As someone interested in theoretical computer science, I'm fairly comfortable with what intuitionistic logic represents. An intuitionistic proof is a proof we can act upon algorithmically. The law of the excluded middle is rejected because we don't have a magic oracle which can decide any proposition, and thus tell us what to do. By contrast, classical logic assumes that there is such an oracle, and our job is to work out what answer it will give us without invoking it.
Minimal logic can be described as intuitionistic logic without the law of non-contradiction, and dual-intuitionistic (Brazillian) logic is made similarly from classical logic. However, I'm lacking intuition as to how the lack of the law of non-contradiction can be justified. My guess is that – as eschewing A ∨ ¬ A gives positive proofs structure – eschewing ¬ (A ∧ ¬ A) gives negative proofs structure. But I don't know what one might want to do with structure in an impossible situation.
Also, I expect that there is an equivalent of the intuitionistic practice of excluding the middle for certain families of propositions. For example, we can usually justify the fact that ∀ x y : ℕ. (x = y) ∨ ¬ (x = y) by saying that equality of natural numbers is algorithmically decidable (and actually proving it via the Curry-Howard correspondence). I don't know what would recover non-contradiction on a family of propositions, but it surely holds for many of the families of propositions considered in ordinary mathematics.