Logic doesn't care about which axioms you choose. If your belief was that sinners are evil, can corrupt the innocent or (worse) other believers and be sent to god ASAP for judgement, then it would be logical to kill them. Theologians used pretty much that train of 'logic' to justify burning witches and other heretics. Logic is just a mathematical system of evaluating the truth of something but relies on axiomatic truths - which are the 'belief'.
- there can be no proof (or disproof) of deities in any meaningful way - so it makes sense to take that on the chin and get on with our lives, mostly following the idea of "do as you would be done by" which seems to cover most of the sensible bits of religious direction..