Well, there are bivalent logics (True/False), trivalent logics (True/Indeterminate/False), Fuzzy logic where truth is a continuum between 1 (true) and 0 (false), Modal logic (Necessarily true, possibly true, possibly false, and necessarily false), and such. That's just propositional logic. Technically, all of math is logic, so you can have any number of axiomatic systems.
It is difficult to say that these logical systems "conflict" with one another. The measures of truth within a given system only have meaning within the system. A true statement formulated according to the rules of one system might be false or even meaningless when evaluated according to the truth criteria of a different system, but that doesn't mean that they "conflict," IMO.
I wouldn't say that the rules of chess "conflict" with the rules of checkers. They're just two different games. Now, if you tried to move your bishop diagonally while playing checkers, you're going to have a hard time, but that isn't a problem with the rules, but rather a problem with the player.
Surely animals reason, but strictly speaking reason isn't the same as logic. Logic is formalized reason. All logic is reason, therefore, but not all reason is logic.