Page 356 - Encyclopedia of Philosophy of Language

Formal Semantics
Each of those present may have committed the
murder. (27)
It is clear what this means. Sentence (27) may be formalized as VxOMx. It certainly does not mean the same thing as OVxMx, which is the translation of (28):
It is possible that each of those present has
committed the murder. (28)
It is not at all clear how a de re modality like (27) could be reduced to a de dicto modality. And besides, possible worlds semantics does provide a clear interpretation for modalities de re.
3. Intensional Type Theory
A type theory has expressions which are interpreted according to the type associated with them. In the particular case of Montague's intensional type theory, the set of types is the smallest set which contains:
(a) Two basic types: e (for: entity or thing) and t (for: truth value);
(b) All types (a, /3), where a and ft are types;
(c) Alltypes(s,a),whereaisatype.
The first two clauses are identical to those for exten- sional types. The third clause is new, and allows one to form intensional types. Note that s itself is not a type; its only purpose is to enable to form composite intensional types. Informally, expressions of type (a, /?) denote functions that map objects of type a to objects of type /?. The expressions of type (s, a) denote intensional entities: functions from possible worlds to objects of type a.
Among the expressions, there are infinitely many variablesx,foreachtypea,buttheremayalsobe constants ca. The variables and constants of type a constitute the basic or noncompound expressions of that type. The compound expressions are defined by:
(a) If tp is an expression of type /? and xa is a variable of type a, then focx.tp is an expression of type (a, 0).
(b) If r(a>w is an expression of type (a, /?) and fa is an expression of type a, then /<a./r)(O is an expression of type 0.
A
v
interpretation function. To this end, with each type a a domain Da of objects of type a is associated. For the basic type e and t the domains are respectively a given nonempty set of individuals D and the set of truth values:
D,=D
D,= {0,1}
The domains for the complex types are so-called func- tion spaces. In case of type (s,a), a given nonempty set W of possible worlds is used:
D,.^, = {f: f is a function from D. to D^}
D(J.«)={f:fisafunctionfrom WtoD.}
AmodelM=<F,[—]>forintensional typetheory consists of a frame: F={Da:a is a type} and an interpretation function [ —J. In extensional type the- ory the interpretation function assigned an element of Da to each constant of type a. However, in case of intensional type theory the extension of a constant should be able to vary from context to context. That is why constants of type a are interpreted as functions of possible worlds onto elements of type a. So [ca](H')eD,I, for each weW. Individual constants c,, however, are often required to be 'rigid designators/ so that [c,] takes the same value at all worlds.
Given an assignment a for a model M, i.e., a func- tion from variables of type a to objects in M of type a, the interpretation function [—] is extended to associate each expression t with its extension in poss- ible world w under a; notation: [/]„.,„ using the fol- lowing recipe:
(c) If ta is an expression of type a, then expression of type (s, a).
/„ is an
(d) If t(,A) is an expression of type (s, a), then is an expression of type a.
/(M)
Some comments may be in order. Clause (ii) stipulates
(e) If r, is an expression of type t, then EUt is an expression of type t.
(f) If t and /' are typed expressions, then t=t' is an expression of type t.
In a term of the form Axa .tf, the variable xa is said to beboundbyAJC,.Variableswhicharenotbound,are free.
In order to interpret the expressions, one first has to introduce the domains in which the expressions take theirsemanticalvalueandthentomaptheexpressions onto objects of the appropriate kind by means of an
that the extension of a constant of type a in a world w is an object of type a. Given the definition of assign- ments, clause (i) stipulates the same for variables. Clause (iii), the most complicated one, makes use of the assignment c^xa:=d]. By definition, this is the assignmentwhichisidenticaltoa,exceptperhapsfor thevalueatx,,whichisgivenby:o(xa:=d](jcj=d.(If a(xa)=d, the assignments are identical.) Since ['/»I«Lr-d].**' is an object of type 0, [Ax. .tfyijv is a function from D. to Dft. Clause (iv)makes f^CO denote an object of type ft in w, for in that world t, is
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