrase-structure
Our
system
as given,
ponent in each such grammar. Instead, we take the meanings
a relation
and regard a grammar as specifying a way to encode meanings:
and certain expressions
between certain meanings
(sequences of sound
we
or
which
will
call
the
of
types
representing relation deter
mark-types)
by the grammar. We might just identify grammars with representing
but I prefer to take grammars as systems which determine
representing relations in a certain way.
mined
relations;
If we were
concerned with nothing but transformation-free
categorial
we
could take a grammar to consist of nothing but a lexicon:
grammars,
? is a
a finite set of triples of the form <e ? 0> where e is an expression,
and
category,
say
that
an
<?>is an intension
expression
e represents
appropriate
or has
for that category. We
a meaning
m
relative
may
to a lexicon
<[ex ct 0X>, ..., <?? ?? </>?>such that, first, e is
the result of concatenating
el9 ..., en (in that order), and second, the
of m are occupied
terminal nodes
..., <?? </>?>(in that
by <?x $!>,
L
iff L contains
items
order).
We could
in two steps. Let us define a (cate
instead have proceeded
as
a
tree
marker
nodes
having categories at its non-terminal
gorial) phrase
a
and expressions at its terminal nodes. Then
phrase marker p represents
to a lexicon L iff p is obtained from m as
of the meaning m occupied by a pair
terminal
node
given any
node
it
another
below
occupied by an expression e such that
<? </>>,place
c
in the lexicon; then remove the intensions,
the item (e
<?> is contained
node by its unaccompanied
replacing the <? (j)} pair at each non-terminal
or has a meaning
m relative
follows:
thus representable
that the set of meanings
relative to
category ?. Note
a lexicon L comprises all and only those meanings
that are generated by
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DAVID
36
LEWIS
the set of simple meanings
of the lexical items themselves;
let us call it
the set of meanings
the
L.
lexicon
generated by
Next, we define the terminal string of a phrase marker p as the expres
sion obtained by concatenating,
in order, the expressions at the terminal
e represents a meaning m
of;?. Thus we see that an expression
to a lexicon L, according
to the definition
iff e is the
above,
terminal string of some phrase marker
that represents m relative to L.
nodes
relative
In the case of a categorially based transformational
grammar, we have
not two steps but three. Such a grammar consists of a lexicon L together
with a transformational
component T. The latter imposes finitely many
on finite sequences of phrase markers. A sequence <[pl.. ./??>
of phrase markers
that satisfies the constraints
imposed by T will be
a
called
derivation ofpnfromp1
in T. An expression e
(transformational)
represents or has a meaning m in a grammar <L T> iff there exists a deri
constraints
in T such that e is the terminal string of pn and px
(Pi-.-Pn)
m
to the lexicon L. If so, we will also call e a meaning
relative
represents
vation
ful
expression,
pn a surface
structure
ofe,pn^1
and
...
and/?2
intermediate
ofe, p? a base structure ofe, and m a meaning ofe (all relative to
the grammar <L T>). However, we will call any phrase marker;?
a base
structure in <L T> iff it represents a meaning relative to L, whether or not it
structures
structure o/any expression;
thus we allow for base structures
are filtered out by not being the first term of any derivation
in T.
The representing
relation given by a grammar <L T> is by no means
a one-to-one
between meanings
and expressions. A given
correspondence
is the base
which
several different meanings.
expression might be ambiguous, representing
it
several
different
but
cointensive
it
(If
represents
however,
meanings,
to
common
call it ambiguous;
for the
notion of
might be inappropriate
seems to hover between our technical notions of meaning
and
meaning
of intension.)
On the other hand, several expressions might be synonymous,
a single meaning. We might also call several expressions
representing
iff they share all their meanings;
completely synonymous
synonymy and
complete synonymy coincide when we are dealing only with unambiguous
If several expressions
represent different but cointensive
we
call
them
If several
may
meanings,
equivalent but not synonymous.
same
not
the
but also have a single
expressions
only represent
meaning
base structure, we may call them not only equivalent and synonymous
but also paraphrases
of one another.
expressions.
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GENERAL
SEMANTICS
37
Given a representing relation, all the semantic relations defined hitherto
for meanings
carry over to expressions
having those meanings.
(If we
over
to
also
the
and
intermediate
base, surface,
like, they may carry
structures between
and the expressions.)
Thus we know
the meanings
what itmeans to speak, relative to a given grammar and qualified in cases
or 'on all meanings',
of the category and
of ambiguity by 'on a meaning'
at a given index
of
the
extension
intension of any meaningful
expression;
of
of
the
of any expression
thing named by a name;
appropriate category;
of the things to which a common noun applies; of the truth at an index,
truth on an occasion,
so
analyticity,
logical
truth, etc. of a sentence;
and
on.
should note an oddity in our treatment of logical truth. A synonym
of a logically true sentence is itself a logical truth, since it represents the
a descendant
as the original. Hence
same logically true meaning
by
We
of a logical
is confined
truth is itself a logical truth if the
to
single lexical items in the base
synonym-substitution
are groundhogs'
comes out
not
'All
otherwise.
woodchucks
structure; but
'All squares are equilateral rectangles' comes out
logically true, whereas
synonym-substitution
merely analytic (in the strongest sense).
A transformational
component may constrain
markers
in two ways. There is the local constraint
sequences of phrase
that any two adjacent
in a derivation must stand in one of finitely many rela
phrase markers
tions; these permitted relations between adjacent phrase markers are the
constraints
transformations. There may also be global derivational
speci
or
relations
between
markers
fying
phrase
non-adjacent
properties of the
as a whole. An example is the constraint requiring transfor
to
mations
apply in some specified cyclic (or partly cyclic) order.
A transformation-free
categorial grammar is a special case of a cate
It has a transformational
grammar.
gorially based transformational
derivation
no transformations
so that the
or global constraints,
therein are all and only those sequences <[p{) consisting of a
single phrase marker.
or a
I will not attempt to say more exactly what a transformation
transformational
is.
definitions
have
component
Mathematically
precise
component
derivations
with
been given (for instance in Peters and Ritchie,
1969), but to choose among
these would
in syntactic
involve taking sides on disputed questions
no
to
I
I
maintain
have
and
my
theory.
prefer
present need
neutrality,
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DAVID
38
for a precise delineation
in mind
have foremost
LEWIS
of the class of transformational
grammars.
I
a sort of
grammar
simplified Aspects-model
to
I
but
have
said
eliminate
various
alternatives.
1965),
(Chomsky,
nothing
to eliminate generative
I have said nothing
I have
semantics. What
to call the 'lexicon'
chosen
is the initial lexicon. Words
not in that lexicon
on the way from base to surface,
might be introduced transformationally
if that seems desirable.
It might even be that none of the initial lexical
items ever reach the surface, and that all surface lexical items (expressions
at terminal nodes of surface structures) are introduced
transfor
In that case it would be appropriate
to use
within derivations.
mationally
found
a standardized
in all grammars, and to rechristen my base
In that case also there might or
representations'.
not be a level between base and surface at which word-introducing
structures
initial lexicon
'semantic
might
transformations
are done and other
have not yet begun.
semantics. This may seem
transformations
I have also said nothing to eliminate surface
are to be determined by
strange, since I have indeed said that meanings
I rely here on the observation
base structures alone. However,
(Lakoff,
rules are indistinguishable
1970, ? 3) that surface-structure
interpretation
from global derivational
constraints
relating three levels: base structures
(regarded as semantic representations),
deep structures (an intermediate
a
structures
structures.
and
surface
Deep
might be ambiguous;
level),
grammar with
two derivations
transformational
mit
base-deep-surface
constraints
might
per
<ph...pD...pi}
<Pb?jPd?/>!>
differing at the base and surface but not at the deep
of the forms
rule out other derivations
level, but
it might
<j>B?PD?p?y
<J>b~-Pd?Ps}
In such a case base
structure (and hence meaning) would be determined
by deep and surface structure together, but not by deep structure alone.
Similarly, we might have constraints
relating base structure not only to
structure
to
structure at various other inter
and
but
surface
also
deep
mediate
I have
levels.
said nothing
to eliminate
a non-trivial
phonological
component;
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GENERAL
SEMANTICS
39
The
relocate it as part of the transformational
component.
derivation might go from the usual
last few steps of a transformational
surface structure to a post-phonological
surface struc
pre-phonological
ture whence
the output expression can be obtained
simply by concate
but I would
nation
of terminal
nodes.
an elaborate
to eliminate
said nothing
system of selection
on the lexical in
as
not
will
but
restrictions
these
appear
restrictions;
and base structures but as transformational
sertions between meanings
I have
filtering later on. There will be base structures representing the meanings
sentences as 'Seventeen eats beans' and 'He sang a
of such questionable
But these base structures need not be the first terms
toothbrush'.
pregnant
so these meanings may be unrepresented
of any derivations,
by sentences.
If we
restrictions, we might match the lexicon to the trans
in such a way as to filter out just those meanings
formational component
that have the null intension.
like selection
I have not stipulated that only sentential meanings may be represented;
that stipulation could be added if there is reason for it.
In fact, the only restriction I place on syntax is that transformational
based. In other words: a transforma
grammars should be categorially
tional component
should operate on a set of categorial phrase markers
a
set
of
meanings
generated by some lexicon. But categorial
representing
are
that
this
restriction
is not at all severe. I claim
bases
varied enough
that whatever
familiar sort of base component
you may favor on syn
tactic grounds, you can find a categorial base (i.e. a suitable part of the
system of meanings,
lexicon) that re
generated by a suitable chosen
sembles the base you favor closely enough to share its attractive proper
ties. Indeed, with a few preliminary
you can
rearranging transformations
go from my categorial base structures to (notational variants of) more
familiar base structures; then you can proceed exactly as before. I shall
not marshall evidence for this claim; but I think that the following explo
treatments of quantification
will exhibit
ration of alternative categorial
the close
treatments and several
categorial
to choose
If it were necessary
alternative
familiar base components.
for semantics and a non
between a categorial base that was convenient
syntax, I might
categorial base that was convenient for transformational
similarities
between
these
the former. But I deny the need to choose.
of my proposed
the exposition
system
completes
still choose
This
of categories,
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DAVID
40
LEWIS
- either
intensions, and meanings. Now I shall consider how this system
as is or slightly revised
might be applied to two difficult areas: the
semantics of quantification
and the semantics of non-declaratives.
The
are intended only as illustrations,
however; many
following
are
further alternatives
possible, and might be more convenient for syntax.
treatments
VII. TREATMENT
OF QUANTIFICATION
AND NOUN
PHRASES
Let us consider
such expressions as 'a pig', 'most pigs', 'seventeen pigs',
'roughly seventeen pigs', 'some yellow pig', 'everything',
'nobody', and
the like. We call these quantifier phrases (presupposing
that they should
in our system is this?
category
belong to a common
category). What
sort of intensions do quantifier phrases have?
What
to make
combine with verb phrases
sentences:
Quantifier
phrases
'Some pig grunts', 'Nobody grunts', 'Roughly seventeen pigs grunt', and
the like. Names
do this, since the category verb phrase is the derived
But
category S/N.
quantifier phrases cannot be names, under our se
treatment
of
mantic
names, because they do not in general name any
thing. ('The pig' could be an exception at indices such that exactly one
pig existed at the world and time given by the index.) The absurd conse
quences of treating 'nobody', as a name, for instance, are well known
(Dodgson,
1871). If a quantifier phrase combines with an S/N to make
an S, and yet is not an N, it must therefore be an S/(S/N).
Except perhaps for one-word
'nobody',
'every
quantifier phrases
common nouns.
thing', and such
quantifier phrases contain constituent
as in 'every
These may be either simple, as in 'some pig' or compound,
a
we
common
nouns
wins
blue
that
ribbon'.
may regard
Indeed,
pink pig
simply as predicates
from
Montague,
used
1970a.)
to restrict quantifiers.
The
expressions
'a',
(This suggestion
'the',
'some',
and so on which
derives
'every',
combine
'no',
with
'most', 'seventeen',
'roughly seventeen',
common nouns (simple or compound)
to make quantifier phrases and
which are variously called quantifiers, determiners, or articles must there
of quantifiers like
fore belong to the category (S/(S/N))/C. And modifiers
to
with
make
which
combine
certain
'roughly'
quantifiers
quantifiers,
must belong to the category ((S/(S/N))/C)/((S/(S/N))/C). Selection re
by means of transformational
of quantifiers
like 'roughly the'.
strictions
filtering
could be used to dispose
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GENERAL
The
SEMANTICS
41
of 'some pig' may be taken as that function (?>from S/N
such that if (?)xis any S/N-intension,
(?)2 is the
*
then
and
is
anY index,
<t>(<t>\),
intension
intensions
S-intension
to S-intensions
*2(0
truth if, for some N-intension
=
(j>3, </>3(/) is a pig and
if 04 is (?>1(4>3)
then 04(O is truth
falsity
otherwise.
be taken as that function
0 from C-inten
such that if (?)1is any C-intension,
(?)2 is the
$4 is the S-intension
S/(S/N)-intension
<H0i)> $3 is anY S/N-intension,
I
is
then
and
anY
index,
02(03)>
The
intension
of
'some' may
sions to S/(S/N)-intensions
*4(0
rrwr/?if, for some N-intension
(?>5,(?)5(i) is a member
of (?)i(i) and if 06 is 03(</>5) then </>6(0 is rntf?
=
falsity
otherwise.
I spare you the intension of 'roughly'.
Other intensions might be specified for 'some pig' and 'some' that
would differ from these only when a quantifier phrase was applied to a
verb phrases
verb phrase. If there are no non-extensional
non-extensional
in English,
is arbitrary.
then the choice among these alternatives
This treatment of quantifier phrases ismotivated
by a desire to handle
as
as
sentences
involving quantifier phrases
straightforwardly
simple
use
of transformations.
But it raises problems.
the
possible, minimizing
Quantifier phrases seemingly occur not only as subjects of sentences but
also as objects of verbs or prepositions. And in all their roles - as subjects
or as objects - they are interchangeable
with names. That is why it is
usual to have a category noun phrase comprising both quantifier phrases
and
names.
We might try the heroic course of doubling all our object-takers. We
could have one word
'loves' which
is an (S/N)/N and takes the object
'Petunia'
another
to make
it
and alongside
'loves Petunia';
phrase
is an (S/N)/(S/(S/N))
and takes the object 'some
the verb phrase 'loves some pig'. But we need not decide
the verb
'loves' which
pig' to make
how much we mind
It would
since it does not work
doubling,
for 'Every boy loves some girl' :
the sentence can be true even if each boy
such extravagant
give us one meaning
anyway.
on which
the weaker meaning,
loves a different girl. But the sentence
is ambiguous;
where
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shall we get
DAVID
42
a stronger meaning,
on which
LEWIS
the sentence
-
is true only if a certain girl (There are those who do not
is loved by all boys?
perhaps
this ambiguity;
but we seek a treatment
Zuleika,
to
general enough
of
do.) The method
doubling object
look to the method
rather we must
of variable
perceive
handle the idiolects
of those who
is a blind
takers
alley;
routinely used in the semantic
binding,
symbolic logic.
The quantifiers
analysis
of standardly
formulated
of symbolic logic belong to the category S/NS, taking
to make a sentence. The name must be a variable;
a name and a sentence
other
could be disposed of by transformational
filtering.
the
'some' takes the variable V
and
instance,
logician's quantifier
the sentence 'grunts x9 to make a sentence translatable
into English as
combinations
For
'something grunts'. The logician's 'some' has as its intension that function
to S-intensions
and S-intensions
such that if (?)l is
<j>from N-intensions
the wth variable
intension for any number
is <t>(<?>i<?>2),
and i is any index, then
n9 <?>2is any S-intension,
03
'
truth if, for some index / that is like / except perhaps
at tne wtn term ?f tne assignment
coordinate,
a. /-\_
(?>2(?) is truth
falsity
otherwise;
that is not a variable intension and
and such that if <f>xis any N-intension
then <?(0i<?2) is the null intension. The intension
<j>2is any S-intension,
of the logician's quantifier
'every' is specified similarly, with 'for every
index
/'...'
It would
replacing
some
index
/'...'.
in a grammar
quantifiers
are unrestricted,
ranging
'Some pig grunts', for instance,
to employ logician's
In the first place, these quantifiers
for English.
over everything.
would
'for
be troublesome
The
base
structure
of
come out as
S/NS
some
S
N
x
S
S/SS
S
and
S/N
N
!
x
pig
S/N
N
Ill
grunts
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x
GENERAL
in which
'pig' and
structures
SEMANTICS
43
there is no constituent
to 'some pig' and in which
corresponding
are
into
the
put
'grunts'
category S/N. (It was with
like this in mind that Ajdukiewicz
saw fit to omit the category
to
with
attempt
dispense
quantifier phrases in favor of unre
alike
C.) This
stricted quantifiers
taking compound
fails entirely for quantifiers
such as
sentences
is clumsy
at best,
and
'most'
(see Wallace,
1965). In the
second place, by having the quantifier
itself do the binding of variables,
we require there to be bound variables wherever
there are quantifiers.
We
get the unnecessarily
base structure
complicated
S/NS
SN
I
x
some
I
/\S/N
N
i,
grunts
x
i
for 'Something grunts', whereas
if we had employed quantifier phrases
take verb phrases and do not bind variables, we could have had
which
something
with
three constituents
grunts
instead of six and no work
for the transformations
to do.
It is not
that the quantifier
itself should bind
necessary,
however,
variables. We can stick with verb-phrase-taking
quantifier phrases of the
category S/(S/N), restricted by constituent common nouns in most cases,
and bind variables when necessary - but only when necessary - by means
of a separate constituent
called a binder: a certain sort of (S/N)/S that
a
sentence and makes an extensional
takes
verb phrase by binding a
at all its free occurrences
(if any) in the sentence. To every
a binder. Suppose
there corresponds
'*' is a variable; we may
write its corresponding
binder as *x9 and read it as 'is something x such
that'. (But presumably
binders may best be treated as base constituents
that never reach the surface; so if the words
'is something x such that'
variable
variable
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44
DAVID
LEWIS
ever appear in a meaningful
expression, they will be derived not from an
'x' in base structure but in some other way.) For instance, the following
base structure using a binder is equivalent to 'grunts' and might be read
loosely as 'is something
x such that x grunts'.
S/N
The
following
McCawley,
might
be a base
structure
for
'Porky
loves himself.
(Cf.
1969.)
(S/N)/S
(Provided there is no ambiguity among our variables, we can use them
in this way to keep track of coreferentiality,
rather than subscripting
the
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GENERAL
names
SEMANTICS
45
in
S/N N
N
(S/N)/N
loves
Porky
Porky
we are dealing with
one Porky or two.)
binder 6x9
intension, then the corresponding
has the nth binder intension : that function
to S/N
from
S-intensions
</>
such that if <?>1is any S-intension,
intensions
is
the
<?>2
S/N-intension
to indicate whether
If 'x9 has the ?th variable
<M0i)> $3 is anY N-intension,
</>4is the S-intension
(?)2((j)3), i is any index,
and /' is that index which has cb?>(i) as the nth term of its assignment
=
coordinate and otherwise
is like i, then <)!>4(/)
It can be verified
0i(Othat this intension justifies the reading of 'x9 as 'is something x such that'.
supply of variables and binders, however large, would lead to
the mistaken
omission of some sentences. To provide an infinite supply
by means of a finite lexicon, we must allow our variables and binders to
A finite
be generated as compounds. We need only three lexical items :one simple
variable having the first variable intension; an N/N having as intension
a function whose value, given as argument the nth variable intension for
is the (n+ l)th variable intension; and an ((S/N)/S)/N
any ?^1,
having
as intension a function whose value, given as argument the nth variable
intension for any ri^\,
is the nth binder intension. The first item gives
us a starting variable;
the second,
the other
iterated, manufactures
binders out of variables. However,
variables; the third manufactures
will continue
to abbreviate
base structures by writing
variables
binders as if they were simple.
we
and
introduces a sort of spurious ambiguity called alpha
Variable-binding
could have not only the structure
betic variance. 'Porky loves himself
'x' and '?' are replaced by 'y9 and 'j>',
shown but also others in which
or V
and 'f, etc. Since different variables may have different intensions,
to infinitely many different but cointensive
these structures correspond
for 'Porky loves himself.
The simplest way to deal with this
meanings
nuisance
is to define an ordering
of any such set of meanings
and employ
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DAVID
46
LEWIS
in the set
filtering to dispose of all but the first meaning
to
the
ordering).
(according
been discussed by logicians, under the name
Binders have occasionally
or
'lambda
'abstraction operators'
operators'.
(Church, 1941; Carnap,
and Stalnaker,
1958, ? 33; Thomason
1968.)
transformational
to complete our account of the category
in a position
of
quantifier phrases, using binders as needed.
verb-phrase-taking
S/(S/N)
The base structure for 'Every boy loves Zuleika' may be simply
Now
we are
(S/(S/N))/C
every
C(S/N)/N
boy
N
loves
Zuleika
to make work for the transfor
no unnecessary
variable-binding
is another base structure with variable
There
mational
component.
we
read
which
may
roughly as 'Every boy is something x such
binding
it represents a different but equivalent meaning.
that x loves Zuleika';
with
can
structure
and another
(but
or
for 'Every boy loves Zuleika'
get rid of them by
equivalent) meaning
transformational
filtering. The base structure for 'Lothario loves some
is
girl'
We
either
let these be another
base
the quantifier phrase which is the surface object of 'loves' is
the variable
treated as subject of a verb phrase obtained by binding
structure
is the base object of 'loves'. To reach an intermediate
which
in which
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GENERAL
SEMANTICS
47
the quantifier phrase is relocated as the object of 'loves', we
that moves the subject of a verb
have recourse to a transformation
phrase made by variable binding into the place of one (the first?) occur
in which
must
rence of the bound
variable and destroys the variable-binding
apparatus.
that, if desired, this transformation might apply beneath an inter
most closely to the ordinary level of deep
mediate
level corresponding
structure. The two base structures for 'Every boy loves some girl' are
Note
for the weak
(S/(S/N))/C
C
some
girl
(S/ND/S
S^
N
S/N
y
(S/N)/N
N
loves
y
x
sense, and
(S/(S/N))/C
C
N
(S/N)/N
every
boy
loves
for the strong - Zuleika - sense.
It may be that quantifier-phrase
'Lothario seeks a girl',
altogether.
phrased as 'Lothario
structure
x
objects should not be abandoned
in the sense in which it can be para
seeks a certain particular
girl', can have
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the base
DAVID
48
LEWIS
(S/(S/N))/C
but what
it the base
about
the sense in which
any old girl would
do? We might
give
structure
S
(S/N)/(S/(S/N)) S/(S/N)
seeks
(S/(S/N))/C
objects. The alternative
using a second 'seeks' that takes quantifier-phrase
to
is
let the word
'seeks' be introduced
rather than
transformationally
so that the
descendant of 'strives-to-find',
lexically, as a transformational
base
structures would
be
S
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GENERAL
for the sense in which
SEMANTICS
a certain particular
49
girl is sought and
S/N
(S/N)/S
i
strives
S/(S/N)
S/N
(S/(S/N))/C
C
a
girl
(S/N)/SS
x
N
S/N
(S/N)/N
N Lothario
finds
for the sense
in which
this treatment
to 'conceives
any old girl would do. But it is controversial
whether we ought to let words be introduced transformationally
in this
way; and (as remarked in Montague
,1969) it is not clear how to apply
of a tree'. Perhaps
to-exist, but perhaps not.
This completes one treatment
conceiving-of
is imagining
of quantifier
phrases, carried out with
no modification
of the system I originally presented. It is straightforward
from the semantic point of view; however,
it might result in excessive
to transformational
syntax. Ordinary bases have a category
complications
noun phrase which combines quantifier phrases and names; and transfor
seem to work well on bases of that sort. By dividing the category
mations
to be doubled (or
of noun phrases, I may require some transformations
are
etc.). Moreover, my structures involving variable-binding
so
remote
and
from
the
with
surface,
by doing away
complicated
quanti
com
lots of work for the transformational
fier-phrase objects I make
ponent. It might be, therefore, that this treatment is too costly to syntax.
quadrupled,
Therefore
phrase.
let us see how we might reinstate the combined category noun
are two methods:
we might
try to assimilate names to
or
we
to
assimilate
try
might
phrases,
quantifier phrases to
There
quantifier
names.
The method
of assimilating
names
to quantifier
phrases
proceeds
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as
DAVID
50
LEWIS
follows. For every name in our lexicon, for instance 'Porky', we add to
our lexicon a corresponding pseudo-name
in the category S/(S/N).
If the
intension of the original name 'Porky' is the N-intension
then
the
</>1?
intension
of
the corresponding
pseudo-name
'Porky*' should be that
to S-intensions
function
such that for any S/N
<?>from S/N-intensions
=
a
a
intension
sentence
As
such as 'Porky
result,
(?)290(02)
02(0i)can
either
of
structures
be
the
base
grunts'
given
S/CS/N)
S/N
Porky*
and will have
grunts
the same
intension either way. The category S/(S/N) may
noun phrase. It contains our former quantifier phrases
our new pseudo-names.
It does not contain names them
now be renamed
together with
selves. Names are now unnecessary as subjects, but still needed as objects;
so the next step is to replace all name-takers except verb phrases by noun
For instance, the category (S/N)/N of transitive verbs is
phrase-takers.
to be replaced by the category (S/N)/(S/(S/N))
of pseudo-transitive
verbs.
are related to the intensions of the
intensions of the replacements
a
I
in
which
shall not bother to specify. Names
way
systematic
originals
now serve no further purpose, having been supplanted both as subjects
so the next step is to remove names
and as objects by pseudo-names;
The
the lexicon. The category N is left vacant.
for noun-phrase
Since we have provided
objects
we
can
have
also
pseudo-names,
quantifier-phrase
from
down
on variable-binding.
For
instance,
for the sake of the
objects
so cut
and
we have
S/(S/N) S/N
(S/(S/N))/C
every
C
boy
S/CS/N)
(S/N)/(S/(S/N))
loves
(S/(S/N))/C C
I
some
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girl
I
GENERAL
as the base
structure
no work
for
SEMANTICS
51
for 'Every boy loves some girl' in the weak sense,
the transformations.
cannot do away with
We
leaving
variable-binding
altogether, however. The base structure
loves some girl' in the strong - Zuleika - sense is now
CS/(S/N))/C
every
C
boy
for 'Every boy
(S/N)/(S/(S/N))
S/CS/N)
loves x*
the seeming noun-phrase
object 'some girl' is treated as subject
noun phrase
of a verb phrase obtained by binding the pseudo-variable
'x*' which is the real object of'loves'. Variables are names, of course, and
in which
are replaced by pseudo-names
just as any other names are; no
in
is
the
binders.
made, however,
corresponding
change
we
not
So far
have
departed from the system I presented originally,
and we could stop here. It is now advantageous,
however, to take the step
therefore
of eliminating the category N altogether and promoting
the category verb
a
a
new
to
from
derived
basic
category S/N
category VP. Accord
phrase
noun
of
becomes
the
category
ingly,
phrases
S/VP; the category of quan
of transitive verbs becomes
the category
(S/VP)/C;
and the category which includes binders becomes VP/S.
VP/(S/VP);
We can also reopen the question of letting verb-phrase
intensions be
rather
than
We
this
compositional.
rejected
Carnapian
simplification
rule which was
before, principally because it would require a projection
tifiers
becomes
not of our general function-and-arguments
form; but that consideration
no longer holds after names and verb-phrase-plus-name
combinations
are done
still applies: the simplification
away with. A lesser objection
If any non-extensional
verb
for
extensional
verb
works
only
phrases.
phrases exist, they cannot go into our new basic category VP with
Carnapian
intensions.
They will have
to go into the category
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S/(S/VP)
DAVID
52
LEWIS
to Carnapian
for the now-basic
intensions
verb
a
I
most
other
intensions
in
which
need
systematic way
phrases changes
not stop to specify.
We turn last to the opposite method,
in which quantifier phrases are
to names to give an undivided
assimilated
category of noun phrases.
instead.
This will
manner
The
switch
require
discussed
and intensions of names in a
revising the extensions
Mates
by
1968) and Montague
(Mates,
(Montague,
1969 and 1970b).
In the dark ages of logic, a story something
like this was told. The
a
we
names
call the existentially
'some
may
strange
thing
phrase
pig'
some
has
which
that
those
generic pig
just
properties
pig has. Since some
pig ismale, some pig (a different one) is female, some pig is pink (all over),
and some pig is grey (all over), the existentially
generic pig is simultane
he (she?) is in the
female, pink, and grey. Accordingly,
ously male,
extensions both of 'is male' and of 'is female', both of 'is pink all over'
and of 'is grey all over'. The phrase 'every pig' names a different strange
thing called the universally generic pig which has just those properties
that every pig has. Since not every pig is pink, grey, or any other color,
the universally generic pig is not of any color. (Yet neither is he colorless,
- indeed not
since not every
any
pig is colorless). Nor is he(?) male or
female (or neuter), since not every pig is any one of these. He is, however,
a pig and an animal, and he grunts; for every pig is a pig and an animal,
and grunts. There are also the negative universally generic pig which has
that no pig has (he is not a pig, but he is both a
just those properties
stone
a number),
that more
the majority
generic pig which has just those
than
half
of
all
pigs have, and many more. A
properties
sentence formed from a name and an extensional verb phrase is true (we
may add: at an index i) if and only if the thing named by the name (at /)
and
belongs to the extension of the verb phrase (at i); and this is so regardless
of whether
the name happens to be a name like 'Porky' of an ordinary
a
or
name
like 'some pig' of a generic thing.
thing
since nothing, however recondite, can possi
story is preposterous
or
one
more
than
of a set of incompatible
and jointly ex
have
less
bly
haustive properties. At least, nothing can have more or less than one of
This
them as its properties. But something, a set, can have any combination
in that.
there is no contradiction
of them as its members;
Let us define the character of a thing as the set of its properties.
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Porky's
GENERAL
SEMANTICS
53
that
is that set which has as members
just those properties
not
as
could
do
and
The
various
has
not,
generic pigs
Porky
properties.
of
the
but
their
characters
The
character
do.
universally
possibly, exist;
just those proper
generic pig, for instance, is the set having as members
ties that every pig has as properties.
character
is any set of properties. A character is individual iff it is a
so that something could possess
set of properties,
compatible
in it; otherwise
is
contained
the character
all and only the properties
A
character
maximal
generic.
Since no two things share all their properties (on a sufficiently inclusive
to their individual
conception of properties) things correspond one-to-one
to replace things by their
characters. We can exploit this correspondence
have even tried to
philosophers
eliminate things altogether in favor of their characters, saying that things
are 'bundles of properties'.
(Such a system is proposed as a formal recon
in Mates,
individuals
struction of Leibniz's
doctrine of possible
1968.)
characters
We
whenever
convenient.
need not go so far. We will
extensions
of
and
names,
Some
replace
as members
things by individual
of
extensions
of
characters
common
as
nouns.
However, we may keep the things themselves as well, taking them to be
related to their names via their characters. Having made this substitution,
we are ready to assimilate quantifier phrases to names by letting them
also
take
characters
-
in most
cases,
generic
characters
-
as
extensions.
'Porky' has as extension Porky's individual character;
'every pig' has as
extension the generic character of the universally generic pig. Even 'no
the set of just those properties that nobody has.
body' has an extension:
as follows. Our basic categories are
We revise the system of meanings
ex
sentence (S), noun phrase (NP), and common noun (C). Appropriate
are truth values
extensions for noun
; appropriate
exten
phrases are characters, either individual or generic; appropriate
sions for common nouns are sets of individual characters. Intensions are
tensions
for sentences
for basic categories, functions from some or all indices to ap
for a derived category (c/c1...cn), functions
from
propriate extensions;
..., and ??-intensions to c-intensions. A name is an NP that
q-intensions,
as before:
never has a generic character
of quantifiers becomes NP/C;
as its extension
at any index. The category
the category of verb phrases becomes S/NP.
take NP objects which may or may not be names. Some
Object-takers
still is required; the two base structures for 'Every boy
variable-binding
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DAVID
54
LEWIS
loves some girl' are
S/NP NP
NP
(S/NP)/NP
loves
C
NP/C
C
every
boy
I
I
some
for the weak
NP/C
girl
sense and
S/NP*"
(S/NP)/S
x
S/NP
NP
(S/NP)/NP
NP
loves
x
C
NP/C
every
boy
for the strong sense. Variables are names: the nth variable intension now
that assigns to every index / the character at
that NP-intension
becomes
the world coordinate of / of the thing that is the nth term of the assign
ment
of i. The
coordinate
VIII.
intensions
TREATMENT
OF
of binders
are revised
to fit.
NON-DECLARATIVES
A meaning
for a sentence, we said initially, was at least that which deter
mines the conditions
under which the sentence is true or false. But it is
sentences that can be called true or false in any straight
only declarative
sentences: commands,
forward way. What of non-declarative
questions,
as they are commonly
and so on? If these do not have truth-values,
supposed not to, we cannot
their truth conditions.
One method
of treating
very well
say that their meanings
non-declaratives
is to analyze
determine
all sentences,
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GENERAL
SEMANTICS
55
a sentence radical
or non-declarative,
into two components:
that specifies a state of affairs and a mood that determines whether
the
declarative
is declaring that the state of affairs holds, commanding
that it
or
it
whether
the
what.
of
hold, asking
holds,
(I adopt
terminology
one
a
are
recent
to
of
such
Stenius, 1967,
view.) We
exposition
regard the
sentences
speaker
It is the case that you are late.
it the case that you are late!
Is it the case that you are late?
Make
or more
idiomatically
You
are late.
Be late!
Are
you
late?
as having a common sentence-radical
specifying the state of affairs con
of
but
im
your being late,
declarative,
differing in their moods:
sisting
structures
be
the
and
base
given
interrogative. They might
perative,
late
be
with
S now understood
will
induce different
the different
sentences
as the category sentence radical. Different moods
transformations
of the sentence radical, leading to
above.
The
sentence
radical
is not a declarative
If it is represented on the surface at all, it should be represented
as the clause 'that you are late'. All that we have said about sentences
sentence.
be taken as applying rather to sentence radicals. It is sentence
as extensions,
from indices to
that have truth-values
functions
as intensions,
truth-values
and meanings
the
with
category S and an
should
radicals
S-intension
at the topmost
node. We may grant that a declarative
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sentence
DAVID
56
LEWIS
radical has the value truth ; if we liked, we
or other non-declarative
call an imperative or interrogative
sentence true iff its sentence radical has the value truth, but we customarily
se
the entire apparatus of referential
do not. Fundamentally,
however,
is called
could
true iff its sentence
also
mantics
or otherwise)
(whether done on a categorial base as I propose,
to sentence radicals and constituents
thereof. The semantics of
pertains
mood
is something entirely different.
such as these (adapted from Stenius,
It consists
of rules of language
use
1967):
an S
of the mood
the combination
declarative
with
representing
m only if m is true on the occasion
in question.
to a sentence
the combination
of the mood
representing
imperative with an
m (if adressed
over
in a suitable
to you by a person
relation
of authority
S-meaning
in such a way as to make m true on the occasion
in question.
you) by acting
Utter
a sentence
meaning
React
the theory of language use, a
a,
be
pair of a mood and an S-mean
simply
identified with some arbitrarily chosen entities).
In abstract
meaning
semantics,
for a sentence
as distinct
from
should
ing (moods being
of sentence
The method
requires a substantial revision of my
for declaratives,
imperatives, and yes-no questions.
radicals
system. It works well
It is hard to see how it could be applied
sentences like 'Hurrah for Porky!'
to other
sorts of questions,
or to
I prefer an alternative method of treating non-declaratives
that requires
no revision whatever
in my system of categories,
intensions, and mean
dis
ings. Let us once again regard S as the category sentence, without
crimination of mood. But let us pay special attention to those sentential
meanings
that are represented
by base structures
of roughly
the following
form.
Such meanings
can be represented
by performative
sentences
these.
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such as
GENERAL
SEMANTICS
57
I command you to be late.
I ask you whether you are late.
but, as will
1962, for the standard account of performatives;
(See Austin,
be seen, I reject part of this account.) Such meanings might also be repre
elaborate
transformational
derivation,
sented, after a more
by non
declaratives.
Be late!
Are you
I propose
of
late?
that these non-declaratives
the corresponding
to be treated as paraphrases
the same base structure,
having
ought
performatives,
intension, and truth-value at an index or on an occasion. And
of these
that there is no difference in kind between the meanings
and
and
of
the
non-declaratives
the
performatives
ordinary
meanings
meaning,
I propose
sentences
declarative
considered
previously.
we would
sentences as
classify the performative
If not, then we can divide sentential meanings
into declara
sentential meanings
and non-declarative
the
sentential meanings,
It is not clear whether
declarative.
tive
latter being
both by performatives
and by imperatives,
represented
as
etc. But if, as I would prefer, we classify performatives
then the distinction between declarative and non-declarative
declarative,
sentences becomes a purely syntactic, surface distinction.
The only dis
questions,
tinction
is the distinction
between
those sentential
among meanings
can
sentences
that
be
and those
declarative
meanings
only
represented by
can
that
be represented either by suitable declarative sentences (performa
thereof. Let us call the latter
tives) or by non-declarative
paraphrases
I need not delineate
the class of per
sentential meanings.
performative
formative sentential meanings precisely, since I am claiming that they do
not need to be singled out for special semantic treatment.
The method
of paraphrased
performatives
those non-declaratives
that resisted treatment
radicals. Not
spond
can easily be extended to
of sentence
by the method
only yes-no questions but other questions
to performative
sentences. The sentences below
I ask who
Sylvia
as well
is.
Who is Sylvia?
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corre
DAVID
58
for
have
instance, might
structure something
LEWIS
a common
meaning
represented
is
Sylvia
by a base
like this.
the sentences
And
I cheer Porky.
Hurrah for Porky!
might have this base structure.
duced transformationally.)
(Thus the word
'Hurrah' would
be intro
s
S/N N
(S/N)/NNI
I
I
cheer
We
may
structures
We
to
also
as
meanings
represented
by
these
base
performative.
truth-values.
common
sentential
at the outset
noted
lack
the
classify
Porky
are commonly
supposed
sentence radicals
this
respects
that non-declaratives
The method
of
to sentence
fundamentally
are under no compulsion
to
a
as
sentence
sentence
of
the
its
truth-value
non-declarative
regard
sharing
radical, and we have chosen not to. The method of paraphrased perform
atives, on the other hand, does call for the assignment of truth-values to
truth-values
opinion by assigning
rather than to whole sentences. We
radicals
is not that of the
assigned
to
sentence
the
radical), however, but
(corresponding
I
of
If
that
the paraphrased
say to you 'Be late!'
performative.
non-declarative
embedded
rather
sentences.
The
truth-value
sentence
and you are not late, the embedded
sentence
is false, but the paraphrased
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GENERAL
SEMANTICS
59
is true because I do command
that you be late. I see no
performative
have the truth-values of the perform
problem in letting non-declaratives
atives they paraphrase;
after all, we need not ever mention
their truth
values if we would rather not.
So far, I have assumed that performatives
themselves do have truth
but
that
also
been
has
denied.
values,
(Austin, 1962, Lecture I.) I would
wish to say that T bet you sixpence it will rain tomorrow'
is true on an
of utterance
iff the utterer does then bet his audience sixpence
rain on the following day; and, if the occasion
is normal in
is true.
certain respects, the utterer does so bet; therefore his utterance
nor
Austin
says it is obviously neither true
false, apparently because to
utter the sentence (in normal circumstances)
is to bet. Granted;
but why
occasion
that it will
is that a reason to deny that the utterance is true? To utter T am speaking'
is to speak, but it is also to speak the truth. This much can be said in
Austin's
defense:
the truth-values
that is inten
(and truth conditions,
are easily ignored just
and their paraphrases
sions) of performatives
to be anything but true on an
because
it is hard for a performative
Hard but possible:
you can be play-acting,
or
an
and say T command
officer
elocution,
practicing
impersonating
that you be late' falsely, that is, say it without
your
thereby commanding
in which
audience to be late. I claim that those are the very circumstances
of
occasion
its utterance.
is
it, like the performative,
falsely say 'Be late!'; otherwise
no
when
it
is
It
if
uttered
because
and
uttered.
is
wonder
the
truth
truly
of the sentences embedded
conditions
in performatives
and their non
you
could
tend to eclipse the truth conditions
of the per
paraphrases
and non-declaratives
themselves.
sentences of
This eclipsing is most visible in the case of performative
or T declare that-'.
the form T state that-'
If someone says T
declarative
formatives
is flat' (sincerely, not play-acting,
etc.) I claim that
so
he has spoken truly: he does indeed
declare. I claim this not only for
sense. Yet one might
the sake of my theory but as a point of common
be tempted to say that he has spoken falsely, because the sentence em
- the content of his
bedded in his performative
the belief he
declaration,
avows - is false. Hence I do not propose to take ordinary declaratives as
declare
that the Earth
in Ross,
that
paraphrased
performatives
(as proposed
1968) because
would get their truth conditions wrong.
If there are strong syntactic
reasons for adopting Ross's proposal,
I would regard it as semantically
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DAVID
60
a version
structures
of the method
that
LEWIS
radicals, even if it employs base
like the base structures employed
in the
of sentence
look
exactly
of paraphrased performatives.
I provide only one meaning
for the sentence T command you to be
late'. Someone might well object that this sentence ought to come out
because it can be used in two ways. It can be used to com
ambiguous,
method
as 'Be late!', and it is true when
it can be paraphrased
because
it is uttered. It can be used
circumstances
just
to describe what I am doing; thus used, it cannot be paraphrased
thus used,
mand;
uttered in normal
instead
as an imperative, and it is likely to be false when uttered because
it is
am
I
that
difficult to issue a command and simultaneously
say
doing so.
I
be
but
the
commanding
by signing my
possible:
might
doing
(Difficult
name on a letter while describing what I am doing by talking.)
I agree that there are two alternative uses of this and other performative
use and the non-performative
sentences : the genuinely performative
self
use.
occur
can
I
non-declarative
also
that
the
agree
paraphrase
descriptive
use. It still does not follow
in the performative
the case of these two sentences.
meanings.
Compare
only
I am talking
In hexameter
The
latter can be used
that there are two
in trochaic hexameter.
trochaic am I talking.
to talk in trochaic
hexameter
and is true on any
former cannot be so
of its correctly accented utterance. The
occasion
used and is false on any occasion of its correctly accented utterance. Yet
a sentence can be
the two sentences are obviously paraphrases. Whether
The
is not a matter of its meaning.
used to talk in trochaic hexameter
using a sentence to talk in trochaic hexameter or not
so using it is one sort of distinction;
between using a
the distinction
it
sentence
and
is
self-descriptively
using
performatively
performative
A
is
instructive.
distinction
sort.
Still
I
think
the
another
parallel
quite
distinction
between
in uses need not
involve a distinction
in meanings
of the sentences
used.
in conversational
in surface form; or distinction
or
some other sort.
of
distinction
and
intentions,
expectations;
setting,
in meanings
I see no decisive reason to insist that there is any distinction
It can involve distinction
associated
with
the difference
uses of performative
sentences,
between
performative
if the contrary
and
assumption
self-descriptive
is theoretically
convenient.
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GENERAL
61
SEMANTICS
of sentence radicals and the
We may ask to what extent the method
are compatible.
In particular: given
of paraphrased performatives
can
can
mood
and
into
be
sentence
that
sentence-radical,
any
analyzed
intension from the meaning
we recover the mood and the sentence-radical
method
of paraphrased performatives?
according to the method
of sentence
can do this, but not quite. On the method
and
the
between
difference
the
self-descriptive uses
radicals,
performative
sentences must be treated as a difference of meanings.
of performative
of the sentence
We
almost
we will get two pairs of a
sentence meaning,
So given a performative
to the two uses.
intension corresponding
mood and a sentence-radical
a
are
we
sentential
meaning
represented by
performative
given
Suppose
a base
structure
like this, for instance.
command
you
(S/N)/(C/C)
be
I
S/N N
C/C
you
I
late
the self-descriptive
use, we do just what we would do for a non
sentence
take the mood to be declarative and the
meaning:
performative
In
intension to be the intension of the entire meaning.
sentence-radical
to
sentence
the
radical
would
be
the
intension
this case, it
corresponding
'that I command you to be late'. For the performative
use, we take the
For
mood
to be determined
the sentence-radical
at node <1, 1>, and
at node <1, 3>. In this
of 'command', which determines
by the (S/N)/NS-intension
intension to be the S-intension
case, these are respectively the intension
sen
of the embedded
is imperative, and the S-intension
that the mood
to the sentence radical 'that you are late'.
tence meaning,
corresponding
of
apart from fineness of individuation,
advantage,
as
rather
than
markers
semantically
interpreted phrase
taking meanings
of constituents
from
as single intensions :we can recover the meanings
Note
here a second
the meanings
of their compounds.
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DAVID
62
indices
appendix:
LEWIS
expanded
Indices are supposed to be packages of everything but meaning
into determining
extensions. Do we have everything? Let me
on several expansions
of the indices that might prove useful.
that goes
speculate
the sentence
'This is older than this9. I might
say it
a
at
I
at
a
1962
when
the
first
'this'
and
1963
say
Volkswagen
pointing
on
I
sentence
the
true
when
second
'this'.
The
should
be
say
Volkswagen
consider
First,
such an occasion; but how can it be? Using the intension of 'this', with
its sensitivity to the indicated-objects
coordinate, we obtain the intension
sentence; then we take the value ofthat intension at an index
and contextual
coordinates
determined
by features of the
over
of utterance.
indices
alike
except at the
(We generalize
of the whole
with
world
occasion
assignment
assignment
but we can consider
any one of these, since the
to the sentence in question.)
This
that the indicated object changes part-way
coordinate;
coordinate
is irrelevant
ignores the fact
the
occasion of utterance. So the sentence
through
on
should
any occasion when the indicated object
procedure
comes
out false, as it
stays the same.
a solution would be
to semantics,
On a more extensional
approach
easy. We could take the two extensions of 'this' on the two occasions of
its utterance and use these, rather than the fixed intension of 'this', to
determine the truth-value of the sentence. The intension and the occasion
of utterance of the sentence as a whole would drop out. But since the
are not in general determined by the extensions
extensions of compounds
of their constituents,
this extensional
solution would preclude a uniform
treatment of semantic projection
rules.
An acceptable
solution has been suggested to me by David Kaplan,
as follows. Let the indicated-objects
be not just one set of
coordinate
objects capable of being pointed at but an infinite sequence of such sets.
the indicated-objects
coordinate
determined
by a given occasion of
of a sentence have as its nth term the set of things pointed to
at the nth utterance of 'this' during the utterance of the sentence so long
Let
utterance
as n does
empty
not exceed
set when
the number
n does
exceed
of such utterances,
and let it be the
that number. Let there be an infinite
'this/, 'this2',... with intensions such that 'thisn'
depends for its extension at an index on the nth term of the assignment
coordinate.
So that the lexicon will remain finite, let all but 'this/ be
sequence
of constituents
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GENERAL
SEMANTICS
63
to 'this/.
compounds generated by iterated application of a suitable N/N
Let all members of the sequence appear as 'this' in surface structure. Use
transformational
filtering to dispose of all base structures except those
an
so arranged that if the
initial
segment of the 'this'-sequence
employing
were
carried to the surface, they would appear in numerical
subscripts
order without
repetition.
than this' will be
is older
Thus
than
the only base
solution must
for 'This is older
this2
which will be true on occasions
The
structure
be modified
of the sort in question.
to allow for the fact that
I omit details.
the only demonstrative;
are
for other
similar solutions
possible,
Similar
'this' is not
difficulties
contextual
arise, and
coordinates:
time,
audience, and perhaps speaker.
Second, consider the sentence 'The door
place,
is open'. This does not mean
that the one and only door that now exists is open; nor does it mean
that the one and only door near the place of utterance, or pointed at,
or mentioned
in previous discourse,
is open. Rather
it means
that the
one and only door among the objects that are somehow prominent
on
the occasion
is open. An object may be prominent because it is nearby,
or pointed at, or mentioned
; but none of these is a necessary condition
of contextual prominence.
So perhaps we need &prominent-objects
coordi
a
new
coordinate
contextual
nate,
independent of the others. It will be
on a given occasion of utterance of a sentence, by mental
determined,
factors
to bring
such as the speaker's expectations
to the attention of his audience.
regarding
the things he is likely
1968; Donnellan,
Third, consider the suggestion
(Kaplan,
1970) that
the extension of a personal name on a given occasion depends partly on
the causal chain leading from the bestowal of that name on some person
to the later use of that name by a speaker on the occasion
in question.
We might wish to accept this theory, and yet wish to deny that the
intension
or meaning
of the name depends,
on the occasion
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in question,
DAVID
64
upon the causal history
to give up the common
LEWIS
of the speaker's use of it; for we might not wish
that the meaning
of an expression
presumption
on
mental
factors
within
him.
We might solve
only
for a speaker depends
in Lewis, 1968b) by including a causal-history
this dilemma (as proposed
coordinate in our indices and letting the intensions
of-acquisition-of-names
their extensions only relative to that
of names for a speaker determine
coordinate.
Fourth, we have so far been ignoring the vagueness of natural language.
Perhaps we are right to ignore it, or rather to deport it from semantics
to the theory of language-use. We could say (as I did in Lewis, 1969,
V) that languages themselves
of a population,
linguistic conventions
a fuzzy region
a
not
but
select
point
Chapter
are free of vagueness but that the
or the linguistic habits of a person,
in the space of precise languages.
it might prove better to treat vagueness within
However,
we could do so as follows. (A related treatment, developed
is to be found in Goguen,
1969.)
and
semantics,
independently,
that the only vagueness
of 'cool' and
is the vagueness
are
'warm' ; and suppose for simplicity that these
extensional adjectives.
Let the indices contain a delineation coordinate: a positive real number,
temperature between cool and warm things.
regarded as the boundary
Pretend
first
at an index i the extension of 'cool' is the set of things at the world
of i having temperatures
and time coordinates
less
(in degrees Kelvin)
or
coordinate of i; the extension of 'warm'
than
equal to the delineation
is the set of such things having temperatures greater than the delineation
coordinate. A vague sentence such as 'This is cool' is true, on a given
at some but not all delineations;
that is, at some but not all
occasion,
Thus
indices that are alike except in delineation and have the world and contex
determined by the occasion of utterance. But sentences
tual coordinates
are not necessarily vague: 'This is cool or warm,
with vague constituents
but not both'
on an occasion
is true at all delineations,
even if the indicated object
is a unique indicated object,
The delineation
coordinate
on which
there
is lukewarm.
is non-contextual.
It resembles the assign
in that we will ordinarily generalize over it rather than
coordinate,
hold it fixed. We may say that a sentence is true over a set s of delineations
at an index /, iff, for any index V that is like / except perhaps at the
ment
the sentence is true at V if and only if the deline
coordinate,
ation coordinate of V belongs to s. Given a normalized measure function
delineation
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GENERAL
SEMANTICS
65
we can say that a sentence is true to degree d at i iff it
over delineations,
is true at i over a set of delineations
of measure d. Note
that the degree
of truth of a truth-functional
of
sentences
not a function
is
compound
of the degrees of truth of its constituent
:
sentences
'x is cool' and 'x is
warm' may both be true to degree .5 at an index /, but 'x is cool or x is
cool' is true at / to degree .5 whereas
'x is cool or x is warm' is true at /
to degree 1.
semantics makes for simple specifications
Treating vagueness within
of the intensions of such expressions as 'in some sense', 'paradigmatic',
than'.
The
idiom
'_ish',
and
instance,
is an S/S related to the delineation
'possibly' is related to the world
operator
'in some
'_er
contemporary
is that function
sense'
coordinate
coordinate.
'in
some
sense',
for
just as the modal
The intension of
<?>such that if (^ is any S-intension,
(?>2
is (H$i), and / is any index, then
=
?2(0
truth if, for some index V that is like i except perhaps
at the delineation
is truth
coordinate, 0i(O
otherwise.
falsity
The comparative
'_er
such that, for instance,
than' is a ((C/C)/N)/(C/C)
having an intension
than y9 is true at an index i iff the set
'x is cooler
'y is cool' is true at i is a proper subset of the
over which
'x is cool' is true at i. It follows that the
than Sirius unless in some sense the sun is cool; but
over which
of delineations
set of delineations
sun is not cooler
seems correct, although I do not know whether to deny
than Sirius or to agree that in some sense the sun
was suggested to me by David
(This analysis of comparatives
that conclusion
that the sun is cooler
is cool.
Kaplan.)
More
generally,
be a sequence of
expressions will depend
for the extensions of
(or, if they are not extensional,
on
terms
different
of
the
delineation. More
compounds)
boundary-specifying
for their extensions
their extensional
the delineation
numbers.
than one term of the delineation
expression.
For
instance,
coordinate
Different
must
vague
might be involved for a single
of 'green' might involve one term
boundary and another regarded as
coordinate
the intension
the blue-green
regarded as delineating
the
boundary. The former but not the latter
delineating
green-yellow
would be one of the two terms involved in the intension of 'blue' ; and
so on around the circle of hues.
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66 DAVID
LEWIS
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