{"id":37,"date":"2023-01-25T16:33:55","date_gmt":"2023-01-25T16:33:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/analyzing-arguments-with-logic-learn-it-page-2\/"},"modified":"2024-10-18T20:50:15","modified_gmt":"2024-10-18T20:50:15","slug":"analyzing-arguments-with-logic-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/analyzing-arguments-with-logic-learn-it-2\/","title":{"raw":"Analyzing Arguments With Logic: Learn It 2","rendered":"Analyzing Arguments With Logic: Learn It 2"},"content":{"raw":"

Analyzing Arguments with Venn\/Euler Diagrams<\/h2>\r\n

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.<\/p>\r\n

\r\n

How To: Analyzing arguments with Venn\/Euler diagrams<\/strong><\/p>\r\n

    \r\n\t
  1. Draw a Venn\/ Euler diagram based on the premises of the argument.<\/li>\r\n\t
  2. If the premises are insufficient to determine what determines the location of an element, indicate that.<\/li>\r\n\t
  3. The argument is valid if it is clear that the conclusion must be true.<\/li>\r\n<\/ol>\r\n<\/section>\r\n
    Analyze the following argument using a Venn\/Euler diagram.\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Premise:<\/td>\r\nAll firefighters know CPR<\/td>\r\n<\/tr>\r\n
    Premise:<\/td>\r\nJill knows CPR<\/td>\r\n<\/tr>\r\n
    Conclusion:<\/td>\r\nJill is a firefighter<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

    [reveal-answer q=\"88483\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"88483\"]
    \r\nFrom the first premise, we know that firefighters all lie inside the set of those 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 if she also is a member of the smaller subset that is firefighters.<\/p>\r\n

    \"A<\/center>\r\n

    Since the conclusion does not necessarily follow from the premises, this is an invalid argument, regardless of whether Jill actually is a firefighter.<\/p>\r\n

    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 only concerned with whether the premises are enough to prove the conclusion.<\/p>\r\n

    [\/hidden-answer]<\/p>\r\n<\/section>\r\n

    In addition to these categorical style premises of the form \u201call ___,\u201d \u201csome ____,\u201d and \u201cno ____,\u201d it is also common to see premises that are implications.<\/p>\r\n

    Analyzing Arguments with Truth Tables<\/h2>\r\n

    Some arguments are better analyzed using truth tables.<\/p>\r\n

    \r\n

    How To: Analyzing Arguments Using Truth Tables<\/strong><\/p>\r\n

      \r\n\t
    1. Represent each of the premises symbolically.<\/li>\r\n\t
    2. Create a conditional statement, joining all the premises with and to form the antecedent, and using the conclusion as the consequent.<\/li>\r\n\t
    3. Create a truth table for that statement. If it is always true, then the argument is valid.<\/li>\r\n<\/ol>\r\n<\/section>\r\n
      Analyze the following argument using a truth table.\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Premise:<\/td>\r\nIf I go to the mall, then I\u2019ll buy new jeans.<\/td>\r\n<\/tr>\r\n
      Premise:<\/td>\r\nIf I buy new jeans, I\u2019ll buy a shirt to go with it.<\/td>\r\n<\/tr>\r\n
      Conclusion:<\/td>\r\nIf I got to the mall, I\u2019ll buy a shirt.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

      [reveal-answer q=\"88486\"]Show Solution[\/reveal-answer]
      \r\n[hidden-answer a=\"88486\"]
      \r\nLet [latex]M =[\/latex] I go to the mall, [latex]J =[\/latex] I buy jeans, and [latex]S =[\/latex] I buy a shirt.<\/p>\r\n

      The premises and conclusion can be stated as:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
      Premise:<\/td>\r\n[latex]M{\\rightarrow}J[\/latex]<\/td>\r\n<\/tr>\r\n
      Premise:<\/td>\r\n[latex]J{\\rightarrow}S[\/latex]<\/td>\r\n<\/tr>\r\n
      Conclusion:<\/td>\r\n[latex]M{\\rightarrow}S[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

       <\/p>\r\n

      We can construct a truth table for [latex]\\left[\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)\\right]{\\rightarrow}\\left(M{\\rightarrow}S\\right)[\/latex]:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      [latex]M[\/latex]<\/th>\r\n[latex]J[\/latex]<\/th>\r\n[latex]S[\/latex]<\/th>\r\n[latex]M{\\rightarrow}J[\/latex]<\/th>\r\n[latex]J{\\rightarrow}S[\/latex]<\/th>\r\n[latex]\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)[\/latex]<\/th>\r\n[latex]M{\\rightarrow}S[\/latex]<\/th>\r\n[latex]\\left[\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)\\right]{\\rightarrow}\\left(M{\\rightarrow}S\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      T<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
      T<\/td>\r\nT<\/td>\r\nF<\/td>\r\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\nF<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
      T<\/td>\r\nF<\/td>\r\nT<\/td>\r\nF<\/td>\r\nT<\/td>\r\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
      T<\/td>\r\nF<\/td>\r\nF<\/td>\r\nF<\/td>\r\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
      F<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
      F<\/td>\r\nT<\/td>\r\nF<\/td>\r\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
      F<\/td>\r\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
      F<\/td>\r\nF<\/td>\r\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

       <\/p>\r\n

      From the truth table, we can see this is a valid argument.<\/p>\r\n

      [\/hidden-answer]<\/p>\r\n<\/section>\r\n

      The previous problem is an example of a syllogism<\/strong>.<\/p>\r\n

      \r\n
      \r\n

      Syllogism<\/h3>\r\n

      A syllogism<\/strong> is an implication derived from two others, where the consequence of one is the antecedent to the other. The general form of a syllogism is:<\/p>\r\n

       <\/p>\r\n\r\n\r\n\r\n\r\n\r\n
      Premise:<\/td>\r\n[latex]p{\\rightarrow}q[\/latex]<\/td>\r\n<\/tr>\r\n
      Premise:<\/td>\r\n[latex]q{\\rightarrow}r[\/latex]<\/td>\r\n<\/tr>\r\n
      Conclusion:<\/td>\r\n[latex]p{\\rightarrow}r[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

       <\/p>\r\n

      This is sometimes called the transitive property for implication.<\/p>\r\n<\/div>\r\n<\/section>\r\n

      The transitive property appears regularly in the various branches of mathematical study. For example, the transitive property of equality states if [latex]a = b[\/latex] and [latex]b = c[\/latex] then [latex]a = c[\/latex].<\/section>\r\n
      [ohm2_question hide_question_numbers=1]2963[\/ohm2_question]<\/section>\r\n

      Logical Inference<\/h2>\r\n

      Suppose we know that a statement of the form [latex]P{\\rightarrow}Q[\/latex] is true. This tells us that whenever [latex]P[\/latex] is true, [latex]Q[\/latex] \u00a0will also be true. By itself, [latex]P{\\rightarrow}Q[\/latex] being true does not tell us that either [latex]P[\/latex] or [latex]Q[\/latex] is true (they could both be false or [latex]P[\/latex]\u00a0 could be false and [latex]Q[\/latex] true). However if in addition, we happen to know that [latex]P[\/latex]\u00a0 is true then it must be that [latex]Q[\/latex] is true.<\/p>\r\n

      This is called a logical\u00a0inference<\/strong>: Given two true statements, we can infer that a third statement is true. In this instance true statements [latex]P{\\rightarrow}Q[\/latex] and [latex]P[\/latex]\u00a0 are \u201cadded together\u201d\u00a0to get [latex]Q[\/latex]. This is described below with [latex]P{\\rightarrow}Q[\/latex] stacked one atop the other with a line separating them from [latex]Q[\/latex]. The intended meaning is that [latex]P{\\rightarrow}Q[\/latex] combined with [latex]P[\/latex]\u00a0 produces [latex]Q[\/latex].<\/p>\r\n

      \"An<\/p>\r\n

      Two other logical inferences are listed above. In each case, you should convince yourself (based on your knowledge of the relevant truth tables) that the truth of the statements above the line forces the statement below the line to be true.<\/p>\r\n

      \r\n
      \r\n

      Logical inference<\/h3>\r\n

      Logical inference<\/strong> refers to the process of deriving new knowledge or conclusions based on existing knowledge or premises.<\/p>\r\n<\/div>\r\n<\/section>\r\n

      It is important to be aware of the reasons that we study logic. There are three very significant reasons. First, the truth tables we studied tell us the exact meanings of the words such as \u201cand,\u201d \u201cor,\u201d \u201cnot,\u201d and so on. For instance, whenever we use or read the \u201cIf\u2026, then\u201d construction in a mathematical context, logic tells us exactly what is meant. Second, the rules of inference provide a system in which we can produce new information (statements) from known information.
      \r\n
      \r\nThus, logic helps us understand the meanings of statements and it also produces new meaningful statements. Logic is the glue that holds strings of statements together and pins down the exact meaning of certain key phrases such as the \u201cIf\u2026, then\u201d or \u201cFor all\u201d constructions. Logic is the common language that all mathematicians use, so we must have a firm grip on it in order to write and understand mathematics.<\/section>","rendered":"

      Analyzing Arguments with Venn\/Euler Diagrams<\/h2>\n

      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.<\/p>\n

      \n

      How To: Analyzing arguments with Venn\/Euler diagrams<\/strong><\/p>\n

        \n
      1. Draw a Venn\/ Euler diagram based on the premises of the argument.<\/li>\n
      2. If the premises are insufficient to determine what determines the location of an element, indicate that.<\/li>\n
      3. The argument is valid if it is clear that the conclusion must be true.<\/li>\n<\/ol>\n<\/section>\n
        Analyze the following argument using a Venn\/Euler diagram.<\/p>\n\n\n\n\n\n
        Premise:<\/td>\nAll firefighters know CPR<\/td>\n<\/tr>\n
        Premise:<\/td>\nJill knows CPR<\/td>\n<\/tr>\n
        Conclusion:<\/td>\nJill is a firefighter<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n