Partial continuations are control operators in functional programming such that a function-like object is abstracted from a part of the rest of computation, rather than the whole rest of computation. Several different formulations of partial continuations have been proposed by Felleisen, Danvy&Filinski, Hieb et al, and others, but as far as we know, no one ever studied logic for partial continuations, nor proposed a typed calculus of partial continuations which corresponds to a logical system through the Curry-Howard isomorphism. This paper gives a simple type-theoretic formulation of a form of partial continuations (which we call delimited continuations), and study its properties. Our calculus does reflect the intended operational semantics, and enjoys nice properties such as subject reduction and confluence. By restricting the type of delimiters to be atomic, we obtain the normal form property. We also show a few examples.