We often reach conclusions partially on the basis that we do not have evidence that the conclusion is false. A newspaper story warning that the local water supply has been contaminated would prevent a person from drinking water from the tap in her home. This suggests that the absence of such evidence contributes to her usual belief that her water is safe. On the other hand, if a reasonable person received a letter telling her that she had won a million dollars, she would consciously consider whether there was any evidence that the letter was a hoax or somehow misleading before making plans to spend the money. All to often we arrive at conclusions which we later retract when contrary evidence becomes available. The contrary evidence defeats our earlier reasoning. Much of our reasoning is defeasible in this way. Since around 1980, considerable research in AI has focused on how to model reasoning of this sort. In this paper, I describe one theoretical approach to this problem, discuss implementation of this approach as an extension of Prolog, and describe some application of this work to normative reasoning, learning, planning, and other types of automated reasoning.

In this paper we present a Gentzen system for reasoning with contrary-to-duty obligations. The intuition behind the system is that a contrary-to-duty is a special kind of normative exception. The logical machinery to formalise this idea is taken from substructural logics and it is based on the definition of a new non-classical connective capturing the notion of reparational obligation. Then the system is tested against well-known contrary-to-duty paradoxes. 1

this paper we introduce a version of a labeled deductive system as it was introduced by Gabbay in [4] to reason about goals. It has some desirable properties not found in other proposals. First, the labeled logics formalize that goals interact and conflict. Goals only impose partial preferences, i.e. preferences given some objective and given some context. As a consequence, goals with overlapping contexts can conflict, because objectives can conflict. For example, to minimize time a tank has to be filled quickly, but to minimize loss it must be filled slowly. This cannot easily be formalized in standard formalisms. For example, the following counterintuitive derivation has to be blocked, where G(ff) is read as `preferably ff.' G(p) G(p q) G(:p) G(q :p)

. In previous work we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propositional modal logics that includes K, D, T, B, S4, S4:2, KD45, and S5. Here we extend this approach to quantified modal logics, providing formalizations for logics with varying, increasing, decreasing, or constant domains. The result is modular with respect to both properties of the accessibility relation in the Kripke frame and the way domains of individuals change between worlds. Our approach has a modular metatheory too; soundness, completeness and normalization are proved uniformly for every logic in our class. Finally, our work leads to a simple implementation of a modal logic theorem prover in a standard logical framework. 1 Introduction Motivation Modal logic is an active area of research in computer science and artificial intelligence: a large number of modal logics have been studied and new ones are frequently proposed. Each new log...

In this paper we present a Gentzen system for reasoning with contraryto -duty obligations. The intuition behind the system is that a contraryto -duty is a special kind of normative exception. The logical machinery to formalize this idea is taken from substructural logics and it is based on the definition of a new non-classical connective capturing the notion of reparational obligation. Then the system is tested against well-known contrary-to-duty paradoxes.

We give a categorial characterization of how labelled deduction systems for logics with a propositional basis behave under unconstrained fibring and under fibring that is constrained by symbol sharing. At the semantic level, we introduce a general semantics for our systems and then give a categorial characterization of fibring of models. Based on this, we establish the conditions under which our systems are sound and complete with respect to the general semantics for the corresponding logics, and establish requirements on logics and systems so that completeness is preserved by both forms of fibring.

Proof Search According to [12], abstract proof search is a process by which, starting from a representation of a problem at a so-called ground level, we construct a new and simpler representation at a so-called abstract level and use it to solve the original problem. That is, we abstract the given goal, prove its abstracted version and then use the information about the resulting abstract proof as an outline to construct the proof at the ground level. Dierent techniques to abstract from details have been studied in the literature. The problem is to nd out which details should be abstracted away. On one hand, if we abstract too much information then we often obtain abstract solutions that cannot be transferred to the ground level. Then, planning at the abstract level is even more dicult than planning at the ground level because the abstraction removes necessary control information, or we obtain only little information from the abstract proof how to guide the proof at the ground leve...

Neural-Symbolic integration has become a very active research area in the last decade. In this paper, we present a new massively parallel model for modal logic. We do so by extending the lang ugF of Modal Prolog [32, 37] to allow modal operators in the head of theclau( s. We thenu se an ensemble of C-IL neu ral networks [14, 15] to encode the extended modal theory (and its relations), and show that the ensemble compu tes a fixpoint semantics of the extended theory. An immediate resui of ou approach is the ability to perform learning from examples e#cientlyu sing each network of the ensemble. Therefore, one can adapt the extended C-IL P system by training possible world representations.

The multiplicative fragment of linear logic has found a number of applications in computational linguistics: in the "glue language" approach to LFG semantics, and in the formulation and parsing of various categorial grammars. These applications call for efficient deduction methods. Although a number of deduction methods for multiplicative linear logic are known, none of them are tabular meth- ods, which bring a substantial efficiency gain by avoiding redundant computation (c.f. chart methods in CFG parsing): this paper presents such a method, and discusses its use in relation to the above applications.

. In this article we introduce Contextual Deontic Logic (Cdl) to analyze the relation between deontic, contextual and defeasible reasoning. The optimal state, and therefore the set of active obligations, can change radically when the violation context changes. In such cases we say that the obligations only in force in the previous violation context are defeated; contextual deontic logic is therefore a defeasible deontic logic. This is expressed by the definition O fl (ffjfi) =def O(ffjfin:fl): `ff ought to be (done) if fi is (done) in the context where fl is (done)' is defined as `ff ought to be (done) if fi is (done) unless :fl is (done).' The unless clause formalizes explicit exceptions and is analogous to the justification in Reiter's default rules. Cdl is a monotonic defeasible deontic logic, because it has factual defeasibility but not overridden defeasibility. 1. Introduction It is well-known in deontic logic that there are striking similarities between deontic reasoning and c...