## Supplement: Burgess-Xu Axiomatic System for Since and Until and Some Extensions

The axiomatic system of Burgess-Xu for the reflexive versions of $$S$$ and $$U$$ extends classical propositional logic with the following axiom schemata and their mirror images, with $$U$$ and $$S$$ as well as $$G$$ and $$H$$ swapped:

• $$G\varphi \rightarrow \varphi$$
• $$G(\varphi \rightarrow \psi)\rightarrow \varphi U\chi \rightarrow \psi U\chi$$
• $$G(\varphi \rightarrow \psi)\rightarrow \chi U\varphi \rightarrow \chi U\psi$$
• $$\varphi \wedge \chi U\psi \rightarrow \chi U(\psi \wedge \chi S\varphi)$$
• $$\varphi U\psi \rightarrow (\varphi \wedge \varphi U\psi)U\psi$$
• $$\varphi U(\varphi \wedge \varphi U\psi)\rightarrow \varphi U\psi$$
• $$\varphi U\psi \wedge \chi U\theta \rightarrow (\varphi \wedge \chi)U(\psi \wedge \theta)\vee (\varphi \wedge \chi)U(\psi \wedge \chi)\vee (\varphi \wedge \chi)U(\varphi \wedge \theta)$$

and the inference rules NEC$$_G$$ and NEC$$_H$$. This axiomatization, translated for the strict versions of $$S$$ and $$U$$, was extended by Venema (1993) to complete axiomatic systems for:

• all discrete linear orderings, by adding $$F\top \rightarrow \bot U\top$$ and its dual $$P\top \rightarrow \bot S\top$$;
• all well-orderings, by further adding $$H\bot \vee PH\bot$$ and $$F\varphi \rightarrow (\lnot \varphi)U\varphi$$;
• $$\left\langle \mathbf{N,\lt}\right\rangle$$, by adding $$F\top$$ to the previous system.
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