Horacio Arló Costa

Conditional Probability and Defeasible Inference

Abstract We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey–Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of non-monotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).‘Expectation’ is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability.

Non-Adjunctive Inference and Classical Modalities

The article focuses on representing different forms of non-adjunctive inference as sub-Kripkean systems of classical modal logic, where the inference from □A and □B to □A∧B fails. In particular we prove a completeness result showing that the modal system that Schotch and Jennings derive from a form of non-adjunctive inference in (Schotch and Jennings, 1980) is a classical system strictly stronger than EMN and weaker than K (following the notation for classical modalities presented in Chellas, 1980). The unified semantical characterization in terms of neighborhoods permits comparisons between different forms of non-adjunctive inference. For example, we show that the non-adjunctive logic proposed in (Schotch and Jennings, 1980) is not adequate in general for representing the logic of high probability operators. An alternative interpretation of the forcing relation of Schotch and Jennings is derived from the proposed unified semantics and utilized in order to propose a more fine-grained measure of epistemic coherence than the one presented in (Schotch and Jennings, 1980). Finally we propose a syntactic translation of the purely implicative part of Jaśkowskis system D2 into a classical system preserving all the theorems (and non-theorems) explicilty mentioned in (Jaśkowski, 1969). The translation method can be used in order to develop epistemic semantics for a larger class of non-adjunctive (discursive) logics than the ones historically investigated by Jaśkowski.

Bounded Rationality: Models for Some Fast and Frugal Heuristics

Herb Simon pioneered the study of bounded models of rationality. Simon famously argued that decision makers typically satisfice rather than optimize. According to Simon, a decision maker normally chooses an alternative that meets or exceeds specified criteria, even when this alternative is not guaranteed to be unique or in any sense optimal. For example, Simon argued that an organism – instead of scanning all the possible alternatives, computing each probability of every outcome of each alternative, calculating the utility of each alternative, and thereupon selecting the optimal option with respect to expected utility – typically chooses the first option that satisfies its “aspiration level.”

Value-based contraction: a representation result

Sven-Ove Hansson and Erik Olsson studied in [6] the logical properties of an operation of contraction first proposed by Isaac Levi in [11]. They provided a completeness result for the simplest version of contraction that they call Levi-contraction but left open the problem of characterizing axiomatically the more complex operation of value-based contraction or saturatable contraction. In this paper we propose an axiomatization for this operation and prove a completeness result for it. We argue that the resulting operation is better behaved than various rival operations of contraction defined in recent years.

Trade-Offs between Inductive Power and Logical Omniscience in Modeling Context

In a series of recent papers the economist Michael Bacharach [2] has pointed out that most approaches in knowledge representation suffer from two complementary defects. Typically they assume that agents know all the logical consequences of information encoded in a finite knowledge base (Cleverness) and only these consequences (Cloistered-ness). In this article we develop first-order and inductive extensions of Montague-Scott’s semantics in order to tackle both problems at once. Three desiderata are put forward for representing epistemic context: (1) expressive adequacy, (2) inductive power, (3) boundedness. An important part of our effort focuses on reconsidering the role of standard formalisms (both logical and probabilistic) in representing the information of bounded agents.In a series of recent papers the economist Michael Bacharach [2] has pointed out that most approaches in knowledge representation suffer from two complementary defects. Typically they assume that agents know all the logical consequences of information encoded in a finite knowledge base (Cleverness) and only these consequences (Cloisteredness). In this article we develop first-order and inductive extensions of Montague-Scotts semantics in order to tackle both problems at once. Three desiderata are put forward for representing epistemic context: (1) expressive adequacy, (2) inductive power, (3) boundedness. An important part of our effort focuses on reconsidering the role of standard formalisms (both logical and probabilistic) in representing the information of bounded agents.

A Representation Result for Value-based Contraction

Sven-Ove Hansson and Erik Olsson studied in Hansson and Olsson (Notre Dame Journal of Formal Logic, 36(1), 103–119 1995) the logical properties of an operation of contraction first proposed by Isaac Levi in Levi (1991). They provided a completeness result for the simplest version of contraction that they call Levi-contraction but left open the problem of characterizing axiomatically the more complex operation of value-based contraction or saturatable contraction. In this paper we propose an axiomatization for this operation and prove a completeness result for it. We argue that the resulting operation is better behaved than various rival operations of contraction defined in recent years.

More foundations of the decision sciences: introduction

This second part of the special issue on the foundations of the decision sciences continues with the treatment of some topics that were discussed in the first part and includes papers on four additional areas of research. The first part of this special issue—which was published as Volume 172, Number 1 of Synthese in 2010—features various papers that focus on models of choice under uncertainty. In this second part we are pleased to present two additional papers that elaborate on this topic. The first of these is by Itzhak Gilboa, Andrew Postlewaite, and David Schmeidler. Their paper is a criticism of the classical Bayesian approach that the authors think is neither necessary nor sufficient for the rationality of belief. Therefore this paper has a rather general scope focusing on the normative limitations of Savage’s approach. The second of these two papers on choice under uncertainty is by Alan Hajek and Michael Smithson. Their paper presents a novel argument for models that allow for doxastic indeterminacy via the use of sets of probabilities. There is some overlap between these two papers because Gilboa et al. think that one of the reasons for the alleged inadequacy of the Bayesian approach is the fact that in many cases there is not sufficient information for a rational agent to form a Bayesian prior. Therefore they also recommend the use of sets of probabilities and discuss in certain detail (in an Appendix) Ellsberg’s paradox.

A theory of contextual propositions for indicatives

A dispenser for a roll of coated abrasive discs having a front face including a cutter bar, a guide means at one side of said cutter bar for positioning a disc edge at the cutter bar, and a recess on the other side of said cutter bar to permit manual access to said roll.A theory of contextual propositions for indicative conditionals is presented. The main challenge is to give a precise account of how the dynamics of possible worlds depends on epistemic context. Robert Stalnaker suggested in [Stalnaker 84] that even when selection functions for evaluating indicatives cannot be defined in terms of epistemic context, they can be importantly constrained by a principle of context dependency that we adopt here. In addition, we show how to define a gradation of possibilities for each point in an epistemic context by taking into account a proposal first introduced byWolfgang Spohn in [Spohn 87] and later refined by Darwiche and Pearl in [Darwiche & Pearl 97]. The resulting theory of contextual propositions (unlike some alternative views) is shown to be compatible with basic qualitative consequences of the Bayesian principle of conditionalization (which is frequently used in probabilistic semantics for indicative conditionals).

Epistemic Context, Defeasible Inference and Conversational Implicature

Recent foundational work on the nature of defeasible inference has appealed to an epistemic context principle (ECP): βfollows defeasibly from α (α β )if and only if follows classically from C(α), where C(α) is the given epistemic context for α. Since nothing requires that C(α) ⊆ C(α ^ γ), the induced notion of consequence is nonmonotonic. We will focus on a particular manner of articulating ECP where C(α) is an autoepistemic (AE) extension of {α}.