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So, I recently started learning about modal logic, and thus also Possible Worlds semantics.

The formulation for a frame I was given is:

A frame is defined as an ordered pair $[G,R]$ where $G$ is the non-empty set of possible worlds, and $R$ is a relation $R \subseteq G \times G$.

So, it's obvious that this formulation uses set theoretic notions, but is this necessary? Are there formulations of modal logic that don't require set theory?

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    $\begingroup$ Most of, if not all mathematics nowadays uses set theory (obviously it is felt differently in different fields - but the axiomatic basis is always set theory) $\endgroup$ – Max Jun 12 '17 at 20:10
  • $\begingroup$ I agree, and I have no issue with using set theory. Really I was just curious if other approaches exist. $\endgroup$ – Conner N. Howell Jun 12 '17 at 20:25
  • $\begingroup$ @Max Most of mathematics operates at a level where the "axiomatic basis" is almost completely irrelevant or it's not set theory (e.g. it's an axiomatic system at a higher level). It's certainly not the case that all mathematics uses set theory or has it as its "axiomatic basis", even with a relatively broad reading of "set theory". I will agree that most mathematicians use aspects of the language of set theory without much concerning themselves with what that entails. Indeed, much of math seems like it would be more at home in a type theoretic context. $\endgroup$ – Derek Elkins Jun 12 '17 at 23:30
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Modal logic is an attempt to axiomatize the intuitive idea of modal thought, where statements are true in some universe and false in others. The set theory definition of "frame" is just a rigorous set theory version of that idea.

However, the formal syntax of Modal Logic does not require set theory, any more than the formal syntax of Peano arithmetic does not require set theory.

Rather, a "frame" is just a starting idea before you get to the formal syntax, so you can understand the "semantics" of an expression in modal logic: "What does this expression 'mean?'"

Often, we construct models in set theory or category theory or number theory, to see the possible consistency or inconsistency of a logical system. Set theory is the most common, because it is more intuitive than category theory, and more powerful than number theory.

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