1.5 Argument Forms Proving Invalidity


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Course lecture I developed over section 1.5 of Patrick Hurley\'s "A Concise Introduction to Logic".

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1.5 Argument Forms Proving Invalidity

  1. 1. 1.5 Argument Forms: Proving Invalidity
  2. 2. Overview <ul><li>Proper form for a deductive argument </li></ul><ul><li>Substitution instances </li></ul><ul><li>Steps needed to prove an argument is invalid </li></ul>
  3. 3. Popular deductive forms <ul><li>Categorical syllogisms </li></ul><ul><ul><li>Three line arguments….two premises followed by a conclusion. </li></ul></ul><ul><ul><li>Use the words “all”, “some”, and “no”. </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>All people are humans. </li></ul></ul></ul><ul><ul><ul><li>All humans are happy. </li></ul></ul></ul><ul><ul><ul><li>---------------------------------- </li></ul></ul></ul><ul><ul><ul><li>Therefore, all people are happy. </li></ul></ul></ul><ul><li>Hypothetical syllogisms </li></ul><ul><ul><li>Three line arguments, also have two premises and a conclusion. </li></ul></ul><ul><ul><li>Use if-then to form an argument using a conditional statement. </li></ul></ul><ul><ul><li>The first line establishes the condition, and the second line makes a claim based on that condition. </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>If humans are happy, then the world is good. </li></ul></ul></ul><ul><ul><ul><li>Humans are happy. </li></ul></ul></ul><ul><ul><ul><li>----------------------------------- </li></ul></ul></ul><ul><ul><ul><li>Therefore, the world is good. </li></ul></ul></ul>
  4. 4. Proper form of an argument <ul><li>For some arguments, it can be unclear how to test for validity since arguments can be expressed in many different forms. </li></ul><ul><li>However, many of these arguments are incomplete since they are missing certain premises. </li></ul><ul><li>Example: </li></ul><ul><ul><li>“ Geese are migratory waterfowl, so they fly south for the winter” (Page 53). </li></ul></ul><ul><ul><li>This argument is missing the premise that “Migratory waterfowl fly south for the winter” (Page 53). </li></ul></ul><ul><li>By including this premise, we are able to create what can be considered an acceptable form of a deductive argument. Each step in the premises can be followed to the conclusion. </li></ul>
  5. 5. Proper form, continued <ul><li>The argument can be rewritten to include the missing premise and its validity becomes clear. </li></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>“ All geese are migratory waterfowl. </li></ul></ul></ul><ul><ul><ul><li>All migratory waterfowl are birds that fly south for the winter. </li></ul></ul></ul><ul><ul><ul><li>Therefore, all geese are birds that fly south for the winter” (Page 53). </li></ul></ul></ul><ul><ul><li>The form of this argument can be illustrated as: </li></ul></ul><ul><ul><ul><li>All A are B. </li></ul></ul></ul><ul><ul><ul><li>All B are C. </li></ul></ul></ul><ul><ul><ul><li>----------------------- </li></ul></ul></ul><ul><ul><ul><li>All A are C. </li></ul></ul></ul><ul><ul><ul><ul><li>This can be considered a valid argument form since the conclusion is supported by its premises. </li></ul></ul></ul></ul><ul><ul><ul><ul><li>B follows from A, and C follows from B. So C follows from A. </li></ul></ul></ul></ul><ul><ul><ul><ul><li>A  B and B  C, therefore A  C. </li></ul></ul></ul></ul>
  6. 6. Proper form, continued <ul><li>Any argument that fits this form can be considered valid, so we can insert any terms we wish into A, B, or C since they only act as placeholders. </li></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>“ All daisies are flowers. </li></ul></ul></ul><ul><ul><ul><li>All flowers are plants. </li></ul></ul></ul><ul><ul><ul><li>----------------------- </li></ul></ul></ul><ul><ul><ul><li>Therefore, all daisies are plants” (Page 53). </li></ul></ul></ul><ul><ul><li>Any argument that fits this form, where different terms can be used in place of the letters, can be called a substitution instance . </li></ul></ul>
  7. 7. Recognizing invalid forms <ul><li>What if we try using a substitution instance with a form that doesn’t work? </li></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>All A are B. </li></ul></ul></ul><ul><ul><ul><li>All C are B. </li></ul></ul></ul><ul><ul><ul><li>----------------------- </li></ul></ul></ul><ul><ul><ul><li>All A are C . </li></ul></ul></ul><ul><ul><li>This form doesn’t work since it’s possible for it to have true premises and a false conclusion. (Refer to page 54). But this can be illustrated more clearly with real world references. </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>“ All cats are animals. (true) </li></ul></ul></ul><ul><ul><ul><li>All dogs are animals. (true) </li></ul></ul></ul><ul><ul><ul><li>----------------------- </li></ul></ul></ul><ul><ul><ul><li>Therefore, all cats are dogs. (false) </li></ul></ul></ul><ul><ul><ul><li>The conclusion does not follow from the premises so this is not a valid argument form. </li></ul></ul></ul>
  8. 8. Exceptions to the rule <ul><li>Every substitution instance of a valid form is a valid argument, but… </li></ul><ul><ul><li>It’s not the case that every substitution of an invalid form is an invalid argument. </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>“ All bachelors are persons. </li></ul></ul></ul><ul><ul><ul><li>All unmarried men are persons. </li></ul></ul></ul><ul><ul><ul><li>All bachelors are unmarried men” (Page 54). </li></ul></ul></ul><ul><ul><ul><li>This example has the same form as the invalid argument we used earlier and here it’s valid, but only because of the definition of the words used. </li></ul></ul></ul><ul><ul><ul><li>Bachelors (A) and unmarried men (C) refer to the same type of person, but different terms can be used here that don’t have that same overlap. </li></ul></ul></ul>
  9. 9. Invalid substitutions <ul><li>A substitution instance having true premises and a false conclusion is called a counterexample. </li></ul><ul><li>The method used to prove a substitution’s invalidity is called the counterexample method. </li></ul><ul><ul><li>It’s done by establishing the form of the argument (All A are B, etc.) and creating a substitution instance where the premises are true and conclusion is false, just like we did in the cat and dog example. </li></ul></ul><ul><ul><li>It can be used to prove that invalid forms are invalid, but not that valid forms are valid. Thus, for the counterexample method to work, it’s best to know or suspect that an argument is invalid in the first place. </li></ul></ul><ul><ul><li>The following categorical syllogism is invalid, so we’ll use a substitution instance to prove it. </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>Since some employees are not social climbers and all vice-presidents are employees, we may conclude that some vice-presidents are not social climbers. </li></ul></ul></ul>
  10. 10. Invalid substitutions, continued <ul><li>This example can be rewritten into the following form: </li></ul><ul><ul><li>Some E are not S. </li></ul></ul><ul><ul><li>All V are E. </li></ul></ul><ul><ul><li>----------------------- </li></ul></ul><ul><ul><li>Some V are not S. </li></ul></ul><ul><li>Now we insert terms that will make the premises true and the conclusion false. </li></ul><ul><ul><li>“ Some animals are not mammals. </li></ul></ul><ul><ul><li>All dogs are animals. </li></ul></ul><ul><ul><li>Therefore, some dogs are not mammals” (Page 55). </li></ul></ul><ul><li>The substitution instance is invalid, thus the form is invalid and the argument itself is invalid. </li></ul>
  11. 11. Substituting tips <ul><li>When dealing with categorical syllogisms, keep in mind the terms “cat”, “dog”, “mammal”, “fish”, and “animal”. </li></ul><ul><li>The connection between these words is easy to understand so they can be used in accurately testing a substitution instance’s validity. </li></ul><ul><li>It sometimes helps to begin with the conclusion first and find two terms that will make the conclusion false, then insert a third term in the premises to make them true. </li></ul><ul><li>Pay attention to key words. In CS’s, look for “all”, “no”, “some”, “are”, and “not”, which are form words. </li></ul><ul><li>In HS’s, look for “if”, “then”, and “not”. These are all form words for this specific form. </li></ul>
  12. 12. Other invalid forms <ul><li>Here’s an example of an invalid hypothetical syllogism: </li></ul><ul><ul><li>“ If the government imposes import restrictions, the price of automobiles will rise. Therefore, since the government will not impose import restrictions, the price of automobiles will not rise” (Page 56). </li></ul></ul><ul><ul><li>The form of this argument is as follows: </li></ul></ul><ul><ul><li>If G then P. </li></ul></ul><ul><ul><li>Not G. </li></ul></ul><ul><ul><li>----------------------- </li></ul></ul><ul><ul><li>Therefore, not P. </li></ul></ul><ul><ul><li>Now we make a substitution to see if this argument can be made false. </li></ul></ul><ul><ul><ul><li>If Abraham Lincoln committed suicide, then Abraham Lincoln is dead. </li></ul></ul></ul><ul><ul><ul><li>Abraham Lincoln did not commit suicide. </li></ul></ul></ul><ul><ul><ul><li>Therefore, Abraham Lincoln is not dead. </li></ul></ul></ul><ul><ul><li>The premises clearly do not support the conclusion so the substitution instance, the form, and the original argument are all invalid. </li></ul></ul>
  13. 13. Deductive fallacies in hypothetical syllogisms <ul><li>There are two logical fallacies that can help you to identify invalid hypothetical arguments. </li></ul><ul><ul><li>Affirming the consequent </li></ul></ul><ul><ul><ul><li>Example: </li></ul></ul></ul><ul><ul><ul><ul><li>If A then B </li></ul></ul></ul></ul><ul><ul><ul><ul><li>B </li></ul></ul></ul></ul><ul><ul><ul><ul><li>----------------------- </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Therefore, A </li></ul></ul></ul></ul><ul><ul><ul><li>This form doesn’t work since it asserts that if the consequent is true (statement after then), then the antecedent (statement after if, before then) also has to be true. </li></ul></ul></ul><ul><ul><li>Denying the antecedent </li></ul></ul><ul><ul><ul><li>Example: </li></ul></ul></ul><ul><ul><ul><ul><li>If A then B </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Not A </li></ul></ul></ul></ul><ul><ul><ul><ul><li>----------------------- </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Therefore, not B </li></ul></ul></ul></ul><ul><ul><ul><li>This form doesn’t work, as it claims that if we deny the statement in the antecedent (statement after if, before then) then the consequent (statement after then) can also be denied. </li></ul></ul></ul>