Abstract Basic notions linked with concept theory can be accounted for by partial order relations. These orders translate the fact that, for an agent, an object may be seen as a better or a more typical exemplar of a concept than anyother. They adequately model notions linked with categorial membership, typicality and resemblance, without any of the drawbacks that are classically encountered in conjunction theory. An interesting consequence of such a concept representation is the possibility of using the tools of non-monotonic logic to address some well-known problems of cognitive psychology. Thus, conceptual entailment and concept induction can be reexamined in the framework of preferential inference relations. This leads to a rigorous definition of the basic notions used in the study of category-based induction.

It is well known that preferential inference relations can be semantically represented through a smooth order defined on the set of states of a preferential model, but it is also possible to represent them via a preferential order, directly defined on the underlying language. In this paper, we study the link that exists between preferential orders and plausibility measures, and prove that any preferential order can be extended, in a canonical way, into a plausibility measure. This extended order has a natural intrepretation in the infinitary framework of deductive inference operations.

In this paper we study negation rationality, an important property for non-monotonic inference relations. We show that the preferential relations that satisfy negation rationality and admit an injective model are exactly the preferential relations that satisfy the property of disjunctive rationality. We also propose a semantic-type characterization for relations that satisfy negation rationality.

In this paper, we prove the existence of a one-to-one mapping between the set of all preferential inference relations defined on a language £ and a family of transitive relations among the elements of this language. Given a preferential inference relation, we use the corresponding transitive relation to define an order among the set of all worlds associated to £. We characterize all the preferential inference relations that can be represented by this ordered set. As a particular case, we study the family of disjunctive relations, that is relations where a conclusion drawn from a disjunction of premisses can be drawn from one at least of those premisses taken alone. We show that for this type of relation, the associated model is injective and filtered: if two worlds m and n satisfy a proposition α and are not minimal for that property, then there exists a world p less than both of them that satisfies α. As we prove, conversely, that any filtered model defines a disjunctive inference relation, we obtain a representation theorem for these relations. Applying finally these results to the family of rational relations, we get a new proof of the representation of these relations by means of ranked injective models.