A very snide Model Theorist would ask you to axiomatically specify your universe of discourse first.
I am a very snide Model Theorist.
Can you axiomatically specify your universe of discourse?
The trouble with using logic to reason about the world around us, is that definitions are true in every world; so they tell us nothing that helps distinguish this one.
No matter what convoluted thought experiment universe you might construe, if I write down the twenty-or-so axioms of PA, and then "Therefore S(S(0)) + S(S(0)) = S(S(S(S(0))))" then I have just correctly stated that 2+2=4.
This, nothing can prevent from being true. Mathematics hitches on the fact that if you start out with certain symbol strings and apply certain well-defined transformations in a well-defined order, you always get the same result.
Mathematics, effectively is a different universe. Inside our brains we happen to have a bundle of neurons which behave as a "universe of discourse," i.e. it obeys the rules of mathematical reasoning; which is why we can reason about mathematics (to an extent.)
Also, there is at one point in the canonical series, two wolverines with adamantium skeletons.
If you want to reason about the real world, might I recommend bayesian statistics in stead?
Sources: Eliezer Yudkowsky's A Human's Guide to Words, 2008