On the Logics with Propositional Quantifiers Extending S5Pi
[eScholarship] Scroggs's theorem on the extensions of S5 is an early landmark in the modern mathematical studies of modal logics. From it, we know that the lattice of normal extensions of S5 is isomorphic to the inverse order of the natural numbers with infinity and that all extensions of S5 are in fact normal. In this paper, we consider extending Scroggs's theorem to modal logics with propositional quantifiers governed by the axioms and rules analogous to the usual ones for ordinary quantifiers. We call them Π-logics. Taking S5Π, the smallest normal Π-logic extending S5, as the natural counterpart to S5 in Scroggs's theorem, we show that all normal Π-logics extending S5Π are complete with respect to their complete simple S5 algebras, that they form a lattice that is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N, that they have arbitrarily high Turing-degrees, and that there are non-normal Π-logics extending S5Π.
The Logic of Comparative Cardinality
[eScholarhip] [Slides] This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections.
Epistemic Logic with Functional Dependency Operator
[arXiv] Epistemic logic with non-standard knowledge operators, especially the "knowing-value'' operator, has recently gathered much attention. With the "knowing-value'' operator, we can express knowledge of individual variables, but not of the relations between them in general. In this paper, we propose a new operator Kf to express knowledge of the functional dependencies between variables. The semantics of this Kf operator uses a function domain which imposes a constraint on what counts as a functional dependency relation. By adjusting this function domain, different interesting logics arise, and in this paper we axiomatize three such logics in a single agent setting. Then we show how these three logics can be unified by allowing the function domain to vary relative to different agents and possible worlds. A multiagent axiomatization is given in this case.
The axiomatization and complexity of Knowing-What-Logic on model class K
[arXiv] Standard epistemic logic studies propositional knowledge, yet many other types of knowledge such as "knowing whether'', "knowing what'', "knowing how'' are frequently and widely used. This paper presents a axiomatization and a tableau for the modal logic of "knowing-what" operator on arbitrary Kripke models. As we are not working on S5 model class, this operator is not technically a "knowing" operator, but the inner structure is clearer in this setting.