document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=975438>Question 975438</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm having real difficulty solving this, can you please help?\r<BR>\n" );
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document.write( "INSTRUCTIONS: Use indirect truth tables to answer the following problems.\r<BR>\n" );
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document.write( "Given the argument:<BR>\n" );
document.write( "Premises: E &#8835; J / B &#8835; Q / D &#8835; (J • &#8764;Q) Conclusion: (E • B) &#8801; D\r<BR>\n" );
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document.write( "This argument is:<BR>\n" );
document.write( "		<BR>\n" );
document.write( "Uncogent.<BR>\n" );
document.write( "		<BR>\n" );
document.write( "Sound.<BR>\n" );
document.write( "		<BR>\n" );
document.write( "Valid.<BR>\n" );
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document.write( "Invalid.<BR>\n" );
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document.write( "Cogent. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #597219 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=597219\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Symbols Note: <BR>\n" );
document.write( "* I'm going to use \"->\" in place of the horseshoe<BR>\n" );
document.write( "* I'm going to use & in place of the dot<BR>\n" );
document.write( "* I'm going to use = in place of the triple equals sign\r<BR>\n" );
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document.write( "First assume that the argument is invalid. Recall that an argument is invalid if all premises are true with a false conclusion\r<BR>\n" );
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document.write( "So in this form\r<BR>\n" );
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document.write( "<tr><th>premise1</th><th>/</th><th>premise2</th><th>/</th><th>premise3</th><th>//</th><th>conclusion</th></tr><tr><TD >T</TD><TD ></TD><TD >T</TD><TD ></TD><TD >T</TD><TD ></TD><TD >F</TD></tr>\n" );
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document.write( "So for this particular argument, I'm going to place F under the outermost connector \"=\" in the conclusion. Also I'm going to place T's under the outermost connectors of each premise (see below)\r<BR>\n" );
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document.write( "<tr><th>E</th><th>-&gt;</th><th>J</th><th>/</th><th>B</th><th>-&gt;</th><th>Q</th><th>/</th><th>D</th><th>-&gt;</th><th>(J</th><th>&amp;</th><th>~</th><th>Q)</th><th>//</th><th>(E</th><th>&amp;</th><th>B)</th><th>=</th><th>D</th></tr><tr><TD ></TD><TD >T</TD><TD ></TD><TD ></TD><TD ></TD><TD >T</TD><TD ></TD><TD ></TD><TD ></TD><TD >T</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD >F</TD><TD ></TD></tr>\n" );
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document.write( "Note: The table form is a bit ugly, but it's the only way I could separate each term enough (to give the proper amount of alignment and spacing). \r<BR>\n" );
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document.write( "Then we try to find the truth values of all the symbols to force it to be invalid\r<BR>\n" );
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document.write( "Now let's assume D is false. If that's the case, then (E&B) would have to be true. Place a T under the & in the conclusion. That forces E and B to both be true. So I'm going to place T's under the E and B in the conclusion\r<BR>\n" );
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document.write( "<tr><th>E</th><th>-&gt;</th><th>J</th><th>/</th><th>B</th><th>-&gt;</th><th>Q</th><th>/</th><th>D</th><th>-&gt;</th><th>(J</th><th>&amp;</th><th>~</th><th>Q)</th><th>//</th><th>(E</th><th>&amp;</th><th>B)</th><th>=</th><th>D</th></tr><tr><TD >T</TD><TD >T</TD><TD ></TD><TD ></TD><TD >T</TD><TD >T</TD><TD ></TD><TD ></TD><TD ></TD><TD >T</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr>\n" );
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document.write( "If E is true, then J has to be true for E -> J to be true. Place a T under the J in premise1<BR>\n" );
document.write( "If B is true, then Q has to be true for B -> Q to be true. Place a T under the Q in premise2<BR>\n" );
document.write( "Place T's under the J and Q in premise3<BR>\n" );
document.write( "Q is true, so ~Q is false. Place a F under the ~ next to the Q in premise3<BR>\n" );
document.write( "overall, J&~Q is false, so place a F under the & in premise3<BR>\n" );
document.write( "premise3 is forced to be true, (J&~Q) is false, so D must also be false (leading us back full circle to the initial assumption)\r<BR>\n" );
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document.write( "So here is what the full updated indirect truth table looks like\r<BR>\n" );
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document.write( "<tr><th>E</th><th>-&gt;</th><th>J</th><th>/</th><th>B</th><th>-&gt;</th><th>Q</th><th>/</th><th>D</th><th>-&gt;</th><th>(J</th><th>&amp;</th><th>~</th><th>Q)</th><th>//</th><th>(E</th><th>&amp;</th><th>B)</th><th>=</th><th>D</th></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD ></TD><TD >T</TD><TD >T</TD><TD >T</TD><TD ></TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD ></TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr>\n" );
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document.write( "You'll notice that there are no contradictions. So assuming D is false leads to no contradictions. <BR>\n" );
document.write( "Let's assume D is true<BR>\n" );
document.write( "Following the same sort of logic, we will have this updated table (the second row of Ts and Fs corresponds to when D is true). I'm skipping showing the detailed step by step picture to save time/space. Hopefully my steps above will be enough to help you see how I'm getting this new row\r<BR>\n" );
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document.write( "<tr><th>E</th><th>-&gt;</th><th>J</th><th>/</th><th>B</th><th>-&gt;</th><th>Q</th><th>/</th><th>D</th><th>-&gt;</th><th>(J</th><th>&amp;</th><th>~</th><th>Q)</th><th>//</th><th>(E</th><th>&amp;</th><th>B)</th><th>=</th><th>D</th></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD ></TD><TD >T</TD><TD >T</TD><TD >T</TD><TD ></TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD ></TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >T or F</TD><TD >T</TD><TD >T</TD><TD ></TD><TD >F</TD><TD >T</TD><TD >F</TD><TD ></TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD ></TD><TD >T or F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr>\n" );
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document.write( "Note: The E in premise1 and E in the conclusion could either be True or False. You'll find it doesn't matter. So that's why I put \"T or F\" under those \"E\"s\r<BR>\n" );
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document.write( "Again I couldn't find any contradictions for either row. Since there are no contradictions, this means that the claim \"the argument is invalid\" initially made goes unchallenged. The claim is true.\r<BR>\n" );
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document.write( "So the argument is indeed <font color=\"red\">invalid</font>. Those two rows prove that there is a way to set up the truth values to where the premises are all true which lead to a false conclusion.<BR>\n" );
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