Recent work has considered the problem of extending to the case of iterated belief change the so-called `Harper Identity' (HI), which defines single-shot contraction in terms of single-shot revision. The present paper considers the prospects of providing a similar extension of the Levi Identity (LI), in which the direction of definition runs the other way. We restrict our attention here to the three classic iterated revision operators--natural, restrained and lexicographic, for which we provide here the first collective characterisation in the (...) literature, under the appellation of `elementary' operators. We consider two prima facie plausible ways of extending (LI). The first proposal involves the use of the rational closure operator to offer a `reductive' account of iterated revision in terms of iterated contraction. The second, which doesn't commit to reductionism, was put forward some years ago by Nayak et al. We establish that, for elementary revision operators and under mild assumptions regarding contraction, Nayak's proposal is equivalent to a new set of postulates formalising the claim that contraction by ¬A should be considered to be a kind of `mild' revision by A. We then show that these, in turn, under slightly weaker assumptions, jointly amount to the conjunction of a pair of constraints on the extension of (HI) that were recently proposed in the literature. Finally, we consider the consequences of endorsing both suggestions and show that this would yield an identification of rational revision with natural revision. We close the paper by discussing the general prospects for defining iterated revision in terms of iterated contraction. (shrink)

The last 50 years of research has taught us that conditionals are non-monotonic in the antecedent. That is, they invalidate Antecedent Strengthening. Many accounts have been developed for such conditionals, starting with Stalnaker and Lewis. These accounts converge roughly to Burgess’ conditional logic $${{\,\mathrm{\textsf{B}}\,}}$$ B or the non-monotonic reasoning system $$\textbf{P}$$ P. The latter two have Cautious Monotonicity as a weak replacement for Antecedent Strengthening. Lewis’ weakest conditional logic $${{\,\mathrm{\textsf{V}}\,}}$$ V or its non-monotonic reasoning counterpart system $$\textbf{R}$$ R are obtained (...) by adding a stronger proxi for Antecedent Strengthening – the law of Rational Monotonicity. I argue that Rational Monotonicity is too much while Cautious Monotonicity is not enough monotonicity. I investigate two other monotonicity postulates, which are jointly weaker than Rational Monotonicity. One of them is related to Disjunctive Rationality, the other one is related to the idea of replacing the preservation law encoded in Rational Monotonicity by a pivoting principle – instead of conserving a conclusion when adding a compatible supposition, the principle recommends pivoting from a disjunction to another disjunction when the reason for the first is undermined by such an addition. This gives rise to three logics in between $${{\,\mathrm{\textsf{B}}\,}}$$ B and $${{\,\mathrm{\textsf{V}}\,}}$$ V and three other logics in between $${{\,\mathrm{\textsf{BN}}\,}}$$ BN and $${{\,\mathrm{\textsf{VN}}\,}}$$ VN obtained by strengthening $${{\,\mathrm{\textsf{B}}\,}}$$ B and $${{\,\mathrm{\textsf{V}}\,}}$$ V by a consistency axiom. I prove soundness, completeness and decidability results for these six new logics, using set-selection semantics, its correspondence theory, and filtrations. In the second follow-up paper, I investigate five other semantics for the new logics: well founded strict order semantics, a new closeness semantics, several weakenings of ranking semantics, and new versions of Lewis’ and Burgess’ similarity semantics. (shrink)