Page 343 - Encyclopedia of Philosophy of Language

kykx.xrespectsy. (7)
Expression (7) denotes the relation of respecting by presenting it as a function that combines with an indi- vidual to form a property (a function from individuals to truth values). Actual variable names are unim- portant but binding patterns matter: (7) and (8) are (logically equivalent) alphabetic variants.
kxky.y respects x. (8) The relation of being respected is denoted by a slightly
different lambda expression (9).
kxky.x respects y. (9)
The distinction between the active and passive voice is reflected in the different binding patterns of (8) and (9).
In the context of lambda operators, quantifiers can be viewed as higher order functions. The quantifier V combines with the expression Ajc.cp, where <p is a sentence, to form a sentence VAx.<p (written here as Vx<p). Observe that the quantifier itself does not have to act as a binder any more. The binding mechanism is taken over by the lambda operator. The sentence Vx<p is true if and only if Ax.<p denotes a function which gives true for every argument. Thus, Vdenotes a function from characteristic functions to truth values. It maps every characteristic function that always gives true to true, and every other charac- teristic function to false. Similarly, 3 denotes the func- tion from characteristic functions to truth values which maps every function that for some value assumes the value true, to true, and the function that assigns false to every argument, to false.
Lambda abstraction and quantifiers make it poss- ible to express what it means to admire an attractive girl (10).
Xx3y(girly &attractivey&xadmiresy). (10)
To be courted by every unmarried man, on the other hand, is something quite different, as expression (11) makes clear.
kx.Vy((man y&—married y) -»y courts x). (11)
It is also possible to abstract over objects of more complex types. Again starting from (5), an abstraction can be made over the transitive verb or over the predi- cate. Abstracting over the predicate yields (12), an expression which combines with a property denoting expression (i.e., a predicate) to form a sentence.
kP.P(John). (12)
Interestingly, (12) is an expression of the same type as that of quantified noun phrases. The quantified noun phrase every man combines with a property denoting expression to form a sentence, so (13) is an appropriate translation.
Combining (13) with (6) gives the expression in (14). In fact, for convenience (6) has been replaced with an alphabetic variant (an expression using different variables to effect the same binding pattern). This kind of conversion is called a conversion.
AP.Vx(man x -> P(x))().y.y respects Bill). (14)
Expression (14) is the result of combining the trans- lation of every man with that of respects Bill. The result should be a sentential expression, that is, an expression denoting a truth value. To see that this is indeed the case, a reduction of the expression is necessary. All expressions of the form Xv.E(A) are reducible to a simpler form; they are called 'redexes' (reducible expressions).
4. ReducingLambdaExpressions
To reduce expression (14), Sect. 3 above to its simplest form, two steps of so-called P conversion are needed. During P conversion of an expression consisting of a functor expression fa/.E followed by an argument expression A, basically the following happens. The prefix hi. is removed, the argument expression A is removed, and finally the argument expression A is substituted in E for all free occurrences of v. The free occurrences of v in E are precisely the occurrences which were bound byfa>in \v.E.
There is a proviso. In some cases, the substitution process described above cannot be applied, because it will result in unintended capturing of variables within the argument expression A. Consider expression (15):
In this expression, y is bound in the functional part kxky.R(y)(x) but free in the argument part y. Reduc- ing (15) by P conversion would result in ky.R(y)(y), with capture of the argument y at the place where it is substituted for x. The problem is sidestepped if ft conversion is performed on an alphabetic variant of the original expression, say on (AxAz./?(z)(x))(y).
Another example where a conversion (i.e., swit- ching to an alphabetic variant) is necessary before P conversion to prevent unintended capture of free variables is the expression (16):
In (16), p is a variable of type truth value, and x one of type individual entity. Then x is bound in the functional part A/?.Vx(y4(x)=p) but free in the argu- ment part B(x). Substituting B(x) for/? in the function expression would cause x to be captured, with failure to preserve the original meaning. Again, the problem is sidestepped if P conversion is performed on an alphabetic variant of the original expression, say on A/?.Vz.(,4(z) =p)(B(x)). The result of P conversion then is Vz.(y4(z)s5(x)), with the argument of B still free, as it should be.
J.P.Vx(manx-+P(x)). (13) Using [A/v] for the substitution operation, the p- 321
Formal Semantics