On Strengthening the Logic of Iterated Belief Revision: Proper Ordinal Interval Operators

Darwiche and Pearl’s seminal 1997 article outlined a number of baseline principles for a logic of iterated belief revision. These principles, the DP postulates, have been supplemented in a number of alternative ways. Most suggestions have resulted in a form of ‘reductionism’ that identifies belief states with orderings of worlds. However, this position has recently been criticised as being unacceptably strong. Other proposals, such as the popular principle (P), aka ‘Independence’, characteristic of ‘admissible’ operators, remain commendably more modest. In this paper, we supplement the DP postulates and (P) with a number of novel conditions. While the DP postulates constrain the relation between a prior and a posterior conditional belief set, our new principles notably govern the relation between two posterior conditional belief sets obtained from a common prior by different revisions. We show that operators from the resulting family, which subsumes both lexicographic and restrained revision, can be represented as relating belief states associated with a ‘proper ordinal interval’ (POI) assignment, a structure more fine-grained than a simple ordering of worlds. We close the paper by noting that these operators satisfy iterated versions of many AGM era postulates, including Superexpansion, that are not sound for admissible operators in general

Extending the Harper Identity to Iterated Belief Change

The field of iterated belief change has focused mainly on revision, with the other main operator of AGM belief change theory, i.e. contraction, receiving relatively little attention. In this paper we extend the Harper Identity from single-step change to define iterated contraction in terms of iterated revision. Specifically, just as the Harper Identity provides a recipe for defining the belief set resulting from contracting A in terms of (i) the initial belief set and (ii) the belief set resulting from revision by ¬A, we look at ways to define the plausibility ordering over worlds resulting from contracting A in terms of (iii) the initial plausibility ordering, and (iv) the plausibility ordering resulting from revision by ¬A. After noting that the most straightforward such extension leads to a trivialisation of the space of permissible orderings, we provide a family of operators for combining plausibility orderings that avoid such a result. These operators are characterised in our domain of interest by a pair of intuitively compelling properties, which turn out to enable the derivation of a number of iterated contraction postulates from postulates for iterated revision. We finish by observing that a salient member of this family allows for the derivation of counterparts for contraction of some well known iterated revision operators, as well as for defining new iterated contraction operators

Preservation, Commutativity and Modus Ponens: Two Recent Triviality Results

In a recent pair of publications, Richard Bradley has offered two novel no-go theorems involving the principle of Preservation for conditionals, which guarantees that one’s prior conditional beliefs will exhibit a certain degree of inertia in the face of a change in one’s non-conditional beliefs. We first note that Bradley’s original discussions of these results—in which he finds motivation for rejecting Preservation, first in a principle of Commutativity, then in a doxastic analogue of the rule of modus ponens —are problematic in a significant number of respects. We then turn to a recent U-turn on his part, in which he winds up rescinding his commitment to modus ponens, on the grounds of a tension with the rule of Import-Export for conditionals. Here we offer an important positive contribution to the literature, settling the following crucial question that Bradley leaves unanswered: assuming that one gives up on full-blown modus ponens on the grounds of its incompatibility with Import-Export, what weakened version of the principle should one be settling for instead? Our discussion of the issue turns out to unearth an interesting connection between epistemic undermining and the apparent failures of modus ponens in McGee’s famous counterexamples

The Irreducibility of Iterated to Single Revision

After a number of decades of research into the dynamics of rational belief, the belief revision theory community remains split on the appropriate handling of sequences of changes in view, the issue of so-called iterated revision. It has long been suggested that the matter is at least partly settled by facts pertaining to the results of various single revisions of one’s initial state of belief. Recent work has pushed this thesis further, offering various strong principles that ultimately result in a wholesale reduction of iterated to one-shot revision. The present paper offers grounds to hold that these principles should be significantly weakened and that the reductionist thesis should ultimately be rejected. Furthermore, the considerations provided suggest a close connection between the logic of iterated belief change and the logic of evidential relevance

Acceptance, Aggregation and Scoring Rules

As the ongoing literature on the paradoxes of the Lottery and the Preface reminds us, the nature of the relation between probability and rational acceptability remains far from settled. This article provides a novel perspective on the matter by exploiting a recently noted structural parallel with the problem of judgment aggregation. After offering a number of general desiderata on the relation between finite probability models and sets of accepted sentences in a Boolean sentential language, it is noted that a number of these constraints will be satisfied if and only if acceptable sentences are true under all valuations in a distinguished non-empty set W. Drawing inspiration from distance-based aggregation procedures, various scoring rule based membership conditions for W are discussed and a possible point of contact with ranking theory is considered. The paper closes with various suggestions for further research

Chemical genetics of seed germination : modulation of a key step in abscisic acid biosynthesis

Cold conditions during imbibition can result in slow or no germination in some maize seed, leading to sub-optimal crop density and uniformity and loss of yield. A novel seed treatment is required that restores germination in seed batches that perform poorly under cold conditions. Germination of seed batches from different varieties was characterised following imbibition under cold conditions which permit no or slow germination. Hydroxamic acid inhibitors of 9-cis-epoxycarotenoid dioxygenase (NCED) stimulate germination through ABA biosynthesis inhibition in other species and had a small significant effect in increasing the proportion of normal seedlings after cold imbibition. This indicated that normal germination of maize may be inhibited by dormancy-related mechanisms during or after imbibition in cold conditions. The maize NCED (ZmNCED) family was characterised. D2 and D4 inhibit other enzymes in the carotenoid cleavage dioxygenase family and exhibit relatively weak inhibition of NCED. ZmNCEDs were cloned for in vitro enzyme inhibition studies to aid identification of NCED-specific inhibitors. An RT-qPCR assay for measuring ZmNCED expression was developed. Seed ZmNCED expression and ABA concentration was elevated under cold conditions, compared to optimal germination conditions. An assay was developed to screen for germination stimulating compounds. 965 of a diverse library of 5074 compounds were identified as potential germination stimulators. Germination stimulating activity was replicated in 171 of these compounds, with some more efficacious than D4. Germination stimulating activity of 88 compounds related to the current lead compound, D4, was assessed at concentrations of 10 ppb to 10 ppm. Compounds were identified that, at less than 10 ppm stimulated germination more than D4 at 312 ppm. The mode-of-action of these compounds will need to be determined and may yield novel targets for germination stimulation. Thus novel seed treatments for improving germination of low vigour maize seed lots under cold conditions could be based on NCED inhibition or the action of the newly identified compounds