I am learning lambda calculus from the book https://www.irif.fr/~mellies/mpri/mpri-ens/biblio/Selinger-Lambda-Calculus-Notes.pdf and do not understand the meaning of the following symbols.

The definition of the lambda terms is:

Lambda terms: $\mathit{M,N} ::= x | (M N) | (λx.M) $

The question is, what do $\mathit{M}$ and $\mathit{N}$ mean?

  • $\begingroup$ @dkaeae I've seen you add Mathjax into titles in a few edits, where I do not think it was nessecary. I recommend against using Mathjax in titles unless nessecary: titles appear in quite a few places where Mathjax is not rendered, for one. Please have a look at my advice on this, thanks. (Do not see this as a discouragement to editing, btw. Your edits usually are good and your effort is much appreciated!) $\endgroup$
    – Discrete lizard
    Feb 22, 2019 at 19:03

1 Answer 1


I suggest you read the page following the supposedly cryptic notation:

The above Backus-Naur Form (BNF) is a convenient abbreviation of the following equivalent, more traditionally mathematical definition:

Definition. [...]

In said definition, it is stated that $M$ and $N$ are simply arbitrary lambda terms. (In a nutshell, it is an inductive definition.)

As to why the author prefers this notation, it is stated right after the definition:

Comparing the two equivalent definitions, we see that the Backus-Naur Form is a convenient notation because: [...] the use of distinct meta-symbols for different syntactic classes ($x$, $y$, $z$ for variables and $M$, $N$ for terms) eliminates the need to explicitly quantify over the sets [of variables and lambda terms].

  • $\begingroup$ Does lambda terms consist of constant, application and lambda abstraction? What does the author try to explain? $\endgroup$ Feb 22, 2019 at 15:02
  • $\begingroup$ "Does lambda terms consist of constant, application and lambda abstraction?" Yes, this follows directly from the definition (since, as I have mentioned, it is an inductive definition). "What does the author try to explain?" I am afraid I do not understand your question. To which part of the text are you referring to? $\endgroup$
    – dkaeae
    Feb 22, 2019 at 15:04
  • $\begingroup$ what is the meaning of the symbol $\mathit{Λ}$ in The set of lambda terms is the smallest subset $Λ ⊆ A∗$ on the page 12 at very beginning ? $\endgroup$ Feb 22, 2019 at 15:09
  • $\begingroup$ $\Lambda$ denotes the set of lambda terms (since $\Lambda$ is a subset of $A^\ast$ and it is the smallest subset with the properties that follow). This kind of phrasing is pretty standard in mathematical texts. $\endgroup$
    – dkaeae
    Feb 22, 2019 at 15:13
  • $\begingroup$ It seems, that I must learn first discrete math to understand all these symbols, because I haven't math background. $\endgroup$ Feb 22, 2019 at 15:17

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