Lambda-operator is an operator which makes it possible to construct expressions which denote predicates or functions. Adding the lambda-operator to predicate logic makes it possible to construct predicates from formulae with free variables.
two expressions with lambda-operators are given in (i):
(i) a lambda x [ kiss(john,x) ] b lambda x [ man(x) & Neg married(x) ]
The lambda-expression in (i)a denotes the property of being kissed by John, the one in (i)b denotes the property of being an unmarried man. The lambda-operator plays an important role in type logic, as a mechanism for making functions. If e is an expression of arbitrary type b and v is a variable of arbitrary type a, then lambda v [ e ] is an expression of type <a,b>, i.e. a function from things of type a to things of type b. The lambda-operator makes it possible to give a logical translation of every expression, including quantified noun phrases:
(ii) a every boy b lambda P [ All(x) [ boy(x) -> P(x) ]]
The noun phrase in (ii)a is translated into a logical expression denoting a function from properties to truth values, assigning the value 1 to those properties that every boy has. When we combine the noun phrase in (ii)a with a predicate like walk, then the expression in (ii)b is applied to the translation of walk. In other words: the translation of (iii)a is (iii)b which is logically equivalent with (iii)c (an equivalence which follows from the semantics of the lambda-operator):
(iii) a Every boy walks b lambda P [ All(x) [ boy(x) -> P(x) ]] (walk) c All(x) [ boy(x) -> walk(x) ]
- Gamut, L.T.F. 1991. Logic, language, and meaning, Univ. of Chicago Press, Chicago.