Qian Chen

This study explored and compared the perspectives of Taiwanese in-service and pre-service high school mathematics teachers regarding ideal teaching behaviours; the perspectives of a nationwide sample of students were taken as the baseline. Fourteen factors contributing to ideal teaching behaviours were identified through exploratory factor analyses. Nine factors, including idea explanation and speedy lecture, were rooted in traditional Chinese culture; five factors, including concrete representation and student activities, were influenced by Western cultures. Three teacher profiles were identified through k-means clustering analysis. The perspectives of in-service teachers were dominated by a painless meaning-emphasised profile; these teachers emphasised meaningful learning for students and avoided the fast pace and demanding requirements that can cause distress in students, whereas pre-service teachers were dominated by an all-round profile, revealing their openness to all factors. Compared with...

This paper presents a logic of preference and functional dependence (LPFD) and its hybrid extension (HLPFD), both of whose sound and strongly complete axiomatization are provided. The decidability of LPFD is also proved. The application of LPFD and HLPFD to modelling cooperative games in strategic form is explored. The resulted framework provides a unified view on Nash equilibrium, Pareto optimality and the core. The philosophical relevance of these game-theoretical notions to discussions of collective agency is made explicit. Some key connections with other logics are also revealed, for example, the coalition logic, the logic of functional dependence and the logic of ceteris paribus preference.

The finite model property (FMP) in weakly transitive tense logics is explored. Let \(\mathbb {S}=[\textsf{wK}_t\textsf{4}, \textsf{K}_t\textsf{4}]\) be the interval of tense logics between \(\textsf{wK}_t\textsf{4}\) and \(\textsf{K}_t\textsf{4}\). We introduce the modal formula \(\textrm{t}_0^n\) for each \(n\ge 1\). Within the class of all weakly transitive frames, \(\textrm{t}_0^n\) defines the class of all frames in which every cluster has at most _n_ irreflexive points. For each \(n\ge 1\), we define the interval \(\mathbb {S}_n=[\textsf{wK}_t\textsf{4T}_0^{n+1}, \textsf{wK}_t\textsf{4T}_0^{n}]\) which is a subset of \(\mathbb {S}\). There are \(2^{\aleph _0}\) logics in \(\mathbb {S}_n\) lacking the FMP, and there are \(2^{\aleph _0}\) logics in \(\mathbb {S}_n\) having the FMP. Then we explore the FMP in finitely alternative tense logics \(L_{n,m}=L\oplus \{\textrm{Alt}_n^F, \textrm{Alt}_m^P\}\) with \(n,m\ge 0\) and \(L\in \mathbb {S}\). For all \(k\ge 0\) and \(n,m\ge 1\), we define intervals \(\mathbb {F}^k_{n,m}\), \(\mathbb {P}^k_{n,m}\) and \(\mathbb {S}^k_{n,m}\) of tense logics. The number of logics lacking the FMP in them is either 0 or \(2^{\aleph _0}\), and the number of logics having the FMP in them is either finite or \(2^{\aleph _0}\).