Logic of metaphors

Does the logic that applies to a mathematical model always carry over to the real world when the model is a metaphor for something in the real world? It is not obvious to me:
Much of (theoretical) economic science works like follows: There is some interesting or important social problem or issue. The economist sets up a mathematical model (a metaphor) which accounts for the most important socioeconomic forces and related phenomena. The economist then performs a mathematical analysis (according to strict, logical rules) on the model to see what happens or to find implications of different possible setups. The inferences are then interpreted and related to the real world. Does this practice hold water? It must, or someone would have questioned the practice earlier, but it is not obvious to me at all! More accurately, I have moments of doubt; I am trained in logic (as a mathematician) and believe in it; I have not yet seen a logical argument I do not believe in. Still, the way economics often work is somehow dissatisfying.

Also, what is logic? I can give examples of logic and illogic, but examples are not enough. Here is the first definition of ‘logic’ from the American Heritage Dictionary, looked up on dictionary.com:

The study of the principles of reasoning, especially of the structure of propositions as distinguished from their content and of method and validity in deductive reasoning.

A field of study? Okey, interesting, but that was not exactly the bull’s eye. Obviously, logic has to do with structure of propositions and pertains to reasoning, but it does not answer what logic is. Maybe the right question is How do logic work? Or Why is something logical? Something is logical because it adheres to the rules of logic. Okey… And why are the rules of logic the way they are? Are they inherent in the (human) brain? Or can we change them if we’d like? Earlier, when I studied math, I was certain that mathematical logic had to be built-in and was sure everyone could understand logical arguments if they would allow themselves. Now, I’ve met so many otherwise brilliant people that does not seem to grasp logic with ease and I got doubt. (Maybe I shouldn’t; logic is always hard to me.) Also, I was certain (one is more certain about things when one is young, or so it seems) that logic was universal. The idea was the following; if there was alien life somewhere, ever so different from earthly life, they still had to have primes. (Note that I resort to examples here: Prime numbers is my favorite example of hard logic.) So, if I were to give up logic, I would have to give up primes (or find another reason for them). Now, I cannot imagine a world without logic, without primes. However, a lot of people seem to get by without much concern for logic, particularly not primes. Am I just mad, questioning logic? No, that is not it. (Or if it is, it does not matter.) It has to be something else. The fourth definition may look more promising:

The relationship between elements and between an element and the whole in a set of objects, individuals, principles, or events: There’s a certain logic to the motion of rush-hour traffic.

Notice that an example is provided to clarify what is meant. Even though it is very common to provide examples in dictionaries it is not always necessary. I also notice that they have not provided an example pertaining to principles, which is what I’m interested in here. I guess it is hard to come up with a simple, short example which anyone can understand. ‘The relationship between principles.’ Hm. Sounds like an empty phrase. (Maybe not. Is logic the relationship between arguments?) I’m not getting anywhere with this, but this is an ongoing debate I have with myself, and now with you. (You didn’t see that coming, did you? The moment of surprise!)

Going back to my initial concern with the application of logic to metaphors and how it carries over to what the metaphor is a metaphor for, obviously one big issue is the logic itself. But what is the logic of the metaphor? Logic is a hard nut to crack. The logic of the metaphor is more elusive to me. Even though something follows from logic applied to a metaphor, what is the logic of the conclusion carrying over to the real world. My experience in economics is that often, when interpreting a logical result in a metaphor, it is often justified with additional explanations, often with an ad hoc flavour, all to make it more easy to accept. The alarm bells are ringing. There is a sales man at the door, and he tries to avert or attention from the crucial matter: Does the logic of the metaphor hold? Does it?