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# Inductive versus Deductive Reasoning

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Arguments and Reasoning

Arguments

• An argument is a series of statements, progressing (usually in order, but not necessarily) from the premises , which are the assumptions (true or untrue), to the conclusion .
• The purpose of an argument is to present the premises in such a way as to support the truth of the conclusion .
• concrete statement is one that provides a specific example of a concept rather than just a generalization.  For instance:
• Generalization : If , then , therefore .
• Concrete : If it rains, I carry an umbrella. It is raining, therefore I am carrying an umbrella.
• An argument is valid if the truth of its premises assures the truth of its conclusion .
• An argument is invalid or fallacious if it is not valid .
• sound argument has both true premises and valid reasoning.

Types of Reasoning

Deductive Reasoning – Begins with the question or theory and works toward specific examples or evidences to support or renounce it.

Example: 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.

Example: 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.

Euler Diagrams

An Euler diagram is similar to a Venn diagram. It is a visual representation of the relationship between sets, subsets, and members.

Generally, one oval is constructed to represent each set described in the argument, and an “X” is used to represent solitary units. Possible relationships can be expressed by the location of the ovals and “X’s”.

Valid Forms

It is common when describing forms of argument to replace sentences or phrases with single letters, such as $P$ and $Q$ . By using letters to generalizean argument form, we can more easily evaluate a concrete argument for validity. It is a common, and useful, practice to replace  $P$ and  $Q$ with statements of your own in order to clarify the use of a particular form.

• Modus ponens (affirm by affirming): If $P$ , then $Q$ . $P$ , therefore $Q$ .
• If water is frozen, then it is below 32 degrees Fahrenheit. This water is frozen, therefore it is below 32 degrees Fahrenheit.
• Modus tollens (denying the consequent): If $P$ , then $Q$ . Not $Q$ , therefore not $P$ .
• If water is frozen, then it is below 32 degrees Fahrenheit. This water is not below 32 degrees Fahrenheit, therefore it is not frozen.
• Hypothetical syllogism (the chain argument): If $P$ , then $Q$ . If $Q$ , then $R$ . Therefore, if  $P$ then $R$ .
• If you wear sunscreen, you won’t get sunburn. If you don’t get sunburn, you will not get skin cancer. Therefore, if you wear sunscreen, you won’t get skin cancer.
• Disjunctive syllogism :  $P$ or $Q$ . Not $P$ . Therefore $Q$ . (also works in reverse)
• You are either dead or alive. You are not dead. Therefore you are alive.
• You are either dead or alive. You are not alive. Therefore you are dead.

Hidden Premises

hidden premise is a premise that is required in order to reach the stated conclusion, but is not itself stated clearly in the argument. Consider the following:

“My bag of candy is better than yours, because mine has more red pieces”.

This is not a valid argument as written, what is wrong with it?

Let’s break it down and see:

Premise 1: My bag of candy has more red pieces

Hidden premise: Red candy pieces are better than other-colored pieces.

Conclusion: My bag of candy is better than yours.

Without the assumption of the hidden premise , the conclusion makes no sense, and the argument is invalid . In order to make a decision about the soundness of the argument, you will need to decide if you accept the premise “red candies are best”. If you agree that “red candies are best” is a viable premise, the argument is sound, and the conclusion is reasonable. If you believe that yellow candies are better than red ones, then you will obviously reject the premise, and the conclusion will no longer seem reasonable. Regardless of your feelings about red candy, however, the important point here is that you must take the hidden premise into account as you evaluate the argument.

Structural Fallacies:

Structural or formal fallacies are fallacies based on the form of the argument. In the case of a formal fallacy, the conclusion may or may not be true, but it does not follow from the premises.

Common Structural Fallacies:

• Affirming the Consequent : If A then B. B, therefore A.
• If it is snowing, I wear my boots.  I am wearing my boots, therefore it is snowing.
• Just because I wear my boots when it is snowing does not mean I don’t also wear my boots for some other reason.
• Appeal to Ignorance : Use the absence of proof for a premise as evidence in favor of the opposing premise.
• There are no fossilized remains of a winged snake, so snakes must not have evolved into birds.
• The lack of proof of winged snakes is not, in and of itself, proof either for or against the evolution of snakes to birds.
• Diversion : Trying to support one premise by arguing for other premise.
• ABC Dog Food is flavored with beef-like flavoring.  According to studies, dogs choose hamburgers 3:2 over chicken tenders, so ABC Dog Food is the best.
• Showing that dogs prefer hamburger to chicken tenders is not evidence that ABC Dog Food tastes better than any other dog food.
• Equivocation : Using one meaning of a word in the premise, and another in the conclusion.
• Criminal actions are illegal.  All murder trials are criminal actions.  Therefore all murder trials must be illegal.
• Coincidental Correlation (also known as “post hoc ergo propter hoc”, which means “after this, therefore because of this” or just “post hoc”):  Falsely assuming that just because one thing occurs after another, it must have been  caused by the other.
• Public school attendance has skyrocketed in the past 10 years, and so has the number of kids in juvenile hall, so school must be corrupting children.

Content Fallacies:

content fallacy or informal fallacy is a logical fallacy based on what is stated in the premises, rather than the form in which they are presented.

Common Content Fallacies:

• Ad Hominem : This fallacy is committed when an argument is based on the perceived failings of an adversary.
• My sister likes that book, and she is annoying. The book must be bad.
• Bandwagon : This is an argument based on the concept that the majority is always right.
• That video has 100,000 hits, it must be really good!
• Begging the Question (circular argument) : An argument that assumes the truth of its conclusion.
• Executions are moral because we must have a death penalty to discourage violent crime.
• False Dilemma : An argument which over simplifies a complex situation into only two possible alternatives.
• Bad people make bad decisions, good people good ones. I lied once, so I must be a bad person.
• Non-Sequitur : An argument where the conclusion is not based on the premises.
• I am a math teacher, so I know all about fashion.
• Straw Man : An argument based on misrepresenting the opponent’s argument so it may be easily defeated.
• “ Straw man has always been a stock-in-trade of advertisers.... A Post Office commercial once pictured competitors trying to deliver packages with rickety old planes that fell apart on camera.” (H. Kahane and N. Cavender, Logic and Contemporary Rhetoric. Wordsworth, 1998)