Print this chapterPrint this chapter

Lesson 4

1. Lesson 4

1.4. Explore 3

Mathematics 20-2 M2 Lesson 4

Module 2: Logic and Geometry

 

Errors in logic can also lead to invalid conclusions. Take a look at the following cartoon.

 

This is a cartoon of a penguin with a thought bubble above its head stating, “Penguins are black and white. Some old TV shows are black and white. Therefore, some penguins are old TV shows.” At the bottom of the cartoon the following statement appears: “Logic: another thing that penguins aren’t very good at.”

© 1997 Randy Glasbergen. www.glasbergen.com

 

It is true that penguins are black and white. It is true that some old television shows are black and white. To deduce from those two statements that penguins are old TV shows is not logical. There is no connection between the two statements. When there is an error in logic (an invalid argument), no reliable conclusion can be made regardless of whether the premises are true.

 

Share 3
 

Consider the following arguments. A Venn diagram can be used to examine each argument.

 

This is a diagram of two Venn diagrams that are used to visualize two arguments. The premise of both arguments reads: Everyone who drives at 90 km/h is breaking the law. The first argument notes that Kai is driving at 90 km/h; therefore, Kai is breaking the law. The second argument states that Rose is breaking the law. The conclusion being that Rose is driving at 90 km/h. Both Venn diagrams show an inner circle that reads “drives at 90 km/h.” The outer circle in both diagrams reads “breaking the law.”

 

With a partner, discuss whether each argument is valid and justify your reasoning. How do the diagrams help you to answer this question? Where does Kai’s name belong in the Venn diagram? Where does Rose’s name belong in the Venn diagram?

 

Self-Check 3
  1. Decide whether this argument is valid or invalid. Justify your response:

    1. Val lives in Sylvan Lake. All people who live in Sylvan Lake like to swim. Therefore, Val likes to swim. Answer
      2.  
    2. A pound of Julia’s Special Blend coffee costs more than $22.75 per pound. John didn’t buy Julia’s Special Blend when he purchased coffee. Therefore, John didn’t spend more than $22.75 per pound. Answer

textbook
  1. Complete “Practising” questions 4 and 6 on pages 42 and 43 of your textbook. If you don’t have a house number (street address number), make one up for question 6. Answer

textbook
For more examples of using reasoning to determine the validity of proofs, you can read either or both “Example 1: Using reasoning to determine the validity of an argument” on page 37 and “Example 5: Using reasoning to determine the validity of a proof” on page 41 of your textbook.

notes organizer

If you haven’t done so already, organize your thoughts on identifying errors in proofs in your notes organizer. You might find it helpful to identify specific types of errors that invalidate a proof, such as mathematical errors like dividing by zero or errors in logic like circular reasoning. Reading “In Summary” on page 41 of your textbook may be useful for helping to organize the key ideas of this lesson.