Probabilistic Logics and Probabilistic Networks

Haenni, Rolf; Romeijn, Jan-Willem; Wheeler, Gregory; Williamson, Jon

doi:10.1007/978-94-007-0008-6

Cited by 50 publications

(36 citation statements)

References 102 publications

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“…, P n }. In contrast, probabilistic logics attach probabilities to the premises and the inference problem is to determine what (set of) probabilities should be attached to the conclusion [22,24]. In coherence based probability logic the inference problem consists in determining the tightest coherent probability bounds on the conclusion [8,18,20,42].…”

Section: Coherence Based Probability Logicmentioning

confidence: 99%

Reasoning About Uncertain Conditionals

Pfeifer

2013

Stud Logica

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Abstract.There is a long tradition in formal epistemology and in the psychology of reasoning to investigate indicative conditionals. In psychology, the propositional calculus was taken for granted to be the normative standard of reference. Experimental tasks, evaluation of the participants' responses and psychological model building, were inspired by the semantics of the material conditional. Recent empirical work on indicative conditionals focuses on uncertainty. Consequently, the normative standard of reference has changed.I argue why neither logic nor standard probability theory provide appropriate rationality norms for uncertain conditionals. I advocate coherence based probability logic as an appropriate framework for investigating uncertain conditionals. Detailed proofs of the probabilistic non-informativeness of a paradox of the material conditional illustrate the approach from a formal point of view. I survey selected data on human reasoning about uncertain conditionals which additionally support the plausibility of the approach from an empirical point of view.Keywords: coherence, conditionals, formal epistemology, paradoxes of the material conditional, probability logic, psychology of reasoning Introduction and overviewThere is a long tradition in formal epistemology and in the psychology of reasoning to investigate indicative conditionals. 1 In the psychology of reasoning, the propositional calculus was taken for granted to be the correct normative standard of reference for investigating conditionals [14]. The chosen normative standard of reference guided the construction of psychological theories like a roadmap [37]. Proof-theoretic semantics, for example, stimulated the emergence of two prominent psychological theories of reasoning: Rips' theory of mental rules [47] and Braine and O'Brien's theory of mental logic [6]. Both are rule-based. Likewise, model-theoretic semantics influenced Johnson-Laird's psychological mental models theory [27].Not only psychological theories but also the experimental paradigms for investigating human conditional reasoning were strongly influenced by the Presented by Name of Editor; Received ** **,**** 1 Indicative conditionals are of the form "If A, then C", where the antecedent A and the consequent C are propositions. The ¬A ∧ C and ¬A ∧ ¬C cases are not irrelevant. They rather make A ⊃ C true. Thus, propositional logic also provided rationality criteria for evaluating human inference: Logical validity and the interpretation of indicative conditionals as material conditionals were the most important rationality criteria for assessing human reasoning performance. The participant's interpretation of indicative conditionals was classified as "rational" if, and only if the participant's interpretation is consistent with the semantics of the material conditional. For about ten years now, empirical work on indicative conditionals focuses on uncertainty. Consequently, the normative standard of reference has changed: More and more psychological studies of reasoning adopted proba...

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Section: Coherence Based Probability Logicmentioning

confidence: 99%

Reasoning About Uncertain Conditionals

Pfeifer

2013

Stud Logica

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“…This is the object of a number of thorough investigations including Paris (1994); Howson (2009);Haenni et al (2011); 41 Suffice it to mention that much of the popularity enjoyed by non-monotonic logics during the 1980s was more or less directly linked to the idea that logic would better serve the (computational) needs of artificial intellingence than probability. Makinson (2010).…”

Section: Resultsmentioning

confidence: 99%

Towards a Bayesian Theory of Second-Order Uncertainty: Lessons from Non-Standard Logics

Hosni

2013

David Makinson on Classical Methods for Non-Classical Problems

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Abstract. Second-order uncertainty, also known as model uncertainty and Knightian uncertainty, arises when decision-makers can (partly) model the parameters of their decision problems. It is widely believed that subjective probability, and more generally Bayesian theory, are illsuited to represent a number of interesting second-order uncertainty features, especially "ignorance" and "ambiguity". This failure is sometimes taken as an argument for the rejection of the whole Bayesian approach, triggering a Bayes vs anti-Bayes debate which is in many ways analogous to what the classical vs non-classical debate used to be in logic. This paper attempts to unfold this analogy and suggests that the development of non-standard logics offers very useful lessons on the contextualisation of justified norms of rationality. By putting those lessons to work I will flesh out an epistemological framework suitable for extending the expressive power of standard Bayesian norms of rationality to secondorder uncertainty in a way which is both formally and foundationally conservative.

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“…The problem of reconciling stochastic independence with convex closure conditions for sets of probabilities is not a new one (Levi 1980, Jeffrey 1987, Kyburg and Pittarelli 1996, Schervish et al 2003, Haenni et al 2011, Cozman 2011). The problem is that moving from a single distribution to a convex set of distributions introduces a plurality of independence concepts, and this splintering of probabilistic independence has ramifications for rational choice (Levi 1980, Seidenfeld and Wasserman 1993, Kyburg and Pittarelli 1996, statistical inference (Walley 1991), and probabilistic logic (Haenni et al 2011), mainly because some inferences from independence conditions which are perfectly sound in the context of a single probability distribution are fallacious in the context of a set of distributions.…”

Section: Probabilitymentioning

confidence: 99%

Objective Bayesian Calibration and the Problem of Non-convex Evidence

Wheeler

2012

The British Journal for the Philosophy of Science

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Jon Williamson's Objective Bayesian Epistemology relies upon a calibration norm to constrain credal probability by both quantitative and qualitative evidence. One role of the calibration norm is to ensure that evidence works to constrain a convex set of probability functions. This essay brings into focus a problem for Williamson's theory when qualitative evidence specifies non-convex constraints.Jon Williamson (2010) provides a spirited defense of a version of Objective Bayesianism which relies upon a calibration norm to constrain credal probability by evidence. According to this norm, an agent's degrees of belief should be constrained by two types of evidence: quantitative evidence, which directly constrains admissible values of chance functions, and qualitative evidence, such as logical or causal constraints on chance variables, which may indirectly constrain chance functions. Once those constraints are in place, the theory maintains that the agent's degrees of belief should be maximally equivocal between the basic outcomes.I have discussed the calibration norm with sympathy (Wheeler and Williamson 2011), although in that setting our aim was to reconcile the reference class reasoning machinery of Evidential Probability (Kyburg and Teng 2001) with Williamson's Objective Bayesian Epistemology (OBE), so we naturally played down the differences between the two theories. Nevertheless, I have reservations about viewing OBE as a general theory for rational belief. So, in response to Williamson's call to go whole hog, I would like to bring into focus a problem for OBE that arises when qualitative evidence prescribes non-convex constraints on a set of chance functions.The mechanics of OBE depend on a convex set of probability functions, and one role the calibration norm plays is to enforce this convexity condition. This can be a reasonable approach when the endpoints of an interval designate upper and lower bounds on admissible degrees of belief and the question to answer is what point within this range represents the most cautious position for an agent to take. That is the question that OBE is set up to answer, and it does so by advising the agent to pick the most equivocal point within the convex hull of admissible options

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Probabilistic Logics and Probabilistic Networks

Cited by 50 publications

References 102 publications

Reasoning About Uncertain Conditionals

Reasoning About Uncertain Conditionals

Towards a Bayesian Theory of Second-Order Uncertainty: Lessons from Non-Standard Logics

Objective Bayesian Calibration and the Problem of Non-convex Evidence

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