Dynamic algebras combine the classes of Boolean (B 0) and regular (R ; *) algebras into a single finitely axiomatized variety (B R ) resembling an R-module with scalar multiplication . The basic result is that * is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result (...) is that free dynamic algebras are residually finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of prepositional dynamic logic, and yet another notion of regular algebra. (shrink)

Gentzen’s approach to deductive systems, and Carnap’s and Popper’s treatment of probability in logic were two fruitful ideas that appeared in logic of the mid-twentieth century. By combining these two concepts, the notion of sentence probability, and the deduction relation formalized in the sequent calculus, we introduce the notion of ’probabilized sequent’ \ with the intended meaning that “the probability of truthfulness of \ belongs to the interval [a, b]”. This method makes it possible to define a system of derivations (...) based on ’axioms’ of the form \, obtained as a result of empirical research, and then infer conclusions of the form \. We discuss the consistency, define the models, and prove the soundness and completeness for the defined probabilized sequent calculus. (shrink)

Constructing an intuitive theory from data confronts learners with a “chicken‐and‐egg” problem: The laws can only be expressed in terms of the theory's core concepts, but these concepts are only meaningful in terms of the role they play in the theory's laws; how can a learner discover appropriate concepts and laws simultaneously, knowing neither to begin with? We explore how children can solve this chicken‐and‐egg problem in the domain of magnetism, drawing on perspectives from computational modeling and behavioral experiments. We (...) present 4‐ and 5‐year‐olds with two different simplified magnet‐learning tasks. Children appropriately constrain their beliefs to two hypotheses following ambiguous but informative evidence. Following a critical intervention, they learn the correct theory. In the second study, children infer the correct number of categories given no information about the possible causal laws. Children's hypotheses in these tasks are explained as rational inferences within a Bayesian computational framework. (shrink)

Textbook for students in mathematical logic. Part 1. Total formalization is possible! Formal theories. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional and predicate logic. Glivenko's theorem and constructive embedding. Axiom independence. Interpretations, models and completeness theorems. Normal forms. Tableaux method. Resolution method. Herbrand's theorem.