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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.

csp2018
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  • is te semantical interpretation of a language : the way to assign meaning to symbols and expressions. It is a "map" from symbols to "objects".
    • – Mauro ALLEGRANZA Sep 09 '19 at 09:25
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  • 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".
    • – Mauro ALLEGRANZA Sep 09 '19 at 09:26
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    @MauroALLEGRANZA thanks as always! But what is the difference between mapping symbols and expressions to objects vs saying T is to stand for Tall and a to stand for Abraham Lincoln? – csp2018 Sep 09 '19 at 09:57
  • It is the same as for natural language : the expression (that belongs to language) "Abraham Lincoln" stands for the object (that belongs to the world) Abraham Lincoln. – Mauro ALLEGRANZA Sep 09 '19 at 12:12
  • Withe the example above, we have an interpretation that maps the predicate symbol $T$ on the "property" tall and the symbol $a$ on the man Abraham Lincoln. Thus, the symbolic expression $T(a)$ is interpreted as meaning the same as the natural language expression "Abraham Lincoln is tall". – Mauro ALLEGRANZA Sep 09 '19 at 12:15
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    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. https://www.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). – Derek Elkins left SE Sep 09 '19 at 21:25
  • @MauroALLEGRANZA In the second quote, is the symbol "a" part of the object language, or outside? Also, this part seems to suggest that interpretation isn't concerned with mappping between langagues: "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.A"... This is what confuses me. – csp2018 Sep 10 '19 at 00:11
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    You can see many related posts : Interpretation vs model and Models vs structures – Mauro ALLEGRANZA Sep 10 '19 at 06:29
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    See also How is “interpretation” used differently in propositional vs. first-order logic? as well as What is a model of a formal system – Mauro ALLEGRANZA Sep 10 '19 at 06:32
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    And see Richard Kaye, The Mathematics of Logic : A guide to completeness theorems and their applications (Cambridge, 2007), page 24-on for simple example of a formal system and its interpretation. – Mauro ALLEGRANZA Sep 10 '19 at 11:51