A sentence, such as a Godel string, can be given any intepretation whatsoever, so in a sense, when communicating, we have to be benevolent about the interpretation function that we infer our co-communicators are using.

I am looking for a formal treatment of the notion of interpretation functions and the fact that there are multiple interpretations possible for any given sentence and formal system.

I believe this is glossed over in Godel's (1931) Incompleteness Proofs and I have found the following two quotes:

"No sentence can be made both true and false by the same interpretation, but it is possible that the truth value of the same sentence can be different under different interpretations." [1]

"Any given formal language can be paired with any of a number of competing semantic rule sets." [2]

[1] http://en.wikipedia.org/wiki/Interpretation_%28logic%29

[2] http://en.wikibooks.org/wiki/Formal_Logic/Sentential_Logic/Formal_Semantics

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    There is an exciting area of mathematical logic known as Model Theory, which treats the notion of interpretation from a formal point of view. I'd recommend you check out any good book on that subject. Check out Hodges' shorter model theory book and his SEP entry on model theory for a start. Marker is also good. Chang and Keisler, of course, is a recommended classic. – Hunan Rostomyan Apr 26 '15 at 23:36
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    The standard example of this kind of interpretation dependence is independence results in logic. We can create interpretations of set theory where the famous 'continuum hypothesis' is true, or false. And we can create interpretations of the limited set theory without he Axiom of Choice, where the usual axiom is false, and instead, its opposite, an Axiom of Determinacy, is true. Goedel did not wholly gloss over interpretation or models. He is the source of the first two different models of set theory, L and V. – jobermark Apr 26 '15 at 23:53
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    It is also possible to outright toy with an interpretation and create real objects that play special roles in a given model, but do so below the level of the axiomatic system. You can create infinitesimal real numbers and rescue Cauchy's proofs in calculus, you can add a range of infinities and clarify various limit theorems, etc. Not to dis @HunanRostomyan, but I feel I almost suffocated under Change and Keisler. Compared to straight Model Theory, I find this more exciting. So you might want to find your introduction to Model Theory via Nonstandard Analysis. – jobermark Apr 27 '15 at 0:00

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