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Chapter 1: Problem 67

Additional information may strengthen or weaken the probability of my inductive arguments.

Short Answer

Expert verified
With inductive arguments, additional information can either strengthen or weaken the argument. This is because inductive reasoning is based on degrees of probability, not certainty. Therefore, new information is essentially a re-calibration of the probability of the truth of the conclusion.

Step by step solution

01

Define the Concept

An inductive argument is a type of argument where the premises are supposed to support the conclusion in such a way that if the premises are true, it is probable that the conclusion is also true.
02

Clarify the Role of Additional Information

In inductive arguments, additional information can either strengthen or weaken the probability of the conclusion being true. This is because inductive arguments depend on degrees of probability rather than certainty. For example, if there is an inductive argument that 'All swans observed so far are white, therefore all swans are white', discovering a black swan would weaken the probability of the conclusion being true, and finding more white swans would strengthen it.
03

Incorporate the Role of Additional Information into Inductive Arguments

When new, relevant information is introduced, it needs to be weighed against the existing premises to reassess their strength and the probability of the conclusion being true. The impact of additional information is not uniform but varies depending on the nature of that information.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is at the heart of inductive reasoning. It measures how likely it is for a certain conclusion to be true based on the given premises. Unlike deductive reasoning, where conclusions are certain if the premises are true, inductive reasoning deals with probabilities.
In everyday situations, probability helps us make decisions based on available information. For instance, if it rains most of the time on cloudy days, you might bring an umbrella when you see clouds. But this is not a guarantee, only a probability.

  • Inductive reasoning relies on generalizations.
  • Probability ranges from 0 (impossible) to 1 (certain).
  • New evidence can alter the probability.
In summary, while not delivering certainty, understanding probability allows us to assess how likely our conclusions are based on the available premises.
Premises
Premises in an inductive argument are the foundation on which probability is built. They are the statements or observations that provide the groundwork for drawing a conclusion. The strength of your conclusion depends heavily on the validity and reliability of your premises.
For instance, observing multiple white swans might lead you to conclude all swans are white. But if your sample is limited, your conclusion could be weak.

  • A premise should be based on accurate information.
  • More premises generally strengthen the argument.
  • Quality of premises affects validity.
Therefore, it鈥檚 important to critically evaluate the premises to gauge their contribution to the overall argument.
Conclusion
A conclusion in inductive reasoning sums up the probable outcome that stems from the premises. It represents what you infer from your observations and collected data.
In the classic swan example, the conclusion that 'all swans are white' is drawn from repeated observations of color-consistent swans. However, inductive conclusions are never perfectly certain, which distinguishes them from deductive ones.

  • Conclusions based on strong premises offer highprobability but not certainty.
  • Additional evidence can alter the conclusion.
  • Conclusions are the final step in inductive reasoning.
To sum up, the goal of inductive reasoning is to arrive at a likely conclusion, always remaining open to adjustments based on new information.

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Most popular questions from this chapter

Five housemates \((A, B, C, D\), and \(E)\) agreed to share the expenses of a party equally. If A spent \(\$ 42, B\) spent \(\$ 10\), C spent \(\$ 26, D\) spent \(\$ 32\), and E spent \(\$ 30\), who owes money after the party and how much do they owe? To whom is money owed, and how much should they receive? In order to resolve these discrepancies, who should pay how much to whom?

In Exercises 9-22, obtain an estimate for each computation by rounding the numbers so that the resulting arithmetic can easily be performed by hand or in your head. Then use a calculator to perform the computation. How reasonable is your estimate when compared to the actual answer? \(39.67 \times 5.5\)

Write your own problem that can be solved using the four-step procedure. Then use the four steps to solve the problem.

In Exercises 9-22, obtain an estimate for each computation by rounding the numbers so that the resulting arithmetic can easily be performed by hand or in your head. Then use a calculator to perform the computation. How reasonable is your estimate when compared to the actual answer? \(359+596\)

In Exercises 57-60, identify the reasoning process, induction or deduction, in each example. Explain your answer. It can be shown that $$ 1+2+3+\cdots+n=\frac{n(n+1)}{2} $$ I can use this formula to conclude that the sum of the first one hundred counting numbers, \(1+2+3+\cdots+100\), is $$ \frac{100(100+1)}{2}=\frac{100(101)}{2}=50(101) \text {, or } 5050 \text {. } $$
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