Notes 2.3 Deductive Reasoning.

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Sec.2-3 Deductive Reasoning

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Notes 2.3 Deductive Reasoning

INDUCTIVE REASONING DEDUCTIVE REASONING draw conclusions from facts, definitions, and properties. Decisions based on scientific evidence, facts, or proof. using common sense or patterns to draw conclusions. Not based on any scientific evidence or fact. Based only on common sense.

Example 1 Is the conclusion a result of inductive or deductive reasoning? There is a myth that you can balance an egg on its end only on the spring equinox. A person was able to balance an egg on July 8, September 21, and December 19. Therefore this myth is false. Since the conclusion is based on a pattern of observations, it is a result of inductive reasoning.

Example 2 Is the conclusion a result of inductive or deductive reasoning? There is a myth that the Great Wall of China is the only man-made object visible from the Moon. The Great Wall is barely visible in photographs taken from 180 miles above Earth. The Moon is about 237,000 miles from Earth. Therefore, the myth cannot be true. The conclusion is based on logical reasoning from scientific research. It is a result of deductive reasoning.

The Law of detachment If a conditional statement is true and the hypothesis is true, then the conclusion must be true. If p  q is a true statement and p is true, then q is true.

Example 2 – law of detachment Determine if the conjecture is valid by the Law of Detachment. Given: In the World Series, if a team wins four games, then the team wins the series. The Red Sox won four games in the 2004 World Series. Conjecture: The Red Sox won the 2004 World Series.

Example 2 continued Identify the hypothesis and conclusion in the given conditional. In the World Series, if a team wins four games, then the team wins the series. The statement “The Red Sox won four games in the 2004 World Series” matches the hypothesis of a true conditional. By the Law of Detachment, the Red Sox won the 2004 World Series. The conjecture is valid.

Law of syllogism If a hypothesis leads to a conclusion and the conclusion is a hypothesis that leads to another conclusion, then the first hypothesis leads you to the last conclusion. If p  q and q  r are true statements, then p  r is a true statement.

Example 3 Determine if the conjecture is valid by the Law of Syllogism. Given: If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal. Conjecture: If an animal is a dog, then it has hair.

Example 3 Continued H1 C1 Animal is a mammal animal has hair H2 C2 Animal is a dog Animal is a mammal The conjecture is valid by Law of Syllogism.