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What follows is what I'm reading from Wikipedia, and my reactions. I hope it helps in clarifying what I'm confused about.

An interpretation is an assignment of meaning to the symbols of a formal language

1) So it's just semantics of formal languages?

In these contexts an interpretation is a function that provides the extension of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate T (for "tall") and assign it the extension {a} (for "Abraham Lincoln"). Note that all our interpretation does is assign the extension {a} to the non-logical constant T, and does not make a claim about whether T is to stand for tall and 'a' for Abraham Lincoln. Nor does logical interpretation have anything to say about logical connectives like 'and', 'or' and 'not'. Though we may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.

2) Oh, so what makes it special is that it maps a symbol in the language to other symbols in the language without concern for what they "mean" outside the language?

A formal language W can be defined with the alphabet A = {u, v}, and with a word being in W if it begins with u and is composed solely of the symbols u and v... A possible interpretation of W could assign the decimal digit '1' to u and '0' to v. Then uvu would denote 101 under this interpretation of W.

3) Okay, so it actually maps symbols in the language to symbols outside the language.

Now I'm confused about what an interpretation actually means, and how the second quote relates to the third quote in a coherent manner.

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  • $\begingroup$ 1) is te semantical interpretation of a language : the way to assign meaning to symbols and expressions. It is a "map" from symbols to "objects". $\endgroup$
    – Mauro ALLEGRANZA
    Sep 9, 2019 at 9:25
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    $\begingroup$ 2) is more a "translation" from one language to another, like translating an English sentence into German. But this one (as per Wiki example) can be subsuned under 1) simply because language is part of the "world". $\endgroup$
    – Mauro ALLEGRANZA
    Sep 9, 2019 at 9:26
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    $\begingroup$ I recommend learning about these topics from books rather than Wikipedia. Wikipedia is good for finding references and usually for refreshing your knowledge of explicit formal definitions, e.g. "what are the axioms of a module", but is not a good source for learning on its own. There are tons of goods texts that are freely and legally available online for learning logic and many good books. logicmatters.net/tyl provides an extensive list of resources (though it's not geared toward resources that are readily available online, it nevertheless indicates this). $\endgroup$
    – Derek Elkins left SE
    Sep 9, 2019 at 21:25
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    $\begingroup$ You can see many related posts : Interpretation vs model and Models vs structures $\endgroup$
    – Mauro ALLEGRANZA
    Sep 10, 2019 at 6:29
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    $\begingroup$ See also How is “interpretation” used differently in propositional vs. first-order logic? as well as What is a model of a formal system $\endgroup$
    – Mauro ALLEGRANZA
    Sep 10, 2019 at 6:32

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For more mathematical examples, an interpretation is a set of objects, to some of which have been assigned the objects mentioned in the axioms, and functions, relations, etc. between those objects as required by the axioms. The ring axioms are a formal language. We have to have a set that includes an object we match with $0$ in the axioms, one we match with $1$, maybe some other objects, and the operations of addition and multiplication. The ring $\Bbb {Z/3Z}$ is one interpretation of those axioms, but we could name the elements $p,q,r$ if we want. We would have to specify which element plays the role of $0$ and which plays the role of $1$. The addition and multiplication would have to be compatible with the axioms, so if $q=0,r=1$ we would have $r+r=p,p\cdot p=r$ and so on.

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