We present a general proof theoretical methodology for default systems. Given a default theory D#, the default rules D are simply understood as restrictions on the tableaux construction of the logic. Di#erent default approaches have their own way of understanding these restrictions and executing them. For each default approach (such as Reiter, Brewka or Lukaszewicz), the allowable default extensions can be obtained from the default tableau construction. The advantage of our approach, besides being simple and neat, is in its generality: it allows for the development of a default theory for any logic with a tableau formulation, such as intuitionistic logic, linear logic or modal logic.

Commonsense reasoning is the reasoning of agents interacting with the real world. Non monotonic reasoning is a well developed research area gathering the logical formalisms that treat commonsense reasoning. One of the best known of such formalisms is Default logic. In this paper we discuss Default logic at both the proof-theoretic and semantics levels and show that Default logic provides a clear and formal framework to understand the logical nature of commonsense reasoning. Fondazione Ugo Bordoni via Baldassarre Castiglione 59, 00142 Roma, Italia. e-mail: gba@fub.it. Work carried out in the framework of the agreement between the Italian PT Administration and Fondazione Ugo Bordoni y Dipartimento di Informatica e Sistemistica Universit`a di Roma "La Sapienza" via Salaria 113, 00198 Roma, Italia e-mail:aiello,pirri@assi.dis.uniroma1.it 1 Introduction Commonsense reasoning is the way an agent reasons about the real world: drawing conclusions from an incomplete information on the cu...

en a formal and symbolic apparatus independent of the instrument or tool that will make it work. This view allowed us to understand an important pattern that links the di#erent forms of commonsense reasoning, i.e. the implicit and explicit contextual definability. We here briefly report our recent results on the logic of defaults, and dedicate them to Ray on his 60th birthday. 1 An overview of Default Logic The fundamental idea of Default Logic is to combine a classical theory W with a set D of special inference rules, called default rules and intended to capture patterns of reasoning like the following one: " Normally, if # was shown to be true, then # can be inferred". Here, " normally" means that the applicability of the default rule itself is not granted: the presence of an exception may prevent applicability. To make exceptions explicit, rules are defined as follows: # : #/#, which reads as " If # is proved and # is consistent (or, equivalently, is not provable; that is: th