First, contradictions must be deemed false. Hence their negations must be deemed true. Alternatively, it may be rescued by the claim that some premises were left unstated but understood in context. Consider the game of the axiomatization of a given system, then. The logic of axiomatization is more strict than logic proper, but is not in any way opposed to it, of course.
Logic permits inductive inferences on the condition that they are viewed as mere surmises.
This was stressed by Bertrand Russell, for example. This extra bit required by Russell, said Popper, is neither possible nor necessary. Suffice it to require, he said, that surmises should be put to empirical test. Only the more promising ones should. Russell said, this is not enough.