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April 14

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Translation in classical logic?

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Is there something like the Gödel–Gentzen translation for a translation from Many-valued logic in classical logic?--82.82.233.47 (talk) 20:05, 14 April 2021 (UTC)[reply]

It depends a bit on what you demand of the translation and which version of many-valued logic is taken as the source logic of the translation. I assume we want the translation to be compositional. To explain what I mean by this, let stand for any logical operator of the source logic – for simplicity I only consider the case of a dyadic (binary) operator, taking two logical formulas and as operands to form a new formula Denoting translation by a superscript "", for this translation to be compositional it should be the case that where denotes an operation that is definable using the target logic, here classical logic. It is a must that the translation is representation insensitive, meaning that it maps equivalent formulas in the source formalism to equivalent results in the target formalism. I also assume that we exclude trivial translations, such as the one that defines for all I'll concentrate on Kleene's The translation has to be a bit more complicated, since a propositional formula in the source logic – excluding trivial translations that map every formula to the same classical formula – cannot simply be mapped compositionally to a formula in the target language. Proof of this impossibility is by showing that a compositional translation is trivial. Let be any compositional translation. Put and Since the operation has a fixpoint. There are three classical logical operation that have a fixpoint, constant constant and the identity. If is the identity, , so Then also The constant operations can likewise be excluded. However, a translation is possible if we translate a source formula to a pair of formulas as follows:
The other operations follow from the usual identities, such as The first component of the pair can be interpreted as "definitely true", and the second as "definitely false". Then means: "neither definitely true nor definitely false".  --Lambiam 00:38, 15 April 2021 (UTC)[reply]