# Church lambda-abstraction

A notation for introducing functions in the languages of mathematical logic, in particular in combinatory logic. More precisely, if in an exact language a term $A$ has been defined, expressing an object of the theory and depending on parameters $x_1,\dots,x_n$ (and possibly also on other parameters), then

$$\lambda x_1,\dots,x_nA\label{*}\tag{*}$$

serves in the language as the notation for the function that transforms the values of the arguments $x_1,\dots,x_n$ into the object expressed by the term $A$. The expression \eqref{*} is also called a Church $\lambda$-abstraction. This Church $\lambda$-abstraction, also called an explicit definition of a function, is used mostly when in the language of a theory there is danger of confusing functions as objects of study with the values of functions for certain values of the arguments. Introduced by A. Church [1].

#### References

 [1] A. Church, "The calculi of $\lambda$-conversion" , Princeton Univ. Press (1941) [2] H.B. Curry, "Foundations of mathematical logic" , McGraw-Hill (1963)