g Circ odal Logic

This paper discusses the logic LKM which extends circumscription into an epistemic domain. This extension will allow us to define circumscription of predicates that appear within the context of a modal operator. In fact, LKM can be seen as a method of extending any first-order nonmonotonic logic whose semantic definition is based on a partial-order among models, into a new nonmonotonic logic defined for a modal language, whose modal operator (K) follows an underlying S5 or weak-S5 semantics. One interesting use of this nonmonotonic logic is to model nonmonotonic aspects of the communication between agents. Introduction We will assume, as McCarthy implicitly does, that the speaker always believes in what she said. Thus there is no need to represent Grice’s maxim as a default and we can automatically assert that the speaker believes the content of her utterance. We will also avoid dealing with the last step in the reasoning: the incorporation of the speaker’s beliefs into the hearer’s beliefs. What is left to be modeled is the reasoning from a) the fact that the speaker knows a proposition, and b) that the hearer can assume that the speaker knows that there is an default that applies to that proposition, to conclude that the speaker believes that the default hold. In a schematic way, and assuming that A’s utterance was that a entity b is a bird: McCarthy [McCarthy, 19861 suggests using a nonmonotonic logic to model conventions in communication. For example, if A tells B about a bird, and A says nothing about the bird’s ability to fly, B should conclude that the bird does fly. McCarthy proposes that circumscription could account for this reasoning. We claim that the reasoning cannot be accounted for by circumscription alone (or any first-order nonmonotonic logic) and that a nonmonotonic logic that combines both circumscription and knowledge is necessary for the task. (1) From KA bird( b) and KA pz [bird(z) A labn(z) fly(z)]] conclude KA fly(b) where KA is the operator that represents the speaker’s knowledge (or belief). The first formula above comes from the assumption that the speaker believes in what she said. The second formula comes from the fact that the default on birds flying condition is mutually known to A and B, and thus A knows it. Let us detail the defaults involved in the reasoning above. First, there is the need to represent the default that birds usually fly, or, at least, that if nothing is mentioned about a bird’s flying condition, one should assume that it can fly. I will assume circumscription or some other first-order nonmonotonic logic can represent such a defau1t.l Furthermore, there is the default that the speaker A is being truthful and she believes in what she says (Grice’s conversational maxim of quality [Grice, 19751). That is, if A utters a proposition Q! then B should assume that A believes o. McCarthy’s proposal implicitly assumes that one can remove the external knowledge operator from the assumptions, perform the default reasoning within a first-order framework (by circumscribing abn), and reinsert (if needed) the knowledge operator in the conclusions. In other words, the reasoning is performed within a single belief space: the propositions known by the speaker. We will call this methodology internal reasoning. This method seems reasonable, but it cannot account for all utterances. For example, if A had uttered: (2) b is a bird, perhaps a penguin. ‘One also has to represent the fact that this default is one would not like to conclude that b flies, or that the mutually known between A and B in order to characterize speaker believes that b flies. In fact, the speaker menit as a convention. But I will leave this detail aside. tioned that she considers it possible that b cannot fly. Finally there is the default that if B knows that A believes o, and CI! does not contradicts B’s beliefs, then B should also believe cy. 648 Representation and Reasoning: Belief From: AAAI-92 Proceedings. Copyright ©1992, AAAI (www.aaai.org). All rights reserved.