Semantics:
Handout 11
Extension vs. Intension
Yılmaz Kılıçaslan
December 17, 2009
1 Introduction
In the previous lecture, we treated tense and modal operators syncategorematically. When
describing the relevant languages (i.e. L2t and L2mt), we did not, for example, define a
denotation for the symbol F directly, but only a denotation of the whole expression F in
terms of the interpretation of the expression . Could we not assign a denotation separately to
F in such a way that the denotation of F could be determined entirely on the basis of the
denotation of F and the denotation of ? This would parallel the way in which we gave a
denotation for it-is-not-the-case-that in L1E in such a way that the denotation of it-is-not-thecase-that  was the result of applying the function denoted by it-is-not-the-case-that to the
denotation of . However, a little thought suffices to see that this cannot be done for F. This
is simply because the denotation of  is a truth value but the truth value of F is independent
of the truth value of .
In the subsequent section, we will see several cases where the denotation of a natural
language phrase cannot be explained in terms of the denotation of its parts.
2 Cases Awaiting Reconciliation with the Principle of Compositionality
2.1
Non-truth functional operators
The denotation of  at any time depends merely on the denotation (truth value) of  at that
same time. For this reason, negation is said to be a truth-functional operator.The tense
operators, on the other hand, are non-truth-functional. F may be true at a given time whether
or not  is true at that time; the same is true of formulas of the form P. The following pairs
of sentences illustrate this point (examples cited by Thomason 1974, to establish the same
point):
(1)
a. Iceland is covered with a glacier.
b. Africa is covered with a glacier.
(2)
a. Iceland was (once) covered with a glacier.
b. Africa was (once) covered with a glacier.
Though (1a) and (1b) are both false at the present time, (2a) is now true but (2b) is now false.
Obviously, the denotations of F and P at a time t will depend on other things besides the
denotation (i.e. truth value) of  at t; specifically they will depend on the denotation of  at
other times. Hence, as long as the semantic interpretation of a sentence is confined to a truth
value, as what we have done up to this point, it is not possible to give a compositional
treatment to the semantics of tensed sentences.
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The first philosopher to confront the problem pointed out above was Gottlob Frege (Frege
1893). But, he noticed the problem with modal operators, not tenses. The way he put the
problem has been known as Frege’s puzzle about identity. Most systems of logic with an
identity predicate contain a law (the so-called Leibnitz Law) that the result of substituting in
any formula one name for another name denoting the same individual results in a formula that
is true if and only if the original formula was true. Thus to take a natural language example,
‘Morning Star’ and ‘Evening Star’ are two different names that denote the same entity: the
planet Venus.1 Therefore, the sentence
(3)
The Morning Star is not visible now.
is true just in case the sentence
(4)
The Evening Star is not visible now.
is true. However, contrast sentence (5) and (6):
(5)
Necessarily, the Morning Star is the Morning Star.
(6)
Necessarily, the Morning Star is the Evening Star.
Sentence (5) is true because
(7)
The Morning Star is the Morning Star.
is a logically true sentence, an instance of the axiom a = a (for any name a) in predicate logic
with identity. However, (6) is not ture according to Frege, since the sentence
(8)
The Morning Star is the Evening Star.
expresses a contingent fact, not a matter of logical necessity. In other words, (7) can be known
to be true simply by understanding the meaning of its words. It is a priori (i.e. describes a
statement which we can know to be true or false independently of sense experience). The
same cannot be said about (8). We have to discover whether it is true by some sort of sensory
observation. It is a posteriori (i.e. describes a statement which we can know to be true or false
only by sense experience).
It follows that in a context created by the necessity operator, , it is not always possible to
substitute a name for another name denoting the same entity. Thus, the operator  produces
another case where the denotation of the whole expression is not strictly a function of the
denotation of the parts. (Recall that in the rules given when describing L2t or L2mt, the
denotation of  did not depend just on the denotation of  at that index but on ’s denotation
at other indices as well.)
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But, as Robert Lane (2004) Points out, it was not always known that this is the case. Sailors would refer to the
first star to appear at night at certain times during the year as the Evening Star, and the last star to disappear in
the morning at other times during the year as the Morning Star, without realizing that the Morning Star and the
Evening Star are actually one and the same object.
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2.2
Opaque Positions
Syntactic constructions where co-designative names cannot be substituted for each other
while preserving truth values (i.e. without violating Leibnitz’ Law) are called oblique
constructions or referentially opaque constructions, as opposed to the more “ordinary”
referentially transparent constructions for which Leibnitz’ Law holds. In addition to
sentences with modals, we have also the complements of verbs like believe, think, hope,
imagine, say etc. that display a referentially opaque behaviour. Consider the following pair of
sentences:
(9)
a. John believes that the Morning Star is the Morning Star.
b. John believes that the Morning Star is the Evening Star.
Assuming that the invidual named ‘John’ is a perfectly rational person when it comes to
principles of identity but is nevertheless somewhat benighted astronomicaly, (9a) might be
true while (9b) is false. Clearly, Leibnitz’ Law does not hold in the given context. If that were
the case, (9b) would have the same truth value as (9a) from which it differs only by ‘the
Morning Star’ substituted for the co-designator ‘the Evening Star’.
Another frequently discussed opaque position is the object position of transitive verbs like
want, seek, need, etc. Even though there is no single individual whose name could be
substituted for the object phrase in this position while preserving truth value, the whole
statement can be true. Consider the sentence below:
(10)
John needs a friend.
In one reading at least, the denotation of the whole sentence cannot be explained in terms of
the denotation of its parts as far as only an individual can serve as the denotation of an
indefinite. This is what linguists would call the non-specific reading of (10), the reading in
which there is no particular friend that John needs. Interestingly, the use of case morphology
in the direct object position of Turkish sentences seems to closely relate to whether the object
phrase is interpreted to be specific or non-specific:
(11)
a. Oya bir avukat ari-yor.
one lawyer seek-prok
‘Oya is seeking a lawyer.’
b. Oya bir
avukat-i ari-yor.
one lawyer-acc seek-prok
‘Oya is seeking a lawyer.’
Of the two sentences above only (a) (the one with an object carrying case morphology) can
receive a specific interpretation, an interpretation where there is a particular lawyer sought by
Oya.2
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Enç (1991) argues that Turkish is a language where specificity is overtly marked in the direct object positions.
According to her, direct objects bearing the accusative case suffix, -yI, are specific, whereas those without case
morphology are non-specific in this language. Taking this classification as a point of departure, she developes a
view of specificity where this notion is equated with Pesetsky’s (1987) notion of D(iscourse)-linking.
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2.3
Some Adjectives Displaying a Different Semantic Behaviour
Even though “referential opacity” is not applicaple where they occur, certain adjectives (and
also adverbs) do not allow for a ‘straightforward’ compositional treatment, either. A
straightforward approach to the semantics of an adjective is to to take its denotation to be a
set, just as the denotation of intransitive verbs and nouns. This seems to work well for either
the predicative or attributive position in many cases. Consider the following sentence:
(12)
Some young minister is talkative.
(12) can be said to be true just in case there is an individual that is a member of the set
denoted by young and of the set of ministers, and that individual is in the set denoted by
talkative. However, the truth conditions for the sentence
(13) Some former minister was talkative.
cannot be given in the same way. The denotation of the subject cannot be defined as the
individual that is both in the set of ministers and in the set of “individuals that are former”.
This latter phrase is simply meaningless. In order to give a compositional treatment for the
semantics of phrases containing adjectives such as former and alleged, we need to consider as
meaning of such adjectives something richer than a mere set of individuals at a certain index.
In fact, a similar step is required for all other cases discussed in this section in order to be
analyzable in line with the Principle of Compositionality. In what follows, we will look at a
distinction proposed to solve similar problems of compositionality.
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Sense vs. Reference (or Intension vs. Extension)
Frege’s solution to problems of the type exemplified above lay in his distinction between
between the sense of an expression and its reference or denominatum. The term intension later
came to be used for sense, and the term extension for reference. The reference of an
expression for Frege corresponded to what we have uniformly termed the denotation of an
expression (e.g. a truth value in the case of sentences, a set in the case in the case of
adjectives, nouns or verbs, etc.). The sense corresponded to what we might intuitively think of
as the “meaning” of the espression as distinct from its denotation. The sense of an expression
is supposed to determine what its reference is in any possible state of affairs.
Armed with this distinnction, Frege attempted to solve the problem of non-referential contexts
(modal contexts, belief contexts, etc.) by saying that expressions of natural language have a
kind of “ambiguity” in that sometimes an expression has a “normal” denotation (i.e. the kind
of denotation we have been assigning to expressions of our formal languages), but in certain
circumstances an expression “denotes” what is ordinarily its sense. The idea is that two
sentences having the same denotation, for instance, in isolation (e.g. (7) and (8)) may have
different senses when embedded in modal or belief contexts (e.g. in (9a) and (9b)) so that the
denotation of the whole sentence can be a function of the “denotations” of its parts. As such
expressions “denote” different senses in non-referential contexts, they cannot be substituted
for each other in such contexts.
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4 Formalization of the Sense vs. Reference Distinction
The first attempt to formalize the notion of sense was made by Rudolf Carnap (1947). Since
the meaning of an utterance is supposed to determine its extension, Carnap suggested that the
sense of an expression (for which he substituted the term intension of an expression) is simply
a function from possible states of affairs which gives, for any particular state of affairs, the
denotation of the expression (which Carnap called the extension of the expression) at that
state of affairs. With the advent of Kripke’s semantics for modal logic (taking possible worlds
as indices), it became possible for the first time to give an unproblematic formal definition of
intension for formalized languages.
For our formal languages that employ coordinate semantics, we will want to take an intension
as a function from indices to extensions; in this way we will be able treat the “meanings” of
indexical expressions uniformly with that of all other expressions. We assume, for simplicity,
that indices are just ordered pairs consisting of a possible world and a time (as in L2mt).
Individual concepts: A name denotes an individual at each index. Hence, the intension of a
name is a function from indices to individuals (intuitively, a function which gives for each
context the person denoted by the name at that context). It is traditional to call such an
intension an individual concept or the concept of a particular name. For example, if Miss
America (or simply m) is a name, then the concept of Miss America is that function which for
each index (i.e. for each world and time) gives the person picked out by the name at that
world and time. For the actual world these are, for each time, the persons who actually were,
are, or will be Miss America, and for different world coordinates, these are persons who
would be or would have been Miss America had the world been different in various ways
from the way it actually is. Since “ordinary” names (unlike Miss America) denote the same
individual at various times and even in different worlds according to some philosophical
views on naming, their sense will simply be a constant function, even though it has been a
matter of some philosophical dispute whether names should have an intension as well as an
extension.
Properties (of individuals): A one-place predicate denotes a set of individuals (or
equivalently, the characteristic function of a set of individuals) at each index. Thus the
intension of a predicate will be the function that gives, for each index, the set (or
characteristic function thereof) denoted at that index. Such an intension will be called a
property. Thus the property corresponding to the English is asleep can be thought of as the
function which gives, for each possible situation, the set of individuals that are asleep in that
situation.
Propositions: A formula denotes a truth value at each index. Hence the intension of a formula
is a function from indices to truth values, and such an intension will be called a proposition.
Since a function from any set into {0, 1} is the characteristic function of some subset of that
set, a proposition as just defined “characterizes” a set of indices; i.e. a proposition maps all the
indices at which a formula is true into 1, and maps all other indices into 0. Thus it
characterizes the set of indices at which the formula is true, and on this view it is common to
equate a proposition with a set of indices (in the same way as the function denoted by a
predicate is equated with the set of individuals that function characterizes).
We should also note that variables, as well as constants, are supposed to be provided with
intensions. But recall that the extension (i.e. denotation) of a variable is always supplied by
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the value assignment g and thus does not differ from one index to the next. Thus the intension
of a variable will be a constant function on indices, in particular, one that gives at any index
whatever g assigns to that variable.
PS: The content of this handout is adopted from Dowty et al (1981).
References
Carnap, R. (1947) Meaning and Necessity. Chicago: University of Chicago Press.
Dowty D.R., Wall R. E. and Peters S. (1981) Introduction to Montague Semantics. Dordrecht,
The Netherlands: Kluwer Academic Publishers.
Enç, M. (1991) The semantics of specificity. Linguistic Inquiry 22, 1-26.
Lane, R. (2004) Lecture notes in analytic philosophy. In Robert Lane’s Homepage
(www.westga.edu/~rlane/analytic/lecture_philLanguage2.html).
Pesetsky, D. (1987) Wh-in-situ: movement and unselective binding. The Representation of
Indefinites
Thomason, R.H. (1974) Introduction. Formal Philosophy: Selected Papers by Richard
Montague, R.H. Thomason (ed.). New Haven: Yale University Press, 1-69.
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