On Halldén completeness of modal logics determined by homogeneous Kripke frames (Q2810126)

A set of propositional modal formulas \(L\) is \textit{Halldén complete} iff for all members \(\varphi\) and \(\psi\) of \(L\) with no common propositional variables, if \(\varphi\vee\psi\in L\) then \(\varphi\in L\) or \(\psi\in L\). After listing a few known results, the author announces that he will give a semantic method for constructing Halldén complete modal logics, with tools borrowed from \textit{J. F. A. K. van Benthem} and \textit{I. L. Humberstone} [Notre Dame J. Formal Logic 24, 426--430 (1983; Zbl 0487.03008)] and \textit{E. J. Lemmon} [Notre Dame J. Formal Logic 7, 296--300 (1966; Zbl 0192.03202)]. It is difficult to summarise the many results in this paper, but here is a selection of some of the main ones. The statement of the first selected result involves the following notation and notions. {\parindent=0.7cm\begin{itemize}\item[--] Given a class \(\mathcal K\) of Kripke frames, \(L({\mathcal K})\) denotes the set of all formulas that are true in all members of \(\mathcal K\). \item[--] Let \({\mathfrak F}_1 = \langle W_1,R_1\rangle\) and \({\mathfrak F}_2 = \langle W_2,R_2\rangle\) be two Kripke frames. A map \(f: W_1\rightarrow W_2\) is a \textit{\(p\)-morphism} from \({\mathfrak F}_1\) to \({\mathfrak F}_2\) iff: {\parindent=0.9cm \begin{itemize}\item[{\(\bullet\)}] \(f\) maps \(W_1\) onto \(W_2\), \item[{\(\bullet\)}] for all \(x, y\in W_1\), \(x R_1 y\) implies \(f(x)R_2f(y)\), and \item[{\(\bullet\)}] for all \(x\in W_1\) and \(a\in W_2\), if \(f(x)R_2 a\) then there exists \(y\in W_1\) with \(xR_1 y\) and \(f(y) = a\). NEWLINENEWLINE\end{itemize}} \item[--] A Kripke frame \({\mathfrak F} = \langle W,R\rangle\) is \textit{homogeneous} iff for all \(x, y\in W\), there exists an automorphism \(f\) of \(\langle W,R\rangle\) with \(f(x) = y\). NEWLINENEWLINE\end{itemize}}NEWLINENEWLINEThe first selected result is then the following.NEWLINENEWLINELet \({\mathfrak F} = \langle W,R\rangle\) be a \(\mathrm{KTB}\)-Kripke frame (which is finite and connected). Then \(L({\mathfrak F})\) is Halldén complete iff \({\mathfrak F}\) is homogeneous.NEWLINENEWLINEThe second selected result is the following.NEWLINENEWLINEEach logic determined by a circular, that is, reflexive and symmetric, Kripke frame is Halldén complete.NEWLINENEWLINEThe last selected results rely on the following notions. Given a modal logic \(L\), denote by \(\mathrm{NEXT}(L)\) the class of all normal extensions of \(L\). Given a normal logic L and a formula \(\varphi\), let \(L\oplus\varphi\) denote the smallest normal extension of \(L\) containing \(\varphi\). For all \(n \geq 0\), let \(\mathrm{alt}_n\) denote the formula NEWLINE\[NEWLINE \square p_1\vee\square (p_1\rightarrow p_2)\vee\dots\vee\square\bigl((p_1\wedge\dots\wedge p_n)\rightarrow p_{n+1})\bigr) NEWLINE\]NEWLINE and for all \(n\geq 1\), let \(4_n\) denote the formula NEWLINE\[NEWLINE \square^n p\rightarrow\square^{n+1}p NEWLINE\]NEWLINE It is then shown that:NEWLINENEWLINE{\parindent=0.7cm \begin{itemize}\item[--] there are countably many Halldén complete as well as countably many Halldén incomplete logics in \(\mathrm{NEXT}(\mathbf{KTB}\oplus\mathrm{alt}_3)\); \item[--] there are countably many Halldén complete as well as countably many Halldén incomplete logics in \(\mathrm{NEXT}(\mathbf{KTB}\oplus\mathrm{alt}_m)\setminus\mathrm{NEXT}(\mathbf{KTB}\oplus\mathrm{alt}_{m-1})\) for all \(m\geq 4\); \item[--] there are countably many Halldén complete as well as countably many Halldén incomplete logics in \(\mathrm{NEXT}(\mathbf{KTB}\oplus4_2)\setminus\mathrm{NEXT}(S5)\). NEWLINENEWLINE\end{itemize}}