More on Reasoning

As you learned in the Introduction, inductive reasoning uses limited observations to draw conclusions, referred to as conjectures. The Warm-Up gave you an opportunity to practice using your inductive reasoning skills by allowing you to observe the image and make a conjecture about why the man on the couch was displaying a particular behavior. In the following interactivities, you will explore methods of deductive reasoning as you investigate the law of detachment, the law of contrapositive, and the law of syllogism.

 

Law of Detachment

The law of detachment states that when given a true conditional statement, if the hypothesis is true, then the conclusion is also true. Explore this form of deductive reasoning in this video. Click the player button to begin.

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Law of Contrapositive

The law of contrapositive states that when given a true conditional statement, if the negation of the conclusion is true, then the negation of the hypothesis is also true. In this video, you will explore this form of deductive reasoning. Click the player button to begin.

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Law of Syllogism

When given two true conditional statements, for which the conclusion of one statement is the hypothesis of the other, a true conclusion can be drawn using the law of syllogism. Investigate this form of deductive reasoning in this video. Click the player button to begin.

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Truth Versus Validity

When using logic to draw a conclusion, it is important to understand that truth and validity are different concepts. An argument is true if it is based in fact. An argument is valid if its conclusion is drawn logically. Consider the following argument.

Example: Determine if the argument below is true, valid, or both.

All U.S. citizens who are eligible to vote are 18 years old.
Mariah is a U.S. citizen who is eligible to vote.
Mariah is 18 years old.

Answer: It may help to begin by representing the first statement in if-then form.

If a U.S. citizen is eligible to vote, then the citizen is 18 years old.    p → q
Mariah is a citizen who is eligible to vote.    p
Mariah is 18 years old.     ∴ q

Notice that the conclusion was drawn using the law of detachment, which is a valid form of deductive reasoning. So, you can conclude that the argument is valid. Notice, however, that the first statement of the argument is not true.

If a U.S. citizen is eligible to vote, then the citizen is 18 years old.

Citizens of the U.S. who are eligible to vote may be 18 years old or older. So although the conclusion is valid because it is based on a valid form of deductive reasoning, it is false because it is not based on a true statement.

 

Inductive and Deductive Reasoning Review

Now that you have learned about deductive reasoning, review your knowledge in this interactivity. Click the player button to get started.