Abstract
A family of prepositional logics is considered to be intermediate between the intuitionistic and classical ones. The generalized interpolation property is defined and proved is the following.
Theorem on interpolation. For every intermediate logic L the following statements are equivalent:
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(i)
Craig's interpolation theorem holds in L,
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(ii)
L possesses the generalized interpolation property,
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(iii)
Robinson's consistency statement is true in L.
There are just 7 intermediate logics in which Craig's theorem holds.
Besides, Craig's interpolation theorem holds in L iff all the modal companions of L possess Craig's interpolation property restricted to those formulas in which every variable is proceeded by necessity symbol.
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Maksimova, L.L. Interpolation properties of superintuitionistic logics. Stud Logica 38, 419–428 (1979). https://doi.org/10.1007/BF00370479
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DOI: https://doi.org/10.1007/BF00370479