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3.3.2: Evaluating Deductive Arguments with Euler Diagrams

  • Page ID
    74311
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    3.3.2 Learning Objectives

    • Evaluate deductive arguments with Euler diagrams

    • Use Euler diagrams to determine if an argument is valid

    We can interpret a deductive argument visually with an Euler diagram, which is essentially the same thing as a Venn diagram. This can make it easier to determine whether the argument is valid or invalid.

    Example 1

    Consider the deductive argument “All cats are mammals and a tiger is a cat, so a tiger is a mammal.” Is this argument valid?

    Solution

    There is a large circle and then a smaller circle within it. The word mammals is in the larger circle only. The words cats and tiger are within the smaller circle.The premises are:

    All cats are mammals.

    A tiger is a cat.

    The conclusion is:

    A tiger is a mammal.

    Both the premises are true. To see that the premises must logically lead to the conclusion, we can use an Euler diagram. From the first premise, we draw the set of cats as a subset of the set of mammals. From the second premise, we are told that a tiger is contained within the set of cats. From that, we can see in the Venn diagram that the tiger must also be inside the set of mammals, so the conclusion is valid.

    Analyzing arguments with Euler diagrams

    To analyze an argument with an Euler diagram:

    1) Draw an Euler diagram based on the premises of the argument

    2) The argument is valid if the diagram cannot be drawn to make the conclusion false

    3) The argument is invalid if there is a way to draw the diagram that makes the conclusion false

    4) The argument is invalid if the premises are insufficient to determine the location of an element or a set mentioned in the conclusion

    Try it Now 1

    Determine the validity of this argument:

    \(\begin{array} {ll} \text{Premise:} & \text{All cats are scared of vacuum cleaners.} \\ \text{Premise:} & \text{Max is a cat.} \\ \text{Conclusion:} & \text{Max is scared of vacuum cleaners.} \end{array}\)

    Answer

    There is a large circle and then a smaller circle within it. The word scared is in the larger circle only. The words cats and Max are within the smaller circle.

     

     

     

    Valid. Cats are a subset of creatures that are scared by vacuum cleaners. Max is in the set of cats, so he must also be in the set of creatures that are scared by vacuum cleaners.

     

    Example 2

    There is a large circle and then a smaller circle within it. The word "Know CPR" and "Jill x?" is in the larger circle only. The letter x with a question mark and the word firefighter are within the smaller circle.\(\begin{array} {ll} \text{Premise:} & \text{All firefighters know CPR.} \\ \text{Premise:} & \text{Jill knows CPR.} \\ \text{Conclusion:} & \text{Jill is a firefighter.} \end{array}\)

    Solution

    From the first premise, we know that firefighters all lie inside the set of those who know CPR. (Firefighters are a subset of people who know CPR.) From the second premise, we know that Jill is a member of that larger set, but we do not have enough information to know whether she also is a member of the smaller subset that is firefighters.

    Since the conclusion does not necessarily follow from the premises, this is an invalid argument. It’s possible that Jill is a firefighter, but the structure of the argument doesn’t allow us to conclude that she definitely is.

    It is important to note that whether or not Jill is actually a firefighter is not important in evaluating the validity of the argument; we are concerned with whether the premises are enough to prove the conclusion.

    Try it Now 2

    Determine the validity of this argument:

    \(\begin{array} {ll} \text{Premise:} & \text{All bicycles have two wheels.} \\ \text{Premise:} & \text{This Harley-Davidson has two wheels.} \\ \text{Conclusion:} & \text{This Harley-Davidson is a bicycle.} \end{array}\)

    Answer

    There is a large circle and then a smaller circle within it. The word "Two Wheels" and "Harley x?" is in the larger circle only. The letter x with a question mark and the word "bicycles," are within the smaller circle.Invalid. The set of bicycles is a subset of the set of vehicles with two wheels; the Harley-Davidson is in the set of two-wheeled vehicles but not necessarily in the smaller circle.

    Try it Now 3

    Determine the validity of this argument:

    \(\begin{array} {ll} \text{Premise:} & \text{No cows are purple.} \\ \text{Premise:} & \text{Fido is not a cow.} \\ \text{Conclusion:} & \text{Fido is purple.} \end{array}\)

    Answer

    There two separate circles. The circle on the left contains the word "Cows." The circle on the right has the word "Purple Things," and the letter x with a question mark. Outside both circles is the name "Fido" along with the letter x and a question mark.Invalid. Since no cows are purple, we know there is no overlap between the set of cows and the set of purple things. We know Fido is not in the cow set, but that is not enough to conclude that Fido is in the purple things set.

    In addition to these categorical style premises of the form “all ___”, “some ____”, and “no ____”, it is also common to see premises that are conditionals.

    Example 3

    \(\begin{array} {ll} \text{Premise:} & \text{If you live in Seattle, you live in Washington.} \\ \text{Premise:} & \text{Marcus does not live in Seattle.} \\ \text{Conclusion:} & \text{Marcus does not live in Washington.} \end{array}\)

    Solution

    From the first premise, we know that the set of people who live in Seattle is inside the set of those who live in Washington. From the second premise, we know that Marcus does not lie in the Seattle set, but we have insufficient information to know whether Marcus lives in Washington or not. This is an invalid argument.

    Try it Now 4

    Determine the validity of this argument:

    \(\begin{array} {ll} \text{Premise:} & \text{If you have lipstick on your collar, then you are cheating on me.} \\ \text{Premise:} & \text{If you are cheating on me, then I will divorce you.} \\ \text{Premise:} & \text{ You do not have lipstick on your collar.} \\ \text{Conclusion:} & \text{I will not divorce you.} \end{array}\)

    Answer

    There are three concentric circles. The word "lipstick"  lives in the smallest inner circle. The middle concentric circle contains the word "charming" and the letter x with a question mark. The largest circle has the word "divorce" and the letter x with a question mark. Not contained within any of the circles is the  word "you" and the letter x with a question mark.Invalid. Lipstick on your collar is a subset of scenarios in which you are cheating, and cheating is a subset of the scenarios in which I will divorce you. Although it is wonderful that you don’t have lipstick on your collar, you could still be cheating on me, and I will divorce you. In fact, even if you aren’t cheating on me, I might divorce you for another reason. You’d better shape up.


    This page titled 3.3.2: Evaluating Deductive Arguments with Euler Diagrams is shared under a CC BY-SA license and was authored, remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.