Suppose we have a consistent theory (that is, a set of formulas where one cannot derive false). We can assume that it's complete, if necessary. Is there a frame where all these formulas are valid? That is, does the consistency of $L$ imply $\mathsf{Frames}(L) \neq\varnothing$?
2026-03-27 06:06:35.1774591595
Does a (complete) consistent theory have a frame?
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No, in general a consistent set of sentences doesn't ever have to be validated.
For example, fix a propositional letter $p$; then $\{p\}$ is consistent but not validated in any frame. Clearly demanding completeness only makes this worse.
(OK fine, you have to require frames to be nonempty, since otherwise the empty frame validates everything; but this is part of the usual definition of frames, so that's fine.)