An Implicative Expansion of Belnap’s Four-Valued Matrix: A Modal Four-Valued Logic Without Strong Modal Lukasiewicz-Type Paradoxes
José Miguel Blanco · Bulletin of Symbolic Logic · 2020
In [3], where Inv(U) was introduced to prove an Ax-Kochen-Eršov-type result, it was claimed that this semigroup is always well-defined and commutative.We disprove both statements, provide ∼ D -invariants, and show independence of S inv (U)/ ∼ D from the choice of U to contradict the Independence Property.Theorem A. There is a supersimple theory of SU-rank 2 in which ∼ D is not a congruence with respect to ⊗, and where ≥ D differs from nonforking-domination.Moreover, in the Random Graph Inv(U) is not commutative. Theorem B.If p 0 ≥ D p 1 and p 0 is definable, finitely satisfiable in some small model, generically stable, or weakly orthogonal to q, then so is p 1 . Theorem C.If there are only boundedly many ∼ D -classes, then T is NIP.Beyond the above results from [4], we reduce the study of Inv(U) in o-minimal context to proving that every invariant type is equivalent to a product of 1-types, and show this to hold in Real Closed Fields.This yields a complete characterisation both in this theory and, using results from [1], in that of Real Closed Valued Fields.We also survey the stable case, compute Inv(U) in several other theories, including that of dense meet-trees, and show its well-definedness in certain expansions of the latter studied in [2].Theorem D. In Real Closed Fields, ( Inv(U),⊗) is well-defined and isomorphic to the semilattice of finite subsets (P fin (X ),∪), where X is the set of convex subrings of U which, for some small A, are fixed by the stabiliser Aut(U/A).Theorem E. In dense meet-trees, Inv(U) has the form P fin (X ) ⊕ |U| N.