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.