On the Logic of Belief and Propositional Quantification

We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is some-thing that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss

Theorem proving for non-normal modal logics

In this work we briefly summarize our recent contributions in the field of proof methods, theorem proving and countermodel generation for non-normal modal logics. We first recall some labelled sequent calculi for the basic system E and its extensions with axioms M, N, and C based on bi-neighbourhood semantics. Then, we present PRONOM, a theorem prover and countermodel generator for non-normal modal logics implemented in Prolog. When a modal formula is valid, then PRONOM computes a proof in the labelled calculi, otherwise it is able to extract a model falsifying it from an open, saturated branch. © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).Peer reviewe

Bayesian Confirmation and Justifications

We introduce a family of probabilistic justification logics that feature Bayesian confirmations. Our logics include new justification terms representing evidence that make a proposition firm in the sense of making it more probable. We present syntax and semantics of our logic and establish soundness and strong completeness. Moreover, we show how to formalize in our logic the screening-off condition for transitivity of Bayesian confirmations