Conjunction, Disjunction and Iterated Conditioning of Conditional Events

Gilio, Angelo; Sanfilippo, Giuseppe

doi:10.1007/978-3-642-33042-1_43

Cited by 35 publications

(53 citation statements)

References 7 publications

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“…Then, as already observed in [45], when HK " H the conditional events A|H and B|K should be looked at as uncorrelated random quantities. However, when HK " H, (the two random quantities) A|H and B|K are (in general) not independent.…”

Section: Independence and Uncorrelationmentioning

confidence: 88%

Section: Conjoined and Iterated Conditionalsmentioning

confidence: 99%

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Probabilities of conditionals and previsions of iterated conditionals

Sanfilippo

Gilio

Over

et al. 2020

International Journal of Approximate Reasoning

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We analyze selected iterated conditionals in the framework of conditional random quantities. We point out that it is instructive to examine Lewis's triviality result, which shows the conditions a conditional must satisfy for its probability to be the conditional probability. In our approach, however, we avoid triviality because the import-export principle is invalid. We then analyze an example of reasoning under partial knowledge where, given a conditional if A then C as information, the probability of A should intuitively increase. In our approach, we explain this intuition by making some implicit background information explicit. We consider several (generalized) iterated conditionals, which allow us to formalize different kinds of latent information. We verify that for these iterated conditionals the prevision is greater than or equal to the probability of A. We also investigate the lower and upper bounds of the Affirmation of the Consequent inference. We conclude our study with some remarks on the supposed "independence" of two conditionals, and we interpret this property as uncorrelation between two random quantities.

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Section: Independence and Uncorrelationmentioning

confidence: 88%

Section: Conjoined and Iterated Conditionalsmentioning

confidence: 99%

Probabilities of conditionals and previsions of iterated conditionals

Sanfilippo

Gilio

Over

et al. 2020

International Journal of Approximate Reasoning

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“…The expected value of the indicators of an event is called its prevision (Augustin et al, 2014;Coletti & Scozzafava, 2002;de Finetti, 1974;Walley, 1991). The indicators are uncertain quantities (de Finetti, 1974;Gilio & Sanfilippo, 2013;Stalnaker & Jeffrey, 1994). In the literature, propositions and indicators are 1 Sometimes the subjective interpretation of probability is called "Bayesian", often however, for no other reason than to avoid the term "subjective".…”

Section: Introductionmentioning

confidence: 99%

“…Conditionals may be nested or iterated if the indicator values are combined with probabilities; degrees of belief are then measured by previsions in conditional random quantities. (Biazzo, Gilio, & Sanfilippo, 2009;Gilio & Sanfilippo, 2013;Gilio, Pfeifer, & Sanfilippo, (2016;Stalnaker & Jeffrey, 1994;Sanfilippo, Pfeifer, Over, & Gilio, 2018;Van Wijnbergen-Huitink, Elqayam, & Over, 2015).…”

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confidence: 99%

A process model of the understanding of uncertain conditionals

Kleiter

Fugard

Pfeifer

2018

Thinking & Reasoning

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To build a process model of the understanding of conditionals we extract a common core of three semantics of if-then sentences: (a) the conditional event interpretation in the coherencebased probability logic, (b) the discourse processingtheory of Hans Kamp, and (c) the game-theoretical approach of Jaakko Hintikka. The empirical part reports three experiments in which each participant assessed the probability of 52 if-then sentencesin a truth table task. Each experiment included a second task: An n-back task relating the interpretation of conditionals to working memory, a Bayesian bookbag and poker chip task relating the interpretation of conditionals to probability updating, and a probabilistic modus ponens task relating the interpretation of conditionals to a classical inference task. Data analysis shows that the way in which the conditionals are interpreted correlates with each of the supplementary tasks. The results are discussed within the process model proposed in the introduction.

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“…We use a notion of conjunction which, differently from other authors, is defined as a suitable conditional random quantity with values in the unit interval (see, e.g. [20,21,23,24,36]). We study the relationship between our notion of conjunction and the notion of Frank tnorm.…”

Section: Introductionmentioning

confidence: 99%

Conjunction of Conditional Events and t-Norms

Gilio

Sanfilippo

2019

Lecture Notes in Computer Science

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We study the relationship between a notion of conjunction among conditional events, introduced in recent papers, and the notion of Frank t-norm. By examining different cases, in the setting of coherence, we show each time that the conjunction coincides with a suitable Frank t-norm. In particular, the conjunction may coincide with the Product t-norm, the Minimum t-norm, and Lukasiewicz t-norm. We show by a counterexample, that the prevision assessments obtained by Lukasiewicz t-norm may be not coherent. Then, we give some conditions of coherence when using Lukasiewicz t-norm.

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Conjunction, Disjunction and Iterated Conditioning of Conditional Events

Cited by 35 publications

References 7 publications

Probabilities of conditionals and previsions of iterated conditionals

Probabilities of conditionals and previsions of iterated conditionals

A process model of the understanding of uncertain conditionals

Conjunction of Conditional Events and t-Norms

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