document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190255>Question 1190255</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Topics In Contemporary Math\r<BR>\n" );
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document.write( " Arguments\r<BR>\n" );
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document.write( "Use truth tables to determine if each of the following arguments are valid or invalid.\r<BR>\n" );
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document.write( "4) If you back up your hard drive, then you are protected.<BR>\n" );
document.write( "Either you are protected or you are daring.<BR>\n" );
document.write( "Therefore, if you are daring, then you won’t back up your hard drive. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #821857 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=821857\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll define the following<BR>\n" );
document.write( "B = you back up your hard drive<BR>\n" );
document.write( "P = you are protected<BR>\n" );
document.write( "D = you are daring\r<BR>\n" );
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document.write( "We have 3 variables that can either take on values of true (T) or false (F). So there will be 2^3 = 8 different combos of those T's and F's.\r<BR>\n" );
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document.write( "Here's one way we can set up the first part of the table<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >B</TD><TD >P</TD><TD >D</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD></tr>\n" );
document.write( "</TABLE>Notice that the first column has four T's followed by four F's<BR>\n" );
document.write( "The next column has two blocks of TTFF<BR>\n" );
document.write( "The third columnn has four copies of TF\r<BR>\n" );
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document.write( "Let's add in a column denoting ~B, which means \"not B\". This negates everything in the B column<BR>\n" );
document.write( "B = you back up your hard drive<BR>\n" );
document.write( "~B = you do not back up your hard drive<BR>\n" );
document.write( "This will be useful later to set up the conclusion.<BR>\n" );
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document.write( "<tr><TD >B</TD><TD >P</TD><TD >D</TD><TD >~B</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr>\n" );
document.write( "</TABLE>Premise #1 is \"If you back up your hard drive, then you are protected.\"<BR>\n" );
document.write( "That translates to B -> P<BR>\n" );
document.write( "The arrow notation is used in conditional statements of the form \"if this, then that\".<BR>\n" );
document.write( "Writing B -> P means \"If B, then P\".<BR>\n" );
document.write( "Sometimes a sideways horseshoe symbol is used in place of the arrow.\r<BR>\n" );
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document.write( "The conditional statement B -> P will be false if B = T and P = F<BR>\n" );
document.write( "Otherwise, B -> P is true.\r<BR>\n" );
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document.write( "Let's add that to our table<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD >Premise 1</TD></tr><tr><TD >B</TD><TD >P</TD><TD >D</TD><TD >~B</TD><TD >B -> P</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD></tr>\n" );
document.write( "</TABLE>The second premise is \"Either you are protected or you are daring\" which translates to P v D<BR>\n" );
document.write( "The \"v\" symbol means \"or\", used for disjunctions.<BR>\n" );
document.write( "Disjunctions are only false when both pieces are false. Otherwise, the entire thing is true.<BR>\n" );
document.write( "So P v D is only false when P = F and D = F<BR>\n" );
document.write( "Otherwise, P v D is true.\r<BR>\n" );
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document.write( "Let's add that to the truth table<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD >Premise 1</TD><TD >Premise 2</TD></tr><tr><TD >B</TD><TD >P</TD><TD >D</TD><TD >~B</TD><TD >B -> P</TD><TD >P v D</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >F</TD></tr>\n" );
document.write( "</TABLE>Finally, the conclusion is \"If you are daring, then you won't back up your hard drive\"<BR>\n" );
document.write( "This translates to the symbolic form of D -> ~B\r<BR>\n" );
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document.write( "Here's the full truth table<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD >Premise 1</TD><TD >Premise 2</TD><TD >Conclusion</TD></tr><tr><TD >B</TD><TD >P</TD><TD >D</TD><TD >~B</TD><TD >B -> P</TD><TD >P v D</TD><TD >D -> ~B</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD ><font color=red>T</font></TD><TD ><font color=red>T</font></TD><TD ><font color=red>F</font></TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >T</TD></tr>\n" );
document.write( "</TABLE>Now to answer the question if this argument is valid or not.<BR>\n" );
document.write( "Notice that in row 1, which I've marked in red above, we have<ul><li>Premise B -> P is true</li><li>Premise P v D is true</li><li>Conclusion D -> ~B is false</li></ul>Invalid arguments are where all true premises lead to a false conclusion. \r<BR>\n" );
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document.write( "So the argument<BR>\n" );
document.write( "B -> P<BR>\n" );
document.write( "P v D<BR>\n" );
document.write( ":. D -> ~B<BR>\n" );
document.write( "<font color=red>is invalid</font>\r<BR>\n" );
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document.write( "The :. means \"therefore\"<BR>\n" );
document.write( "It could be written as .:<BR>\n" );
document.write( "It's supposed to represent an equilateral triangle of dots \r<BR>\n" );
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document.write( "--------------------------------------------------------\r<BR>\n" );
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document.write( "Here's a shortcut\r<BR>\n" );
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document.write( "Let's assume that the argument is invalid. <BR>\n" );
document.write( "If so, then conclusion D -> ~B must be false and the goal is to try to get all premises to be true.\r<BR>\n" );
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document.write( "If D -> ~B is false, then that leads to D = T and ~B = F<BR>\n" );
document.write( "~B = F flips to B = T\r<BR>\n" );
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document.write( "If B = T, then B -> P is true if and only if P = T as well<BR>\n" );
document.write( "Put another way: if B = T and P = F, then B -> P would be false.\r<BR>\n" );
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document.write( "Since we found P = T and D = T, premise 2 P v D is true<BR>\n" );
document.write( "Though we technically didn't need the truth value of P to figure out that P v D is true because we found that D = T earlier.<BR>\n" );
document.write( "P v D = T v T = T<BR>\n" );
document.write( "P v D = F v T = T\r<BR>\n" );
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document.write( "With these values<BR>\n" );
document.write( "B = T<BR>\n" );
document.write( "P = T<BR>\n" );
document.write( "D = T<BR>\n" );
document.write( "we have shown that all the premises are true but they lead to a false conclusion.<BR>\n" );
document.write( "Therefore, the entire argument posed by your teacher/textbook is invalid.\r<BR>\n" );
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document.write( "While this shortcut is nice to use (perhaps in an exam environment), it's still helpful practice to construct the full truth table when it comes to homework assignments. <BR>\n" );
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