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Trying to better understand how an "interpretation" works in logic: https://en.wikipedia.org/wiki/Interpretation_(logic)

For the most part it seems to be the idea that we have some syntactical symbol and we can assign it a semantic meaning of our choosing.

To my understanding, and also based on reading this Wiki article, that would mean assigning syntactic symbols like $p, q, r$, etc, with semantic meanings like "true" or "false" or I suppose "unknown" or "$67$% true" or whatever you'd want to do.

For this reason I get a little confused when we talk about certain concepts/requirements (such as validity) that something is true "under every interpretation" since it seems conceivable we could make up an interpretation that has nothing to do with true/false at all.

Or is an interpretation strictly an assignment of true/false? What about other logics? Does it even make sense to talk about logics that have nothing to do with true/false at all? Or is the concept of true/false a necessary component in any logic system for it to even be a logic system?

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  • $\begingroup$ "Interpretations" in this context is a technical term, a synonym for the notion of models (or structures). Each logic has it's own notion of model/structure; in the usual logics (classical propositional and classical first-order), everything is indeed true and false. The notion of validity is made with respect to a logic, which pins it down to something fairly narrow. $\endgroup$ – Noah Schweber Sep 6 '18 at 13:49
  • $\begingroup$ @NoahSchweber So validity in one logical calculus could be something completely different compared to that of another, depending on what that system is meant to model/accomplish -- which in turn means that "interpretations" are sort of bound to whatever makes sense in that system? $\endgroup$ – user525966 Sep 6 '18 at 13:54
  • $\begingroup$ Yup, that's right. Would you like me to expand this into an answer? $\endgroup$ – Noah Schweber Sep 6 '18 at 14:05
  • $\begingroup$ The accepted answer to this question is an example of a time an arbitrary interpretation using 3 values for propositional logic was useful: math.stackexchange.com/q/2436136/97045 $\endgroup$ – DanielV Sep 6 '18 at 15:30
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"Interpretations" in this context is a technical term, a synonym for the notion of models (or structures). So when we say "every interpretation," we don't mean "everything you could possibly think of;" we mean "every structure in the relevant logic." Each logic has it's own notion of model/structure; in the usual logics (classical propositional and classical first-order), everything is indeed true and false.

The notion of validity is made with respect to a logic, which pins it down to something fairly narrow. Changing the logic can indeed change the notion of validity.

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We may follow Wiki's entry carefully [emphasis added]:

An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.

The most commonly studied formal logics are [classical] propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of symbols of an object language.

An interpretation often (but not always) provides a way to determine the truth values of sentences in a language. If a given interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentence or theory.

Things are different for the so-called non-classical logic.

See e.g. Intuitionistic Logic :

Intuitionistic systems have inspired a variety of interpretations, including Beth’s tableaux, Rasiowa and Sikorski’s topological models, Heyting algebras, formulas-as-types, Kleene’s recursive realizabilities, the Kleene and Aczel slashes, and models based on sheafs and topoi. Of all these interpretations Kripke’s possible-world semantics, with respect to which intuitionistic predicate logic is complete and consistent, most resembles classical model theory.

According, for example, to Brouwer–Heyting–Kolmogorov interpretation the "correct" interpretation of Intuitionistic logic is in term of proof (instead of truth-value).


As discussed before, in the context of classical propositional logic,

where the language consists of formulas built up from propositional symbols (or variables) and logical connectives, [...] the standard kind of interpretation is a function that maps each propositional symbol to one of the truth values true and false.

This function is known as a truth assignment or valuation function.

Given any truth assignment for a set of propositional symbols, there is a unique extension to an interpretation for all the propositional formulas built up from those variables. This extended interpretation is defined inductively, using the truth-table definitions of the logical connectives discussed above.

But also for classical logic we may have different kind of interpretations, like the algebraic models.


In conclusion :

is an interpretation strictly an assignment of true/false?

NO.

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  • $\begingroup$ I did see these sections -- it's why I ultimately made this particular question $\endgroup$ – user525966 Sep 6 '18 at 13:48
  • $\begingroup$ The only thing I would add to this is that the "truth" values have no philosophical significance on their own. Everywhere "true" and "false" is used above could be replaced with $1$/$0$ or $0$/$1$ or $\mathsf{dog}$/$\mathsf{cat}$ or any other two distinct things. Saying "true" and "false" where what we mean are meaningless symbols is, in my opinion, one of the sources of a lot of this confusion. A formula $P$ or a "truth" value $v(P)$ where $v$ is some interpretation is not true or false any more than $3$ is true or false, but $v(P)=\mathsf{dog}$ can be called true or false. $\endgroup$ – Derek Elkins Sep 6 '18 at 19:16
  • $\begingroup$ @DerekElkins That is precisely what I am trying to ask about, thank you for clarifying -- I get a little lost in that "true/false" seems arbitrary. What stops us from talking about interpretations/validity/soundness/etc in terms of k-valued logic where none of the k values are true/false/etc, but rather {dog, cat, fish}? $\endgroup$ – user525966 Sep 6 '18 at 23:35
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By definition, interpretation is valuation of given sentence when variables are given to be - true or false. So, if you have a sentence with variables $p,q,r,...$ those $p,q,r,....$ can be either true or false.

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