Forcing with restricted condition: it's definition What does it mean to apply the symbol $\Vdash$ to a condition $q$ restricted to $\xi$:
$q\upharpoonright \xi\Vdash\ldots$ 
as used on the page 5 above lemma 2.4 here; is this
$\upharpoonright$ there specific to the context of that paper or, on the contrary, this restriction can be defined in general, with just using the names, $\mathbb P$,conditions and $G$,the $\Vdash$ symbol etc.?
 A: I had already mentioned this in the comments, but here in is as an answer. 
It is explained earlier in the paper that they follow standard notation such as in Jech's "Set Theory", with the major difference being that Shelah reverses the order (thus uses a bottom element, and lets stronger conditions be larger).
Above Observation 2.2 it is stated that the forcing poset is a $\zeta^*$-iteration, and further back (start of page 4) it is stated that the conditions of iterated forcing are viewed as functions on the length of the iteration that are trivial for all except at most as many steps as are in the support. This should explain how to read the restriction to some $\xi<\zeta^*$: the condition is restricted to the first $\xi$ steps of the iteration.
If you're unfamiliar with iterated forcing, this paper is probably not very readible. Iterated forcing can be found in: 


*

*Chapter 16 of Jech's "Set Theory", 

*part V.3-5 of the new edition of Kunen's "Set Theory", or 

*Ch VIII $\S$5 of Kunen's old edition.

