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We study groups definable in tame expansions of -stable theories. Assuming several tameness conditions, we obtain structural theorems for groups definable and interpretable in these expansions. As our main example, by characterizing independence in the pair , where is an algebraically closed field and is a multiplicative subgroup of with the Mann property, we show that the pair satisfies the assumptions. In particular, this provides a characterization of definable and interpretable groups in in terms of algebraic groups in and interpretable groups in . Furthermore, we compute the Morley rank and the -rank in and both ranks agree.
In this article, we develop and clarify some of the basic combinatorial properties of the new notion of -dependence (for ) recently introduced by Shelah. In the same way as dependence of a theory means its inability to encode a bipartite random graph with a definable edge relation, -dependence corresponds to the inability to encode a random -partite -hypergraph with a definable edge relation. We characterize -dependence by counting -types over finite sets (generalizing the Sauer–Shelah lemma, answering a question of Shelah), and in terms of the collapse of random ordered -hypergraph indiscernibles down to order-indiscernibles (which implies that the failure of -dependence is always witnessed by a formula in a single free variable).
Mereotopology is the discipline obtained from combining topology with the formal study of parts and their relation to wholes, or mereology. This article develops a mereotopological theory of time, illustrating how different temporal topologies can be effectively discriminated on this basis. Specifically, we demonstrate how the three principal types of temporal models—namely, the linear ones, the forking ones, and the circular ones—can be characterized by differently combining two sole mereotopological constraints: one to denote the absence of closed loops, and the other one to denote the absence of branches.
Given an uncountable regular cardinal , we study the structural properties of the class of all sets of functions from to that are definable over the structure by a -formula with parameters. It is well known that many important statements about these classes are not decided by the axioms of together with large cardinal axioms. In this paper, we present other canonical extensions of that provide a strong structure theory for these classes. These axioms are variations of the Maximality Principle introduced by Stavi and Väänänen and later rediscovered by Hamkins.
Inspired by the supervenience-determined consequence relation and the semantics of agreement operator, we introduce a modal logic of supervenience, which has a dyadic operator of supervenience as a sole modality. The semantics of supervenience modality very naturally correspond to the supervenience-determined consequence relation, in a quite similar way that the strict implication corresponds to the inference-determined consequence relation. We show that this new logic is more expressive than the modal logic of agreement, by proposing a notion of bisimulation for the latter. We provide a sound proof system for the new logic. We lift onto more general logics of supervenience. Related to this, we address an interesting open research direction listed in the literature, by comparing propositional logic of determinacy and noncontingency logic in expressive powers and axiomatizing propositional logic of determinacy over various classes of frames. We also obtain an alternative axiomatization for propositional logic of determinacy over universal models.
In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5 extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5 with respect to its most natural algebraic semantics, with propositional quantifiers interpreted by meets and joins over all elements in a complete Boolean algebra. In this note, we give such a proof. This result raises the question: For which normal modal logics can one axiomatize the quantified propositional modal logic determined by the complete modal algebras for ?
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