Not necessarily, you have competing theories based on the same kind of logic - it is just that every theory would be based on different set of postulates and will reside in a different logical space (i.e. a different category). But the fact that that two theories don't agree with each other does not mean that one of them is illogical. Actually, I'd argue that they have to both be logical in order not to agree with each other at all i.e. logic is the common ground which enables you to express the differences.

Also, I do think that physics is pretty axiomatic right from it's roots. Newton's main inspiration for forming his theory is the work of Euclid, i.e. first formal system that was ever created.