Free lattices proof-theoretically

A sequent system is used to give alternative proofs of two well known properties of free lattices: Whitman’s condition and semidistributivity. It demonstrates usefulness of such proof systems outside logic

Free lattices proof-theoretically

A sequent system is used to give alternative proofs of two well known properties of free lattices: Whitman’s condition and semidistributivity. It demonstrates usefulness of such proof systems outside logic

The automorphism group of the Fra\"iss\'e limit of finite Heyting algebras

Roelcke non-precompactness, non-simplicity, and non-amenability of the automorphism group of the Fra\"iss\'e limit of finite Heyting algebras are examined among others.Comment: Submitted to Journal of Symbolic Logic; 15 page

Algebraic Characterization of the Local Craig Interpolation Property

The sole purpose of this paper is to give an algebraic characterization, in terms of a superamalgamation property, of a local version of Craig interpolation theorem that has been introduced and studied in earlier papers. We continue ongoing research in abstract algebraic logic and use the framework developed by Andréka–Németi and Sain.

The Modelwise Interpolation Property of Semantic Logics

In this paper we introduce the modelwise interpolation property of a logic that states that whenever $\models\phi\to\psi$ holds for two formulas $\phi$ and $\psi$, then for every model $\mathfrak{M}$ there is an interpolant formula $\chi$ formulated in the intersection of the vocabularies of $\phi$ and $\psi$, such that $\mathfrak{M}\models\phi\to\chi$ and $\mathfrak{M}\models\chi\to\psi$, that is, the interpolant formula in Craig interpolation may vary from model to model. We compare the modelwise interpolation property with the standard Craig interpolation and with the local interpolation property by discussing examples, most notably the finite variable fragments of first order logic, and difference logic. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic

Pupils' understanding of scientific evidence in the context of investigative work

The research reported in this study is based on a detailed observation study of pupils carrying out investigative work in primary and secondary schools. The thesis examines the question of the type and level of understanding of scientific evidence which is employed in a number of different types of tasks associated with two underlying substantive concepts. The results suggest that the characteristics of each task make different demands on pupils of different ages and show that some pupils, in both primary and secondary schools, are capable of applying and synthesising ideas about evidence effectively in a problem-solving situation. Most pupils, however, even in Year 9, show a weak understanding of the application of many of these ideas, such as repeatability, range and interval or the most appropriate type of graph. It is argued that if pupils are to understand scientific evidence, then these ideas need to be taught and their synthesis in investigative work regularly reinforced. The procedural understanding demonstrated by pupils in practical investigations is compared with the understanding revealed orally in an interview situation and in written tasks. While some pupils appear to use tacit understanding in practical tasks which cannot be accessed readily in other ways, the results also show that most pupils who can express ideas about evidence explicitly are more likely to apply and synthesise them in a problem-solving situation. It is likely that if ideas about evidence were taught explicitly, then the ability of pupils to apply and synthesise them in a problem-solving situation would improve. It is also argued that, because problem-solving skills are required by employers in science-based industry and because understanding evidence is important in everyday life, then these ideas need to be formally taught and assessed within science education