# Lambda-operator

**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.

### Example

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) ]

### Link

Utrecht Lexicon of Linguistics

### References

- Gamut, L.T.F. 1991.
*Logic, language, and meaning,*Univ. of Chicago Press, Chicago.