The numerically definite syllogistic is the fragment of English obtained by extending the language of the classical syllogism with numerical quantifiers. The numerically definite relational syllogistic is the fragment of English obtained by extending the numerically definite syllogistic with predicates involving transitive verbs. This paper investigates the computational complexity of the satisfiability problem for these fragments. We show that the satisfiability problem (= finite satisfiability problem) for the numerically definite syllogistic is strongly NP-complete, and that the satisfiability problem (= finite satisfiability problem) for the numerically definite relational syllogistic is NEXPTIME-complete, but perhaps not strongly so. We discuss the related problem of probabilistic (propositional) satisfiability, and thereby demonstrate the incompleteness of some proof-systems that have been proposed for the numerically definite syllogistic.

This paper deals with the problem of giving a principled characterization of the class of logical constants. According to the so-called Tarski—Sher thesis, an operation is logical iff it is invariant under permutation. In the model-theoretic tradition, this criterion has been widely accepted as giving a necessary condition for an operation to be logical. But it has been also widely criticized on the account that it counts too many operations as logical, failing thus to provide a sufficient condition. Our aim is to solve this problem of overgeneration by modifying the invariance criterion. We introduce a general notion of invariance under a similarity relation and present the connection between similarity relations and classes of invariant operations. The next task is to isolate a similarity relation well-suited for a definition of logicality. We argue that the standard arguments in favor of invariance under permutation, which rely on the generality and the formality of logic, should be modified. The revised arguments are shown to support an alternative to Tarski’s criterion, according to which an operation is logical iff it is invariant under potential isomorphism.

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, $\varepsilon$, such that the question of membership in this orbit is $\Sigma_{1}^{1}$-complete. This result and proof have a number of nice corollaries: the Scott rank of $\varepsilon$ is $\omega_{1}^{CK}+1$; not all orbits are elementarily definable; there is no arithmetic description of all orbits of $\varepsilon$; for all finite $\alpha \geq$ 9, there is a properly $\Delta_{\alpha}^{0}$ orbit (from the proof).

In continuation of [JOH04, OH07], we prove that existentially closed CSA-groups have the independence property. This is done by showing that there exist words having the independence property relative to the class of torsion-free hyperbolic groups.