Combinator terminology: What is "strong" about strong composition? A strong composition operator seems to be very fundamental. (Hindley and Seldin use the notation S for &quot;strong&quot; composition combinator). It abstracts the pattern $f(x, g(x))$, i.e. a direct and a $g$-mediated effect of $x$. It's probably my inexperience but, why is this &quot;strong&quot;? Is it because normal (weak?) composition looks like a special case of this (where $f$ just ignores the argument $x$)?
I'm probably cheating and squeezing in another question here, but is there an intuition for why this form of composition seems to be the more fundamental one in combinatory logic?
I'm not familiar with the &quot;strong&quot; terminology. However, the 'obvious' answer to the other question is that it both duplicates the $x$ argument and exchanges the relative positions of (one copy of) the $x$ and $g$ arguments:
$$\mathsf s\ f \ g\ x = f\ x\ (g\ x)$$
The other combinator it's usually paired with:
$$\mathsf k\ x\ y = x$$
discards an argument. Together, this is all three of the structural rules for a variable context (contraction, exchange, weakening), and it includes composition as well.