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Probabilistic Reasoning

Assistant editor: Joshua Luczak (University of Western Ontario, Georgetown University)
About this topic
Summary What principles govern uncertain reasoning?  And how do they apply to other philosophical problems; like whether a decision is rational, or whether one thing is a cause of another? Most philosophers think uncertain reasoning should at least obey the axioms of the mathematical theory of probability; though some prefer other axioms, like those of Dempster-Shafer theory or ranking theory.  Many also endorse principles governing beliefs about physical probabilities (chance-credence principles), and principles for responding to new evidence (updating principles).  Some also endorse principles for reasoning in the absence of relevant information (indifference principles).  A perennial question is how many principles we should accept: how "objective" is probabilistic reasoning? Probabilistic principles have traditionally been applied to the study of scientific reasoning (confirmation theory) and practical rationality (decision theory).  But they also apply to more traditional epistemological issues, like foundationalism vs. coherentism, and to metaphysical questions, e.g. about the nature of causality and our access to it.
Key works Key works defending the probability axioms as normative principles are Ramsey 2010, Finetti 1989, Savage 1954, and Joyce 1998.  Locus classici for additional probabilistic principles are Lewis 1980 (chance-credence), Fraassen 1984 (reflection), Carnap 1962, Jaynes 1973 (indifference), and Lewis 2010 (updating). Alternative axiomatic frameworks originate with Shafer 1976 (Dempster-Shafer theory) and Spohn 1988 (ranking theory). Some classic applications of probabilistic principles to epistemological and other problems are Good 1960 (the raven paradox), Pearl 2000 (causal inference), and Elga 2000 (sleeping beauty and self-location). 
Introductions Skyrms 1975 is an excellent and gentle introduction for non-initiates.  A next step up is Jeffrey 1983.  More advanced introductions are Howson & Urbach 1993 and Earman 1992.  More recently, Halpern 2003 provides an excellent overview of the mathematical options.  A recent overview of the more philosophical issues can be found in Weisberg manuscript.
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  1. William Forbes Cooley (1912). The Principles of Science a College Text-Book. H. Holt and Company.
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  2. J. Acacio de Barros, Carlos Montemayor & Leonardo De Assis (forthcoming). Contextuality in the Integrated Information Theory. In J. A. de Barros, B. Coecke & E. Pothos (eds.), Lecture Notes on Computer Science.
    Integrated Information Theory (IIT) is one of the most influential theories of consciousness, mainly due to its claim of mathematically formalizing consciousness in a measurable way. However, the theory, as it is formulated, does not account for contextual observations that are crucial for understanding consciousness. Here we put forth three possible difficulties for its current version, which could be interpreted as a trilemma. Either consciousness is contextual or not. If contextual, either IIT needs revisions to its axioms to include contextuality, (...)
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  3. Luis J. Flores (2015). Therapeutic Inferences for Individual Patients. Journal of Evaluation in Clinical Practice 21 (3):440-447.
    RATIONALE, AIMS AND OBJECTIVES: Increased awareness of the gap between controlled research and medical practice has raised concerns over whether the special attention of doctors to probability estimates from clinical trials really improves the care of individuals. Evidence-based medicine has acknowledged that research results are not applicable to all kinds of patients, and consequently, has attempted to overcome this limitation by introducing improvements in the design and analysis of clinical trials. METHODS: A clinical case is used to highlight the premises (...)
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  4. Patrick Forber (2012). Modeling Scientific Evidence: The Challenge of Specifying Likelihoods. In Henk W. de Regt (ed.), Epsa Philosophy of Science: Amsterdam 2009. Springer 55--65.
    Evidence is an objective matter. This is the prevailing view within science, and confirmation theory should aim to capture the objective nature of scientific evidence. Modeling an objective evidence relation in a probabilistic framework faces two challenges: the probabilities must have the right epistemic foundation, and they must be specifiable given the hypotheses and data under consideration. Here I will explore how Sober's approach to confirmation handles these challenges of foundation and specification. In particular, I will argue that the specification (...)
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  5. Malcolm Forster (2007). A Philosopher's Guide to Empirical Success. Philosophy of Science 74 (5):588-600.
    The simple question, what is empirical success? turns out to have a surprisingly complicated answer. We need to distinguish between meritorious fit and ‘fudged fit', which is akin to the distinction between prediction and accommodation. The final proposal is that empirical success emerges in a theory dependent way from the agreement of independent measurements of theoretically postulated quantities. Implications for realism and Bayesianism are discussed. ‡This paper was written when I was a visiting fellow at the Center for Philosophy of (...)
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  6. James G. Greeno (1970). Theoretical Entities in Statistical Explanation. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1970:3 - 26.
  7. Theo A. F. Kuipers (2009). Empirical Progress and Truth Approximation by the 'Hypothetico-Probabilistic Method'. Erkenntnis 70 (3):313 - 330.
    Three related intuitions are explicated in this paper. The first is the idea that there must be some kind of probabilistic version of the HD-method, a ‘Hypothetico-Probabilistic (HP-) method’, in terms of something like probabilistic consequences, instead of deductive consequences. According to the second intuition, the comparative application of this method should also be functional for some probabilistic kind of empirical progress, and according to the third intuition this should be functional for something like probabilistic truth approximation. In all three (...)
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  8. Theo A. F. Kuipers (1983). Non-Inductive Explication of Two Inductive Intuitions. British Journal for the Philosophy of Science 34 (3):209-223.
    In section I the notions of logical and inductive probability will be discussed as well as two explicanda, viz. degree of confirmation, the base for inductive probability, and degree of evidential support, Popper's favourite explicandum. In section II it will be argued that Popper's paradox of ideal evidence is no paradox at all; however, it will also be shown that Popper's way out has its own merits.
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  9. Johannes Lenhard (2006). Models and Statistical Inference: The Controversy Between Fisher and Neyman–Pearson. British Journal for the Philosophy of Science 57 (1):69-91.
    The main thesis of the paper is that in the case of modern statistics, the differences between the various concepts of models were the key to its formative controversies. The mathematical theory of statistical inference was mainly developed by Ronald A. Fisher, Jerzy Neyman, and Egon S. Pearson. Fisher on the one side and Neyman–Pearson on the other were involved often in a polemic controversy. The common view is that Neyman and Pearson made Fisher's account more stringent mathematically. It is (...)
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  10. Patrick Maher (1993). Howson and Franklin on Prediction. Philosophy of Science 60 (2):329-340.
    Evidence for a hypothesis typically confirms the hypothesis more if the evidence was predicted than if it was accommodated. Or so I argued in previous papers, where I also developed an analysis of why this should be so. But this was all a mistake if Howson and Franklin (1991) are to be believed. In this paper, I show why they are not to be believed. I also identify a grain of truth that may have been dimly grasped by those Bayesians (...)
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  11. Ilkka Niiniluoto (1978). High Probability and Inductive Systematization. Journal of Philosophy 75 (12):737-739.
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  12. Jeanne Peijnenburg & David Atkinson, Probabilistic Justification.
    We discuss two objections that foundationalists have raised against infinite chains of probabilistic justification. We demonstrate that neither of the objections can be maintained.
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  13. Michael Redhead (1985). On the Impossibility of Inductive Probability. British Journal for the Philosophy of Science 36 (2):185-191.
  14. Ferdinand Schoeman (1987). Statistical Vs. Direct Evidence. Noûs 21 (2):179-198.
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  15. Marc Meléndez Schofield (2011). Probabilidad, causalidad y explicación. Theoria 26 (1):109-112.
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  16. Jie Zheng, Marcelline R. Harris, Anna Maria Masci, Lin Yu, Alfred Hero, Barry Smith & Yongqun He (2016). The Ontology of Biological and Clinical Statistics (OBCS) for Standardized and Reproducible Statistical Analysis. Journal of Biomedical Semantics 7 (53).
    Statistics play a critical role in biological and clinical research. However, most reports of scientific results in the published literature make it difficult for the reader to reproduce the statistical analyses performed in achieving those results because they provide inadequate documentation of the statistical tests and algorithms applied. The Ontology of Biological and Clinical Statistics (OBCS) is put forward here as a step towards solving this problem. Terms in OBCS, including ‘data collection’, ‘data transformation in statistics’, ‘data visualization’, ‘statistical data (...)
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Bayesian Reasoning
  1. David Cox & Deborah G. Mayo (2010). Objectivity and Conditionality in Frequentist Inference. In Deborah G. Mayo & Aris Spanos (eds.), Error and Inference: Recent Exchanges on Experimental Reasoning, Reliability, and the Objectivity and Rationality of Science. Cambridge University Press 276.
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Bayesian Reasoning, Misc
  1. Peter Achinstein (1992). The Evidence Against Kronz. Philosophical Studies 67 (2):169-175.
  2. Peter Achinstein (1963). Confirmation Theory, Order, and Periodicity. Philosophy of Science 30 (1):17-35.
    This paper examines problems of order and periodicity which arise when the attempt is made to define a confirmation function for a language containing elementary number theory as applied to a universe in which the individuals are considered to be arranged in some fixed order. Certain plausible conditions of adequacy are stated for such a confirmation function. By the construction of certain types of predicates, it is proved, however, that these conditions of adequacy are violated by any confirmation function defined (...)
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  3. Peter Achinstein (1963). Variety and Analogy in Confirmation Theory. Philosophy of Science 30 (3):207-221.
    Confirmation theorists seek to define a function that will take into account the various factors relevant in determining the degree to which an hypothesis is confirmed by its evidence. Among confirmation theorists, only Rudolf Carnap has constructed a system which purports to consider factors in addition to the number of instances, viz. the variety manifested by the instances and the amount of analogy between the instances. It is the purpose of this paper to examine the problem which these additional factors (...)
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  4. Daniel Acuna & Paul Schrater (2008). Bayesian Modeling of Human Sequential Decision-Making on the Multi-Armed Bandit Problem. In B. C. Love, K. McRae & V. M. Sloutsky (eds.), Proceedings of the 30th Annual Conference of the Cognitive Science Society. Cognitive Science Society 100--200.
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  5. Ernest W. Adams (1988). Confirming Inexact Generalizations. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988:10 - 16.
    I suppose that 'ravens are black' is an inexact generalization having a degree of truth measured by the proportion of ravens that are black, and a probability measured by its expected degree of truth in different 'possible worlds.' Given this, 'ravens are black' differs in truth, probability, and confirmation from 'non-black things are not ravens', and this suggests a new approach to Hempel's Paradox as well as to other aspects of confirmation. Basic concepts of a formal theory developing this approach (...)
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  6. Arif Ahmed (2015). Hume and the Independent Witnesses. Mind 124 (496):1013-1044.
    The Humean argument concerning miracles says that one should always think it more likely that anyone who testifies to a miracle is lying or deluded than that the alleged miracle actually occurred, and so should always reject any single report of it. A longstanding and widely accepted objection is that even if this is right, the concurring and non-collusive testimony of many witnesses should make it rational to believe in whatever miracle they all report. I argue that on the contrary, (...)
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  7. P. M. Ainsworth (2012). In Defence of Objective Bayesianism. Analysis 72 (4):832-843.
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  8. Jean Aitchison (1995). Free or Ensnared? The Hidden Nets Of. In E. Barker (ed.), Lse on Freedom. Lse Books 75.
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  9. Laith Al-Shawaf & David Buss (2011). Evolutionary Psychology and Bayesian Modeling. Behavioral and Brain Sciences 34 (4):188-189.
    The target article provides important theoretical contributions to psychology and Bayesian modeling. Despite the article's excellent points, we suggest that it succumbs to a few misconceptions about evolutionary psychology (EP). These include a mischaracterization of evolutionary psychology's approach to optimality; failure to appreciate the centrality of mechanism in EP; and an incorrect depiction of hypothesis testing. An accurate characterization of EP offers more promise for successful integration with Bayesian modeling.
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  10. Casper J. Albers, Barteld P. Kooi & Willem Schaafsma (2005). Trying to Resolve the Two-Envelope Problem. Synthese 145 (1):89 - 109.
    After explaining the well-known two-envelope paradox by indicating the fallacy involved, we consider the two-envelope problem of evaluating the factual information provided to us in the form of the value contained by the envelope chosen first. We try to provide a synthesis of contributions from economy, psychology, logic, probability theory (in the form of Bayesian statistics), mathematical statistics (in the form of a decision-theoretic approach) and game theory. We conclude that the two-envelope problem does not allow a satisfactory solution. An (...)
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  11. Max Albert (2005). Should Bayesians Bet Where Frequentists Fear to Tread? Philosophy of Science 72 (4):584-593.
  12. Max Albert (2003). Bayesian Rationality and Decision Making: A Critical Review. Analyse & Kritik 25 (1):101-117.
    Bayesianism is the predominant philosophy of science in North-America, the most important school of statistics world-wide, and the general version of the rational-choice approach in the social sciences. Although often rejected as a theory of actual behavior, it is still the benchmark case of perfect rationality. The paper reviews the development of Bayesianism in philosophy, statistics and decision making and questions its status as an account of perfect rationality. Bayesians, who otherwise are squarely in the empiricist camp, invoke a priori (...)
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  13. J. McKenzie Alexander (2009). Social Deliberation: Nash, Bayes, and the Partial Vindication of Gabriele Tarde. Episteme 6 (2):164-184.
    At the very end of the 19th century, Gabriele Tarde wrote that all society was a product of imitation and innovation. This view regarding the development of society has, to a large extent, fallen out of favour, and especially so in those areas where the rational actor model looms large. I argue that this is unfortunate, as models of imitative learning, in some cases, agree better with what people actually do than more sophisticated models of learning. In this paper, I (...)
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  14. J. McKenzie Alexander (2006). The Stag Hunt and the Evolution of Social Structure, Brian Skyrms. Cambridge University Press, 2004, 149 Pages. [REVIEW] Economics and Philosophy 22 (3):441-448.
  15. Moorad Alexanian (forthcoming). Nature, Science, Bayes 'Theorem, and the Whole of Reality‖. Zygon.
    A fundamental problem in science is how to make logical inferences from scientific data. Mere data does not suffice since additional information is necessary to select a domain of models or hypotheses and thus determine the likelihood of each model or hypothesis. Thomas Bayes’ Theorem relates the data and prior information to posterior probabilities associated with differing models or hypotheses and thus is useful in identifying the roles played by the known data and the assumed prior information when making inferences. (...)
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  16. Mark Alfano (2007). A Critical Discussion Of The Compatibility Of Bayesianism And Inference To The Best Explanation. Philosophical Writings 34 (1).
    In this paper I critique Peter Lipton’s attempt to deal with the threat of Bayesianism to the normative aspect of his project in Inference to the Best Explanation. I consider the five approaches Lipton proposes for reconciling the doxastic recommendations of Inference to the Best Explanation with BA’s: IBE gives a ‘boost’ to the posterior probability of particularly ‘lovely’ hypotheses after the Bayesian calculation is performed; IBE helps us to set the likelihood of evidence on a given hypothesis; IBE helps (...)
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  17. Nilufa Ali, Anne Schlottman, Abigail Shaw, Nick Chater, & Oaksford & Mike (2010). Causal Discounting and Conditional Reasoning in Children. In Mike Oaksford & Nick Chater (eds.), Cognition and Conditionals: Probability and Logic in Human Thinking. OUP Oxford
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  18. Nilufa Ali, Anne Schlottmann, Abigail Shaw, Nick Chater & Mike Oaksford (2010). Causal Discounting and Conditional Reasoning in Children. In M. Oaksford & N. Chater (eds.), Cognition and Conditionals: Probability and Logic in Human Thought. Oxford University Press
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  19. Maurice Allais (1979). The So-Called Allais Paradox and Rational Decisions Under Uncertainty. In Maurice Allais & Ole Hagen (eds.), Expected Utility Hypotheses and the Allais Paradox. D. Reidel
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  20. Maurice Allais & Ole Hagen (eds.) (1979). Expected Utility Hypotheses and the Allais Paradox. D. Reidel.
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  21. Ronald J. Allen (2001). Artificial Intelligence and the Evidentiary Process: The Challenges of Formalism and Computation. [REVIEW] Artificial Intelligence and Law 9 (2-3):99-114.
    The tension between rule and judgment is well known with respect to the meaning of substantive legal commands. The same conflict is present in fact finding. The law penetrates to virtually all aspects of human affairs; irtually any interaction can generate a legal conflict. Accurate fact finding about such disputes is a necessary condition for the appropriate application of substantive legal commands. Without accuracy in fact finding, the law is unpredictable, and thus individuals cannot efficiently accommodate their affairs to its (...)
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  22. Mair Allen-Williams & Nicholas R. Jennings (2009). Bayesian Learning for Cooperation in Multi-Agent Systems. In L. Magnani (ed.), Computational Intelligence. 321--360.
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  23. Peter Allmark (2005). Bayes and Health Care Research. Medicine, Health Care and Philosophy 7 (3):321-332.
    Bayes’ rule shows how one might rationally change one’s beliefs in the light of evidence. It is the foundation of a statistical method called Bayesianism. In health care research, Bayesianism has its advocates but the dominant statistical method is frequentism. There are at least two important philosophical differences between these methods. First, Bayesianism takes a subjectivist view of probability (i.e. that probability scores are statements of subjective belief, not objective fact) whilst frequentism takes an objectivist view. Second, Bayesianism is explicitly (...)
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  24. Sarah Allred (2012). Approaching Color with Bayesian Algorithms. In Gary Hatfield & Sarah Allred (eds.), Visual Experience: Sensation, Cognition, and Constancy. OUP Oxford 212.
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  25. Paul Anand (2005). Bayes's Theorem (Proceedings of the British Academy, Vol. 113), Edited by Richard Swinburne, Oxford University Press, 2002, 160 Pages. [REVIEW] Economics and Philosophy 21 (1):139-142.
  26. Barton L. Anderson (2011). The Myth of Computational Level Theory and the Vacuity of Rational Analysis. Behavioral and Brain Sciences 34 (4):189-190.
    I extend Jones & Love's (J&L's) critique of Bayesian models and evaluate the conceptual foundations on which they are built. I argue that: (1) the part of Bayesian models is scientifically trivial; (2) theory is a fiction that arises from an inappropriate programming metaphor; and (3) the real scientific problems lie outside Bayesian theorizing.
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  27. Arthur Isak Applbaum (2014). Bayesian Inference and Contractualist Justification on Interstate 95. In Andrew I. Cohen & Christopher H. Wellman (eds.), Contemporary Debates in Applied Ethics. Wiley Blackwell 219.
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  28. Mr István A. Aranyosi, The Doomsday Simulation Argument. Or Why Isn't the End Nigh, and You're Not Living in a Simulation.
    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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  29. Horacio Arló-Costa (2001). Bayesian Epistemology and Epistemic Conditionals: On the Status of the Export-Import Laws. Journal of Philosophy 98 (11):555-593.
    Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use.
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  30. Horacio Arló-Costa (2001). Bayesian Epistemology and Epistemic Conditionals. Journal of Philosophy 98 (11):555-593.
    The notion of probability occupies a central role in contemporary epistemology and cognitive science. Nevertheless, the classical notion of probability is hard to reconcile with the central notions postulated by the epistemological tradition.
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  31. Frank Arntzenius, Adam Elga & John Hawthorne (2004). Bayesianism, Infinite Decisions, and Binding. Mind 113 (450):251 - 283.
    We pose and resolve several vexing decision theoretic puzzles. Some are variants of existing puzzles, such as 'Trumped' (Arntzenius and McCarthy 1997), 'Rouble trouble' (Arntzenius and Barrett 1999), 'The airtight Dutch book' (McGee 1999), and 'The two envelopes puzzle' (Broome 1995). Others are new. A unified resolution of the puzzles shows that Dutch book arguments have no force in infinite cases. It thereby provides evidence that reasonable utility functions may be unbounded and that reasonable credence functions need not be countably (...)
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  32. Jerrold L. Aronson (1989). The Bayesians and the Raven Paradox. Noûs 23 (2):221-240.
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  33. Alfonso Arroyo-Santos, Mark E. Olson & Francisco Vergara-Silva (2014). Practice-Oriented Controversies and Borrowed Epistemic Support in Current Evolutionary Biology: Phylogeography as a Case Study. Biology and Philosophy 29:833-850.
    Although there is increasing recognition that theory and practice in science are often inseparably intertwined, discussions of scientific controversies often continue to focus on theory, and not practice or methodologies. As a contribution to constructing a framework towards understanding controversies linked to scientific practices, we introduce the notion of borrowed epistemic credibility, to describe the situation in which scientists exploit fallacious similarities between accepted tenets in other fields to garner support for a given position in their own field. Our proposal (...)
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