G: An algebraic non-monotonic logic for reasoning with graded propositions

We present [Log.sub.A]G, a weighted algebraic non-monotonic logic for reasoning with graded beliefs. [Log.sub.A]G is algebraic in that it is a language of only terms, some of which denote propositions and may be associated with ordered grades. The grades could be taken to represent a wide variety of phenomena including preference degrees, priority levels, trust ranks, and uncertainty measures. Reasoning in [Log.sub.A]G is non-monotonic and may give rise to contradictions. Belief revision is, hence, an integral part of reasoning and is guided by the grades. This yields a quite expressive language providing an interesting alternative to the currently existing approaches to non-monotonicity. We show how [Log.sub.A]G can be utilised for modelling resource-bounded reasoning; simulating inconclusive reasoning with circular, liar-like sentences; and reasoning about information arriving over a chain of sources each with a different degree of trust. While there certainly are accounts in the literature for each of these issues, we are not aware of any single framework that accounts for them all like [Log.sub.A]G does. We also show how [Log.sub.A]G captures a wide variety of non-monotonic logical formalisms. As such, [Log.sub.A]G is a unifying framework for non-monotonicity which is flexible enough to admit a wide array of potential uses. Keywords Non-Monotonicity * Weighted logics * Uncertainty * Graded propositions * Unified framework for Non-Monotonicity Mathematics Subject Classification (2010) 68T27