A Cognitive View of Relevant Implication Claudio Masolo and Daniele Porello Laboratory for Applied Ontology, ISTC-CNR, Trento, Italy Abstract. Relevant logics provide an alternative to classical implication that is capable of accounting for the relationship between the antecedent and the consequence of a valid implication. Relevant implication is usu- ally explained in terms of information required to assess a proposition. By doing so, relevant implication introduces a number of cognitively rel- evant aspects in the definition of logical operators. In this paper, we aim to take a closer look at the cognitive feature of relevant implication. For this purpose, we develop a cognitively-oriented interpretation of the semantics of relevant logics. In particular, we provide an interpretation of Routley-Meyer semantics in terms of conceptual spaces and we show that it meets the constraints of the algebraic semantics of relevant logic. 1 Introduction Paradoxes of classical material implication often show a mismatch between our intuitions concerning valid patterns of reasoning and the formalization of impli- cation provided by classical logic. Debates on the nature of implication can be traced back to the very origin of modern logic, involving for instance Brentano, Husserl, and Frege. Turning to contemporary developments of mathematical logic, the problem of the logical properties of implication has been approached by providing systems that aims to mend classical logic from inference patterns that are not motivated on the basis of a specific view of reasoning. Since in any logical system, the implication has the important role of encoding the properties of logical inference, by rejecting the properties of classical implica- tion, one is often lead to rejecting classical logic. For instance, intuitionistic logic criticizes the non-constructive nature of classical implication. For that reason, intuitionists designed an alternative logic that rejects inference by contradiction and the law of the excluded middle. Moreover, relevant logic criticizes the lack of connection between the premises and the conclusion of a logical inference made explicit by some valid formula of classical logic, e.g., A → (B → A)—once A holds, one can infer that any B entails A—or (A → B) ∨ (B → A)—every pair of propositions can be connected by means of an implication. By keeping track of the antecedent-consequent connection, relevant logic prevents these paradoxes. Furthermore, classical implication does not model any sort of relationship between the knowing subject and the matter of the proposition. The truth- conditional definition of the classical implication A → B is given in terms of those states of affairs such that either the state of affairs corresponding to A does not hold or the state of affairs corresponding to B holds. Prosaically, A → B is true