Chapter 9 528
We are interested here more in a general point that stems from
the considerations above than in this or that specific result. The
truth-conditional and model-theoretic approach to meaning we
have presented is not just an exercise in applied logic. It has real
empirical bite and a profound relevance for linguistic theory.
Without it, it would seem, there are nontrivial properties of language that we would just miss. The present kind of semantics thus
seems capable of contributing in a fundamental way to the attempt
to characterize what a human language is. Such a semantics might
well be limited in its scope. It might well need to be put in a
broader perspective and perhaps changed in fundamental ways.
But it has empirical payoffs that linguistic theory cannot disregard.
7
Truth-conditional semantics is here to stay.
Appendix Set· Theoretic Notation and Concepts
This appendix provides a brief and highly selective introduction to
the basic set-theoretic concepts, terminology, and notation assumed
in the text.
Sets and Set Membership
A set is a collection of entities of any kind. The members of a set
need not share any properties.
A set can be finite: the set of people in room 220, Morrill Hall,
Cornell University, at 2 P.M. on Tuesday, April 20, 1999. A set can
be infinite: the set of integers greater than one million.
A finite set can in principle be specified by enumerating or listing
its members, for example, the set consisting of Orwell's 1984, the
square root of 2, Noam Chomsky, and the muffin Sally McConnellGinet ate for breakfast on Sunday, September 4, 1988. This set can
be designated as in (1):
(1) {1984, Noam Chomsky, J2, Sally's breakfast muffin for 9/4/88}
When we use the notation in (1), the order in which things are
listed inside the curly brackets makes no difference. So (1), (2a),
and (2b) all refer to the same set.
(2) a. {Noam Chomsky, 1984, J2, Sally's breakfast muffin for
9/4/88}
b. { 1984, J2, No am Chomsky, Sally's breakfast muffin for
9/4/88}
Some sets can be specified by description: the set of all redwood
trees chopped down in California during 1984. This set can be
designated as in (3):
(3) {x: xis a redwood tree chopped down in California during
1984}
A minor notational variant is to use a vertical bar instead of a colon
when designating a set descriptively. Hence, we can designate the
setin (3) as in (4):
Appendix 530
(4) {xI xis a redwood tree chopped down in California during
1984}
A set is completely defined by its members, the entities that
belong to it. Different descriptions that happen to specify the same
entities thus specify the same set. For example, as the authors of
this book were the only semanticists at Cornell during 1986/1987,
each of the three expressions in ( 5) designates the same set.
(5) a. {Gennaro Chierchia, Sally McConnell-Ginet}
b. {x: xis one of the authors of Meaning and Grammar}
c. {x: x was a semanticist at Cornell during 1986/1987}
This notion of set is extensional: only the members of a set matter
and not how they are chosen or identified. If two sets have the same
members they are the same entity. Properties, on the other hand,
are intensional: two properties can well be true of the very same
objects and yet be distinct; consider the properties of being bought
and being sold. We sometimes, however, identify properties with
extensional sets.
A set may contain other sets as its members. For example,
{{a,b},c} has two members: c and the set {a,b}.
Below are some notations and their definitions:
a E P The element a is a member of set P or belongs to P. For
example, j E {x: xis a linguist} = j belongs to the set of linguists;
j E { j, a}
j belongs to the set consisting of j and a.
a ¢ P The element a is not a member of set P or does not belong
to P. For example, m ¢ {x: xis an actor} m does not belong to
the set of actors, and m ¢ {a, m} = m does not belong to the set
consisting of m and a (which is false).
A = B Where A and B both designate sets, this says that they are
identical. This happens just in case whenever a E A, then a E B and
whenever bE B, then bE A. This is the basic identity condition for
sets. Two sets are said to· be disjoint if they have no members in
common.
{a} A unit set or singleton set to which only a belongs. For example, {x: x has resigned the U.S. presidency while in office}=
{Nixon}, and {1} =the set that contains just the number 1. Note
that {a} =1= a. Nixon is different from the set of those who have
resigned; he simply happens to be the sole member of that set at the
time of writing.
Set-Theoretic Notation and Concepts 531
0 The empty set or null set. The set containing no elements. Note
that there is only one such set. The set of female United States
presidents (as of 1999) is the same as the set of males who have
won two Nobel prizes in the sciences (as of 1999). Each has the
same members, namely none.
P <;; R The set P is a subset of R or is included in R. This means
that every member of Pis also a member of R. For example, the set
of U.S. Senators (P) is a subset of the U.S. population over 30 (R);
{a, b, c} s; {a, b, c, 1, m} = the set that contains a, b, and c is a subset
of the set that contains a, b, c, 1, and m; {a, b} <;; {a, b} = the set that
contains a and b is a subset of itself.
P c R The set P is a proper subset of R or is properly included in
R. This means that P s; Rand P =1= R, that is, that all members of P
also belong to R but R has at least one member that does not belong
to P. The last subset example in the previous definition is not a
proper subset; the other are.
P cj;_ R The set Pis not included in R; there is at least one member
of P that does not belong to R. For example, the set of mathematicians (P) is not included in the set of men (R), because some
mathematicians are not men, and {a, b} cj;_ {a, c, d} = the set that
contains a and b is not included in the set that contains a, c, and d,
because b is in the former but not the latter set.
PuR The union of P and R (also read "P union R") or the join of P
and R. The union of P and R is a set that contains all the elements of
P and all the elements of R. If something is either a member of P
or a member of R, then it is a member of PuR. The set consisting of
the union of Italians (P) and syntacticians (R) consists of those
belonging to either group. Pavarotti is an Italian, so he is included,
and Chomsky is a syntactician, so he is included. To give a further
example, {a,b}u{a,c,d} ={a,b,c,d}. Note that PuR=R if and
only if P <;; R. (Convince yourself of this by using the definitions.)
UP Where Pis a set of sets, UP, the generalized union over P, is
the set that contains all the elements of each member of P. That is,
UP= {x: for some BE P,x E B}. For example, U{ {a, b}, {c}, {b, d}}
= {a,b,c,d}, and U{{a},{b},0} = {a,b}.
P 11 R The intersection of P and R (also read "P intersection R") or
the meet of P and R. The intersection of P and R is a set whose
members contains all and only the elements shared by P and R. If
something is a member of P and also a member of R, then it is a
member of P 11 R. For example, the set consisting of Italians (P)
Appendix 532
who are also syntacticians (R). Pavarotti is not in the set, because
he is not a syntactician. Chomsky is not a member, because he
is not an Italian. But Luigi Rizzi does belong because he is both
Italian and a syntactician. To give two further examples, {a, b, c} n
{f,g,c} {c}, and {a,b}n{c}=0. Note that PnR=P if and
only if P <;: R, and that P n R = 0 if and only if P and R are disjoint.
(Again, use the definitions to show this.)
np Where P is a set of sets, nP, the generalized intersection
over P, is the set that contains the elements that belong to
every member of P. That is, np = {x: for every A E P,x E A}.
Here are two examples: n{{a,b,c,d},{a,b,d},{a,d}} = {a,d}, and
n{{a, b}, {c, d}} = 0.
P- R The difference of P and R or the complement of R relative to
P. This set consists of those members of P that are not also members
of R. For example, the set of linguists (P) who are not French (R).
This set contains Luigi Burzio, who belongs to the set of linguists,
but not N. Ruwet, who belongs to the set of French linguists.
Here are some other examples: {a,b,c}- {a,b} = {c}, and {a,b,c}
-{a, d} = {b, c}, and {a,b, c}- {d} {a, b, c}. Note that P- R = 0
if and only if P <;: R. (See the appropriate definitions.)
R- The complement of R consists of everything that does not
belong to R relative to some understood domain or universe of discourse D; that is, R- = D - R. For example, if the domain D is the
set of dogs and R is the set of spaniels, then D-R = all non-spaniel
dogs (highland terriers, golden retrievers, mongrels, etc.). Again,
relative to the set of integers N, {x: xis odd}- = {x: xis even}.
Union, intersection, and complementation are all operations that
take two sets and form a third set. Generalized union and generalized intersection take a family of sets and form a set.
Set-Theoretic Notation and Concepts 533
Let us now distinguish sets from ordered structures.
{ a1, . .. , an}
This designates a set of n elements if each of a1, . .. , an
is different. If some elements appear more than once in the designation of the set, then the set has fewer than n elements: what
matters for set identity is not how the' elements are listed but which
individual elements are listed. For example {a b c} == {a a b b
'
' '
' ' ' '
b,c,c,c,c}. Repetition is redundant, since the identity of a set
depends only on which elements belong to it. For the same reason,
it does not matter in what order the elements are enumerated:
{a,b,c} = {b,c,a}.
<a1, ... , an) This designates an ordered n-tuple. Here the same
element may recur nonredundantly, for order is critical. The identity of an n-tuple depends on the identity of elements in each of the
distinct n positions. This means that <a1, ... , an)= <b1 , ... , bn) if
and only if a1 = b1 and a 2 = b2 and ... and an = bn. This is the
identity condition for ordered n-tuples. For instance, <a, b) i=
<b, a), and <a, a) i= <a, a, a).
Although intuitively we can think of distinct positions and elements that fill each position (with the same element in principle
able to fill more than one position), it is possible to define ordered
n-tuples with only set-theoretic notions and without introducing
order directly. If, for example, we identify <a,b) with {{a},{a,b}}
and <a,b,c) with {{a},{a,b},{a,b,c}} and so on, then we can use
our identity conditions for sets to show that we have reproduced
the identity conditions for ordered pairs and triples and so on. We
mention the possibility of this reduction just to indicate why
ordered n-tuples are considered set-theoretic objects. It is because
they can be regarded as sets of a special kind.
Let us now turn to the notion of a Cartesian product.
A xB
2 Power Sets and Cartesian Products
The notion of power set is defined as follows.
&(A) The power set of A is the set of all subsets of A. If A contains
n elements, then &(A) contains 2n elements. The set A itself
and the null set are always members of &(A). For example, let
A={Chris,1}. Then &(A) is {{Chris,1},{Chris},{1},0}. Also,
&({a,b,c})
{{a,b,c},{a,b},{a,c},{b,c},{a},{b},{c},0}.
From the definition of power set it follows that A
c;: B
iff A
E
&(B).
The Cartesian product of A and B is the set of all ordered
pairs whose first member belongs to A and whose second member
belongs to B. Thus, Ax B = { <x, y): x E A andy E B}. The Cartesian product of the set of real numbers with itself is used to define
points on a plane; the first number usually represents the horizontal axis, and the second the vertical. As an example of a Cartesian
product we have {a, b} x {1, 2} = {<a, 1), <a, 2), <b, 1), <b, 2) }.
More generally, if A 1, ... , An are sets, A 1 x · · · x An (the Cartesian
product of A 1, . .. , An) is the set that contains all the ordered ntuples <a1, ... , an) such that a 1 E A 1 and ... and an E An.
Appendix 534
Set-Theoretic Notation and Concepts 535
The power set of a Cartesian product A x B will be the set containing all the sets of ordered pairs that one can build out of A and
B. In symbols, a E &(A x B) iff a s; A x B.
Then B = {<Aspects, Chomsky), <Little Women, Alcott), ("The
Coiled Alizarine," Hollander), <"Remarks on Nominalization,"
Chomsky), ... }. Hence, <On Raising, Postal) E B means that On
Raising was written by Postal. More generally, <a, b) E B means that
a was written by b.
A binary relation R is the converse of another such relation S if
whenever <a, b) belongs to R (whenever aRb), then <b, a) belongs
to S (bSa). As defined above, is the author of (relation A) is the
converse of was written by (relation B).
A relation R is reflexive iff every element in the domain bears the
relation to itself, that is, iff for all a in the domain, <a, a) E R (aRa).
For example, being the same age as designates a reflexive relation.
Set inclusion is a reflexive relation between sets (for every set
A, A s; A), whereas proper set inclusion is not (for every set A, it is
not the case that A c A). A relation that is not reflexive is nonreflexive. The transitive verb like is associated with a nonreflexive
relation, since individuals do not always like themselves (though
some do). A relation is irreflexive if nothing stands in that relation
to itself. Proper inclusion is irreflexive since no set is properly included in itself. In most set theories, membership is irreflexive (no
set belongs to itself).
A relation R is transitive iff whenever aRb and bRc (or <a, b) E R
and <b, c) E R), then aRc (or <a, c) E R). Both ordinary set inclusion
and proper set inclusion are transitive relations; being older than
also designates a transitive relation. A relation that is not transitive
is nontransitive. The membership relation between elements and
the sets to which they belong is nontransitive. For example, Mary
might belong to the Task Force for Battered Women (TFBW), and
the TFBW might belong to the United Way, but it does not follow
that Mary belongs to the United Way. Again, the transitive verb like
designates a relation that is nontransitive. Joan may like Linda, and
Linda may like Bill, but we may find that Joan does not like Bill. If
R is a relation such that whenever aRb and bRc, it is not the case
that aRc (<a, c) ¢ R), then R is said to be intransitive. For example,
being the mother of is associated with an intransitive relation (if we
confine our attention to biological motherhood). (Note that a transitive verb need not designate a transitive relation, nor is intransitivity of relations associated with intransitive verbs. Grammatical
and mathematical terminology must be kept distinct here.)
A relation R is symmetric if whenever aRb (<a, b) E R), then bRa
( <b, a) E R). The relation being five miles from is symmetric. A
3 Relations
Sets of ordered n-tuples and Cartesian products are useful in characterizing relations from a set-theoretic perspective. In set theory a
two-place or binary extensional relation is a set of ordered pairs, a
three-place or ternary relation is a set of ordered triples, and in
general an n-place or n-ary relation is a set of ordered n-tuples. A
one-place or unary relation is just a set of individuals.
Let A and B be two sets. A binary relation R between members of
A and members of B will be a subset of the Cartesian product A x B.
In symbols, R s; A x B. Another way of expressing this is to say that
a binary relation R between members of A and members of B is a
member of the power set of A x B. In symbols, R E &(A x B).
The definition above can be extended to n-place relations. If
A 1 , . .. , An are sets, an n-place relation K among A 1, . .. , An will be a
subset of A 1 x · · · x An (that is, K <;::; A1 x · · · x An)·
Two-place or binary relations are particularly important. If R is a
two-place relation between sets A and B, the set of elements from
which the first members of the pairs in relation R are drawn is the
domain of R, and the second members are drawn from the range or
codomain of R. Relations are often notated as follows: aRb. This is
just another notation that says the pair <a, b) belongs to relation R;
that is, <a, b) E R. We often read this as "a stands in relation R to
b." We can think of the incomplete VP is the author of as designating a binary relation that holds between an author and something she or he wrote, as designating the set of ordered pairs whose
first member is an author and whose second is something the
author wrote. Call this set A. Then A = {<Chomsky, Aspects),
<Alcott, Little Women), <Hollander, "The Coiled Alizarine"),
<Chomsky, "Remarks on Nominalization"), ... }. Hence, <Austen,
Emma) E A means that Austen is the author of Emma;
<Shakespeare, Syntactic Structures)¢ A means that Shakespeare is
not the author of Syntactic Structures. Generally, <a, b) E A means
that a is the author of b. We can think of the incomplete VP
was written by as designating the set of ordered pairs whose first
member is a piece of writing and whose second member is the
person who wrote the work in question. Call this set B.
Appendix 536
Set-Theoretic Notation and Concepts 537
relation that is not symmetric is nonsymmetric. Being the sister of
is a nonsymmetric relation, since Joan may be the sister of Lee and
Lee may be the brother (not sister) of Joan. A relation is asymmetric
if it is never the case both that aRb and bRa. Being the mother of is
an asymmetric relation, unless we go beyond standard biological
parenthood (which allows all sorts of more complex scenarios: my
(step) daughter might marry my father and then become my (step)
mother, making me my own grandmother). Proper set inclusion
and set membership are asymmetric.
An equivalence relation is a relation that is reflexive, transitive,
and symmetric. An equivalence relation R partitions a set A into
equivalence classes, which are disjoint and whose union is identical with A. For each a in the domain of R, let S(a) ={bE A: aRb}.
Then S(a) is the equivalence class to which a belongs. Being the
same age as is an equivalence relation, and each equivalence class
consists of a cohort: those who are some particular age.
Relation R is one-one if each element in the domain is paired
with exactly one element in the range and vice versa. The relation
between individuals and their fingerprints is thought to be one-one.
Relation R is one-many if some members of the domain are paired
with more than one member of the range. Being the mother of is
associated with a one-many relation, since some mothers have several children. Relation R is many-one if different members of the
domain can be paired with the same member of the range. Being the
child of is associated with a many-one relation, since sometimes
several children have the same parent. A relation is many-many if
it is both many-one and one-many. Has visited designates a manymany relation between people and cities, since a person may visit
many cities and a particular city may be visited by many people.
with its converse, is written by (we restrict our attention to singleauthored works). Here once the first slot is filled in, there is a single
value for the second slot. In other words, was written by can be associated with a function that assigns books, essays, poems, etc., to
the individual who wrote them. Thf,'l expression was born in designates a function that assigns a unique year to each person, the year
in which the person was born. (Of course, the same year will occur
as second member of different ordered pairs, since many different
people are born in a single year.)
The first member of an ordered pair in a function is its argument;
arguments belong to the domain of the function. The second member of an ordered pair in a function is its value; values belong to the
function's range.
To indicate that <x,y) belongs to the function f, we often write
f(x) = y. A function of this type is a one-place function, for it takes
only one argument. Thus a one-place function is a one-one or a
many-one two-place relation. Examples of such functions are the
following:
A two-place relation R is a function just in case any element a in
the domain of R is the first member of only one ordered pair, that is,
just in case if aRb and aRc, then b = c. A function is a relation that
is not one-many but either one-one or many-one. The relation is the
author of that we discussed in the last section is not a function: it
holds between Chomsky and Syntactic Structmes and also between
Chomsky and "Remarks on Nominalization." That is putting
Chomsky in the first slot of " __ is the author of __ " does not
uniquely determine a value to fill in the second slot. Contrast this
This is just an alternative way of representing x's spouse(Alan) =
Alan's spouse =Nancy, etc.
By means of our definitions, we can designate the same function
in different ways; the notion of function is extensional like the
notion of set that helps define it. Suppose that f1 (x) = x 2 - 1 and
f2 (x) (x- 1)(x + 1). We have specified f 1 and f2 using different
rules for determing the value of the function, but the rules yield
exactly the same value for any numerical argument. Thus f 1 = f 2 ;
that is, we have not two functions but only one.
4 Functions
was-born-in(Alan) = 1962
was-born-in(Blanche) = 1897
the-height-of(Sally) = 5 feet, 4 inches
the-senior-senator-from(N.Y.) =Patrick Moynihan (as of spring
1999)
When a function has a small domain we often represent it in tabular form. So, for example, if the domain is {Alan, Lisa, Greg} and
the function is that expressed by x's spouse, we might display the
pairings in an array:
Alan
Lisa
[ Greg
__.,
__.,
__.,
Nancy]
Bob
Jill
Appendix 538
Some helpful notation is the following:
f : A ~ B In words, f maps A, its domain, onto B, its range.
BA The set of functions with domain A and range B; that is, the set
of functions from A to B.
From the above definition it follows that (CB) A is the set of all
functions from A onto cB. Thus a member g of (CB)A is a functionvalued function. For any a E A,g(a) is a function from B onto C, or
for any bE B, g(a)(b) E C. This should not be confused with C(BA),
which is the set of all functions from BA to C. That is, a function h
in C(BA) will map each function dE BA onto a member of C; that is,
h(d) E C.
A complete function assigns values to every member of its domain; every member of the domain is a first member of an ordered
pair belonging to the function. A function that fails to assign values
to some members of its domain is a partial function. Generally,
when reference is made simply to functions, it is complete functions that are meant.
Any set can be associated with a particular distinguished function called its characteristic function or its membership function
relative to a given universe or domain D. The characteristic function of A relative to D is the function ftA with domain D such that
ftA(x) = 1 iff x E A and flA(x) = 0 iff x ¢A.
It is conventional to choose {0, 1} as the range of characteristic
functions, although any two-membered set will do the job. What is
crucial is that the ;:;pecification of a set defines a unique characteristic function and the specification of a complete function from
universe D to {0, 1} defines a unique subset of D. The characteristic
function sorts the members of D into those that belong to A and
those that do not. The fact that we can go from a set to a unique
function and from a function to a unique set allows us for certain
purposes to identify sets with their characteristic functions.
The characteristic functions associated with the members of
t?P(A), the power set of set A, are just the family {0, 1}A. Any characteristic function of a subset of A belongs to this family, and any
member of this family is the characteristic function of some subset
of A.
So far we have considered only one-place (unary) functions.
However, we can extend our approach to n-place functions (functions that take n arguments). An n-place function g from A 1 , ... , An
to B is an (n + 1)-place relation such that for any a1 E A1, ... ,
OnE An and any b, b' E B, if <a1, ... , On, b) E g and <a1, ... , On, b') E g,
Set-Theoretic Notation and Concepts 539
then b = b'. We write g(a 1, ... , an)= b for <a 1, ... , an, b) E g. For
example, addition over the positive integers is a two-place function
mapping two numbers onto their sum. It is the infinite set
{ <x, y, z) : x and y are positive integers and z = x + y} =
{ <1, 1, 2), <1, 2, 3), <2, 1, 3), ... }.
It turns out that n-place functions can always be reduced to the
successive application of one-place functions. The reduction is
often called currying a function, in recognition of logician H. B.
Curry, who adapted the technique from M. Schi:infinkel, in whose
honor Kratzer and Heim (1998) speak of schonfinkelization. The
basic idea is simple, and we will illustrate it using the addition
function that maps any two integers onto their sum. That is, we
begin with f(x, y) = x + y. The trick is to let functions assign other
functions as values. So for any x, let f' (x) be the function that when
applied to y yields f(x, y) = x + y. In other words f'(x)(y) =
f(x, y) = x + y. Exactly which function is assigned as f'(x) depends, of course, on the argument x. So if x = 2, f'(2) is the function that when applied to argument y yields the value 2 + y; i.e.,
f'(2)(y) = 2 + y and thus f'(2)(3) = [2 + y](3) = 2 + 3 = f(2, 3). If
x = 4 then f' (4) is the function that when applied to argument y
yields the value 4+ y; i.e., f'(4)(y) 4+ y and thus f'(4)(2) =
[4 + y](2) = 4 + 2. Or we can take the second argument first and let
f" (y) be the function that when applied to x yields f(x, y) = x + y.
In this case, we have f"(y)(x) = x + y = f(x, y). So f"(2)(x) =
x + 2 and f"(2)(3) = [x + 2](3) = 3 + 2 = f(3, 2). And so on.
5 Boolean Algebras
Finally, we need to define the algebraic notion of a Boolean algebra, which we mention at several points in the text. An algebra is
just a set together with operations that map an element or elements
of the set onto some unique element of the same set. Boolean algebras have three operations: two binary operations (which we will
denote with n and u) and a unary operation (denoted by-). There
are also two distinguished elements of the set (denoted by 1 and 0).
Call the set B.
Then the elements of the algebra satisfy the following axioms:
(1) For any A, BE B,
a. AuB=BuA
b. BnA = AnB
This axiom says that both the binary operations are commutative.
Appendix 540
Notes
(2) For any A, B, C E B,
a. Au (Bu C)= (A uB) u C
b. A n (B n C) = (A n B) n C
This says that both binary operations are associative.
(3) ForanyA,B,CEB,
a. A u (B n C) = (Au B) n (Au C)
b. A n (B u C) = (An B) u (A n C)
Chapter 1
1.
This says that each binary operation is distributive over the other.
(4) For any A E B, A- is the unique element satisfying the
conditions Au A- 1 and An A- = 0.
2.
Our choice of symbols for the operations is motivated by the fact
that the set-theoretic operations in a particular domain form a
Boolean algebra. A nonempty set D generates a Boolean algebra as
follows. Let B = &(D), the power set of D or the set whose members
are all the subsets of D. Interpret the operations as ordinary set
union ( u ), intersection ( n ), and complementation(-), and let 1 = D
and 0 = 0 (the null set). The resulting structure is then a Boolean
algebra. Propositional logic also forms a Boolean algebra under a
suitable interpretation if we identify v with u, 1\ with n, and -,
with -. Intuitively, 0 and 1 correspond to falsity and truth, respectively. A bit more has to be said to make this precise, but the point
is that the axioms above are indeed theorems of propositional logic.
Further easy-to-read resources for linguists with little or no
formal background are Wall (1972) and Allwood, Andersson, and
Dahl (1971). Partee, ter Meulen, and Wall (1990) cover a much
wider range of basic mathematical material relevant for work in
formal linguistics. The reader might also find it useful to consult an
elementary introduction to set theory like Halmos (1960) or Stoll
(1963).
3.
4.
5.
6.
7.
8.
9.
See for example, work in generalized phrase-structure grammar (GPSG) as described in
Gazdar, Klein, Pullum, and Sag (1985) and related approaches, such as those outlined in
Pollard and Sag (1 988).
See van Riemsdijk and Williams (1 986) for an introduction to the "principles and
parameters" framework in current Chomskyan syntactic theory. Chomsky (1986), especially 3.4, discusses how parametric approaches shed new light on the structpre of the
human language faculty. (Chomsky 1995 and other recent work in the minimalist
framework does not really use the "parameter" idea directly.) Within semantics, explicitly comparative work is more recent. Although still not very extensive, research on
cross linguistic semantics is rapidly increasing. Chierchia (1 998a, 1998b) argues that
parameters belong in semantic theory, whereas other recent work on crosslinguistic
semantics takes a somewhat different approach to the question of universals; see, e.g.,
Bach eta! (1 995) and Bittner (1995).
See Chomsky (1 965) for the distinction between formal and material universals of language and J. D. Fodor (1 977) for an excellent account of the contributions of McCawley
and others in generative semantics and more generally of semantic research in the generative tradition into the early seventies.
The illustration comes from Cresswell (1985, 3).
See esp. Grice (1975, 1978), both of which are reprinted in Grice (1 989). Grice also suggested that some inferences are licensed by virtue of noninformational aspects of linguistic meaning rather than by either entailment or conversational principles; we defer
discussion of these so-called conventional implicatures until chapter 4.
See Evans (1980) for detailed discussion of this issue.
See, for example, Higginbotham (1983).
See J.D. Fodor (1977) for a discussion of this essentially syntactic account of anomaly and
of the observations that led linguists to explore more semantic approaches.
Hollander's poem, "Coiled Alizarine: For Noam Chomsky," is reprinted from John
Hollander, The Night Mirror (Atheneum Publishers, 1971) on page 1 of Harman (1 974).
Chapter 2
See J. A. Fodor (1975) for a discussion of the notion of a "language of thought" and its
semantic relevance.
2. A clear and more extensive formulation of these criticisms can be found in Lewis
(1972).
3. A problem that clearly shows the direct semantic relevance of these questions is the mass/
count distinction, the semantic (and syntactic) differences between nouns like gold, furniture, equipment, etc. versus boy, cot, choir, etc. See, for example, Pelletier (1979).
4. This argument is already implicit in Frege (1892). More recent versions of it can be found
in Church (1956) and Davidson (1967a). Barwise and Perry (1983) dub it "the slingshot."
The argument might be a little hard to grasp at first. A full understanding of it is not a
necessary prerequisite for subsequent material.
1.