document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204214>Question 1204214</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ~p ^ r<BR>\n" );
document.write( "p v q<BR>\n" );
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document.write( "∴ q v r </SPAN>\n" );
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document.write( "Method 1\r<BR>\n" );
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document.write( "Informal approach:\r<BR>\n" );
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document.write( "If ~p ^ r is the case then ~p is the case and r is also the case.\r<BR>\n" );
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document.write( "Using ~p and p v q, we find that q comes out of that.<BR>\n" );
document.write( "p v q means \"p or q\". We then know that p isn't the case since ~p is, so q must be the case.\r<BR>\n" );
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document.write( "If q is true then so is q v r. <BR>\n" );
document.write( "In fact, we can replace r with any other logical statement. There's nothing really special about the r.\r<BR>\n" );
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document.write( "Method 2\r<BR>\n" );
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document.write( "Truth Table<BR>\n" );
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document.write( "<tr><TD >p</TD><TD >q</TD><TD >r</TD><TD >~p</TD><TD >~p ^ r</TD><TD >p v q</TD><TD >q v r</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</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 >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><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</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 >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><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD></tr>\n" );
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document.write( "Notice that we do not have a situation where all premises are true but the conclusion is false.<BR>\n" );
document.write( "Therefore, this is a valid argument.\r<BR>\n" );
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document.write( "Here is a review of various truth table rules<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.algebra.com/algebra/homework/Conjunction/truth-table1.lesson\">https://www.algebra.com/algebra/homework/Conjunction/truth-table1.lesson</a>\r<BR>\n" );
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document.write( "Method 3\r<BR>\n" );
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document.write( "Logic Derivation<BR>\n" );
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document.write( "<tr><TD >Number</TD><TD >Statement</TD><TD >Line(s) Used</TD><TD >Reason</TD></tr><tr><TD >1</TD><TD >~p ^ r</TD><TD ></TD><TD ></TD></tr><tr><TD >2</TD><TD >p v q</TD><TD ></TD><TD ></TD></tr><tr><TD >:.</TD><TD >q v r</TD><TD ></TD><TD ></TD></tr><tr><TD >3</TD><TD >~p</TD><TD >1</TD><TD >Simplification</TD></tr><tr><TD >4</TD><TD >q</TD><TD >2,3</TD><TD >Disjunctive Syllogism</TD></tr><tr><TD >5</TD><TD >q v r</TD><TD >4</TD><TD >Addition</TD></tr>\n" );
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document.write( "Refer to these rules of inference and replacement<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://logiccurriculum.com/2019/02/09/rules-for-proofs/\">https://logiccurriculum.com/2019/02/09/rules-for-proofs/</a>\r<BR>\n" );
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