3,620 research outputs found
The Topology-Free Construction of the Universal Type Structure for Conditional Probability Systems
We construct the universal type structure for conditional probability systems
without any topological assumption, namely a type structure that is terminal,
belief-complete, and non-redundant. In particular, in order to obtain the
belief-completeness in a constructive way, we extend the work of Meier [An
Infinitary Probability Logic for Type Spaces. Israel Journal of Mathematics,
192, 1-58] by proving strong soundness and strong completeness of an infinitary
conditional probability logic with truthful and non-epistemic conditioning
events.Comment: In Proceedings TARK 2017, arXiv:1707.0825
On the proof theory of infinitary modal logic
The article deals with infinitary modal logic. We first discuss the difficulties related to the development of a satisfactory proof theory and then we show how to overcome these problems by introducing a labelled sequent calculus which is sound and complete with respect to Kripke semantics. We establish the structural properties of the system, namely admissibility of the structural rules and of the cut rule. Finally, we show how to embed common knowledge in the infinitary calculus and we discuss first-order extensions of infinitary modal logic
A parametrised axiomatization for a large number of restricted second-order logics
By limiting the range of the predicate variables in a second-order language
one may obtain restricted versions of second-order logic such as weak
second-order logic or definable subset logic. In this note we provide an
infinitary strongly complete axiomatization for several systems of this kind
having the range of the predicate variables as a parameter. The completeness
argument uses simple techniques from the theory of Boolean algebras
Uniform Inductive Reasoning in Transitive Closure Logic via Infinite Descent
Transitive closure logic is a known extension of first-order logic obtained by introducing a
transitive closure operator. While other extensions of first-order logic with inductive definitions
are a priori parametrized by a set of inductive definitions, the addition of the transitive closure
operator uniformly captures all finitary inductive definitions. In this paper we present an
infinitary proof system for transitive closure logic which is an infinite descent-style counterpart
to the existing (explicit induction) proof system for the logic. We show that, as for similar
systems for first-order logic with inductive definitions, our infinitary system is complete for the
standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive
closure operator allows semantically meaningful complete restrictions to be defined using simple
syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides
the basis for an effective system for automating inductive reasoning
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