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  1. added 2018-12-25
    The Brier Rule Is Not a Good Measure of Epistemic Utility.Don Fallis & Peter J. Lewis - 2016 - Australasian Journal of Philosophy 94 (3):576-590.
    Measures of epistemic utility are used by formal epistemologists to make determinations of epistemic betterness among cognitive states. The Brier rule is the most popular choice among formal epistemologists for such a measure. In this paper, however, we show that the Brier rule is sometimes seriously wrong about whether one cognitive state is epistemically better than another. In particular, there are cases where an agent gets evidence that definitively eliminates a false hypothesis, but where the Brier rule says that things (...)
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  2. added 2018-12-18
    A Dutch Book Theorem and Converse Dutch Book Theorem for Kolmogorov Conditionalization.Michael Rescorla - 2018 - Review of Symbolic Logic 11 (4):705-735.
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  3. added 2018-11-12
    On Difference-Splitting and the Equal Weight View.Randall G. McCutcheon - manuscript
    Dawid, DeGroot and Mortera showed, a quarter century ago, that any agent who regards a fellow agent as a peer--in particular, defers to the fellow agent's prior credences in the same way that she defers to her own--and updates by split-the-difference is prone to diachronic incoherence. On the other hand one may show that there are special scenarios in which Bayesian updating approximates difference splitting, so it remains an important question whether it remains a viable response to ``generic" peer update. (...)
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  4. added 2018-10-11
    Abominable KK Failures.Kevin Dorst - forthcoming - Mind:fzy067.
    KK is the thesis that if you can know p, you can know that you can know p. Though it’s unpopular, a flurry of considerations have recently emerged in its favor. Here we add fuel to the fire: standard resources allow us to show that any failure of KK will lead to the knowability and assertability of abominable indicative conditionals of the form, ‘If I don’t know it, p.’ Such conditionals are manifestly not assertable—a fact that KK defenders can easily (...)
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  5. added 2018-10-04
    Higher-Order Uncertainty.Kevin Dorst - forthcoming - In Mattias Skipper & Asbjørn Steglich Petersen (eds.), Higher-Order Evidence: New Essays.
    You have higher-order uncertainty iff you are uncertain of what opinions you should have. I defend three claims about it. First, the higher-order evidence debate can be helpfully reframed in terms of higher-order uncertainty. The central question becomes how your first- and higher-order opinions should relate—a precise question that can be embedded within a general, tractable framework. Second, this question is nontrivial. Rational higher-order uncertainty is pervasive, and lies at the foundations of the epistemology of disagreement. Third, the answer is (...)
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  6. added 2018-09-10
    Higher-Order Defeat and Doxastic Resilience.Asbjørn Steglich-Petersen - forthcoming - In Mattias Skipper & Asbjørn Steglich-Petersen (eds.), Higher-Order Evidence: New Essays. Oxford University Press.
    It seems obvious that when higher-order evidence makes it rational for one to doubt that one’s own belief on some matter is rational, this can undermine the rationality of that belief. This is known as higher-order defeat. However, despite its intuitive plausibility, it has proved puzzling how higher-order defeat works, exactly. To highlight two prominent sources of puzzlement, higher-order defeat seems to defy being understood in terms of conditionalization; and higher-order defeat can sometimes place agents in what seem like epistemic (...)
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  7. added 2018-07-30
    Accuracy, Conditionalization, and Probabilism.Peter J. Lewis & Don Fallis - manuscript
    Accuracy-based arguments for conditionalization and probabilism appear to have a significant advantage over their Dutch Book rivals. They rely only on the plausible epistemic norm that one should try to decrease the inaccuracy of one's beliefs. Furthermore, it seems that conditionalization and probabilism follow from a wide range of measures of inaccuracy. However, we argue that among the measures in the literature, there are some from which one can prove conditionalization, others from which one can prove probabilism, and none from (...)
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  8. added 2018-07-30
    Three Aspects of Typicality in Multiverse Cosmology.Feraz Azhar - unknown
    Extracting predictions from cosmological theories that describe a multiverse, for what we are likely to observe in our domain, is crucial to establishing the validity of these theories. One way to extract such predictions is from theory-generated probability distributions that allow for selection effects---generally expressed in terms of assumptions about anthropic conditionalization and how typical we are. In this paper, I urge three lessons about typicality in multiverse settings. Because it is difficult to characterize our observational situation in the multiverse, (...)
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  9. added 2018-06-25
    Credence for Conclusions: A Brief for Jeffrey’s Rule.John R. Welch - forthcoming - Synthese:1-22.
    Some arguments are good; others are not. How can we tell the difference? This article advances three proposals as a partial answer to this question. The proposals are keyed to arguments conditioned by different degrees of uncertainty: mild, where the argument’s premises are hedged with point-valued probabilities; moderate, where the premises are hedged with interval probabilities; and severe, where the premises are hedged with non-numeric plausibilities such as ‘very likely’ or ‘unconfirmed’. For mild uncertainty, the article proposes to apply a (...)
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  10. added 2018-06-25
    Imaging All the People.Hannes Leitgeb - 2017 - Episteme 14 (4):463-479.
    It is well known that aggregating the degree-of-belief functions of different subjects by linear pooling or averaging is subject to a commutativity dilemma: other than in trivial cases, conditionalizing the individual degree-of-belief functions on a piece of evidence E followed by linearly aggregating them does not yield the same result as rst aggregating them linearly and then conditionalizing the resulting social degree- of-belief function on E. In the present paper we suggest a novel way out of this dilemma: adapting the (...)
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  11. added 2018-05-30
    Central Limit Theorem for Functional of Jump Markov Processes.Nguyen Van Huu, Quan-Hoang Vuong & Minh-Ngoc Tran - 2005 - Vietnam Journal of Mathematics 33 (4):443-461.
    Some conditions are given to ensure that for a jump homogeneous Markov process $\{X(t),t\ge 0\}$ the law of the integral functional of the process $T^{-1/2} \int^T_0\varphi(X(t))dt$ converges to the normal law $N(0,\sigma^2)$ as $T\to \infty$, where $\varphi$ is a mapping from the state space $E$ into $\bbfR$.
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  12. added 2018-05-23
    A Puzzle About Experts, Evidential Screening-Off and Conditionalization.Ittay Nissan-Rozen - forthcoming - Episteme:1-9.
    I present a puzzle about the epistemic role beliefs about experts' beliefs play in a rational agent's system of beliefs. It is shown that accepting the claim that an expert's degree of belief in a proposition, A, screens off the evidential support another proposition, B, gives to A in case the expert knows and is certain about whether B is true, leads in some cases to highly unintuitive conclusions. I suggest a solution to the puzzle according to which evidential screening (...)
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  13. added 2018-05-07
    Bayesian Argumentation and the Value of Logical Validity.Benjamin Eva & Stephan Hartmann - 2018 - Psychological Review 125 (5):806-821.
    According to the Bayesian paradigm in the psychology of reasoning, the norms by which everyday human cognition is best evaluated are probabilistic rather than logical in character. Recently, the Bayesian paradigm has been applied to the domain of argumentation, where the fundamental norms are traditionally assumed to be logical. Here, we present a major generalisation of extant Bayesian approaches to argumentation that utilizes a new class of Bayesian learning methods that are better suited to modelling dynamic and conditional inferences than (...)
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  14. added 2018-05-07
    Curve-Fitting for Bayesians?Gordon Belot - 2016 - British Journal for the Philosophy of Science:axv061.
    Bayesians often assume, suppose, or conjecture that for any reasonable explication of the notion of simplicity a prior can be designed that will enforce a preference for hypotheses simpler in just that sense. Further, it is often claimed that the Bayesian framework automatically implements Occam's razor—that conditionalizing on data consistent with both a simple theory and a complex theory more or less inevitably favours the simpler theory. But it is shown here that there are simplicity-driven approaches to curve-fitting problems that (...)
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  15. added 2018-02-12
    Ideal Counterpart Theorizing and the Accuracy Argument for Probabilism.Clinton Castro & Olav Vassend - 2018 - Analysis 78 (2):207-216.
    One of the main goals of Bayesian epistemology is to justify the rational norms credence functions ought to obey. Accuracy arguments attempt to justify these norms from the assumption that the source of value for credences relevant to their epistemic status is their accuracy. This assumption and some standard decision-theoretic principles are used to argue for norms like Probabilism, the thesis that an agent’s credence function is rational only if it obeys the probability axioms. We introduce an example that shows (...)
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  16. added 2018-01-30
    Foley’s Threshold View of Belief and the Safety Condition on Knowledge.Michael J. Shaffer - 2018 - Metaphilosophy 49 (4):589-594.
    This paper introduces a new argument against Richard Foley’s threshold view of belief. His view is based on the Lockean Thesis (LT) and the Rational Threshold Thesis (RTT). The argument introduced here shows that the views derived from the LT and the RTT violate the safety condition on knowledge in way that threatens the LT and/or the RTT.
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  17. added 2018-01-08
    Deference and Uniqueness.Christopher Meacham - forthcoming - Philosophical Studies:1-24.
    Deference principles are principles that describe when, and to what extent, it’s rational to defer to others. Recently, some authors have used such principles to argue for Evidential Uniqueness, the claim that for every batch of evidence, there’s a unique doxastic state that it’s permissible for subjects with that total evidence to have. This paper has two aims. The first aim is to assess these deference-based arguments for Evidential Uniqueness. I’ll show that these arguments only work given a particular kind (...)
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  18. added 2018-01-08
    On Quantum Conditional Probability.Bobo Isabel Guerra - 2013 - Theoria : An International Journal for Theory, History and Fundations of Science 28 (1):115-137.
    We argue that quantum theory does not allow for a generalization of the notion of classical conditional probability by showing that the probability defined by the Lüders rule, standardly interpreted in the literature as the quantum-mechanical conditionalization rule, cannot be interpreted as such.
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  19. added 2017-12-06
    Coherence and Common Causes: Against Relevance-Sensitive Measures of Coherence.Jakob Koscholke & Michael Schippers - forthcoming - British Journal for the Philosophy of Science:axx003.
    Changing weather conditions and barometer changes usually coincide. Accordingly, the propositions that my barometer falls and that the weather conditions deteriorate are quite coherent—especially under the assumption that there is a drop in atmospheric pressure. Nevertheless, scenarios like these involving common causes turn out to be highly problematic for a prominent class of probabilistic coherence measures, namely, those explicating coherence based on the idea of relevance-sensitivity. In this article, we show that none of these measures accords with the intuition that (...)
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  20. added 2017-12-06
    Inductive Explanation and Garber–Style Solutions to the Problem of Old Evidence.David Kinney - 2017 - Synthese:1-15.
    The Problem of Old Evidence is a perennial issue for Bayesian confirmation theory. Garber famously argues that the problem can be solved by conditionalizing on the proposition that a hypothesis deductively implies the existence of the old evidence. In recent work, Hartmann and Fitelson :712–717, 2015) and Sprenger :383–401, 2015) aim for similar, but more general, solutions to the Problem of Old Evidence. These solutions are more general because they allow the explanatory relationship between a new hypothesis and old evidence (...)
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  21. added 2017-11-30
    Generalised Reichenbachian Common Cause Systems.Claudio Mazzola - forthcoming - Synthese.
    The principle of the common cause claims that if an improbable coincidence has occurred, there must exist a common cause. This is generally taken to mean that positive correlations between non-causally related events should disappear when conditioning on the action of some underlying common cause. The extended interpretation of the principle, by contrast, urges that common causes should be called for in order to explain positive deviations between the estimated correlation of two events and the expected value of their correlation. (...)
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  22. added 2017-11-28
    What is Justified Credence?Richard Pettigrew - manuscript
    In this paper, we seek a reliabilist account of justified credence. Reliabilism about justified beliefs comes in two varieties: process reliabilism (Goldman, 1979, 2008) and indicator reliabilism (Alston, 1988, 2005). Existing accounts of reliabilism about justified credence comes in the same two varieties: Jeff Dunn’s is a version of process reliabilism (Dunn, 2015) while Weng Hong Tang offers a version of indicator reliabilism (Tang, 2016). As we will see, both face the same objection. If they are right about what justification (...)
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  23. added 2017-10-16
    Basic‐Know And Super‐Know.Anna Mahtani - forthcoming - Philosophy and Phenomenological Research.
    Sometimes a proposition is ‘opaque’ to an agent: he doesn't know it, but he does know something about how coming to know it should affect his or her credence function. It is tempting to assume that a rational agent's credence function coheres in a certain way with his or her knowledge of these opaque propositions, and I call this the ‘Opaque Proposition Principle’. The principle is compelling but demonstrably false. I explain this incongruity by showing that the principle is ambiguous: (...)
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  24. added 2017-10-16
    Updating Probability: Tracking Statistics as Criterion.Bas C. van Fraassen & Joseph Y. Halpern - 2016 - British Journal for the Philosophy of Science:axv027.
    ABSTRACT For changing opinion, represented by an assignment of probabilities to propositions, the criterion proposed is motivated by the requirement that the assignment should have, and maintain, the possibility of matching in some appropriate sense statistical proportions in a population. This ‘tracking’ criterion implies limitations on policies for updating in response to a wide range of types of new input. Satisfying the criterion is shown equivalent to the principle that the prior must be a convex combination of the possible posteriors. (...)
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  25. added 2017-09-27
    Situation et probabilité chez Saint Thomas d'Aquin.Edmund F. Byrne - 1966 - Revue Philosophique De Louvain 64:525-549.
    Il s'agit ici de la dimension existentielle de la theorie morale de S. Thomas d'Aquin. Pour lui, le domaine de l'incertain est generalement coextensif a celui de la contingence, de ce qui peut etre autre qu'il n'est. En general, S. Thomas envisage la contingence de la meme maniere qu'Aristote, mais dans une perspective totalement differente. Theologien, il s'interesse au monde physique surtout comme manifestation de la sagesse divine vers laquelle il desire monter. Il ne dedaigne pas pour autant les outils (...)
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  26. added 2017-09-14
    Conditioning Using Conditional Expectations: The Borel-Kolmogorov Paradox.Z. Gyenis, G. Hofer-Szabó & M. Rédei - unknown
    The Borel-Kolmogorov Paradox is typically taken to highlight a tension between our intuition that certain conditional probabilities with respect to probability zero conditioning events are well defined and the mathematical definition of conditional probability by Bayes' formula, which looses its meaning when the conditioning event has probability zero. We argue in this paper that the theory of conditional expectations is the proper mathematical device to conditionalize, and this theory allows conditionalization with respect to probability zero events. The conditional probabilities on (...)
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  27. added 2017-09-11
    An Accuracy Based Approach to Higher Order Evidence.Miriam Schoenfield - 2018 - Philosophy and Phenomenological Research 96 (3):690-715.
    The aim of this paper is to apply the accuracy based approach to epistemology to the case of higher order evidence: evidence that bears on the rationality of one's beliefs. I proceed in two stages. First, I show that the accuracy based framework that is standardly used to motivate rational requirements supports steadfastness—a position according to which higher order evidence should have no impact on one's doxastic attitudes towards first order propositions. The argument for this will require a generalization of (...)
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  28. added 2017-03-27
    Quinean Holism, Analyticity, and Diachronic Rational Norms.Brett Topey - 2018 - Synthese 195 (7):3143-3171.
    I argue that Quinean naturalists’ holism-based arguments against analyticity and apriority are more difficult to resist than is generally supposed, for two reasons. First, although opponents of naturalism sometimes dismiss these arguments on the grounds that the holistic premises on which they depend are unacceptably radical, it turns out that the sort of holism required by these arguments is actually quite minimal. And second, although it’s true, as Grice and Strawson pointed out long ago, that these arguments can succeed only (...)
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  29. added 2017-03-27
    The Faulty Signal Problem: Counterfactual Asymmetries in Causal Decision Theory and Rational Deliberation.Daniel Listwa - 2018 - Synthese 195 (6):2717-2739.
    A decision theory can be useful not only as a tool for determining which action, given your desires and beliefs, is most preferable, but also as a means for analyzing the nature of rational deliberation. In this paper, I turn to two classic proposals for a causal decision theory, that of Lewis and that of Sobel :407–437, 1986. doi: 10.1080/00048408612342621). As Rabinowicz revealed, Lewis’ proposal is unable to be applied to as broad a set of decision problems as a version (...)
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  30. added 2017-03-08
    The Doomsday Simulation Argument. Or Why Isn't the End Nigh, and You're Not Living in a Simulation.Mr István A. Aranyosi - manuscript
    According to the Carter-Leslie Doomsday Argument, we should assign a high probability to the hypothesis that the human species will go extinct very soon. The argument is based on the application of Bayes’s theo-rem and a certain indifference principle with respect to the temporal location of our observed birth rank within the totality of birth ranks of all humans who will ever have lived. According to Bostrom’s Simulation Argument, which appeals to a weaker indifference principle than the Doomsday Argument, at (...)
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  31. added 2017-03-06
    Reward Versus Risk in Uncertain Inference: Theorems and Simulations.Gerhard Schurz & Paul D. Thorn - 2012 - Review of Symbolic Logic 5 (4):574-612.
    Systems of logico-probabilistic reasoning characterize inference from conditional assertions that express high conditional probabilities. In this paper we investigate four prominent LP systems, the systems _O, P_, _Z_, and _QC_. These systems differ in the number of inferences they licence _. LP systems that license more inferences enjoy the possible reward of deriving more true and informative conclusions, but with this possible reward comes the risk of drawing more false or uninformative conclusions. In the first part of the paper, we (...)
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  32. added 2017-02-21
    An Epistemic Principle Which Solves Newcomb's Paradox.Keith Lehrer & Vann Mcgee - 1991 - Grazer Philosophische Studien 40:197-217.
    If it is certain that performing an observation to determine whether P is true will in no way influence whether P is tme, then the proposition that the observation is performed ought to be probabilistically independent of P. Applying the notion of "observation" liberally, so that a wide variety of actions are treated as observations, this proposed new principle of belief revision yields the result that simple utihty maximization gives the correct solution to the Fisher smoking paradox and the two-box (...)
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  33. added 2017-02-13
    The Theory of Spectrum Exchangeability.E. Howarth & J. B. Paris - 2015 - Review of Symbolic Logic 8 (1):108-130.
  34. added 2017-02-13
    Pure Inductive Logic.J. B. Paris & A. Vencovska - 2015 - Cambridge, UK: Cambridge University Press.
    Pure Inductive Logic is the study of rational probability treated as a branch of mathematical logic. This monograph, the first devoted to this approach, brings together the key results from the past seventy years, plus the main contributions of the authors and their collaborators over the last decade, to present a comprehensive account of the discipline within a single unified context.
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  35. added 2017-02-13
    The Twin Continua of Inductive Methods.Alena Vencovská & Jeff B. Paris - 2015 - In Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics. De Gruyter. pp. 355-366.
  36. added 2017-02-13
    Predicate Exchangeability and Language Invariance in Pure Inductive Logic.M. S. Kliess & J. B. Paris - 2014 - Logique Et Analyse 57 (228):513-540.
    In Pure Inductive Logic, the rational principle of Predicate Exchangeability states that permuting the predicates in a given language L and replacing each occurrence of a predicate in an L-sentence phi according to this permutation should not change our belief in the truth of phi. In this paper we study when a prior probability function w on a purely unary language L satisfying Predicate Exchangeability also satisfies the principle of Unary Language Invariance.
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  37. added 2017-02-13
    Principles of Remembering and Forgetting.E. Howarth & J. B. Paris - 2014 - Logique Et Analyse 57 (228):489-511.
    We propose two principles of inductive reasoning related to how observed information is handled by conditioning, and justify why they may be said to represent aspects of rational reasoning. A partial classification is given of the probability functions which satisfy these principles.
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  38. added 2017-02-13
    Second Order Inductive Logic and Wilmers' Principle.M. S. Kließ & J. B. Paris - 2014 - Journal of Applied Logic 12 (4):462-476.
  39. added 2017-02-13
    An Observation on Carnapʼs Continuum and Stochastic Independencies.J. B. Paris - 2013 - Journal of Applied Logic 11 (4):421-429.
  40. added 2017-02-13
    An Analogy Principle in Inductive Logic.A. Hill & J. B. Paris - 2013 - Annals of Pure and Applied Logic 164 (12):1293-1321.
    We propose an Analogy Principle in the context of Unary Inductive Logic and characterize the probability functions which satisfy it. In particular in the case of a language with just two predicates the probability functions satisfying this principle correspond to solutions of Skyrmsʼ ‘Wheel of Fortune’.
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  41. added 2017-02-13
    Symmetry in Polyadic Inductive Logic.J. B. Paris & A. Vencovská - 2012 - Journal of Logic, Language and Information 21 (2):189-216.
    A family of symmetries of polyadic inductive logic are described which in turn give rise to the purportedly rational Permutation Invariance Principle stating that a rational assignment of probabilities should respect these symmetries. An equivalent, and more practical, version of this principle is then derived.
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  42. added 2017-02-13
    A Survey of Some Recent Results on Spectrum Exchangeability in Polyadic Inductive Logic.J. Landes, J. B. Paris & A. Vencovská - 2011 - Synthese 181 (S1):19 - 47.
    We give a unified account of some results in the development of Polyadic Inductive Logic in the last decade with particular reference to the Principle of Spectrum Exchangeability, its consequences for Instantial Relevance, Language Invariance and Johnson's Sufficientness Principle, and the corresponding de Finetti style representation theorems.
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  43. added 2017-02-13
    Reasoning by Analogy in Inductive Logic.Alexandra Hill & J. B. Paris - 2011 - In Michal Peliš & Vít Punčochář (eds.), The Logica Yearbook. College Publications. pp. 63--76.
  44. added 2017-02-13
    A Note on Irrelevance in Inductive Logic.Jeff B. Paris & Alena Vencovská - 2011 - Journal of Philosophical Logic 40 (3):357 - 370.
    We consider two formalizations of the notion of irrelevance as a rationality principle within the framework of (Carnapian) Inductive Logic: Johnson's Sufficientness Principle, JSP, which is classically important because it leads to Carnap's influential Continuum of Inductive Methods and the recently proposed Weak Irrelevance Principle, WIP. We give a complete characterization of the language invariant probability functions satisfying WIP which generalizes the Nix-Paris Continuum. We argue that the derivation of two very disparate families of inductive methods from alternative perceptions of (...)
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  45. added 2017-02-05
    Combining Analogical Support in Pure Inductive Logic.J. B. Paris & A. Vencovská - 2017 - Erkenntnis 82 (2):401-419.
    We investigate the relative probabilistic support afforded by the combination of two analogies based on possibly different, structural similarity within the context of Pure Inductive Logic and under the assumption of Language Invariance. We show that whilst repeated analogies grounded on the same structural similarity only strengthen the probabilistic support this need not be the case when combining analogies based on different structural similarities. That is, two analogies may provide less support than each would individually.
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  46. added 2017-02-05
    An Examination of the SEP Candidate Analogical Inference Rule Within Pure Inductive Logic.E. Howarth, J. B. Paris & A. Vencovská - 2016 - Journal of Applied Logic 14:22-45.
  47. added 2017-02-03
    Ancient Indian Logic and Analogy.J. B. Paris & A. Vencovska - 2017 - In S. Ghosh & S. Prasad (eds.), Logic and its Applications, Lecture Notes in Computer Science 10119. Springer. pp. 198-210.
    B.K.Matilal, and earlier J.F.Staal, have suggested a reading of the `Nyaya five limb schema' (also sometimes referred to as the Indian Schema or Hindu Syllogism) from Gotama's Nyaya-Sutra in terms of a binary occurrence relation. In this paper we provide a rational justification of a version of this reading as Analogical Reasoning within the framework of Polyadic Pure Inductive Logic.
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  48. added 2017-02-03
    Second Order Inductive Logic and Wilmers' Principle.M. S. Kliess & J. B. Paris - 2014 - Journal of Applied Logic 12 (4):462-476.
    We extend the framework of Inductive Logic to Second Order languages and introduce Wilmers' Principle, a rational principle for probability functions on Second Order languages. We derive a representation theorem for functions satisfying this principle and investigate its relationship to the first order principles of Regularity and Super Regularity.
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  49. added 2017-02-03
    The Counterpart Principle of Analogical Support by Structural Similarity.A. Hill & J. B. Paris - 2014 - Erkenntnis 79 (S6):1-16.
    We propose and investigate an Analogy Principle in the context of Unary Inductive Logic based on a notion of support by structural similarity which is often employed to motivate scientific conjectures.
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  50. added 2017-02-03
    An Observation on Carnapʼs Continuum and Stochastic Independencies.J. B. Paris - 2013 - Journal of Applied Logic 11 (4):421-429.
    We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnson's Sufficientness Principle holds, thus giving Carnap's Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnap's Continuum for the language with just a single predicate.
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