The Jellybean Club and the Case of the Puzzling Package

The Jellybean Club and the Case of the Puzzling Package
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Dirty vs. From , Mike wrote and illustrated most of the Berenstain Bears books with his mother, Jan. With Stan and Jan Berenstain. Hop, Waddle, Swim! Western Publishing, Wind Monkey, the: and other stories. Random House, Authored by Stan, Jan, and Mike Berenstain.

Formal Epistemology

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The Berenstain Bears and the Trouble with Friends. The Berenstain Bears and the Ghost of the Forest.

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The Berenstain Bears Earthsaver Calendar for The Berenstain Bears and the Trouble with Grownups. This weakness of the probability axioms generates the famous problem of the priors , the problem of saying where initial probabilities come from.

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Are they always based on evidence previously collected? If so, how does scientific inquiry get started? Formal epistemologists are split on this question.

The so-called objectivists see the probability axioms as incomplete, waiting to be supplemented by additional postulates that determine the probabilities with which inquiry should begin. The Principle of Indifference PoI is the leading candidate here. See the entry on the interpretation of probability. In later sections the problem of the priors will return several times, illustrating its importance and ubiquity.

We also saw that it raises a problem though, the problem of priors, which formal epistemologists are divided on how to resolve. These and other problems have led to the exploration and development of other approaches to scientific reasoning, and reasoning in general. Some stick to the probabilistic framework but develop different methodologies within it Fisher ; Neyman and Pearson a,b; Royall ; Mayo ; Mayo and Spanos ; see entry on the philosophy of statistics. Others depart from standard probability theory, like Dempster-Shafer theory Shafer ; see entry on formal representations of belief , a variant of probability theory meant to solve the problem of the priors and make other improvements.

Ranking theory Spohn , ; again see entry on formal representations of belief also bears some resemblance to probability theory but draws much inspiration from possible-world semantics for conditionals see entry on indicative conditionals. Bootstrapping theory Glymour ; Douven and Meijs leaves the probabilistic framework behind entirely, drawing inspiration instead from the deduction-based approach we began with.

Still other approaches develop non-monotonic logics see entry , logics for making not only deductive inferences, but also defeasible, inductive inferences Pollock , ; Horty Formal learning theory provides a framework for studying the long-run consequences of a wide range of methodologies. A lot of our reasoning seems to involve projecting observed patterns onto unobserved instances. What justifies this kind of reasoning? Hume famously argued that nothing can justify it. Appealing to an inductive argument would be unacceptably circular.

While a deductive argument would have to show that unobserved instances will resemble observed ones, which is not a necessary truth, and hence not demonstrable by any valid argument. So no argument can justify projecting observed patterns onto unobserved cases. Russell and Restall offer a formal development.

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Haack discusses the supposed asymmetry between induction and deduction here. Can probability come to the rescue here? How can that be? Of course, only two of them begin with 9 tails in a row, namely the last two.

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More formally, applying the definition conditional probability gives us:. So it looks like the axioms of probability entail that the first 9 tosses tell us nothing about the 10 th toss. As we saw earlier when we encountered the problem of the priors 1. We can exploit this freedom and get more sensible, induction-friendly results if we assign prior probabilities using a different scheme advocated by Carnap And so on.

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The Jellybean Club and the Case of the Puzzling Package book. Read reviews from world's largest community for readers. Adventure and. Adventure and suspense meet again in this sequel to the first Jellybean book! This time the Jellybean Club is caught by surprise in the most unlikely place and .

However, this scheme is not mandated by the axioms of probability. But the axioms are also compatible with skepticism about induction. Why only on our way? This brings us to a classic debate in formal epistemology.

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If you had to bet on a horserace without knowing anything about any of the horses, which one would you bet on? This kind of reasoning is common and is often attributed to the Principle of Indifference : [ 5 ]. PoI looks quite plausible at first, and may even have the flavor of a conceptual truth.

And yet, the PoI faces a classic and recalcitrant challenge. Consider the first horse listed in the race, Athena. There are two possibilities, that she will win and that she will lose. But suppose there are three horses in the race: Athena, Beatrice, and Cecil. The source of the trouble is that possibilities can be subdivided into further subpossibilities.

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The possibility of Athena losing can be subdivided into two subpossibilities, one where Beatrice wins and another where Cecil wins. What we need, it seems, is some way of choosing a single, privileged way of dividing up the space of possibilities so that we can apply the PoI consistently. But we can actually divide things further—infinitely further in fact.

For example, Athena might win by a full length, by half a length, by a quarter of a length, etc. So the possibility that she wins is actually infinitely divisible. But the same problem we were trying to solve still persists, in the form of the notorious Bertrand paradox Bertrand []. The paradox is nicely illustrated by the following example from van Fraassen Once again, the probabilities given by the PoI seem to depend on how we describe the range of possible outcomes. Described in terms of length, we get one answer; described in terms of volume, we get another.

We face essentially this problem when we frame the problem of induction in probabilistic terms. Earlier we saw two competing ways of assigning prior probabilities to sequences of coin tosses. So one way of applying the PoI leads to inductive skepticism, the other yields the inductive optimism that seems so indispensable to science and daily life. Can it be clarified and justified? Here again we run up against one of the deepest and oldest divides in formal epistemology, that between subjectivists and objectivists.