document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178947>Question 1178947</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. ~(U v R)<BR>\n" );
document.write( "2. (~R v N) ⊃ (P • H)<BR>\n" );
document.write( "3. Q ⊃ ~H			/~Q<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850277 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=850277\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze this logical argument to determine its validity. We'll use a proof by deduction to see if we can derive ~Q from the given premises.\r<BR>\n" );
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document.write( "**1. Premises:**\r<BR>\n" );
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document.write( "1.  ~(U v R)<BR>\n" );
document.write( "2.  (~R v N) ⊃ (P • H)<BR>\n" );
document.write( "3.  Q ⊃ ~H\r<BR>\n" );
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document.write( "**2. Derivations:**\r<BR>\n" );
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document.write( "4.  ~U • ~R (De Morgan's Law on premise 1)<BR>\n" );
document.write( "5.  ~R (Simplification on line 4)<BR>\n" );
document.write( "6.  ~R v N (Addition on line 5)<BR>\n" );
document.write( "7.  P • H (Modus Ponens on lines 2 and 6)<BR>\n" );
document.write( "8.  H (Simplification on line 7)<BR>\n" );
document.write( "9.  ~Q (Modus Tollens on lines 3 and 8)\r<BR>\n" );
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document.write( "**Explanation of Steps:**\r<BR>\n" );
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document.write( "* **Step 4:** We applied De Morgan's Law to premise 1. De Morgan's Law states that ~(A v B) is equivalent to ~A • ~B.<BR>\n" );
document.write( "* **Step 5:** We used simplification on line 4. If a conjunction (A • B) is true, then both A and B are true individually.<BR>\n" );
document.write( "* **Step 6:** We used addition on line 5. If A is true, then (A v B) is also true, regardless of the truth value of B.<BR>\n" );
document.write( "* **Step 7:** We used Modus Ponens on lines 2 and 6. Modus Ponens states that if (A ⊃ B) is true and A is true, then B is true.<BR>\n" );
document.write( "* **Step 8:** We used simplification on line 7, the same as step 5.<BR>\n" );
document.write( "* **Step 9:** We used Modus Tollens on lines 3 and 8. Modus Tollens states that if (A ⊃ B) is true and ~B is true, then ~A is true.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "We were able to derive ~Q from the given premises. Therefore, the argument is **valid**.<BR>\n" );
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