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A model is an interpretation of a given formal language under which any wff in a given set of wffs of this formal language is true.

Why are models called models? What's the reasoning behind the name?

user132181
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    The idea is to detach semantic and syntax. The formal language defines syntax, its models are possible meaning of the sentences of the language. For example, you can have euclidean geometry. A model can be what you normally understand as points and lines. But we can also consider as points spheres of radius $r$ and lines to be cylinders of radius $r$. –  Nov 19 '14 at 21:24
  • @user192614 you're trying to answer a question I didn't ask. – user132181 Nov 19 '14 at 21:26
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    Patience grasshopper, I haven't finished. –  Nov 19 '14 at 21:27
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    In this way, the formal language defines structural conditions (like classes in computer science) that must be satisfied. Then models are like instances of the class that satisfy those structural conditions and at the same time can be taken as a possible meaning of the language. –  Nov 19 '14 at 21:30
  • @user192614 please notify me when you've finished. – user132181 Nov 19 '14 at 21:32
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    That's all. It is called model because it models (common language meaning of the word [in the manner of, with the measures of]) or it is modeled as the formal language prescribes. –  Nov 19 '14 at 21:36
  • @user192614 I don't seem to understand you. Would you kindly compile your comments into an answer at the same time elaborating just a tiny bit? – user132181 Nov 19 '14 at 22:04
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    To the person who voted to close: don't forget to vote to close this question as well. – user132181 Nov 19 '14 at 22:47

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I want to paint the portrait of a female, brunette with blue eyes. Regardless of which model I use I will need blue paint. I could use Miranda Kerr or Adrianne Palicki as models. I can't use my hat as a model. I may not need blue to paint my hat.

We have the formal language with alphabet $\{x,y\}$ and words $\{\emptyset, x,xy,xyy,xyyy,...\}$. Regardless of which model I use we will never find a word beginning with a $y$. I could interpret $x$ as the digit $1$ and $y$ as the digit $0$. Then we get all powers of $10$. We could interpret $x$ as $<$ and $y$ as $-$ and then we get arrows of any length pointing to the left. We will never write $01$ or $-<$ in these models. Just because they are models of this language and it is a property of this language that $yx$ is not a word.

So, the model serves as an interpretation of the model, or a way to give meaning to it (semantic).

It is called model because model means 'with the likeness of/ the measures of'.