2-3 Laws of Logic The laws of logic can be used to provide a system for reaching logical conclusions. This system is called deductive reasoning. Deductive reasoning is one of the cornerstones of the study of geometry.

Law of Detachment  If p → q is a true conditional statement and p is true, then q is true.  p → q pp qq

Example 1  1. If a student is enrolled at East Clarendon High, then the student has an ID number.  2. Brandon Shaw is enrolled at East Clarendon High.  3. Brandon Shaw has an ID number.

Example 2  1. If an angle measures 123, then it is obtuse.  2. mC = 123.  3. C is obtuse.

 Determine if statement 3 follows from statements 1 and 2 by the law of detachment.  1. If you are not satisfied with a tape, then you can return it within a week for a full refund.  2. Joe is not satisfied with a tape.  3. Joe can return the tape within a week for a full refund.

 1. If a rectangle has four congruent sides, then it is a square.  2. A square has diagonals that are perpendicular.  3. A rectangle has diagonals that are perpendicular.

Law of Syllogism  If p → q and q → r are true conditional statements, then p → r is true.  p → q  q → r  p → r

Example 1  1. If a number is a whole number, then it is an integer.  2. If a number is an integer, then it is a rational number.  3. If a number is a whole number, then it is a rational number.

Example 2  1. If fossil fuels are burned, the acid rain is produced.  2. If acid rain falls, wildlife suffers.  3. If fossil fuels are burned, then wildlife suffers.

Determine if statement 3 follows from statements 1 and 2 by the law of syllogism.  1. If Matt lives in Tampa, then he lives in Florida.  2. If Matt lives in Florida, then he lives in the USA.  3. If Matt lives in Tampa, then he lives in the USA.

 1. A vocalist can read music.  2. Ann can read music.  3. Ann is a vocalist.

Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism.  If you plan to attend the University of Notre Dame, then you need to be in the top 10% of your class.  Rosita Nathan plans to attend Notre Dame.  Rosita Nathan needs to be in the top 10% of her class.  If an angle has a measure less than 90, it is acute.  If an angle is acute, then its supplement is obtuse.  If an angle has a measure less than 90, then its supplement is obtuse.

 Vertical angles are congruent.  <1 = <2  <1 and <2 are vertical

Determine if a valid conclusion can be reached from the two true statements using the Law of Detachment or the law of Syllogism.  If you want good health, then you should get 8 hours of sleep each day.  Patricia Gorman wants good health.  Patricia Gorman should get 8 hours of sleep each day.  Detachment

 If AB = BC and BC = CD, then AB =CD.  AB = CD  No conclusion

 If the measure of an angle is less than 90, then it is acute.  If an angle is acute, then it is not obtuse.  If the measure of an angle is less than 90, then it is not obtuse.  Syllogism

 If there are two points, then there is exactly one line that contains them.  There exist two points A and B.  There is exactly one line that contains points A and B.  Detachment.