Page 372 - Encyclopedia of Philosophy of Language

Formal Semantics
A man walks.
Every man walks.
(37a)
(38a)
denoted by formulas in the language of intensional logic. The meaning of John will be denoted by:
APfP(john)] (39) To explain this formula, the variable
P (40) is a variable over properties of individuals: P may be
replaced, for example, by the property 'walking.' The expression
"P (41)
denotes the extension of the property expressed by any predicate P in any given world at a given moment of time. Thus, for the actual world now,
'P(john) (42)
is true if the predicate P holds for John now, in this world, and false otherwise. Any property denoted by P can be abstracted by means of the lambda operator: AP. Lambda abstraction of P gives:
AP["P(john)] (43)
which is the same as (39) above. This expression denotes a function which says, for any property a, given a world-time pair <w, t>, whether John belongs to the extension of that property in <w,t>. Let this function be called Xj. Some properties of this function will now be investigated.
These sentences are very simple, and it is easy to present their interpretation in traditional predicate logic:
walk(john) 3x[man(x) Awalk(x)] Vx[man(x) -»walk(x)]
(36b) (37b) (38b)
One immediately sees that these three sentences are syntactically much alike (a subject and a verb), but the formulas in predicate logic are rather different: two with a quantifier (different ones!), and in each case another main operator (-*, A, and predicate application). What makes it interesting to present these examples here is not the meaning assigned to them, but the fact that these very different meanings can be derived compositionally from the cor- responding sentences.
All three sentences are built from a singular noun phrase and the verb walks, and for this reason one can design a syntactic rule that combines a noun phrase and a verb. Since in Montague grammar there is a correspondence between syntactic rules and semantic rules, one also has to design a rule that combines the meaning of the verb with the meaning of the noun phrase. This in turn requires that there be meanings for verbs and for noun phrases. These meanings will be discussed first, then the semantic rule will be designed.
As explained in Sect. 5, predicates are given inten- sions as meanings: functions from possible worlds and moments of time to sets of individuals. Thus, the intension, or meaning, of the verb walk is a function from possible worlds and moments of time to sets: for each possible world and each moment of time, there is a set (possibly empty) of individuals who walk. Such an intension is called a 'property/ For the noun phrases, it is more difficult to select a format that allows these meanings to be rendered uniformly. In order to keep the calculus going in a maximally uni- form way, all noun phrase meanings should preferably be of the same type. This requirement is easily seen to be nontrivial, as, for example, the expression every man extends over sets of possibly many individuals, where John seems to refer to one individual only.
Montague, proposed (1973) to model the meaning of a noun phrase as a set ('bundle') of properties. An individual is, in this approach, characterized by the set of all its properties. This is possible since no two individuals share all their properties at the same time, if only because the two cannot be at the same place at the same time. This approach permits generalization over all noun phrases referring to (sets of) individuals (see also Lewis 1970).
Meanings of linguistic expressions (functions from world-time pairs to specific world extensions) are
According to the definition of Xj,
Xj(ot) is true iff a now holds for the individual John, i.e., iff a(John) is true, and false otherwise.
(I)
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Xj is, therefore, the characteristic function of the set of properties that can be predicated of John. As usual in logic, this characteristic function can be identified with the set of properties that John has. Xj can there- fore be seen as a formalization of the idea that the meaning of John is a set of properties.
This function Xj can be evaluated with, as argu- ment, the property of being a man, that is, the function that yields for each index (i.e., for each world and time) the extension of the predicate man. The notation for this argument is:
"man (44)
The symbol " translates the predicate man into the language of intensional logic, where * man denotes the intension associated with man. Xj is now applied to this argument to obtain, using result (I),
Xj(" man) is true iff *man now holds for the individual John, i.e., iff * * man(john) is true, and falseotherwise.
(ID
j
In the expression X ("man), Xj can be replaced by
its original definition (viz. (43)). Then, (45) is obtained: