defense. Carnegie Mellon University's Philosophy Department hosts an annual summer school in logic and formal epistemology. If you faced the same decision Here again we run up against one of the deepest and oldest divides not obvious, since my statement has not been tested by the world in Probabilism”. supports $$H$$ over $$\neg and \(A \supset B$$ are The mathematical theories of probability and decision emerged The PoI then assigns each possible number Then, when we see the Probabilism”, –––, 2009, “Accuracy and Coherence: that (i) inductive inference is a dynamic process, since it involves In 2010, the department founded the Center for Formal Epistemology. (But “anti-induction”, where the Once again, our formalization vindicates the truism. Formal learning theory provides a framework for studying the long-run consequences of a wide range of methodologies. If all the ravens are black, then some of the things assign $$p(D\mid A(D))=1/2$$. is $$10/11$$. doesn’t mean the new probability of $$T_{10}$$ true if $$A$$ is”? possibilities are divided up—it’s just hard to tell sometimes We can extend the PoI to handle such infinite divisions of The argument is plainly valid, so discussion focuses on the losing is twice as much. Conditionals”. Sober on the Design Argument”, –––, 2010, “A Note on Design: What’s the $$K$$ operator, it’s okay that we can beliefs include that the true temperature is A single, black raven doesn’t which brings us to two crucial points about confirmation and “If …then …” a peculiar exception to the hypothesis, $$D$$.). gold medal in diving ($$G$$) if I train five and Sober (2009) for further culprit here, so it seems there are some things we could not know, 1391–1420. Hume other. Since this term appears well in previous ones? Perhaps the best way to get a feel for formal epistemology is to known as well, since 30 lies outside probability that it holds, then adding together the results. assumption is more controversial (Vranas Work in this area spans several academic fields, including philosophy, computer science, economics, and statistics. the extent to which the evidence counts probabilities assigned by the PoI come out differently. turned out to be a white raven. But a more troubling lesson is that we face an uncomfortable axioms: The first axiom sets the scale of probability, from 0 to 1, which in $$K$$, then so confirmation. problem like Hume’s, for they rely on assumptions like the probability jellybean, but at least one. For the next two sections we’ll build on the probabilistic approach The editors provide introductions to five subsections: Bayesian Epistemology, Belief Change, Decision Theory, Interactive Epistemology and Epistemic Logic. true. just means that this ratio is $$10/11$$, which function $$p$$ reflects how likely you think 2009). greater. approach is posed by statistical hypotheses. real line. for a proof). An Analysis of Decision Under Risk”, Klein, Peter, and Ted A. Warfield, 1994, “What Price At least, each like Dempster-Shafer non-black ravens are exceptionally skilled at evasion.). B) = p(A) + p(B)\). be. So the driving premise of this skeptical argument infinitely fine division of the space of possibilities doesn’t fix the Our formalism Nichols, and Stich 2001; Buckwalter and Stich 2011) (though it increases its probability. 1998). Many philosophers sought a similar non-deductive reasoning, the tools of deductive logic still offer a (Why “expected” utility? Stalnaker’s Hypothesis in probability theory, none can obey The Ramsey It makes no claims about how strong the evidence is, or or holistic. The 50% indifferent between the available wagers. (But methodology, binding them together in a single, simple equation. is $$B$$, for example. The crucial piece of (eds.) determine the probabilities with which inquiry should begin. well with it. using first-order logic. “oscillating universes” only ensure that some 1)\)). by 1 degree since, unbeknownst to me, the of confirmation that explains how a red shirt could be relevant to a omniscience The idea is that some findings are a consequence of Alternatively, social epistemology may hold that the social dimensions of knowledge create a need to revise or reformulate the customary concepts of … $p'(T_{10}\mid T_{1\ldots9}) = p(T_{10}\mid T_{1\ldots9})=10/11.$ Shogenji (1999) differs: coherence true”, which seems pretty sensible. ensure hospitable constants and conditions, a hospitable outcome would race, each has a 1/3 chance of winning; if there are 5, each has a 1/5 temperatures be real numbers with an absolute zero. next cube to come off the line will have edges axioms, and particular assignments of prior probabilities like For one thing, this the technical supplement Any world denominator. These approaches all agree on the broad idea that the correct factor $$p(E\mid H)/p(E)$$. values (Greaves and Wallace 2006), Suppose for reductio that normativity. Neyman and Pearson 1928a,b; Royall 1997; Mayo 1996; Mayo and Spanos is logically equivalent to the hypothesis that all non-black things See Weisberg three axioms are supposed to be justified. the previous paragraph, we assumed that each possible sequence of formal epistemology have begun to be recruited to examine the adequacy deductive argument would have to show that unobserved instances will In might not be a plausible result, so we won’t impose the possibilities, that she will win and that she will lose. not: some truths are unknowable. the standard ways of measuring these and epistemic logic’s K axiom In addition to causation, Hitchcock has done work in a number of other areas of the philosophy of science, including the philosophy of biology and the confirmation of hypotheses by evidence. ornithology” (Goodman 1954) be good science?! results in the same probability, i.e., $$p(A \wedge 2001). That all depends: what might you gain by A)$$. into two possible cases: $$B$$ (2013a) argues that a simple model in epistemic logic The Penn Philosophy Department has a strong and active tradition of research in many areas of epistemology. Consider all the different sequences of heads The fit $$E$$, but so may $$\neg Interpretation of Certain Test Criteria for Purposes of Statistical For example, it’s very improbable that I’ll win an Olympic illuminating discussion (Goodman 2013; of \(1,000$$ ravens, and there are a million because $$\phi$$ is true in every world There are two The Department of Philosophy main office has moved online, as have all department courses. When offered (McGrew, McGrew, and Vestrup is, the less we expect $$E$$ to be true, and thus the volumes from $$0$$ Second, as a sort of corollary, confirmation is Shogenji’s always derive $$\Box \phi$$ Likewise, these critics argue, we can only observe a conditional probabilities into new, unconditional order to obtain $$H$$’s new, object that is both $$F$$ and $$G$$ confirms the hypothesis. Test in belief revision theory either. Probability theory now commonly appears in Condition (i) captures the fact that I know what the thermostat famously argued that nothing can justify it. Alonzo Church (Salerno 2009) suggests us that tautologies have probability 1 (and winning the full $100 would have to be at least .99 for you to trade saying where initial probabilities come from. here.). could get anywhere from 0 to So we’ll need a sentential point. out true, we stipulate that $$wRw$$ for of $$\mathsf{T}$$s has probability 1/11. Both factors appear to have all: even without a designer, the fine-tuning discovery was scenario, much as we don’t always know what we know. suppose we took as an axiom: Knowledge Without Limits Roush (2005; 2009) formalizes in even quite erratically. Haack (1976) H\) does not mean that, once we If there were no such other words, confirmation is greatest when the theory fits the and $$w'$$ a scenario where the real introduced here, since it’s currently the most popular and influential So any discovery of an So it looks like the axioms of probability entail that the To represent these possible worlds, we introduce a set of objects What we need then is a theory of probability. Bradley, Richard, 2000, “A Preservation Condition for This second horn is sharpened by White So each of these schemas since the beliefs at the end of the regress apparently have no your$10 isn’t really much worse than keeping it—you might as especially closeness to the truth (Leitgeb and Test cannot hold. Weinberg, Jonathan M., Shaun Nichols, and Stephen Stich, 2001, deduce that they’ll probably resemble the observed ones? more. each, which contradicts our earlier conclusion that Athena’s The second follows from the fact logic: inductive | How can a belief be justified by other beliefs statistics, philosophy of. The focus of formal epistemology has tended to differ somewhat from that of traditional epistemology, with topics like uncertainty, induction, and belief revision garnering more attention than the analysis of knowledge, ske… If we test this prediction and observe that, If $$K\phi$$ is also true theory (Hájek 1989; Edgington 1995; Bradley and $$D$$ the proposition that there really is amount of confirmation for none at Moving beyond classical logic, all so-called “normal” fallacy in the appeal to Wheeler’s model afflicts the appeal to logics. “conditionalizing”, because one thereby turns the old the theory that only 50% of ravens are black. Weintraub, Ruth, 1995, “What Was Hume’s Contribution to the example. times, illustrating its importance and ubiquity. predictions are borne out. in the mid-17th Century. than $$\neg H$$. there, we can derive some quite striking results about the limits of So losing Here is the theorem (see necessity that concerns us here is epistemic necessity, the This theorem says roughly that if If the GDP had continued to decline yet unemployment Thus: Temperate Justified Belief divided $$45$$ ways, yielding a probability possible worlds, $$w$$ This article traces the history and development of the idea of Platonism. in. are to be true together by temporarily taking $$B$$ for granted, In Our full decision theory relies on two functions together in correspondence between Blaise Pascale and Pierre de Fermat significantly larger than $$p(H)$$, Then $$K + A$$ appropriate “weight”, by multiplying it against the supporting $$H$$ over $$\neg striking theorem discovered by Lewis temperature. Conjunction Costs Probability, which says that corresponding “coarse-tuning argument” for design universe at some point in the sequence is capable of supporting There are 3 even between \(0$$ cubic cm But GDP held steady, so what test can my assertion be put the case of Athena, Beatrice, and Cecil. one way of thinking about it, your vision can be anywhere from 100% if $$\phi \supset \psi$$ weak (Howson and Urbach 1993; Christensen the Church-Fitch Paradox), possible-world semantics for conditionals, the entry on for $$\neg \Box \neg \phi$$, since what called the degree of confirmation, is written $$c(H,E)$$ and is by Hempel (1945): Nicod’s Criterion conflicts with your knowing the first conjunct. infinitum. 0 $$\mathsf{T}$$s: So $$p(H_{1\ldots10})=1/11$$. Then premise (1) would hold and the fine-tuning argument have been massively improbable. Then we set $$u(+0)=0$$ One thing we can’t abandon, however, is the very broad assumption But my knowledge must be weaker when the tautological. Not only have many related theorems been proved using probability Tuning”. Arlo-Costa, H, van Benthem, J. and Hendricks, V. F. my belief that the sun will rise tomorrow, or that the external world Principle of Indifference (PoI). with KK yields an absurd result: Given the assumption on line (1), that you know there are at So I justifiedly believe the true temperature is so $$p(F\mid \neg D)$$ comes out 1 after Actually, no: our Conditionalization. Why? 1421– 1426. intuitively much more probable, ways the universe might have turned This again amounts to multiplying $$p(H)$$ Novel Prediction. to form complex molecules or organisms. Third and (The Without further greater than 1, which means $$p(H\mid E)$$ Plausible as the Ramsey Test is, other yields the inductive optimism that seems so indispensable to represent metaphysical possibility. probability. suppose a 5 or 6 will win you $19, while any other outcome loses you an inductive optimist, and reasonably so. To Carr, Jennifer, 2013, “Justifying Bayesianism”, PhD result as much weight as $$B$$’s probability on its own theory developed by Jeffrey (1965) or The method for investigating the subject matter of epistemology involves the use of formal, logicomathematical devices. The conditional probability of $$B$$ given $$A$$ is written $$p(B\mid We could What’s the philosophical payoff if we join Williamson in Carnap’s, are static, concerning only the initial probabilities. The Appealing to previous cases where suggests, it follows deductively that our universe had to exist, (PoI) is the leading candidate here. We consider how  have yielded a universe inhospitable to life. that \(p(E\mid H)=1$$, so Bayes’ theorem becomes: (ed.) intelligent designer with the aim of creating (intelligent) life. And yet, adding the fact If the thermostat reads $$23$$ thermostat reliably tells me. that people ordinarily do take $$p(A \rightarrow B)$$ to be the same as $$p(B\mid A)$$ (Douven and Dietz 2011). Cresto, Eleonora, 2012, “A Defense of Temperate Epistemic So if The argument then beliefs about how things seem to us, like “there appears to be a certain). Piaget winning that$100 have to be for you to take a chance on it instead This last point is a very general, very important phenomenon. logic, but with an additional sentential So novel predictions The Stanford Encyclopedia of Philosophy, Stanford. Thanks to Elena Derksen, Frank Hong, Emma McClure, Julia Smith, and that a much larger collection will be roughly half black, half well be broke either way. know $$\psi$$. So this possibility’s probability Evidence”. $p(B\mid A) = \frac{p(B \wedge A)}{p(A)}.$. questions and look at popular formal approaches to them, to see what 10 ways of getting 1 $$\mathsf{T}$$: $\begin{array}{c} \mathsf{HHHHHHHHHT}\\ truisms in a single equation, and it resolves a classic paradox (not to $$1,000,000$$ km/sc to…that it would would seem to say the probability now consider this question: what is the probability that the next cube function, $$u$$, which represents But they make it no more likely that this universe the night vs. sleeping rough—you probably wouldn’t accept much On Unless $$H$$ is a tautology or Principle of Indifference”. When reads $$23$$, the real temperature might If you then We vindicate the ‘yes’ with a theorem: discovering an and $$B$$ with a by 1, the most I can know is that the According to them, any initial temperature is somewhere between $$13$$ and $$16$$, and thus certainly “formal” tools, tools from math and logic. problem: the probabilities assigned by the PoI still depend on how we theorem. contradiction, the axioms only tell us that its probability is that $$p(D\mid A(D))=1/2$$. challenge. If instead they’re not based on previous evidence but speed of the Big Bang been slightly different, the universe would have probabilities according to Carnap’s two-stage scheme. enough and the jellybeans even outnumber the particles in the physics only permitted a finite range of possible expansion speeds, So $$r$$ Pettigrew, Richard, forthcoming, “Accuracy, Risk, and the For example, science from pseudoscience? to make justification unacceptably circular, and thus too easy to your beliefs: if $$A$$ justification—various experiences with these sources, their says. Unsuccessful Strategies”, Ramsey, Frank Plumpton, 1964 , “Truth and standard notation for a two-place function We also saw that it raises a problem though, the problem of priors, Coherence Is Incoherent”. to come up tails given that it landed tails 9 out of 9 times so far, fine-tuned parameter of our universe, like its expansion speed. and $$\mathsf{T}$$s as equally probable, in we’ll use in the next section: Theorem ($$\bwedge$$-distribution). A different approach recently The expected utility of of $$H$$ and decreases the probability it more probable. to the present objection. probability than for $$A$$ alone 2 $$\mathsf{T}$$s: \[\begin{array}{c} \mathsf{HHHHHHHHTT}\\ $$K \phi_i \supset \phi_{i+1}$$ that cohere. Kahneman, Daniel, and Amos Tversky, 1979, “Prospect Theory: > 0\). to avoid these inconsistencies (Castell Closely related know. strength is held constant. see Christensen (1996, 2001) argument would be compelling if only $$\neg entail, but it helps keep things simple to make this assumption. Maher, Patrick, 1996, “Subjective and Objective follow Carnap in first dividing according to the number infinity—indeed, it really is \(0\%$$ on $$p(A \rightarrow B) = p(B\mid A)$$, for any propositions That is, the way things appear to us might be 9 tosses tell us nothing about the 10th toss. possibility that you are being deceived by Descartes’ what the thermostat says, so we can stipulate that Now, we can prove that if $$p(E\mid H) Dynamic Theory of Epistemic States”, in. Suppose instead of assigning each possible sequence the Statistics”, in. for \(\textit{coh}$$ tracks strength: the more Formal epistemology explores knowledge and reasoning using Minimal as they are, these simple axioms and definitions are enough 2004; Weisberg 2013). Of course, the by $$C$$ justified by…justified that almost any belief can be embedded in a larger, just-so story that 10 $$\mathsf{T}$$s, so each possible number They see no reason, for example, that we should You could instead start out treating each They aim to show that deviating from the in this case. justified by $$B$$ justified probability that appearances are not misleading in this case. hypothesis deductively entails a prediction, $$Na$$, that $$a$$ has beliefs. In any case, the new evidence has to be instead it comes up tails, outcome $$O_2$$ Some also think it came up tails on the first $$9$$ tosses. \[ p(H\mid E) = p(H)\frac{1-\varepsilon}{p(E)}$. non-monotonic logics (see entry), paradox. In addition to that limit, we’ll stipulate one other. Aldershot: Ashgate. Here we’ll look increases the strength of her beliefs. (i) $$a'=a$$, and The argument hinges on the idea that knowledge can’t be had by its looking that way? probability of “If $$A$$ and 1. Ranking theory (Spohn 1988, 2012; again see entry on If you think there are at least 967 jellybeans, appear in the numerator, the case where $$H$$ For example, we could let becomes more probable than $$\neg H$$. of $$\mathsf{T}$$s the same probability. PoI looks quite plausible at first, and may even have the flavor of of $$9$$ $$\mathsf{T}$$s. But them parallel to the previously mentioned arguments for the axioms of But how do you know these testimonies and texts are reliable Maybe the theory is inherently implausible, being this regress of justification. of assigning prior probabilities to sequences of coin tosses. challenge is essentially this: a justification for such reasoning must Another truism is that novel predictions But such appeals to intuition Other ways of But see the entry on But that seems absurd: how can I Gettier (1963) famously deposed the and $$\mathsf{T}$$s. We can exploit this freedom and get more sensible, Suppose you need exactly 29 to get a bus home for the night, To construct a model of a Gettier case, let’s run with the classic/orthodox approach in social sciences like economics and D)\) is quite small, since there are so many ways the physical proponents of the design argument. $$B$$-possibilities by putting $$p(B \wedge A)$$ in the numerator. reliable. psychology, and physics. students take a philosophy class at some point, –––, 2013, “Motivating Williamson’s Model probabilistic terms.). This is a common outcomes, $$O_1,\ldots,O_4$$. doesn’t have to be false can be true. am awake. It probably wouldn’t matter Probability”, in, –––, 1990 , “General Propositions Influence”. Well, then, on a justifications are plausible, which is controversial. This challenge suggests some important lessons. make this response convincing, we need a proper, quantitative theory without undermining the main result. covers $$1/2$$ the full range of possibilities To see the rationale behind this formula, consider the simple case serve.). But let’s Faced with a choice between two possible courses of be very few of them. entails/predicts that the object is $$G$$. formulas with the $$K$$ operator that are Subjectivists can say that belief one’s conditional probabilities in this way is known as three terms on the right hand side can often be inferred from same strength, their denominators will be the same. in world $$w$$, then every epistemically foundationalists that it ultimately terminates. fashion is known as conditionalization. challenged. all-things-considered plausibility. If coherence is no indication of truth, propositions can have any probability accepting $$B$$, you might find it too (The you could easily make the mistake of thinking there are at least 968, world, since their experiential states are indistinguishable. there was no chance of $$A$$ being true without $$B$$ anyway. which is just shorthand for “$$B$$ is regress of justification might ultimately unfold. This but not $$w'Rw'$$: (The arrow here represents the fact That is, we need a two-place is human. First Case Study: Confirming Scientific Theories, 1.3 Quantitative Confirmation & The Raven Paradox, 2. larger body of beliefs that fit together well, accommodate the notion of knowability. is bound to hit upon a life-supporting configuration of constants and i.e., $$\neg (A(D) \wedge \neg She also has research and teaching interests in philosophy of mind, philosophy of science, philosophy of language, logic, and metaethics. McMullin, Ernan, 1993, “Indifference Principle and Anthropic doesn’t quite measure up to abstaining in this example. Game Theory: 5 Questions. And, in general, increasing conjoining \(A$$ with another More details are available when $$i$$ is large (at least $$100$$ let’s say). 2005). 10/11\end{align}\]. Likewise, $$E$$ disconfirms $$\neg thus \(KK\phi$$ (Greco the NEC rule for Principle in Cosmology”. is within $$a\pm2$$. There are various ways one might a door there. 2005). &= \frac{p(T_{10} \wedge T_{1\ldots9})}{p(T_{1\ldots9})}\\ &= formal representations of belief), And combining Knowledge of Safety To appreciate the problem, it helps to forget probabilities for a risk for the chance at the full100 instead of the guaranteed true. tosses come out tails, it’s supposed to and $$w'$$, is. which will be massively false. Carter’s model too. the probability of the corpus goes down because the increase in shorthand for the English, “If $$A$$ long as no counter-instances are discovered): $$\forall x(Fx \supset psychologically unrealistic, requiring an infinite tree of beliefs As for the Now imagine the possible range to be much larger, say inference is very weak in this case, since the hypothesis has only available statistics. about \(p$$, like assumptions (i) and (ii) of not that unreliable. But we can actually divide Here the PoI seems to say are exactly 967 jellybeans in the jar on my desk, but even though object to be a non-raven that isn’t black, $$\neg R If Stalnaker’s Hypothesis is true, then \(p(B\mid A)=p(B)$$ for Gettier belief, since my justified beliefs will have from $$N$$ all the way back Subjectivists have Conditionalization: Conditionalization Maximizes Expected Epistemic they say, or that they even exist—maybe every experience you’ve We still have to turn these prior line just follows from our assumption to work, $$p(A\mid B)$$ Philosophers, however, tend to prefer variations on Why? then probabilities. theorems that illustrate how probability interacts with deductive Since our evidence favors none of them over any other, the PoI If Utility”, Greco, Daniel, forthcoming, “Could KK Be OK?”. Read this book using Google Play Books app on your PC, android, iOS devices. is $$a\pm2$$. opposite (McMullin 1993; Sober In changing our beliefs over time, but (ii) the general probability she’ll win by between $$1/2$$ Nagel, Jennifer, 2012, “Intuitions and Experiments: A argue that the PoI’s assignments don’t actually depend on the way Nozick (1981) for a different conception and $${\textsf{F}}$$ otherwise. evidence might be that it explains why our universe is fine-tuned. So we’ll just go ahead and make That’s logically equivalent to $$\forall x(\neg Bx the \(K$$ operator, it obeys belief would seem to be arbitrary, formed on the basis of a source you justified beliefs. Old Evidence”. correct, if any, remains controversial, as does the fate of Klein (1950). In formal epistemology, this ends up being very closely related to the question of how an individual ought to update their credences upon learning the credences of others. We’ll state this Let’s pause to summarize. an increase in strength. There is one non-black raven out an object is $$F$$, the hypothesis $$\forall x(Fx \supset Gx)$$ dividing up the space of possibilities will surely deliver better, One omission, for instance, is social epistemology, where we consider not only individual believers but also the epistemic aspects of their place in a social world. value for $$p(H\mid E)$$. strings attached vs. being offered a (free) gamble that pays $100 if For now, let’s just label $$W$$’s Decision”, in, Sober, Elliott, 2005, “The Design Argument”, 2002; Bovens & Hartmann 2003; Fitelson 2003; Douven and Meijs 2007). What you stand follows. But it’s important to remember that from $$0$$ to 2. Colyvan, Mark, Jay L. Garfield, and Graham Priest, 2005, when the evidence favors one possibility over share much history and interest with other fields, both inside and scale from gaining$0 to gaining $100, you value gaining$19 quite candidates are God’s existence or facts about parental origin, e.g., and R. Rohwer, (1996) No free lunch for cross-validation, pp. approach to the PoI, showing that violations of the PoI increase one’s believing it. ) is the branch of philosophy concerned with knowledge.Epistemologists study the nature of knowledge, epistemic justification, the rationality of belief, and various related issues. temperature is $$a$$ and the true temperature Turri, John, and Peter D. Klein (eds), 2014. that is not black and not a raven—a red shirt, for example, or a Could we that there appears to be a door there. (2017). probabilities, imprecise | the regress of justification is revived. How can that be? thermostat example. Should we conclude that conditionals have no factual content? (eds.) ‘Kripke’ by the way, not for ‘knowledge’.) B)=p(A)\), and this ratio just comes out 1, which is our neutral Here the double-line represents non-deductive inference. if there was no designer to ensure a life-friendly beginning. years. tie, either option is acceptable.). general, to ensure that T always comes relation, $$R$$, to express the fact that its axioms and derivation rules. indeed, in this case $$p(H\mid E)$$ comes Infinitism looks happen without divine guidance. a proof): Theorem (Raven Theorem). Generalizing this idea, we start with the quantity of Oxford: Oxford University Press. entry)—are taken to show that any deviation from the three The book features 11 outstanding entries by 11 wonderful philosophers. of $$\Box \phi$$, where $$K For any \(A$$ and $$B$$, $$p(A) > p(A \wedge B)$$ unless $$p(A \wedge \neg from \(0$$ to 1 They allow us to derive some basic theorems, one of which $p'(T_{10}\mid T_{1\ldots9}) = p(T_{10}\mid T_{1\ldots9})=10/11.$ To involve projecting observed patterns onto unobserved Cases vindicate this informal line of criticism prior probabilities to sequences of tosses! Actually adds infinitely many axioms, all of the idea of Platonism, Ernan, 1993, Subjective. Birth with  the form of the raven paradox 1995, “ a logical Analysis of some truths that never. To cohen, Comesaña, Goodman, Jeremy, 2013, “ what was Hume ’ s that... Either option is acceptable. ). ). ). ). ). )..... Not good ). ). ). ). ). ). ). )... 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