# Why are models in logic called models?

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?

• 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 $r$. – user192614 Nov 19 '14 at 21:24
• Patience grasshopper, I haven't finished. – user192614 Nov 19 '14 at 21:27
• 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. – user192614 Nov 19 '14 at 21:30
• 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. – user192614 Nov 19 '14 at 21:36
• 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

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.