document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191579>Question 1191579</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You are on the island of knights and knaves, where (a) every local is either<BR>\n" );
document.write( "a knight or a knave, (b) knights always tell the truth, and (c) knaves always<BR>\n" );
document.write( "lie. Using a symbolic technique (truth table or natural deduction), can you<BR>\n" );
document.write( "determine who is a knight and who is a knave? (10 pts. for translation, 10 pts.<BR>\n" );
document.write( "for truth table/proof and verdict)\r<BR>\n" );
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document.write( " You meet three locals: Al, Bob, and Carol. Al says, “I’m a knave only<BR>\n" );
document.write( "if Carol is a knight.” Bob says, “I’m a knight if Carol is.” Carol says,<BR>\n" );
document.write( "“Neither Al nor Bob is a knight.” </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849157 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849157\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how we can analyze the statements using symbolic logic and determine who is a knight and who is a knave:\r<BR>\n" );
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document.write( "**1. Translation into Propositional Logic:**\r<BR>\n" );
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document.write( "Let A, B, and C represent the statements \"Al is a knight,\" \"Bob is a knight,\" and \"Carol is a knight,\" respectively.  Therefore, ¬A, ¬B, and ¬C mean \"Al is a knave,\" \"Bob is a knave,\" and \"Carol is a knave,\" respectively.\r<BR>\n" );
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document.write( "* **Al's statement:**  ¬A ↔ C  (Al is a knave if and only if Carol is a knight)<BR>\n" );
document.write( "* **Bob's statement:** B ↔ C (Bob is a knight if and only if Carol is a knight)<BR>\n" );
document.write( "* **Carol's statement:** ¬A & ¬B (Neither Al nor Bob is a knight)\r<BR>\n" );
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document.write( "**2. Truth Table:**\r<BR>\n" );
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document.write( "We need to consider all possible truth values for A, B, and C, and see which combinations satisfy all three statements.\r<BR>\n" );
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document.write( "| A | B | C | ¬A | ¬A ↔ C | B ↔ C | ¬A & ¬B |<BR>\n" );
document.write( "|---|---|---|---|---|---|---|<BR>\n" );
document.write( "| T | T | T | F | F | T | F |<BR>\n" );
document.write( "| T | T | F | F | T | F | F |<BR>\n" );
document.write( "| T | F | T | F | F | F | F |<BR>\n" );
document.write( "| T | F | F | F | T | T | F |<BR>\n" );
document.write( "| F | T | T | T | T | T | F |<BR>\n" );
document.write( "| F | T | F | T | F | F | F |<BR>\n" );
document.write( "| F | F | T | T | T | F | T |<BR>\n" );
document.write( "| F | F | F | T | F | T | T |\r<BR>\n" );
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document.write( "**3. Analysis and Verdict:**\r<BR>\n" );
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document.write( "Looking at the truth table, we're searching for a row where *all three* statements (¬A ↔ C, B ↔ C, and ¬A & ¬B) are true.  Only the last row (A=F, B=F, C=F) satisfies all conditions.\r<BR>\n" );
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document.write( "* **Row 8:** A is False, B is False, and C is False.\r<BR>\n" );
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document.write( "This means:\r<BR>\n" );
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document.write( "* Al is a knave (¬A is true).<BR>\n" );
document.write( "* Bob is a knave (¬B is true).<BR>\n" );
document.write( "* Carol is a knave (¬C is true).\r<BR>\n" );
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document.write( "Therefore, Al, Bob, and Carol are all knaves.<BR>\n" );
document.write( " </SPAN>\n" );
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