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In the lambda calculus, are there commonly accepted names for $x$ and $M$ when they appear in $\lambda x [M]$ ? Something along the lines of "binder" and "bindee"?

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$x$ is a "bound variable" and $M$ is the "scope" in which it is bound.

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  • $\begingroup$ I would say that the scope of $x$ is the whole expression $\lambda x.M$. $\endgroup$ – Emil Jeřábek supports Monica Feb 21 '12 at 18:58
  • $\begingroup$ @Emil Jeřábek Since $x$ can only be evaluated in $M$, I'm not sure what adding the lambda adds to the notion of scope. But hey, if that is how you like it... ;-) $\endgroup$ – Marc Hamann Feb 21 '12 at 19:17
  • $\begingroup$ Personally I couldn’t care less. However, the question is not how I like it, but what is the commonly accepted terminology, and I believe it is what I wrote above. $\endgroup$ – Emil Jeřábek supports Monica Feb 21 '12 at 20:05
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    $\begingroup$ I agree with Marc here. The scope is the body $M$. BTW, expressions like "I couldn't care less" come across as very rude, though I'm sure that's not your intention. $\endgroup$ – Dave Clarke Feb 21 '12 at 21:02
  • $\begingroup$ I apologize, I didn’t intend to be rude. $\endgroup$ – Emil Jeřábek supports Monica Feb 22 '12 at 13:01
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In logic the $\phi$ in $\forall x . \phi$ is called the matrix, and $x$ the bound variable and $\forall x$ the quantifier. By this analogy I would call $M$ in $\lambda x . M$ the matrix, $x$ is the bound variable, and $\lambda x$ is the abstraction, or maybe the "abstractor"?

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