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<!DOCTYPE html>
<html data-require="math word-problems graphie">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Logical arguments and deductive reasoning</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="detachment">
<div class="vars">
<var id="QUESTIONS">
[[person(1)+" misses the bus", he(1)+" will be late for school",
[person(1)+" missed the bus",
person(1)+" is late for school",
person(1)+" did not miss the bus",
person(1)+" is not late for school"]],
["it is Tuesday", "I will have a hamburger for lunch",
["it is Tuesday",
"I will have a hamburger for lunch today",
"it is not Tuesday",
"I will not have a hamburger for lunch today"]],
["Wiggles are walking", "Tiggles are talking",
["Wiggles are walking",
"Tiggles are talking",
"Wiggles are not walking",
"Tiggles are not talking"]],
["I go to practice today", "I will play in the game tomorrow",
["I went to practice today",
"I will play in the game tomorrow",
"I did not go to practice today",
"I will not play in the game tomorrow"]]]
</var>
<var id="Q_TYPE">randRange(0, QUESTIONS.length - 1)</var>
<var id="IF_CLAUSE">QUESTIONS[Q_TYPE][0]</var>
<var id="THEN_CLAUSE">QUESTIONS[Q_TYPE][1]</var>
<var id="TYPE">randRange(0, 3)</var>
<var id="IMPLICATION">QUESTIONS[Q_TYPE][2]</var>
<var id="CONCLUSION">[IMPLICATION[1], IMPLICATION[0], IMPLICATION[3], IMPLICATION[2]]</var>
<var id="SOLUTION">(TYPE === 1 || TYPE === 2) ? "No logical conclusion possible" : capitalize(CONCLUSION[TYPE])</var>
</div>
<p class="problem">Use the given information to make a logical conclusion, if possible. If a logical conclusion is not possible, choose "no logical conclusion possible."</p>
<p class="question">If <span id="if-clause"><var>IF_CLAUSE</var></span>, then <span id="then-clause"><var>THEN_CLAUSE</var></span>. <span id="second-sent"><var>capitalize(IMPLICATION[ TYPE ])</var></span>. </p>
<p class="solution"><var>SOLUTION</var></p>
<ul class="choices" data-category="true">
<li><var>capitalize(CONCLUSION[TYPE])</var></li>
<li>No logical conclusion possible</li>
</ul>
<div class="hints">
<div>
<p>Identify the <span class="hint_blue">hypothesis</span>, the <span class="hint_green">conclusion</span> of the first sentence, and the <span class="hint_purple">second sentence</span>.</p>
<div class="graphie">
$("#if-clause").addClass("hint_blue");
$("#then-clause").addClass("hint_green");
$("#second-sent").addClass("hint_purple");
</div>
</div>
<p>Does the <span class="hint_purple">second sentence</span> refer to the <span class="hint_blue">hypothesis</span> of the first sentence, or the <span class="hint_green">conclusion</span> of the first sentence?</p>
<div data-if="TYPE === 0 || TYPE === 2">
<p>The <span class="hint_purple">second sentence</span> refers to the <span class="hint_blue">hypothesis</span> of the first sentence, because they both talk about whether or not <var>IMPLICATION[0]</var>.</p>
<p>Does the <span class="hint_purple">second sentence</span> state the <span class="hint_blue">hypothesis</span>, or the opposite of the <span class="hint_blue">hypothesis</span>?</p>
</div>
<div data-else>
<p>The <span class="hint_purple">second sentence</span> refers to the <span class="hint_green">conclusion</span> of the first sentence, because they both talk about whether or not <var>IMPLICATION[1]</var>.</p>
<p>Does the <span class="hint_purple">second sentence</span> state the <span class="hint_green">conclusion</span>, or the opposite of the <span class="hint_green">conclusion</span>?</p>
</div>
<div data-if="TYPE === 0" data-unwrap>
<p>The <span class="hint_purple">second sentence</span> states the <span class="hint_blue">hypothesis</span> of the first sentence.</p>
<p>Because the <span class="hint_purple">second sentence</span> states the <span class="hint_blue">hypothesis</span> of the first sentence, the <span class="hint_purple">second sentence</span> implies the first sentence.</p>
<p>Since we are implying the original statement, we can conclude the <span class="hint_green">conclusion</span> of the first statement.</p>
</div>
<div data-if="TYPE === 1" data-unwrap>
<p>The <span class="hint_purple">second sentence</span> states the <span class="hint_green">conclusion</span> of the first sentence.</p>
<p>Because the <span class="hint_purple">second sentence</span> states the <span class="hint_green">conclusion</span> of the first sentence, the <span class="hint_purple">second sentence</span> implies the converse of the first sentence.</p>
<p>Converses are not logically equivalent to their original statements, so we cannot form a logical conclusion.</p>
</div>
<div data-if="TYPE === 2" data-unwrap>
<p>The <span class="hint_purple">second sentence</span> states the opposite of the <span class="hint_blue">hypothesis</span> of the first sentence.</p>
<p>Because the <span class="hint_purple">second sentence</span> states the opposite of the <span class="hint_blue">hypothesis</span> of the first sentence, the <span class="hint_purple">second sentence</span> implies the inverse of the first sentence.</p>
<p>Inverses are not logically equivalent to their original statements, so we cannot form a logical conclusion.</p>
</div>
<div data-if="TYPE === 3" data-unwrap>
<p>Because the <span class="hint_purple">second sentence</span> states the opposite of the <span class="hint_green">conclusion</span> of the first sentence, the <span class="hint_purple">second sentence</span> implies the contrapositive of the first sentence.</p>
<p>Since the contrapositive is implied by the first sentence, the <span class="hint_purple">second sentence</span> implies the opposite of the <span class="hint_blue">hypothesis</span>.</p>
</div>
<p>Thus, the answer is "<var>SOLUTION</var>"</p>
</div>
</div>
<div id="syllogism">
<div class="vars">
<var id="QUESTIONS">
[[true, true, ["a figure is a square", "it is a rectangle", "a figure is a rectangle", "it has four right angles"]],
[false, true, ["you play basketball", "you are athletic", "you play volleyball", "you are athletic"]],
[true, true, ["it is Saturday", "you don't have to go to school", "you don't have to go to school", "you can play in the park"]],
[false, true, ["you live in Los Angeles", "you live in California", "you live in Sacramento", "you live in California"]],
[true, true, ["a ray bisects an angle", "it creates two congruent angles", "there are two congruent angles", "the two angles have the same measure"]],
[false, false, ["a shape is a pentagon", "the shape has five sides", "a shape is a pentagon", "the shape has five angles"]],
[true, true, ["a student is in the twelfth grade", "he or she is in high school", "a student is in high school", "he or she is not in college"]],
[false, true, ["you have a picnic", "you will see ants", "it rains a lot", "you will see ants"]]]</var>
<var id="Q_TYPE">randRange(0, QUESTIONS.length - 1)</var>
<var id="CONC_POSSIBLE">QUESTIONS[Q_TYPE][0]</var>
<var id="HYP_DIFF">QUESTIONS[Q_TYPE][1]</var>
<var id="HYP_A">QUESTIONS[Q_TYPE][2][0]</var>
<var id="CONC_A">QUESTIONS[Q_TYPE][2][1]</var>
<var id="HYP_B">QUESTIONS[Q_TYPE][2][2]</var>
<var id="CONC_B">QUESTIONS[Q_TYPE][2][3]</var>
<var id="SOLUTION">CONC_POSSIBLE ? ("If "+HYP_A+", then "+CONC_B+".") : "No logical conclusion possible."</var>
</div>
<p class="problem">Use the given information to make a logical conclusion, if possible.</p>
<p class="question">If <span id="hyp_a"><var>HYP_A</var></span>, then <span id="conc_a"><var>CONC_A</var></span>. If <span id="hyp_b"><var>HYP_B</var></span>, then <span id="conc_b"><var>CONC_B</var></span>.</p>
<p class="solution"><var>SOLUTION</var></p>
<ul class="choices" data-category="true">
<li data-if="CONC_POSSIBLE">If <var>HYP_A</var>, then <var>CONC_B</var>.</li>
<li data-else-if="HYP_DIFF">If <var>HYP_A</var>, then <var>HYP_B</var>.</li>
<li data-else>If <var>CONC_A</var>, then <var>CONC_B</var>.</li>
<li>No logical conclusion possible.</li>
</ul>
<div class="hints">
<div>
<p>Identify the <span class="hint_blue">first hypothesis</span>, the <span class="hint_green">first conclusion</span>, the <span class="hint_red">second hypothesis</span>, and the <span class="hint_purple">second conclusion</span>.</p>
<div class="graphie">
$( "#hyp_a" ).addClass( "hint_blue" );
$( "#conc_a" ).addClass( "hint_green" );
$( "#hyp_b" ).addClass( "hint_red" );
$( "#conc_b" ).addClass( "hint_purple" );
</div>
</div>
<div>
<p>Do the two sentences come in the form "If <span class="hint_blue">P</span>, then <span class="hint_green">Q</span>. If <span class="hint_red">Q</span>, then <span class="hint_purple">R</span>", where <span class="hint_green">first conclusion</span> and <span class="hint_red">second hypothesis</span> are the same?</p>
<p>In other words, do the sentences look like <code>\blue{P}\implies \green{Q}</code>. <code>\red{Q}\implies \purple{R}</code>?</p>
</div>
<div data-if="CONC_POSSIBLE" data-unwrap>
<p>Yes. Because the middle two statements both say <span class="hint_green"> <var>HYP_B</var></span>, we can chain the statements together: <code>\blue{P}\implies\green{Q}\implies\purple{R}</code> or <span class="hint_blue">"<var>HYP_A</var>"</span><code>\implies</code><span class="hint_green">"<var>CONC_A</var>"</span><code>\implies</code><span class="hint_purple">"<var>CONC_B</var>"</span>.</p>
<p>We can now remove the middle statement, and arrive at the conclusion <span class="hint_blue">"<var>HYP_A</var>"</span><code>\implies</code><span class="hint_purple">"<var>CONC_B</var>"</span>. So, the answer is "If <span class="hint_blue"> <var>HYP_A</var></span>, then <span class="hint_purple"> <var>CONC_B</var></span>."</p>
</div>
<p data-else>No. So, we cannot form a logical conclusion.</p>
</div>
</div>
</div>
</div>
</body>
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