ATOCHA ALISEDA MATHEMATICAL REASONING VS. ABDUCTIVE REASONING: A STRUCTURAL APPROACH 1. INTRODUCTION From a logical perspective, mathematical reasoning may be identified with classical, deductive inference. Two aspects are characteristic of this type of reasoning, namely its certainty and its monotonicity. The first of these is exemplified by the fact that the relationship between premises and con- clusion is that of necessity; a conclusion drawn from a set of premises, necessarily follows from them. The second aspect states that conclusions reached via deductive reasoning are non-defeasible. That is, once a the- orem has been proved, there is no doubt of its validity regardless of further addition of axioms and theorems to the system. There are however, several other types of formal non-classical reas- oning, which albeit their lack of complete certainty and monotonicty, are nevertheless rigorous forms of reasoning with logical properties of their own. Such is the case of inductive and abductive reasoning. As a first approximation, Charles S. Peirce distinction seems useful. According to him (Peirce (1931–1935), there are three basic types of logical reason- ing: deduction, induction and abduction. While deductive reasoning is for making predictions, inductive reasoning is for verifying those predictions; and abductive reasoning is for constructing hypotheses for puzzling phe- nomena. Concerning their certainty level, while deductive reasoning is completely certain, inductive and abductive reasoning are not. Induction must be validated empirically with tests and experiments, therefore it is de- feasible; and abductive reasoning can only offer hypotheses which may be refuted with additional information. For example, a generalization reached by induction (e.g., all birds fly), remains no longer valid after the addition of a premise which refutes the conclusion (e.g., penguins are birds). As for abduction, a hypothesis (e.g., it rained last night) which explains an obser- vation (e.g., the lawn is wet), may be refuted when additional information is incorporated into our knowledge base (e.g., it is a drought period). Synthese 134: 25–44, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.