Chapter 3 Introduction to Logic © 2008 Pearson Addison-Wesley. All rights reserved

Chapter 3: Introduction to Logic 3.1 Statements and Quantifiers 3.2 Truth Tables and Equivalent Statements 3.3 The Conditional and Circuits 3.4 More on the Conditional 3.5 Analyzing Arguments with Euler Diagrams 3.6 Analyzing Arguments with Truth Tables © 2008 Pearson Addison-Wesley. All rights reserved

Section 3-5 Chapter 1 Analyzing Arguments with Euler Diagrams © 2008 Pearson Addison-Wesley. All rights reserved

Analyzing Arguments with Euler Diagrams Logical Arguments Arguments with Universal Quantifiers Arguments with Existential Quantifiers © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Logical Arguments A logical argument is made up of premises (assumptions, laws, rules, widely held ideas, or observations) and a conclusion. Together, the premises and the conclusion make up the argument. © 2008 Pearson Addison-Wesley. All rights reserved

Valid and Invalid Arguments An argument is valid if the fact that all the premises are true forces the conclusion to be true. An argument that is not valid is invalid. It is called a fallacy. © 2008 Pearson Addison-Wesley. All rights reserved

Arguments with Universal Quantifiers Several techniques can be used to check the validity of an argument. One of these is a visual technique based on Euler Diagrams. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Using an Euler Diagram to Determine Validity (Universal Quantifier) Is the following argument valid? All cats are animals. Figgy is a cat. Figgy is an animal. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Using an Euler Diagram to Determine Validity (Universal Quantifier) All cats are animals. Figgy is a cat. Figgy is an animal. Solution The diagram shows that Figgy is inside the region for “animals”. The argument is valid. Animals Cats x x represents Figgy. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Using an Euler Diagram to Determine Validity (Universal Quantifier) Is the following argument valid? All sunny days are hot. Today is not hot Today is not sunny. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Using an Euler Diagram to Determine Validity (Universal Quantifier) All sunny days are hot. Today is not hot Today is not sunny. Solution The diagram shows that today is outside the region for “sunny”. The argument is valid. Hot days x Sunny days x represents today © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Using an Euler Diagram to Determine Validity (Universal Quantifier) Is the following argument valid? All cars have wheels. That vehicle has wheels. That vehicle is a car. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Using an Euler Diagram to Determine Validity (Universal Quantifier) All cars have wheels. That vehicle has wheels. That vehicle is a car. Solution The diagram shows “that vehicle” can be inside the region for “Cars” or outside it. The argument is invalid. Things that have wheels x ? Cars x ? x represents “that vehicle” © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Using an Euler Diagram to Determine Validity (Existential Quantifier) Is the following argument valid? Some students drink coffee. I am a student . I drink coffee . © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Using an Euler Diagram to Determine Validity (Universal Quantifier) Some students drink coffee. I am a student . I drink coffee . Solution The diagram shows that “I” can be inside the region for “Drink coffee” or outside it. The argument is invalid. People that drink coffee I ? Students I ? © 2008 Pearson Addison-Wesley. All rights reserved