document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190300>Question 1190300</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Topics In Contemporary Math<BR>\n" );
document.write( "  Modus Ponens and Modus Tollens\r<BR>\n" );
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document.write( "Another invalid argument form is the Fallacy of the Inclusive “or”, which has the <BR>\n" );
document.write( "argument form<BR>\n" );
document.write( "𝑝 𝑉 𝑞<BR>\n" );
document.write( "𝑝<BR>\n" );
document.write( "∴ ~𝑞<BR>\n" );
document.write( "Create a truth table to prove that this argument form is invalid.<BR>\n" );
document.write( "Translate each of the following into symbols, then determine whether or not the argument <BR>\n" );
document.write( "is valid by providing the appropriate name for the argument form. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #821916 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=821916\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Premise 1: P v Q<BR>\n" );
document.write( "Premise 2: P<BR>\n" );
document.write( "Conclusion: ~Q\r<BR>\n" );
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document.write( "One way to form the truth table<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD ></TD><TD >Premise 1</TD><TD >Premise 2</TD><TD >Conclusion</TD></tr><tr><TD >P</TD><TD >Q</TD><TD >P v Q</TD><TD >P</TD><TD >~Q</TD></tr><tr><TD >T</TD><TD >T</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 >F</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr>\n" );
document.write( "</TABLE>Row 1, marked in red, shows all true premises lead to a false conclusion. This is sufficient to prove the argument is invalid.\r<BR>\n" );
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document.write( "-------------------------------------------------------------\r<BR>\n" );
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document.write( "Here's an alternative way to form the truth table<BR>\n" );
document.write( "What we do is conjunct the list of premises to form the antecedent, and this will lead to the conclusion.<BR>\n" );
document.write( "(P v Q) & P is the antecedent while ~Q is the conclusion<BR>\n" );
document.write( "This forms the conditional [ (P v Q) & P ] -> ~Q<BR>\n" );
document.write( "If that is ever false, for any row, then we have proven the argument is invalid.<BR>\n" );
document.write( "This is because we have true premises point to a false conclusion. \r<BR>\n" );
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document.write( "This is what the truth table looks like using this alternative method<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >P</TD><TD >Q</TD><TD >P v Q</TD><TD >(P v Q) & P</TD><TD >~Q</TD><TD >[ (P v Q) & P ] -> ~Q</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD ><font color=red>F</font></TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD></tr>\n" );
document.write( "</TABLE>We have \"F\" at the very end of the first row (in red) to show that [ (P v Q) & P ] -> ~Q is false when P = T and Q = T<BR>\n" );
document.write( "This confirms what the other table is showing (also in row 1).\r<BR>\n" );
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document.write( "Some side notes:<ul><li>P v Q is false if both P and Q are false, otherwise it's true.</li><li>P & Q is true if both P and Q are true, otherwise it's false.</li><li>P -> Q is false if P = T and Q = F, otherwise it's true.</li></ul><BR>\n" );
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document.write( " </SPAN>\n" );
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