Nonmonotonic Logics and Semantics
<|reference_start|>Nonmonotonic Logics and Semantics: Tarski gave a general semantics for deductive reasoning: a formula a may be deduced from a set A of formulas iff a holds in all models in which each of the elements of A holds. A more liberal semantics has been considered: a formula a may be deduced from a set A of formulas iff a holds in all of the "preferred" models in which all the elements of A hold. Shoham proposed that the notion of "preferred" models be defined by a partial ordering on the models of the underlying language. A more general semantics is described in this paper, based on a set of natural properties of choice functions. This semantics is here shown to be equivalent to a semantics based on comparing the relative "importance" of sets of models, by what amounts to a qualitative probability measure. The consequence operations defined by the equivalent semantics are then characterized by a weakening of Tarski's properties in which the monotonicity requirement is replaced by three weaker conditions. Classical propositional connectives are characterized by natural introduction-elimination rules in a nonmonotonic setting. Even in the nonmonotonic setting, one obtains classical propositional logic, thus showing that monotonicity is not required to justify classical propositional connectives.<|reference_end|>
@article{lehmann2002nonmonotonic,
    title={Nonmonotonic Logics and Semantics},
    author={Daniel Lehmann},
    journal={Journal of Logic and Computation, Vol. 11 No.2, pp.229-256 2001},
    year={2002},
    number={Leibniz Center for Research in Computer Science TR-98-6},
    archivePrefix={arXiv},
    eprint={cs/0202018},
    primaryClass={cs.AI cs.LO math.LO}
}