Semantics: Handout 12
Montague’s Intensional Logic
Yılmaz Kılıçaslan
December 17, 2009
1 Introduction
We are now ready to see Montague’s work itself in formal semantics. We will start with his
intensional logic. Having also equipped with this logic, we will be able to proceed to the
exploration of the system outlined in his paper entitled ‘The Proper Treatment of
Quantification in Ordinary English’ (commonly abbreviated PTQ), which represents the
culmination of his efforts to apply the techniques developed within mathematical logic to the
semantics of natural languages (cf. Handout 12).
2 The intension and extension operators
The distinction we have made between intension and extension is so far confined entirely to
the meta-language. We have shown how to define the intension and extension of a given
expression, but we have not changed any of our object languages in any way so that the
distinction can be properly handled both in the syntax and semantics at the object level. One
way of achieving this effect is to modify the object language to include expressions which
“denote” the intensions of certain other expressions, where “denote” is used as synonymous
with “have as extension”. In Montague’s intensional logic, for any expression α there is
another expression denoting the intension of α. Since the expressions of the language are
infinite in number, it is convenient to form these intension-naming expressions syntactically
out of the expressions whose intension they name. Montague’s device for doing this is quite
simple:
(1)
If α is any expression, then ˆα is an expression which denotes the intension of α.
Henceforth, as a notational convention we will use [α]¢M,g for the intension of any expression
α with respect to a model M and a value assignment g. [ˆα]M,w,t,g will be [α]¢M,g for all indices
<w, t>.
Since the notion of intension is defined for every expression of the language, expressions like
ˆα will themselves have an intension as well as an extension. Notice also that (1) allows to
recursively form expression like ˆˆα, ˆˆˆα, ˆˆˆˆα, etc. In fact, all such expressions count as
well-formed in Montague’s intensional language and are assigned a well-defined intension
and extension by the principles we have already outlined (cf. Handout 10). But these higherorder intensions turn out to be rather uninteresting in the long run. As there is one and only
one [α]¢M,g, [ˆα]¢M,g will be a constant function on indices which gives for any index the value
[α]¢M,g. Likewise, [ˆˆα]¢M,g will be yet another constant function which gives for any index the
constant function [ˆα]¢M,g, and similarly for the more complex iterated intensions.
Montague introduced a second syntactic device into his intensional language as a sort of
converse to the intension operator “ˆ”. This is the extension operator “ˇ”, which can be applied
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to any expression α which denotes an intension to give a new well-formed expression “ˇα”. At
any index <w, t>, the expression “ˇα” denotes the result of applying the function [α]M,g to the
argument <w, t>. Stated formally, if α denotes an intension, then for any index <w, t>, [α]M,w,t,g
is defined as [α]M,g(<w, t>). The result of combining ˇ with an expression of the form ˆα has the
effect of “cancelling” out of the two operators ˇ and ˆ. That is, we have the following equation
that licenses a rule of “down-up cancellation”:
(2)
[ˇˆα]M,w,t,g = [α]M,w,t,g
for all expressions α, and all models M and all w, t, and g.
3 The intensional logic of PTQ
Montague’s intensional logic (“IL”) is a formal system that combines all the syntactic and
semantic resources we have been gradually developing. This system therefore employs a type
hierarchy (with expressions for each of the denumerably infinite set of types), higher-order
quantification (variables and quantifiers for each type), lambda-abstraction for all types,
tenses, modal operators, and finally, means for forming ˆα from any expression α and ˇβ from
any intension-denoting expression .
The syntactic definitions of IL begin, as in Ltype, with the recursive definition of the set of
types. Since we will now be dealing with expressions denoting intensions as well as
extensions for the various types, the definition of the set of types will include a new recursive
clause to define for each type a, a new type <s, a> for the intensions (or senses) corresponding to
each type a. These new types will be explained more fully when the semantic definitions are
introduced.
Syntax of IL
Let t, e, and s be any fixed objects. Then the set of types is defined recursively as follows:
1. t is a type
2. e is a type
3. If a and b are any types, then <a, b> is a type.
4. If a is any type, then <s, a> is a type.
A. Basic Expressions
1. For each type a, IL contains a denumerably infinite set of non-logical constants (or
simply constants) cn,a, for each natural number n. The set of all (non-logical) constants
of type a is called Cona.
2. For each type a, IL contains a denumerably infinite set of variables vn,a, for each
natural number n. The set of all variables of type a is called Vara.
B. Syntactic rules of IL
The set of meaningful expressions of type a, called MEa, is defined recursively as follows:
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1. Every variable of type a is in MEa.
2. Every constant of type a is in MEa.
3. If α  MEa and u is a variable of type b, then λuα  ME<b, a>.
4. If α  ME<b, a> and β  MEa, then α(β)  MEb.
5. If α and β are both in MEa, then α = β  MEt.
6.-10. If  and ψ are in MEt, then the following are also in MEt:
6. ¬
7. [  ψ]
8. [  ψ]
9. [  ψ]
10. [  ψ]
11. If   MEt and u is a variable of any type, then u  MEt.
12. If   MEt and u is a variable of any type, then u  MEt.
13. If   MEt, then □  MEt.
14. If   MEt, then F  MEt.
15. If   MEt, then P  MEt.
16. If α  MEa, then ˆα  ME<s, a>.
17. If α  ME<s, a>, then ˇα  MEa.
Semantics of IL
A. Model
A model for IL is an ordered quintuple <A, W, T, <, F> such that A, W and T are non-empty sets,
< is a linear ordering on the set T, and F is a function that assigns to each non-logical constant
of IL of type a an intension. The set of possible denotations of type a is defined as follows
(where a and b are any types):
1. De is A
2. Dt is {1, 0}
3. D<a, b> = DbDa
4. D<s, a> = DaWxT
B. Semantic rules of IL
The function F will assign to each non-logical constant of IL of type a a member of D<s, a>.
The semantic rules of IL define recursively for any expression a, the extension of a with
respect to model M, w  W, t  T and value assignment g, denoted [a]M,w,t,g, as follows:
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1. If a is a non-logical constant, then [a]M,w,t,g = [F(a)]( <w, t>).
2. If a is a variable, then [a]M,w,t,g = g(a).
3. If α  MEa and u is a variable of type b, then [λuα]M,w,t,g is that function h with domain
Db such that for any object x in that domain, h(x) = [α]M,w,t,g’ where g’ is that value
assignment exactly like g with the possible difference that g’(u) is the object x.
4. If α  ME<a, b>, and β  MEa, then [α(β)] M,w,t,g is [α]M,w,t,g([β]M,w,t,g).
5. If α and β are in MEa, then [α = β]M,w,t,g is 1 if and only if [α] M,w,t,g is the same as
[β]M,w,t,g.
6. If   MEt, then [¬ ]M,w,t,g is 1 if and only if [ ]M,w,t,g is 0, and [¬ ]M,w,t,g is 0
otherwise.
7.-10.
If  and ψ are in MEt, then [  ψ]M,w,t,g is 1 if and only if both [ ]M,w,t,g and
[ψ]M,w,t,g are 1. (The definitions for the other truth-functional connectives of [  ψ], [  ψ],
and [  ψ] are the usual ones, paralling this definition.)
11. If   MEt and u is a variable, then [u ]M,w,t,g is 1 if and only if [ ]M,w,t,g’ is 1 for all g’
exactly like g except possibly for the value assigned to u.
12. If   MEt and u is a variable, then [u ]M,w,t,g is 1 if and only if [ ]M,w,t,g’ is 1 for some
value assignment g’ exactly like g except possibly for the value assigned to u.
13. If   MEt, then [□ ]M,w,t,g is 1 if and only if [ ]M,w’,t’,g is 1 for all w’ in W and all t’ in
T.
14. If   MEt, then [F ]M,w,t,g is 1 if and only if [ ]M,w,t’,g is 1 for some t’ in T such that
t<t’.
15. If   MEt, then [P ]M,w,t,g is 1 if and only if [ ]M,w,t’,g is 1 for some t’ in T such that
t’<t.
16. If α  MEa, then [ˆα]M,w,t,g is that function h with domain W x T such that for all
<w’, t’> in W x T, h(<w’, t’>) is [α]M,w’,t’,g.
17. If α  ME<s, a>, then [ˇα]M,w,t,g is [α]M,w,t,g(<w, t>).
B. Further definitions
1. If  is a formula (i.e. a member of MEt) then  is true with respect to M and to <w, t> if
and only if [ ]M,w,t,g is 1 for all value assignments g.
2. If α is any expression, then the intension of α with respect to M and g, denoted [α]¢M,g,
is that function h, with domain W x T such that for all <w, t> in W x T, h(<w, t>) is
[α]M,w,t,g.
4 Examples of ‘Oblique Contexts’ as Represented in IL
Let us now see how some of the oblique constructions that are seemingly in conflict with the
Principle of Compositionality can be reconciled with this principle (cf. Handout 10).
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First note that sentences of the form Necessarily  could indeed be given an analysis in which
the extension of the whole is a function of the extensions of the parts if we treated the word
Necessarily as a functor which combined not with  but rather with ˆ. This means that the
operator in IL corresponding to necessarily would be treated not as an expression of type
<t, t> but rather of type <<s, t>, t>. Let us call such an operator Nec. Its denotation would be a
function from propositions to truth values. We could give the semantic rule for Nec by saying that at
any index <w, t>, [Nec]M,w,t,g is that function from propositions into truth values that gives the
value 1 when applied to any proposition p just in case p itself maps every index <w, t> into 1;
otherwise p will be mapped into 0.
Consider now the oblique context formed by adjectives such as former. The extension of
phrases like former senator can now be defined in terms of the extensions of its parts if we
combine former not with an expression denoting a predicate, but one denoting a property.
That is, we introduce For as an expression of type <<s, <e, t>>, <e, t>>, whose extension will
therefore be a function from properties (of individuals) to predicates. Thus, [For]M,w,t,g is a functor
which, so to speak, has “access” not only to the current extension of the one-place predicate it
combines with, but also “access” to the extension of that predicate at other indices as well. In
particular, it has access to the extension of that predicate at indices with earlier time
coordinates. Thus [For]M,w,t,g may be thought of as a function which takes as input a property
(say, the property of being a minister) and then gives as output the set of individuals that are
not now but were at earlier times in the extension of that property, i.e. the set of former
ministers.
Next consider the treatment of verbs like believe. It was Montague’s view that such verbs
could be successfully treated as denoting a relation between individuals (specifically, persons)
and propositions. Thus in intensional logic we would treat a constant Bel as an expression of
type <<s, t>, <e, t>>. A sentence like John believes that Miss America is bald would then be
rendered in intensional logic as Bel(ˆ(B(m)))(j). Since the so-called “relational notation”
γ(α, β) is familiar for stating that “α stands in the relation γ to β”, Montague introduces γ(α, β)
as an alternative notation for γ(β)(α). Thus Bel(ˆ(B(m)))(j) is also written as Bel(j, ˆ(B(m))).
Now let us see how Frege’s puzzle about substituting names in belief contexts is treated.
Suppose that m and n denote the same individual at the current index. So it follows that B(m)
and B(n) share the same truth value at this index. In this sense Leibnitz’ Law holds in IL;
substituting another name for the same individual in the expression B(m) preserves truth
value. However, it does not follow from this fact that the proposition denoted by ˆB(m) is
identical to the proposition denoted by ˆB(n) at the current index (or at any other index in our
model). In fact, we can construct a perfectly consistent model where there are some indices at
which m and n denote different individuals and, hence, B(m) and B(n) may differ in truth
value. If B(m) and B(n) do not have the same truth value at an index, ˆB(m) and ˆB(n) will
differ in their denotation and, therefore, it will be nothing surprising for Bel(j, ˆ(B(m))) to be
true while Bel(j, ˆ(B(n))) is false at the same index or vice versa. We might translate these
expressions into ordinary English (using Norma as the name n) and say that it can be true that
Norma is Miss America even though the sentence John believes that Miss America is bald is
true while the sentence John believes that Norma is bald is false.
We should also note that belief sentences like our example John believes that Miss America is
bald can be understood in two ways, though so far we have treated only one of them. This
example might be understood to assert that John has a belief about a certain individual who
happens to be Miss America at the “current” index, the belief being that this individual is
bald. On this reading the example could describe a true belief of John’s about a certain person
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even if John does not know that she is Miss America, does not know any name for that
individual, or is mistaken about her name. Among philosophers this reading is commonly
referred to as the “de re” reading. Linguists have sometimes described this reading as one in
which the name Miss America is the “speaker’s description” of the person in question. The
other reading, called the de dicto reading, is the one in which John believes that whoever is
named by the name Miss America is bald, and on this reading the sentence can describe a
belief of John’s even though he does not know which individual the name actually denotes
and thus there is no specific person of whom he can be said to hold any such belief. This
reading is sometimes described as the one in which Miss America is “John’s description” of
the person in question. It is this second reading, the de dicto reading, which we have
formalized by the formula Bel(j, ˆ(B(m))). The de re reading can also be formalized in IL,
with the aid of the lambda-abstraction operator as x(Bel(j, ˆ(B(x))))(m). The crucial
difference between Bel(j, ˆ(B(m))) and x(Bel(j, ˆ(B(x))))(m) is that the “non-rigiddesignator” m stands within the scope of the oblique-context-creating operator ˆ in the first
formula, but outside the scope of it in the second formula.
Though non-rigid names such as Miss America are not common in natural language, there are
a number of de dicto/de re ambiguities involving quantified noun phrases, and IL is capable of
representing these as well. For example, (3) involves this ambiguity with respect to the
phrase a Republican:
(3)
John believes that a Republican will win.
On the de re reading, this example asserts that John believes of some particular person (who is in fact
a Republican) that he will win. On the de dicto reading it asserts that John’s belief is simply that,
regardless of who the winner turns out to be, it will be a Republican. The de re reading is
fomalized as (4), the de dicto as (5): (Here, R and W are to be constants of type <e, t>
understood as “Republican” and “win” respectively.)
(4)
x(R(x)  Bel(j, ˆ(FW(x))))
(5)
Bel(j, ˆFx(R(x) W(x)))
As before the crucial difference is that in the first formula the existential quantifier and the
constant R occur outside the scope of ˆ. Here, what John believes is the proposition that some
particular individual who now actually is a Republican will win. In the second formula the
quantifier and constant R occur within the scope of ˆ, and here what John believes is the
logically weaker proposition that it will be the case that some Republican or other will win.
PS: The content of this handout is adopted from Dowty et al (1981).
References
Dowty D.R., Wall R. E. and Peters S. (1981) Introduction to Montague Semantics. Dordrecht,
The Netherlands: Kluwer Academic Publishers.
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