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Lambda Calculus semantics are defined over a formal structure of values that are partially ordered with respect a sort of "more defined" relation. The least element is the completely undefined value, bottom. Expressions of the Lambda Calculus are defined with respect to an interpretation that can, for recursive expressions, produce multiple values for a single expression, so we define that the correct interpretation is the least upper bound of the set of values (you have to define the poset in such a way that the LUB is guaranteed to exist).

I was recently reading a description of Kripke's solution to the Liar Paradox that is based on having a "neither true nor false" value (which is reminiscent of bottom) and a hierarchy of interpretations which is reminiscent of the poset of Lambda Calculus semantics. It's not obvious to me that there is any relationship between these two other than the two points I mentioned and the fact that they are both trying to resolve the semantics of self-reference, but the description is a bit vague, so I was wondering if there is a connection between these two things that I'm not seeing.

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    comment: the domain interpretation of lambda calculus allows for partial definition, fixed point allows for an interpretation of recursion. Kripke's solution appears to used fixed points as well to deal with the "recursion" of self reference. So fixed points give at least a third connection Commented Sep 13, 2024 at 4:11
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    In LUB domain theory you compute more defined values until converging to the least fixed point with the bottom element represents non-terminating 'undefined' values. In Kripke's grounded theory of truth you build successive interpretations of truth values until a stable, fixed interpretation is reached, with the 'undefined' truth value serves a similar role for paradoxical sentences. However, the former is to provide a framework for understanding computation, while the latter is to avoid paradoxes via a specific philosophical theory of truth out of many other possible alternatives... Commented Sep 13, 2024 at 4:41
  • I think these can help: ncatlab.org/nlab/show/domain+theory ncatlab.org/nlab/show/Kripke-Joyal+semantics Commented Sep 13, 2024 at 16:48

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