Compatible Logic

Perhaps there could be a system of logic where instead of proposition only by deduction, we allow propositions that are compatible.

That way, we don't need axioms. Instead the concept of abundance can be observed, where as long as things don't contradict, they exist and are all true.

It may be the case that such systems require a minimum level of inclusiveness to work. Or perhaps it implies a minimum level of inclusiveness.

An interesting exercise would be to consider how many related things can be put in such a system before you start getting contradictions?