Today’s Lesson is on DEDUCTIVE REASONING

In Chapter 1 you learned that inductive reasoning is based on observing what has happened and then making a conjecture about what will happen. In this lesson, you will study deductive reasoning. Deductive reasoning (or logical reasoning) is the process of reasoning logically from given statements to a conclusion. If the given statements are true, deductive reasoning produces a true conclusion. Many people use deductive reasoning in their jobs. A physician diagnosing a patient’s illness uses deductive reasoning. A carpenter uses deductive reasoning to determine what materials are needed at a work site.

Law of Detachment Auto Maintenance An auto mechanic knows that if a car has a dead battery, the car will not start. A mechanic begins work on a car and finds the battery is dead. What conclusion can she make? Exactly: The mechanic can conclude that the car will not start. Critical Thinking Suppose that a mechanic begins work on a car and finds that the car will not start. Can the mechanic conclude that the car has a dead battery? Explain.

The mechanic is using a law of deductive reasoning called The Law of Detachment consequent antecedent

P Q P _______ Q

Examples If a baseball player is a pitcher, then that player can not pitch a complete game two days in a row. Vladimir Nuñez is a pitcher. On Monday, he pitches a complete game.What can you conclude? For the given true statements, what can you conclude? Given: If M is the midpoint of a segment, then it divides the segment into two congruent segments. M is the midpoint of segment AB.

Does the following argument illustrate the Law of Detachment? Given: If it is snowing, then the temperature is less than or equal to 32°F. The temperature is 20°F. You conclude: It must be snowing. You are given that a conditional and its conclusion are true. You cannot apply the Law of Detachment and conclude that the hypothesis is true. You cannot come to any conclusion about whether it is snowing from the information given.

The error illustrated is sometimes called the fallacy of the converse. A fallacy is an error in logical thinking. Why is this error a “fallacy of the converse”? The fallacy is concluding that the converse is true because the conditional is true. Q x x P PQ Q Can you assume that P is true based on this law of detachment?

If possible, use the Law of Detachment to draw a conclusion If possible, use the Law of Detachment to draw a conclusion. If it is not possible to use this law, explain why. Given: If a road is icy, then driving conditions are hazardous. Driving conditions are hazardous.

Another law of deductive reasoning is The Law of Syllogism The Law of Syllogism allows you to state a conclusion from two true conditional statements when the consequent of one statement is the antecedent of the other statement.

Algebra Use the Law of Syllogism to draw a conclusion from the following true statements. If a number is prime, then it does not have repeated factors. If a number does not have repeated factors, then it is not a perfect square. What conclusion can you draw from this information? 31 25 2 5 3

If possible, state a conclusion using the Law of Syllogism If possible, state a conclusion using the Law of Syllogism. If it is not possible to use this law, explain why. If a number ends in 0, then it is divisible by 10. If a number is divisible by 10, then it is divisible by 5. If a number ends in 6, then it is divisible by 2. If a number ends in 4, then it is divisible by 2

Try this: and explain your conclusions by stating which law(s) were used If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle. PQ, QR, Then PR

If the circus is in town, then Paul is working as a night watchman. Try this: and explain your conclusions by stating which law(s) were used If the circus is in town, then there are tents at the fairground. If there are tents at the fairground, then Paul is working as a night watchman. The circus is in town. If the circus is in town, then Paul is working as a night watchman.

Look at the symbolic representation If the circus is in town, then there are tents at the fairground. If there are tents at the fairground, then Paul is working as a night watchman. If the circus is in town, then Paul is working as a night watchman. The circus is in town. CT TP This means that using the Law of Syllogism you can get that CP C Using the Law of Detachment, you can get P Therefore, Paul is working as a night watchman.

Deductive reasoning is a process of reasoning logically from given facts to a conclusion. The Law of Detachment is found in almost every line of two-column proofs, where q is the “conclusion” and p  q is the “justification” for q. It is p, the given, that is sometimes lost when thinking about each line.

The use of the word Given before the two statements is an important concept that I just figured you would understand. However, in mathematics, the statements following Given are always considered true. What do you call statements that are assumed to be true without proof?