Scientific ideas neither arise nor develop in a vacuum. They are always nutured against a background of prior, partially conflicting ideas. Systemic hypothesistesting is the problem of testing scientific hypotheses relative to various systems of background knowledge. This paper shows how the problem of systemic hypothesis-testing (Sys HT) can be systematically expressed as a constrained maximimization problem. It is also shown how the error of the third kind (E III) is fundamental to the theory of Sys HT.The error of the (...) third kind is defined as the probability of having solved the ‘wrong’ problem when one should have solved the ‘right’ problem. This paper shows howE III can be given both a systematic as well as a systemic treatment. Sys HT gives rise to a whole host of new decision problems, puzzles, and paradoxes. (shrink)