# 2.4: Reasoning Types

- Page ID
- 2142

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Deductive versus Inductive Reasoning. Deductive is from the general to the specific. Inductive is the reverse. It is impossible for the premises to be true and the conclusion to be false.

## Inductive and Deductive Reasoning

Suppose you were given the task of collecting data from each class in your school on the ratio between male and female students. After reviewing the M:F ratios of each classroom, would you use inductive reasoning or deductive reasoning to come up with a hypothesis regarding an average M:F ratio for the school? What kind of reasoning would be involved if your friend asked you to review your data to see if her theory about ratios being different in different grades was supported by your observations?

## Inductive and Deductive Reasoning

One of the primary uses of probability and statistics is to learn about parameters of a population, and to do that, one must be able to reason from a sample to a population. Either a person observes something and tries to explain it by collecting and distilling data into a conclusion, or else he/she begins with a hypothesis and seeks data to support or renounce it. In this lesson, we will discuss these two types of reasoning, Inductive and Deductive.

**Deductive Reasoning**– Begins with the question or theory and works toward specific examples or evidences to support or renounce it.- Every morning, I eat eggs for breakfast. Every day, I am not hungry again until lunchtime. This morning if I eat eggs for breakfast, I will not be hungry until lunchtime.

**Inductive Reasoning**– Begins with specific observations or data and works toward a general statement to explain it.- This morning I ate eggs for breakfast and was not hungry until lunchtime. As long as I eat eggs for breakfast, I’ll never be hungry until lunchtime.

In scientific study, both sorts of reasoning are used, often in conjunction and to support each other. However, as you will see over the next few lessons, there are a lot of ways to make errors in reasoning (called **fallacies**), and knowing what type of reasoning you are using will help you to learn which fallacies to watch out for!

## Choosing the Applicable Reasoning

1. What sort of reasoning is applicable to finding the solution to a five-step linear equation such as the one below?

\(2(x+3)−7=x+4\)

\(2(x+3)=x+11\)

\(2x+6=11\)

\(2x−x=11−6\)

\(x=5\)

This is deductive reasoning, since we started with a statement or theory: \(2(x+3)−7=x+4\), and used a step-by-step process to find a specific example supporting it, namely that if \(x=5\), then \(2(5+3)−7=5+4\), so the original statement is supported by a specific example.

Since we progressed from general to specific, this was deductive reasoning.

2. Assuming the sequence below, what type of reasoning would you use to conjecture the next number in the sequence?

\(1, 4, 10, 19, 31, 46, 64, …\)

This is an example of inductive reasoning, since we started with a number of specific observations, namely the \(1^{st}, 2^{rd}, 3^{rd}, 4^{nd}\), and so on numbers in a sequence, and use the observations to make the statement that the pattern is to add 3n, where n is the count, to each number to get the next: \(1+3(1)=4, 4+3(2)=10, 10+3(3)=19, 19+3(4)=31\), and so on. That tells us that the next number in the series should be: \(64+3(7)=85\).

Since we progressed from specific to general, this was inductive reasoning.

## Determining What Type of Reasoning is Expressed

What sort of reasoning is expressed in the following statements?

Chloe took her umbrella to work today, and it rained.

Every time Chloe takes her umbrella, it will rain.

This is inductive reasoning, beginning with the specific statement about a specific day and action, and progressing to a general statement about * all* days with the same action.

### Earlier Problem Revisited

*Suppose you were given the task of collecting data from each class in your school on the ratio between male and female students. After reviewing the M**:F**ratios of each classroom, would you use inductive reasoning or deductive reasoning to come up with a hypothesis regarding an average M:F ratio for the school? What kind of reasoning would be involved if your friend asked you to review your data to see if her theory about ratios being different in different grades was supported by your observations?*

First, you begin with specific examples of the ratios of males and females and use them to create a general statement about the ratio of the entire school. That was inductive reasoning: specific to general. Second, you started with the general statement that the ratios are different in different grades, and considered the specific data to support or not support the statement. That was deductive reasoning: general to specific.

For Examples \(\PageIndex{1}\)-\(\PageIndex{4}\), describe the type of reasoning demonstrated in each passage.

Example \(\PageIndex{1}\)

Scott leaves for school at 8:15 in the morning every day, it takes him 15 minutes to get to school, and he arrives on time. If Scott leaves at 8:15 this morning, he will arrive at school on time.

**Solution**

This is deductive reasoning, starting with a general statement about Scott's actions everyday and progressing to the specific occurrence of today.

Example \(\PageIndex{2}\)

On Monday, Sophie went to lunch at the local fast-food joint on her lunch break and arrived back at school in time for class. On Tuesday, she did the same thing and was on time again. If Sophie goes to the same fast-food place for lunch on every day, she will be back in time for class.

**Solution**

This is inductive reasoning, starting with specific examples of actions and progressing to a general statement about every similar action.

Example \(\PageIndex{3}\)

\(3(x−4)−7=6x\), therefore, \(x=−6.3\overline{3}\).

**Solution**

Deductive reasoning, from a general statement to a specific example of the statement being true.

Example \(\PageIndex{4}\)

If \(y=7\), and \(x=4\), therefore \(x \times \dfrac{7}{4}=y\).

**Solution**

Inductive reasoning, from specific stated values of \(x\) and \(y\) to a general statement about them both.

## Review

For each question, state whether the reasoning is an example of inductive or deductive logic.

- All housecats are felines. All felines have claws. Therefore all housecats have claws.
- My dog has fleas. My neighbor’s dog has fleas. Therefore all dogs must have fleas.
- All cows like hay. My cow will like hay.
- My Mac laptop is fast. All Mac laptops are fast.
- My tennis shoes are comfortable. My friend’s tennis shoes are comfortable. All tennis shoes are comfortable.
- The scalloped potatoes I took from the oven were cheesy. The enchiladas I took from the oven were cheesy. If I take cookies from the oven, they will be cheesy.
- Everything cooked on the stove gets hot. If I cook macaroni on the stove, it will get hot.
- iPads are popular. iPhones are popular. Every phone or tablet is popular.
- Roses are red. Tomatoes are red. All red things come from plants.
- Rock music is loud. Sayber listens to rock music. Sayber’s music is loud.
- Milk is good with cookies. Snicker doodles are cookies. Milk is good with snicker doodles.
- Hummers use a lot of gas. Suburbans use a lot of gas. Large SUV’s use a lot of gas.
- My garden has pumpkins. My dad’s garden has pumpkins. All gardens have pumpkins.
- Prob and Stats students are smart. You are a Prob and Stats student. You are smart.
- Students who study hard get good grades. You are a student who studies hard. You will get good grades.

## Vocabulary

Term | Definition |
---|---|

fallacies |
Fallacies are errors in reasoning. |

## Additional Resource

Practice: Reasoning Types