KSL-06-11

## Any $ist_{AO}$ Quantified Context Logic has a First-Order Semantics.

Reference: Makarios, S.; Heuer, K. Any $ist_{AO}$ Quantified Context Logic has a First-Order Semantics. 2006.

Abstract: Context logics of the $ist(c,\phi)$ type of Guha/McCarthy [] can be classified according to the distributivities of $ist$ over the logical connectives and quantifiers. We place subscripts on $ist$ according to the various distributivities; subscript $A$ stands for distributivity of $ist$ over conjunction and universal quantification, that is, $ist(c,\phi \wedge \psi) \leftrightarrow ist(c,\phi) \wedge ist(c,\psi)$ and $ist(c, \forall x \phi) \leftrightarrow \forall x ist(c,phi)$. Likewise $O$ stands for disjunction and existential quantification, and $N$ stands for negation. This work recounts a formal language discussed in [] for quantified context logic, and the formal semantics presented for it. In [], it was established that any $ist_{AON}$ quantified context logic has a first-order semantics. The current work improves upon that result, establishing that the characterization via a first-order semantics is possible even without the restriction of distributivity of $ist$ over negation -- that is -- first-order semantics can be given for any context logic that is merely $ist_{AO}$. This considerably broadens the applicability of the results, and should allow theories over many conceptions of context be treated via pre-compilation into a first-order form.

Notes:

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