ABSTRACT. The aim of this paper is twofold: (i) to introduce the framework of update semantics and to explain what kind of phenomena may successfully be analysed in it; (ii) to give a detailed analysis of one such phenomenon: default reasoning.

As axioms for IL we take the usual axioms A→ A and (A→A)→ A (Löb's Axiom) for the provability logic L and its rules, modus ponens and necessitation, plus the axioms: (1) (A→B)→(A| >B) (2) (A| > B) ∧ (B| > C) → (A| > C) (3) (A| > C) ∧ (B| > C)→(A∨B| > C) (4) (A| > B)→(A→ B) (5) A| >A With respect to priority of parentheses | > is treated as →. Furthermore, we will consider the following extensions of IL: ILM = IL + M, where M is the axiom (A| > B)→(A∧ C| > B∧ C) ILP = IL + P, where P is the axiom (A| >B)→ (A| >B) 2 We will write | _ IL for derivability in IL, similarly for the other systems, but sometimes we may leave the subscript off. 1 We want to thank Albert Visser who inspired these investigations by asking us to try and find a useful semantics for the system ILM. We also thank Rineke Verbrugge for a number of corrections. 2 The scheme M is named after Franco Montagna who showed its soundness with respect to PA, even in the more general case when C is replaced by a Σ-formula. The background of the names of the schemes P and W is semantic and will be explained in the next section.

This paper provides an update semantics for counterfactual conditionals.

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