Weak logic is a partial logic for program verification and system specification. In this article a sequent calculus for weak logic is first defined. Then the completeness theorem, a cut elimination theorem and some interpolation theorems are proved. In the proofs of the theorems, strategies from first order logic are used with some additions to adapt them to weak logic. One of the interpolation theorems is used for a deeper study of one of the main criterias for the definition of weak logic. In order to find effective strategies for doing proofs within weak logic, a resolution principle for weak logic is given. The relationship between provability in weak logic and provability in first order logic is also studied to make it possible to use all known first order logic proof methods more directly.
