Systems with the directedness axiom

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Systems with the directedness axiom

The following theorem states some results for specific systems which will clarify the role played by the directedness axiom (TL3). Observe that the formula $\langle F_i \rangle K_i\alpha \land \langle F_i \rangle K_i(\alpha\to \beta) \to \langle F_i \rangle K_i\beta$ is not provable in DEK, i.e., it may be the case that both $\langle F_i \rangle K_i\alpha$ and $\langle F_i \rangle K_i(\alpha\to \beta)$ are true but $\langle F_i \rangle K_i\beta$ is not true. Generally, if a valid inference rule has at least 2 premises, and if each of these premises will be known after some course of thought, then it is not necessarily the case that the conclusion will be known. Such situations are precluded in the presence of the directedness axiom.

$\begin{theorem} \par Let $\alpha$\ and $\beta$\ be objective formulae. In logics... ... \langle F_i \rangle K_i(\alpha\land \beta)$\par\end{enumerate}\par\end{theorem}$

$\begin{proof} % latex2html id marker 2429\par See appendix \ref{app:proof}. \par\end{proof}$

Utilizing the previous result we can establish an embedding relation between K and DEK. Similar relations obtain between other normal modal systems and their dynamic-epistemic counterparts which contain schema (TL3).

$\begin{theorem} \par Let $\alpha \in \mathcal{L}_{N}^{K}$\ be a formula whose mo... ...only if $\alpha{^\prime}$\ is a theorem of \textbf{DEK$_N^*$}. \par\end{theorem}$

$\begin{proof} % latex2html id marker 2447\par Let $\alpha$\ be a \textbf{K$_N$... ...DEK$_N^*$}-derivable then $\alpha$\ is \textbf{K$_N$}-derivable. \par\end{proof}$

2001-04-05