Todd asked “Why [did Church choose] lambda and not some other Greek letter?”. Here are three answers:

### 1

The story is that in the 10s and 20s, mathematicians and logicians used ^ as a notation for set abstraction, as in ^i : i is prime. Church used ^` (i.e., a primed version of this symbol) for function abstraction, because functions are just sets with extra properties. The first type setter/secretary read it as λ and Church was fine with. True or not? I don’t know but it’s fun.

### 2

This paper (link provided by Dave Herman here):

(By the way, why did Church choose the notation “λ”? In [Church, 1964, §2] he stated clearly that it came from the notation “xˆ” used for class-abstraction by Whitehead and Russell, by first modifying “xˆ” to “ˆx” to distinguish function abstraction from class-abstraction, and then changing “ˆ” to “λ” for ease of printing. This origin was also reported in [Rosser, 1984, p.338]. On the other hand, in his later years Church told two enquirers that the choice was more accidental: a symbol was needed and “λ” just happened to be chosen.)

### 3

This paper (link provided by Dave Herman here):

We end this introduction by telling what seems to be the story how the letter ‘λ’ was chosen to denote function abstraction. In [100] Principia Mathematica the notation for the function f with f(x) = 2x + 1 is 2xˆ +1. Church originally intended to use the notation xˆ .2x+1. The typesetter could not position the hat on top of the x and placed it in front of it, resulting in ˆx.2x + 1. Then another typesetter changed it into λx.2x + 1.

Your quote for #2 seems to be broken; it should be ‘x̂ ‘.

See also this email which gives more support to #2 (although again from J. Rosser):

Thanks Porges, more evidence and I fixed the quote.