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FAILURE OF BETH’S THEOREM IN RELEVANCE LOGICS
- Part of
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- Journal:
- The Review of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 26 June 2025, pp. 1-14
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- Article
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Beth’s theorem equating explicit and implicit definability fails in all logics between Meyer’s basic logic
${\mathbf B}$ and the logic 
${\mathbf R}$ of Anderson and Belnap. This result has a simple proof that depends on the fact that these logics do not contain classical negation; it does not extend to logics such as 
$\mathbf{KR}$ that contain classical negation. Jacob Garber, however, showed that Beth’s theorem fails for 
$\mathbf{KR}$ by adapting Ralph Freese’s result showing that epimorphisms may not be surjective in the category of modular lattices. We extend Garber’s result to show that the Beth theorem fails in all logics between 
${\mathbf B}$ and 
$\mathbf{KR}$.
