The support which the premises provide for the conclusion is dependent on the number of individuals in the sample group compared to the number in the population, and the randomness of the sample. The hasty generalization and biased sample are fallacies related to generalization. Statistical syllogism  A statistical syllogism proceeds from a generalization to a conclusion about an individual. A proportion Q of population P has attribute A. An individual I is a member of P. Concl ...

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Induction philosophy, Induction philosophy - Validity, Induction philosophy - Types of inductive reasoning, Induction philosophy - Bayesian inference

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Mathematical induction - Start at b. This type of proof can be generalized in several ways. For instance, if we want to prove a statement not for all natural numbers but only for all numbers greater than or equal to a certain number b then: Showing that the statement holds when n = b. Showing that if the statement holds for n = m ≥ b then the sam ...

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Mathematical induction, Mathematical induction - Example, Mathematical induction - Proof, Mathematical induction - Generalizations, Mathematical induction - Start at b, Mathematical induction - Assume true for all lesser values, Mathematical induction - Transfinite induction, Mathematical induction - Proof or reformulation of mathematical induction

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Consider a dynamic game in which the players are an incumbent firm in an industry and a potential entrant to that industry. As it stands, the incumbent has a monopoly over the industry and does not want to lose some of its market share to the entrant. If the entrant chooses not to enter, the payoff to the incumbent is high (it maintains its monopoly) and the entrant neither loses nor gains (its payoff is zero). If the entrant enters, the incumbent can fight or accommodate the entrant. It will fight by lowering its price, running the entrant ...

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Backward induction, Backward induction - Example of backward induction, Backward induction - Backward induction and economic entry, Backward induction - A paradox of backward induction

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If the rate of change of current in a circuit is one ampere per second and the resulting electromotive force is one volt, then the inductance of the circuit is one henry. 1 H = Wb/A = 1 m2·kg·s–2·A–2 ...

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Henry inductance, Henry inductance - Definition, Henry inductance - SI multiples, Henry inductance - Explanation

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If the rate of change of current in a circuit is one ampere per second and the resulting electromotive force is one volt, then the inductance of the circuit is one henry. 1 H = Wb/A = 1 m2·kg·s–2·A–2 = 1 V·s/A ...

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Henry inductance, Henry inductance - Definition, Henry inductance - SI multiples, Henry inductance - Explanation

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Formal logic as most people learn it is deductive rather than inductive. Some philosophers claim to have created systems of inductive logic, but it is controversial whether a logic of induction is even possible. In contrast to deductive reasoning, conclusions arrived at by inductive reasoning do not necessarily have the same degree of certainty as the initial premises. For example, a conclusion that all swans are white is obviously wrong, but may have been thought correct in Europe until the settlement of Australia. Inductive arguments are n ...

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Induction philosophy, Induction philosophy - Validity, Induction philosophy - Types of inductive reasoning, Induction philosophy - Bayesian inference

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Suppose we wish to prove the statement: for all natural numbers n. This is a simple formula for the sum of the natural numbers up to the number n. The proof that the statement is true for all natural numbers n proceeds as follows. Mathematical induction - Proof. Check if it is true for n = 0. Clearly, the sum of the first 0 natural numbers is 0, and 0(0 + 1) / 2 = 0. So the statement is true for n = 0. We could define the statement ...

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Mathematical induction, Mathematical induction - Example, Mathematical induction - Proof, Mathematical induction - Generalizations, Mathematical induction - Start at b, Mathematical induction - Assume true for all lesser values, Mathematical induction - Transfinite induction, Mathematical induction - Proof or reformulation of mathematical induction

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Consider the following property of lists: length (L ++ M) = length L + length M [EQ] Here ++ denotes the list append operation. In order to prove this, we need definitions for length and for the append operation. length [] = 0 [LEN1] length (h:t) = 1 + length t [LEN2] [] ++ list = list [AP ...

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Structural induction, Structural induction - Example, Structural induction - Well-ordering, Structural induction - Structural recursion

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Definition and meaning of inductive reasoning:

inductive reasoning - the process of observing data, recognizing patterns, and making generalizations from the observations; reasoning from particular facts to a general conclusion

(Source: US National Oceanic and Atmospheric Administration (NOAA) )

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Transfinite recursion is a notion closely related to transfinite induction, but whereas the latter is a method of proof, the former is a method of definition or construction. In the simple form, indexed by ordinals, one defines a sequence of sets--say, Aα for every ordinal α, or perhaps every α less than some bound γ--by specifying three things: What A0 is How to determine Aα+1 from Aα (or possibly from the entire s ...

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Transfinite induction, Transfinite induction - Transfinite recursion, Transfinite induction - Relationship to AC

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There is a popular misconception that transfinite induction, or transfinite recursion, or both, require the axiom of choice (AC). This is incorrect. However it is very often the case that proofs or constructions using the technique do use AC. For example, consider the following construction of the Vitali set: First, wellorder the reals, say into a sequence <rα | α<c >, where c is the cardinality of the continuum. Let v0 equal r0. Then let v1 equa ...

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Transfinite induction, Transfinite induction - Transfinite recursion, Transfinite induction - Relationship to AC

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Inductance - Mutual inductance. Mutual inductance is the voltage induced in one circuit (the secondary circuit) when the current in another circuit (the primary circuit) changes by a unit amount in unit time. It is important as the mechanism by which transformers work, but it can also cause unwanted coupling between conductors in a circuit. The mutual inductance (in SI) by circuit i on circuit j is given by the double integral Neumann ...

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Inductance, Inductance - Definition, Inductance - Properties of inductance, Inductance - Permeability, Inductance - Coupled inductors, Inductance - Vector field theory derivations, Inductance - Mutual inductance, Inductance - Self-inductance, Inductance - Usage

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We want the dimension of a point to be 0, and a point has empty boundary, so we start with Then inductively, ind(X) is the smallest n such that, for every x∈X and every open set U containing x, there is an open V containing x, where the closure of V is a subset of U, such that the boundary of V has small inductive dimension less than or equal to n − 1. (In the case above, where X is Euclidean n-dimensional space, V will be chosen ...

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Inductive dimension, Inductive dimension - Formal definition, Inductive dimension - Relationship between dimensions

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Of the candidate systems of inductive logic, the most influential is Bayesianism, which uses probability theory as a framework for induction. Bayes theorem is used to calculate how much the strength of one’s belief in a hypothesis should change, given some evidence. There is debate around what it is that informs the original degree of belief. Objective Bayesians seek an objective value for the degree of probability of a hypothesis being correct, and so do not avoid the philosophical criticisms of objectivism. Subjective Bayesians ho ...

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Induction philosophy, Induction philosophy - Validity, Induction philosophy - Types of inductive reasoning, Induction philosophy - Bayesian inference

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Just as standard mathematical induction is equivalent to the well-ordering principle, structural induction is also equivalent to a well-ordering principle. If the set of all structures of a certain kind admits a well-founded partial order, then every nonempty subset must have a minimal element. (This is the definition of "well-founded".) The significance of the lemma in this context is that it allows us to deduce that if there are any counterexamples to the theorem we want to prove, then there must be a minimal counterexample. If we can show ...

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Structural induction, Structural induction - Example, Structural induction - Well-ordering, Structural induction - Structural recursion

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This sense of definition allowed Poincaré to argue with Bertrand Russell over Giuseppe Peano's axiomatic theory of natural numbers. Peano's fifth axiom states: Allow that; zero has a property P; And; if every natural number less than a number x has the property P then x also has the property P. Therefore; every natural number has the property P. This is the principle of complete induction, it establishes the property of induction as necessary to the system. Since Peano's axiom is as infini ...

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Preintuitionism, Preintuitionism - The introduction of natural numbers, Preintuitionism - The principle of complete induction, Preintuitionism - Arguments over the excluded middle, Preintuitionism - Other Pre-Intuitionists

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When n = 1, . For the inductive step, assume it holds for m. Then for n = m + 1, (a + b)m + 1 = a(a + b)m + b(a + b)m = by the inductive hypot ...

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Binomial theorem, Binomial theorem - Newton's generalized binomial theorem, Binomial theorem - Binomial type, Binomial theorem - Proof inductive, Binomial theorem - Trivia

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In his book Fact, Fiction, and Forecast, Goodman introduced the "new riddle of induction", so-called by analogy with Hume's classical problem of induction. He accepted Hume's observation that inductive reasoning (i.e. inferring from past experience about events in the future) was based solely on human habit and regularities to which our day to day existence has accustomed us. Goodman argued, however, that Hume overlooked the fact that some regularities establish habits (a given piece of copper conducting electricity increases the cred ...

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Nelson Goodman, Nelson Goodman - Career, Nelson Goodman - Induction and grue, Nelson Goodman - Nominalism and mereology, Nelson Goodman - Bibliography

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Main article: Backward induction There are games that have multiple Nash equilibria, some of which are unrealistic. In the case of dynamic games, unrealistic Nash equilibria might be eliminated by applying backward induction, which assumes that future play will be rational. It therefore elimates noncredible (or incredible) threats because such threats would be irrational ...

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Solution concept, Solution concept - Rationalizability & Iterated Dominance, Solution concept - Nash equilibrium, Solution concept - Backward induction, Solution concept - Subgame perfect Nash equilibrium, Solution concept - Perfect Bayesian equilibrium, Solution concept - Forward induction

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From a population containing items of which are special, a sample containing items of which are special can be chosen in ways (see binomial coefficient). Fixing , this expression is the unnormalized deduction distribution function of . Fixing , this expression is the unnormalized induction distribution function of . ...

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Inferential statistics, Inferential statistics - Deduction and induction, Inferential statistics - Mean ± standard deviation, Inferential statistics - Limiting cases, Inferential statistics - Binomial and Beta, Inferential statistics - Poisson and Gamma

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Aside from the method of coupling energy into the mercury vapor, these lamps are very similar to conventional fluorescent lamps. Mercury vapor in the discharge vessel is electrically excited to produce short-wave ultraviolet light, which then excites the phosphors to produce visible light. While still relatively unknown to the public, these lamps have been available since 1990. The most common form has the shape of an incandescent light bulb. Unlike an incandescent lamp or conventional fluorescent lamps, there is no electrical connection going inside the glass bulb; the energy is transferred t ...

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Electrodeless lamp, Electrodeless lamp - Fluorescent induction lamps, Electrodeless lamp - Direct-radiating sulfur lamps

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