Scartol: What can we say is the purpose of the pincer? From what I can tell, it appears to be an attempt at miai Ė if the white stone tries to run (in the first example), Black goes beneath and connects. If White tries to settle, Black secures influence on the outside. Is there anything Iím missing or misunderstanding? (Probably a safe bet..)
Adamzero: When I pincer my rationale is this: a) by squirming, my opponent will hopefully let me solidify my corner territory in sente b) prevent my opponent from settling, and by chasing him out in the center build territory and/or influence/thickness.
In the case of the one-space pincer from hoshi, Iím inviting my opponent to jump to the san-san, in order to build influence in a direction that I get to choose. (Because jumping out from a one-space pincer is rarely very good: I get to solidify my corner more, and there is no way to slide under my stones to get eye space.)