Horacio Arlo-Costa

(1956 - 2011)

Following the pioneer work of Bruno De Finetti [12], conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable as Popper functions, and that every Popper function is representable in terms of the standard real values of some infinitesimal measure. Our main goal in this article is to study the constraints on (qualitative and probabilistic) change imposed by an extended version of McGee's result. We focus on an extension capable of allowing for iterated changes of view. Such extension, we argue, seems to be needed in almost all considered applications. Since most of the available axiomatizations stipulate (definitionally) important constraints on iterated change, we propose a non-questionbegging framework, Iterative Probability Systems (IPS) and we show that every Popper function can be regarded as a Bayesian IPS. A generalized version of McGee's result is then proved and several of its consequences considered. In particular we note that our proof requires the imposition of Cumulativity, i.e. the principle that a proposition that is accepted at any stage of an iterative process of acceptance will continue to be accepted at any later stage. The plausibility and range of applicability of Cumulativity is then studied. In particular we appeal to a method for defining belief from conditional probability (first proposed in [42] and then slightly modified in [6] and [3]) in order to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic supposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30].

It is now well known that, on pain of triviality, the probability of a conditional cannot be identified with the corresponding conditional probability [25]. This surprising impossibility result has a qualitative counterpart. In fact, Peter Gärdenfors showed in [13] that believing ‘If A then B’ cannot be equated with the act of believing B on the supposition that A — as long as supposing obeys minimal Bayesian constraints. Recent work has shown that in spite of these negative results, the question ‘how to accept a conditional?’ has a clear answer. Even if conditionals are not truth-carriers, they do have precise acceptability conditions. Nevertheless most epistemic models of conditionals do not provide acceptance conditions for iterated conditionals. One of the main goals of this essay is to provide a comprehensive account of the notion of epistemic conditionality covering all forms of iteration. First we propose an account of the basic idea of epistemic conditionality, by studying the conditionals validated by epistemic models where iteration is permitted but not constrained by special axioms. Our modeling does not presuppose that epistemic states should be represented by belief sets (we only assume that to each epistemic state corresponds an associated belief state). A full encoding of the basic epistemic conditionals (encompassing all forms of iteration) is presented and a representation result is proved. In the second part of the essay we argue that the notion of change involved in the evaluation of conditionals is suppositional, and that such notion should be distinguished from the notion of updating (modelled by AGM and other methods). We conclude by considering how some of the recent modellings of iterated change fare as methods for iterated supposing.

Carlos Alchourrón, Peter Gärdenfors and David Makinson published in 1985 a seminal article on belief change in the Journal of Symbolic Logic. Researchers from various disciplines, from computer science to mathematical economics to philosophical logic, have continued the work first presented in this seminal paper during the last two decades. This paper explores some salient foundational trends that interpret the act of changing view as a decision. We will argue that some of these foundational trends are already present, although only tacitly, in the original article by the AGM trio. Other accounts decidedly depart from the view of contraction and revision presented in this seminal paper. I shall survey various types of theories that progressively depart form the axiomatic treatment defended by AGM. First, I consider theories where rational agents are considered as maximizers as opposed to optimizers ). Second, I consider which feasible set to use in contraction understood as a cognitive decision. This leads to rethink the very notion of what minimal change in contraction is. I shall conclude with some philosophical reflections concerning the sort of epistemological voluntarism that is tacit in seeing change in view as a rational choice. Carlos Alchourrón, Peter Gärdenfors y David Makinson publicaron en 1985 un artículo seminal sobre cambio de creencias en Journal of Symbolic Logic. Investigadores de varias disciplinas, desde la ciencia de la computación hasta la economía matemática y la lógica filosófica, han continuado en las dos últimas décadas esta línea de investigación. Este trabajo explora algunos aspectos fundacionalmente salientes que interpretan el acto de cambio de vista como una decisión. Argumentaremos que algunos de esos aspectos fundacionales ya estaban presentes, aunque solo tácitamente, en el artículo original del trío AGM. Otros abordajes parten decididamente de la contracción y revisión tal como fueran presentadas en el trabajo seminal. Inspeccionaré varios tipos de teorías que progresivamente parten del tratamiento axiomático defendido por AGM. Primero, considero teorías donde los agentes racionales aparecen como maximizadres opuestos a los optimiadores ). Segundo, me pregunto cuál conjunto derrotable debe usarse en una contracción entendida como una decisión cognitiva, lo cual lleva a repensar la importante cuestión de en qué consiste la noción de cambio mínimo en la contracción. Concluiré con algunas reflexiones filosóficas acerca de la suerte de voluntarismo epistemológico que está tácito en la concepción del cambio como una opción racional.