By J. R. Ennals

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One important feature of CS, for which there is a complete and sound axiomatisation, is that non-monotonicity is not treated as the side-effect of some logic as is the case in NMLI, NMLII and |x, but rather it is an inherent property of the actual setting. A non-monotonic theory for commonsense reasoning (IS) IS is similar to CS regarding the setting, except that to each b G B, V assigns a valuation Vb which is neither complete nor satisfies the monotonicity condition. 19 An acceptance relation is defined as: 1 2 3 4 5 M,b| = p M,b| = AAB M,b| = AVB M,b|=MA M,b| = (A->B) iff iff iff iff iff Vb(p) = t (for atomic p).

For instance, we may distinguish between two forms of ignorance at a moment of time t according to whether the agent is ignorant of some actual facts at t or has a complete representation of a changing world. Obeid in (BEL20) distinguishes partiality and impreciseness, where in both cases the underlying logic is shown to exhibit some non-monotonic behaviour. In this section we present a review of some non-monotonic theories, that is theories of reasoning about knowledge/belief which exhibit non-monotonic behaviour, following closely the distinction drawn in (BEL20).

Default reasoning and non-monotonic logics Default reasoning is an important form of human commonsense reasoning of particular interest in AI*. It is characterised by inferences of the form 'infer A if there is no evidence to the contrary' or, more generally, 'infer A if B cannot be inferred', made in situations where A is typically the case. Default reasoning is defeasible; further evidence may lead us to abandon previously established conclusions. First-order Logic (FOL), however, is monotonict so it cannot be used to represent default reasoning.

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