relative to the actual world) of a propositional theory are defined. A theory is ‘closer to the truth’ the logically stronger its positive content and the logically weaker its negative content. This proposal delivers the same verisimilar preordering of theories that has been defined by Brink and Heidema as a ‘power ordering’. The preordering may be collapsed to a partial ordering and then embedded into a complete distributive lattice. The preordering may also be refined to a partial ordering by employing (...) the ‘convex content’ and the ‘non-convex content’ of each theory. Philosophical implications and historical relations are discussed. (shrink)

Although AGM theory contraction (Alchourrón et al., 1985; Alchourrón and Makinson, 1985) occupies a central position in the literature on belief change, there is one aspect about it that has created a fair amount of controversy. It involves the inclusion of the postulate known as Recovery. As a result, a number of alternatives to AGM theory contraction have been proposed that do not always satisfy the Recovery postulate (Levi, 1991, 1998; Hansson and Olsson, 1995; Fermé, 1998; Fermé and Rodriguez, 1998; (...) Rott and Pagnucco, 1999). In this paper we present a new addition, systematic withdrawal, to the family of withdrawal operations, as they have become known. We define systematic withdrawal semantically, in terms of a set of preorders, and show that it can be characterised by a set of postulates. In a comparison of withdrawal operations we show that AGM contraction, systematic withdrawal and the severe withdrawal of Rott and Pagnucco (1999) are intimately connected by virtue of their definition in terms of sets of preorders. In a future paper it will be shown that this connection can be extended to include the epistemic entrenchment orderings of Gärdenfors (1988) and Gärdenfors and Makinson (1988) and the refined entrenchment orderings of Meyer et al. (2000). (shrink)

Epistemic entrenchment, as presented by Gärdenfors and Makinson (1988) and Gärdenfors (1988), is a formalisation of the intuition that, when forced to choose between two beliefs, an agent will giveup the less entrenched one. While their formalisation satisfactorilycaptures the intuitive notion of the entrenchment of beliefs in a number ofaspects, the requirement that all wffs be comparable has drawn criticismfrom various quarters. We define a set of refined versions of theirentrenchment orderings that are not subject to the same criticism, andinvestigate (...) the relationship between the refined entrenched orderings,the entrenchment orderings of Gärdenfors and Makinson, and AGM theorycontraction (Alchourrón et al., 1985). To conclude, we compare refinedentrenchment with two related approaches to epistemic entrenchment. (shrink)

In this article, we redefine classical notions of theory reduction in such a way that model-theoretic preferential semantics becomes part of a realist depiction of this aspect of science. We offer a model-theoretic reconstruction of science in which theory succession or reduction is often better - or at a finer level of analysis - interpreted as the result of model succession or reduction. This analysis leads to 'defeasible reduction', defined as follows: The conjunction of the assumptions of a reducing theory (...) T with the definitions translating the vocabulary of a reduced theory T' to the vocabulary of T, defeasibly entails the assumptions of reduced T'. This relation of defeasible reduction offers, in the context of additional knowledge becoming available, articulation of a more flexible kind of reduction in theory development than in the classical case. Also, defeasible reduction is shown to solve the problems of entailment that classical homogeneous reduction encounters. Reduction in the defeasible sense is a practical device for studying the processes of science, since it is about highlighting different aspects of the same theory at different times of application, rather than about naive dreams concerning a metaphysical unity of science. (shrink)

The intuitive notion of a binary relation on information-bearers, comparingthem with respect to their closeness to the available information, is oftenconstrued in terms of comparing their symmetric difference with, orcompositional similarity to, the available information. This happens forinstance in some treatments of verisimilitude. We expound an abstractmathematical rendering of the relevant data-dependent relation in theframework of Boolean algebras. For every element t of a Boolean algebra B we construct the t-modulated Boolean algebra Btin which the order relation represents `is at (...) most as compatible with t as' or `is at best as similar to t as'. In the case of Lindenbaum-Tarskialgebras, t expresses the available information, and the compatibilityrelation turns out to be an entwinement of inferential and conjecturalrelations. It is just classical entailment when no information is available(i.e., when t is logically true) and becomes more boldly abductive themore information is available. The rich algebraic structures of a Boolean algebra –- including its Boolean group structures –- play a significant role in this combination of deduction and abduction and also induce cautious anddaring variants of the compatibility relation. Links with the literature onverisimilitude, abduction, and related topics are indicated. (shrink)

In logic, including the designer logics of artificial intelligence, and in the philosophy of science, one is often concerned with qualitative, comparative orderings on the states of a system, or on theories expressing information about the system. States may be compared with respect to normality, or some preference criterium, or similarity to some given (set of) state(s). Theories may be compared with respect to logical power, or to truthlikeness, or to how well they capture certain information. We explain a number (...) of these relations, study their properties, and unravel some of their interrelationships. (shrink)

There are various contexts in which it is not pertinent to generate and attend to all the classical consequences of a given premiss—or to trace all the premisses which classically entail a given consequence. Such contexts may involve limited resources of an agent or inferential engine, contextual relevance or irrelevance of certain consequences or premisses, modelling everyday human reasoning, the search for plausible abduced hypotheses or potential causes, etc. In this paper we propose and explicate one formal framework for a (...) whole spectrum of consequence relations, flexible enough to be tailored for choices from a variety of contexts. We do so by investigating semantic constraints on classical entailment which give rise to a family of infra-classical logics with appealing properties. More specifically, our infra-classical reasoning demands (beyond ${\alpha\models\beta}$ ) that Mod(β) does not run wild, but lies within the scope (whatever that may mean in some specific context) of Mod(α), and which can be described by a sentence ${\bullet\alpha}$ with ${\beta\models\bullet\alpha}$ . Besides being infra-classical, the resulting logic is also non-monotonic and allows for non-trivial reasoning in the presence of inconsistencies. (shrink)

We introduce and explore the notion of duality for entailment relations induced by preference orderings on states. We discuss the relationship between these preferential entailment relations from the perspectives of Boolean algebra, inference rules, and modal axiomatisation. Interpreting the preference relations as accessibility relations establishes modular Gödel-Löb logic as a suitable modal framework for rational preferential reasoning.

Unrestricted use of the axiom schema of comprehension, ?to every mathematically (or set-theoretically) describable property there corresponds the set of all mathematical (or set-theoretical) objects having that property?, leads to contradiction. In set theories of the Zermelo?Fraenkel?Skolem (ZFS) style suitable instances of the comprehension schema are chosen ad hoc as axioms, e.g.axioms which guarantee the existence of unions, intersections, pairs, subsets, empty set, power sets and replacement sets. It is demonstrated that a uniform syntactic description may be given of acceptable (...) instances of the comprehension schema, which include all of the axioms mentioned, and which in their turn are theorems of the usual versions of ZFS set theory. Well then, shall we proceed as usual and begin by assuming the existence of a single essential nature or Form for every set of things which we call by the same name? Do you understand? (Plato, Republic X.596a6; cf. Cornford 1966, 317). (shrink)

We explore the psychological foundations of Logic and Artificial Intelligence, touching on representation, categorisation, heuristics, consciousness, and emotion. Specifically, we challenge Dennett's view of the brain as a syntactic engine that is limited to processing symbols according to their structural properties. We show that cognitive psychology and neurobiology support a dual-process model in which one form of cognition is essentially semantical and differs in important ways from the operation of a syntactic engine. The dual-process model illuminates two important events in (...) Logic and Artificial Intelligence, namely the emergence of non-monotonicity and of embodiment, events that changed the traditional paradigms of ‘Logic = the study of deductive inference' and ‘Symbolic AI'. S. Afr. J. Philos. Vol.24(2) 2005: 137-151. (shrink)

Generalisations of theory change involving operations on arbitrary sets ofwffs instead of on belief sets (i.e., sets closed under a consequencerelation), have become known as base change. In one view, a base should bethought of as providing more structure to its generated belief set, whichmeans that it can be employed to determine the theory contraction operationassociated with a base contraction operation. In this paper we follow suchan approach as the first step in defining infobase change. We think of an infobase (...) as a finite set of wffs consisting of independently obtainedbits of information. Taking AGM theory change (Alchourrón et al. 1985) as the general framework, we present a method that uses the structure of aninfobase B to obtain an AGM theory contraction operation for contractingthe belief set Cn(B). Both the infobase and the obtained theory contraction operation then play a role in constructing a unique infobasecontraction operation. Infobase revision is defined in terms of an analogueof the Levi Identity, and it is shown that the associated theory revisionoperation satisfies the AGM postulates for revision. Because every infobaseis associated with a unique infobase contraction and revision operation, the method also allows for iterated base change. (shrink)