Nonmonotonic logics and semantics (Q2720313)

Tarski's general semantics for logical deduction is well known: a formula ``\(a\)'' is deduced from a set of formulas \(A\) iff ``\(a\)'' holds in ``all'' the models such that all the formulas of \(A\) hold. In Tarski's semantic framework, the consequence operators satisfy inclusion, idempotence and monotony properties. NEWLINENEWLINENEWLINEIn this work, Daniel Lehmann considers two different semantics generalizing Tarski's one. NEWLINENEWLINENEWLINEThe first one involves a choice function on the set of models. The choice function is inspired by a function used in research in Social Choice. This function allows to consider not all the models of a set of formulas, but a subset of them (the more relevant models). The choice function satisfies ``natural'' properties. NEWLINENEWLINENEWLINEThe second semantics considered by the author is based on comparing the ``importance'' (qualitative measure) of the set of models by defining a strict partial order relation that satisfies certain properties. NEWLINENEWLINENEWLINEBoth of them are presented as ``natural'' semantics. NEWLINENEWLINENEWLINEEven when these semantics are different, it is proved that they are equivalent (under a simplifying assumption). NEWLINENEWLINENEWLINEIn this semantic framework, the consequence operations satisfy inclusion and idempotency properties. They are nonmonotonic; the usual monotony property is replaced by three properties of weak monotony. NEWLINENEWLINENEWLINEMoreover, an additional requirement on the choice function (a laziness property) is equivalent to a new property of the quality measure and it ensures a new property of weak monotony of the consequence operations. NEWLINENEWLINENEWLINEFinally, the classical propositional connectives are characterized by using properties of the nonmonotonic deduction operations. That is, it is shown that the nonmonotonic logic of the classical connectives is the classical propositional logic.