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3.3 Deductive & Inductive Reasoning

3.3 Deductive & Inductive Reasoning
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  • 9. Undefined Terms

    Consider the following postulates.

    Postulate 1: Given any two towns, a road passes through them.
    Postulate 2: Given any road, there is at least one town that the road does not pass through.
    Postulate 3: There are at least two towns.

    What are the undefined terms? (See Example 3 and Example 4.)

    • Worked-Out Solution

      To answer this question, you need to consider that a logical system is being developed. In this system, you can assume that ordinary English words and concepts are known. However, words and concepts that seem special to the system must be considered undefined. In this postulate system, it is reasonable to assume that the following are undefined.

      • The term "town"
      • The term "road"

      This is comparable to the postulate system for Euclidean Geometry in which the following terms are undefined.

      • The term "point"
      • The term  "line"
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               _    _      _____     ___      _____   
      ____    | || | ||   / ___//   / _ \\   |__  //  
     |    \\  | || | ||   \___ \\  | / \ ||    / //   
     | [] ||  | \\_/ ||   /    //  | \_/ ||   / //__  
     |  __//   \____//   /____//    \___//   /_____|| 
     |_|`-`     `---`   `-----`     `---`    `-----`  
     `-`                                              
    
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  • 10. Explaining Roads and Towns

    Consider the following postulates.

    Postulate 1: Given any two towns, a road passes through them.
    Postulate 2: Given any road, there is at least one town that the road does not pass through.
    Postulate 3: There are at least two towns.

    Does a road need to be a straight line? Explain your reasoning. (See Example 3 and Example 4.)

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     ______    __   __    _  __   _    _     _____   
    |      \\  \ \\/ //  | |/ // | || | ||  |__  //  
    |  --  //   \ ` //   | ' //  | || | ||    / //   
    |  --  \\    | ||    | . \\  | \\_/ ||   / //__  
    |______//    |_||    |_|\_\\  \____//   /_____|| 
    `------`     `-`'    `-` --`   `---`    `-----`  
                                                     
    
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  • 11. Writing a Syllogism Involving Roads and Towns

    Consider the following postulates.

    Postulate 1: Given any two towns, a road passes through them.
    Postulate 2: Given any road, there is at least one town that the road does not pass through.
    Postulate 3: There are at least two towns.

    Write a syllogism that involves the first postulate and illustrate it. (See Example 3 and Example 4.)

    • Worked-Out Solution

      Here is one way to write a syllogism that involves Postulate 1.

      Premise: If there are two towns, then there exists a road that passes through them.
      Premise: There exist two towns.
      Conclusion: There is a road that passes through the two towns.

      Here is one way to illustrate this syllogism.

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      ______   _    _    __   _    __   __   _____    
     /_   _// | || | || | || | ||  \ \\/ // |  __ \\  
       | ||   | || | || | '--' ||   \ ` //  | |  \ || 
      _| ||   | \\_/ || | .--. ||    | ||   | |__/ || 
     /__//     \____//  |_|| |_||    |_||   |_____//  
     `--`       `---`   `-`  `-`     `-`'    -----`   
                                                      
    
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  • 12. Writing a Syllogism Involving Roads and Towns

    Consider the following postulates.

    Postulate 1: Given any two towns, a road passes through them.
    Postulate 2: Given any road, there is at least one town that the road does not pass through.
    Postulate 3: There are at least two towns.

    Write a syllogism that involves the second postulate and illustrate it. (See Example 3 and Example 4.)

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      ______    _____   __    __     ___      ______  
     /_   _//  |  ___|| \ \\ / //   / _ \\   /_   _// 
       | ||    | ||__    \ \/ //   | / \ ||    | ||   
      _| ||    | ||__     \  //    | \_/ ||   _| ||   
     /__//     |_____||    \//      \___//   /__//    
     `--`      `-----`      `       `---`    `--`     
                                                      
    
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  • 13. Using Deductive Reasoning

    Consider the following postulates.

    Postulate 1: Given any two towns, a road passes through them.
    Postulate 2: Given any road, there is at least one town that the road does not pass through.
    Postulate 3: There are at least two towns.

    Use deductive reasoning to explain why there must be at least three towns. (See Example 3 and Example 4.)

    • Worked-Out Solution

      Using all three premises, you can reason that there exists at least three towns.

      Step 1: There exist two towns. Premise 3
      Step 2: There exists a road passing through the two towns. Premise 1
      Step 3: There exists a town which the road does not pass through. Premise 2
      Step 4: The town in Step 3 is different from the towns in Step 1. A road can't pass through a town and also not pass through it.
      Step 5: So, there exist at least three towns. Follows from steps 1 and 4.

      Do you remember this type of deductive reasoning from high school geometry? Notice that each step must be justified using a premise, logic, or a previously used step.

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      _____    __   __    ______    ______    ______  
     /  ___||  \ \\/ //  /_   _//  /_   _//  /_____// 
    | // __     \ ` //     | ||     -| ||-   `____ `  
    | \\_\ ||    | ||     _| ||     _| ||_   /___//   
     \____//     |_||    /__//     /_____//  `__ `    
      `---`      `-`'    `--`      `-----`   /_//     
                                             `-`      
    
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  • 14. Determining Validity of Models

    Consider the following postulates.

    Postulate 1: Given any two towns, a road passes through them.
    Postulate 2: Given any road, there is at least one town that the road does not pass through.
    Postulate 2: There are at least two towns.

    Determine whether each model is valid. If a model is not valid, identify the postulate(s) that it violates. Explain your reasoning. (See Example 3 and Example 4.)

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      _  __   _    _      _____   __   __            
     | |/ // | || | ||   / ___//  \ \\/ //     ___   
     | ' //  | || | ||   \___ \\   \ ` //     /   || 
     | . \\  | \\_/ ||   /    //    | ||     | [] || 
     |_|\_\\  \____//   /____//     |_||      \__ || 
     `-` --`   `---`   `-----`      `-`'       -|_|| 
                                                `-`  
    
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  • 15. Analyzing Postulates about Roads and Towns

    Consider the following postulates.

    Postulate 1: Given any two towns, a road passes through them.
    Postulate 2: Given any road, there is at least one town that the road does not pass through.
    Postulate 3: There are at least two towns.

    Do the postulates guarantee a town at every intersection of two roads? Explain your reasoning. (See Example 3 and Example 4.)

    • Worked-Out Solution

      To decide whether the postulates guarantee the existence of a town at the intersection of any two roads, try drawing a system in which this is not true. Then, check the system to see whether the postulates are still valid. If they are, then you can conclude that the postulates do not guarantee the existence of a town at the intersection of any two roads.

      The above model satisfies all three postulates. It has an intersection of two roads at which there is no town. So, you can conclude that the postulates do not guarantee the existence of a town at every intersection of two roads.

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      ______     ___      _  _    __   __    ____    
     /_   _//   / _ \\   | \| ||  \ \\/ //  |  _ \\  
     `-| |,-   / //\ \\  |  ' ||   \ ` //   | |_| || 
       | ||   |  ___  || | .  ||    | ||    | .  //  
       |_||   |_||  |_|| |_|\_||    |_||    |_|\_\\  
       `-`'   `-`   `-`  `-` -`     `-`'    `-` --`  
                                                     
    
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  • 16. Changing a Postulate

    Consider the following replacement for Postulate 2.

    Postulate 2: Given any town, there is at least one road that does not pass through the town.
    1. Write a syllogism that involves the postulate and illustrate it.
    2. At least how many towns must exist? Explain your reasoning.
    3. At least how many roads must exist? Explain your reasoning.

    (See Example 3 and Example 4.)

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      _  __     ___     _____      _____     _  __  
     | |/ //   / _ \\  |  __ \\   |  ___||  | |/ // 
     | ' //   | / \ || | |  \ ||  | ||__    | ' //  
     | . \\   | \_/ || | |__/ ||  | ||__    | . \\  
     |_|\_\\   \___//  |_____//   |_____||  |_|\_\\ 
     `-` --`   `---`    -----`    `-----`   `-` --` 
                                                    
    
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