{"id":3374,"date":"2023-05-24T15:15:22","date_gmt":"2023-05-24T15:15:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=3374"},"modified":"2024-10-18T20:50:12","modified_gmt":"2024-10-18T20:50:12","slug":"logic-basics-learn-it-4","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/logic-basics-learn-it-4\/","title":{"raw":"Logic Basics: Learn It 4","rendered":"Logic Basics: Learn It 4"},"content":{"raw":"

Implications<\/h2>\r\n

When we discussed conditions earlier, we discussed the type where we take an action based on the value of the condition. We are now going to talk about a more general version of a conditional, sometimes called an implication<\/strong>.<\/p>\r\n

\r\n
\r\n

implications<\/h3>\r\n

Implications<\/strong> are logical conditional sentences stating that a statement [latex]p[\/latex], called the antecedent<\/strong>, implies a statement [latex]q[\/latex], called the consequence<\/strong>.<\/p>\r\n

 <\/p>\r\n

Implications are commonly written as [latex]p\\rightarrow{q}[\/latex] and is translated as \"if [latex]p[\/latex], then [latex]q[\/latex] .\"<\/p>\r\n<\/div>\r\n<\/section>\r\n

Implications are similar to the conditional statements we looked at earlier; [latex]p\\rightarrow{q}[\/latex] is typically written as \u201cif [latex]p[\/latex] then [latex]q[\/latex],\u201d or \u201c[latex]p[\/latex] therefore [latex]q[\/latex].\u201d The difference between implications and conditionals is that conditionals we discussed earlier suggest an action\u2014if the condition is true, then we take some action as a result. Implications are a logical statement that suggest that the consequence must logically follow if the antecedent is true.<\/p>\r\n

The English statement \u201cIf it is raining, then there are clouds is the sky\u201d is a logical implication. Is this a valid argument, why or why not?
\r\n[reveal-answer q=\"913754\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"913754\"]It is a valid argument because if the antecedent \u201cit is raining\u201d is true, then the consequence \u201cthere are clouds in the sky\u201d must also be true.[\/hidden-answer]<\/section>\r\n

Notice that the statement tells us nothing of what to expect if it is not raining. If the antecedent is false, then the implication becomes irrelevant.<\/p>\r\n

A friend tells you that \u201cIf you upload that picture to Facebook, you\u2019ll lose your job.\u201d Describe the possible outcomes related to this statement, and determine whether your friend's statement is invalid.
\r\n[reveal-answer q=\"463067\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"463067\"]There are four possible outcomes:\r\n\r\n
    \r\n\t
  1. You upload the picture and keep your job.<\/li>\r\n\t
  2. You upload the picture and lose your job.<\/li>\r\n\t
  3. You don\u2019t upload the picture and keep your job.<\/li>\r\n\t
  4. You don\u2019t upload the picture and lose your job.<\/li>\r\n<\/ol>\r\n

    There is only one possible case where your friend was lying\u2014the first option where you upload the picture and keep your job. In the last two cases, your friend didn\u2019t say anything about what would happen if you didn\u2019t upload the picture, so you can\u2019t conclude their statement is invalid, even if you didn\u2019t upload the picture and still lost your job.<\/p>\r\n

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

    In traditional logic, an implication is considered valid (true) as long as there are no cases in which the antecedent is true and the consequence is false. It is important to keep in mind that symbolic logic cannot capture all the intricacies of the English language.<\/p>\r\n

    \r\n
    \r\n

    truth values for implications<\/h3>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    [latex]p[\/latex]<\/td>\r\n[latex]q[\/latex]<\/td>\r\n[latex]p \u2192 q[\/latex]<\/td>\r\n<\/tr>\r\n
    T<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
    T<\/td>\r\nF<\/td>\r\nF<\/td>\r\n<\/tr>\r\n
    F<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
    F<\/td>\r\nF<\/td>\r\nT<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/section>\r\n

    Again, if the antecedent [latex]p[\/latex] is false, we cannot prove that the statement is a lie, so the result of the third and fourth rows is true.<\/p>\r\n

    Construct a truth table for the statement [latex]\\left(m\\wedge\\sim{p}\\right)\\rightarrow{r}[\/latex][reveal-answer q=\"6001\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"6001\"]We start by constructing a truth table for the antecedent.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    [latex]m[\/latex]<\/td>\r\n[latex]p[\/latex]<\/td>\r\n[latex]\\sim{p}[\/latex]<\/td>\r\n[latex]m\\wedge\\sim{p}[\/latex]<\/td>\r\n<\/tr>\r\n
    T<\/td>\r\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\n<\/tr>\r\n
    T<\/td>\r\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
    F<\/td>\r\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\n<\/tr>\r\n
    F<\/td>\r\nF<\/td>\r\nT<\/td>\r\nF<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

     <\/p>\r\n

    Now we can build the truth table for the implication.<\/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
    [latex]m[\/latex]<\/td>\r\n[latex]p[\/latex]<\/td>\r\n[latex]\\sim{p}[\/latex]<\/td>\r\n[latex]m\\wedge\\sim{p}[\/latex]<\/td>\r\n[latex]r[\/latex]<\/td>\r\n[latex]\\left(m\\wedge\\sim{p}\\right)\\rightarrow{r}[\/latex]<\/td>\r\n<\/tr>\r\n
    T<\/td>\r\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
    T<\/td>\r\nF<\/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\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
    F<\/td>\r\nF<\/td>\r\nT<\/td>\r\nF<\/td>\r\nT<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
    T<\/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\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\n<\/tr>\r\n
    F<\/td>\r\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\nF<\/td>\r\nT<\/td>\r\n<\/tr>\r\n
    F<\/td>\r\nF<\/td>\r\nT<\/td>\r\nF<\/td>\r\nF<\/td>\r\nT<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

     <\/p>\r\n

    In this case, when [latex]m[\/latex] is true, [latex]p[\/latex] is false, and [latex]r[\/latex] is false, then the antecedent [latex]m\\wedge\\sim{p}[\/latex] will be true but the consequence false, resulting in an invalid implication; every other case gives a valid implication.<\/p>\r\n

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

    [ohm2_question hide_question_numbers=1]7063[\/ohm2_question]<\/section>\r\n

    For any implication, there are three related statements, the converse<\/strong>, the inverse<\/strong>, and the contrapositive<\/strong>.<\/p>\r\n

    \r\n
    \r\n

    related statements<\/h3>\r\n

    The original implication is \u201cif [latex]p[\/latex] then [latex]q[\/latex]\u201d: [latex]p\\rightarrow{q}[\/latex]<\/p>\r\n

     <\/p>\r\n

    The converse<\/strong> is \u201cif [latex]q[\/latex] then [latex]p[\/latex]\u201d: [latex]q\\rightarrow{p}[\/latex]<\/p>\r\n

     <\/p>\r\n

    The inverse <\/strong>is \u201cif not [latex]p[\/latex] then not [latex]q[\/latex]\u201d: [latex]\\sim{p}\\rightarrow\\sim{q}[\/latex]<\/p>\r\n

     <\/p>\r\n

    The contrapositive<\/strong> is \u201cif not [latex]q[\/latex] then not [latex]p[\/latex]\u201d: [latex]\\sim{q}\\rightarrow\\sim{p}[\/latex]<\/p>\r\n<\/div>\r\n<\/section>\r\n

    Consider again the valid implication \u201cIf it is raining, then there are clouds in the sky.\u201d Write the related converse, inverse, and contrapositive statements.
    \r\n[reveal-answer q=\"746956\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"746956\"]The converse would be \u201cIf there are clouds in the sky, it is raining.\u201d This is certainly not always true. The inverse would be \u201cIf it is not raining, then there are not clouds in the sky.\u201d Likewise, this is not always true. The contrapositive would be \u201cIf there are not clouds in the sky, then it is not raining.\u201d This statement is valid, and is equivalent to the original implication.[\/hidden-answer]<\/section>\r\n
    [ohm2_question hide_question_numbers=1]7064[\/ohm2_question]<\/section>","rendered":"

    Implications<\/h2>\n

    When we discussed conditions earlier, we discussed the type where we take an action based on the value of the condition. We are now going to talk about a more general version of a conditional, sometimes called an implication<\/strong>.<\/p>\n

    \n
    \n

    implications<\/h3>\n

    Implications<\/strong> are logical conditional sentences stating that a statement [latex]p[\/latex], called the antecedent<\/strong>, implies a statement [latex]q[\/latex], called the consequence<\/strong>.<\/p>\n

     <\/p>\n

    Implications are commonly written as [latex]p\\rightarrow{q}[\/latex] and is translated as “if [latex]p[\/latex], then [latex]q[\/latex] .”<\/p>\n<\/div>\n<\/section>\n

    Implications are similar to the conditional statements we looked at earlier; [latex]p\\rightarrow{q}[\/latex] is typically written as \u201cif [latex]p[\/latex] then [latex]q[\/latex],\u201d or \u201c[latex]p[\/latex] therefore [latex]q[\/latex].\u201d The difference between implications and conditionals is that conditionals we discussed earlier suggest an action\u2014if the condition is true, then we take some action as a result. Implications are a logical statement that suggest that the consequence must logically follow if the antecedent is true.<\/p>\n

    The English statement \u201cIf it is raining, then there are clouds is the sky\u201d is a logical implication. Is this a valid argument, why or why not?<\/p>\n