If you mean interpretation as in Logic. Usually this relates to the use of interpretation functions that represent pre-conditions we give a formal system. For example, if I wanted to say.

I have a pie, it is delicious.

If I was concerned with modelling this as (Pie or no pie), and (delicious or not delicious), I could use the following two evaluation functions:

P(x) where x is in {0,1}: P("no pie")=0, P("pie")=1.

and

T(y) where y is in {0,1}: T("delicious")=1, T("not delicious")=0.

Now, I can make sentences (maybe not very grand statements, but is an example) in logic to say things about these two things based on my simple model. This is where formal systems and certain classes of Logic and Axioms come into play. Notice that I set up the formal system based on defining some values, then the reasoning follows from the structure of logical statements, and the conclusion only follows from what I assume at the start. This is a very elementary example, but it should hopefully make the point clear. Now, I don't know exactly why logic (maybe it is some form of common sense, or intuition) coincides with reality, but it seems very reliable for developing models. I mean, if our model is accurately represented, we can say a lot about maybe it behaves in the real world. Much of the applications of formal sciences lay on this fact. It is our accuracy of developing accurate pre-conditions that helps give great post-conditions.