Difference between revisions of "Church lambda-abstraction"
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A notation for introducing functions in the languages of mathematical logic, in particular in [[Combinatory logic|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 | A notation for introducing functions in the languages of mathematical logic, in particular in [[Combinatory logic|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 | ||
| − | $$\lambda x_1,\dots,x_nA\tag{*}$$ | + | $$\lambda x_1,\dots,x_nA\label{*}\tag{*}$$ |
| − | 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 \ | + | 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 [[#References|[1]]]. |
====References==== | ====References==== | ||
Latest revision as of 17:18, 14 February 2020
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) |
Comments
References
| [a1] | H.P. Barendrecht, "The lambda-calculus, its syntax and semantics" , North-Holland (1978) |
Church lambda-abstraction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Church_lambda-abstraction&oldid=33147