document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179864>Question 1179864</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I. Use an ordinary proof (not conditional or indirect) to solve the following arguments.\r<BR>\n" );
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document.write( "1)\r<BR>\n" );
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document.write( "1. I v (N • F)\r<BR>\n" );
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document.write( "2. I ⊃ F /F\r<BR>\n" );
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document.write( "2)\r<BR>\n" );
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document.write( "1. P ⊃ ~M\r<BR>\n" );
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document.write( "2. C ⊃ M\r<BR>\n" );
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document.write( "3. ~L v C\r<BR>\n" );
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document.write( "4. (~P ⊃ ~E) • (~E ⊃ ~C)\r<BR>\n" );
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document.write( "5. P v ~P /~L\r<BR>\n" );
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document.write( "3)\r<BR>\n" );
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document.write( "1. O ⊃ (Q • N)\r<BR>\n" );
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document.write( "2. (N Ú E) ⊃ S / O ⊃ S\r<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 #850161 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=850161\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down each argument step-by-step using ordinary proofs:\r<BR>\n" );
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document.write( "**1) Argument:**\r<BR>\n" );
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document.write( "1.  I v (N • F)<BR>\n" );
document.write( "2.  I ⊃ F / F\r<BR>\n" );
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document.write( "**Proof:**\r<BR>\n" );
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document.write( "3.  ~I ∨ F  (2, Implication)<BR>\n" );
document.write( "4.  I ∨ (N • F) (1, Copy)<BR>\n" );
document.write( "5.  ~I ∨ F  (3, Copy)<BR>\n" );
document.write( "6.  F ∨ ~I (5, Commutation)<BR>\n" );
document.write( "7.  F ∨ (N • F) (4, Resolution, 6)<BR>\n" );
document.write( "8.  F ∨ (N • F) (7, Copy)<BR>\n" );
document.write( "9.  F ∨ N (8, Distribution)<BR>\n" );
document.write( "10. F (9, Simplification)\r<BR>\n" );
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document.write( "**Therefore, F is proven.**\r<BR>\n" );
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document.write( "**2) Argument:**\r<BR>\n" );
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document.write( "1.  P ⊃ ~M<BR>\n" );
document.write( "2.  C ⊃ M<BR>\n" );
document.write( "3.  ~L v C<BR>\n" );
document.write( "4.  (~P ⊃ ~E) • (~E ⊃ ~C)<BR>\n" );
document.write( "5.  P v ~P / ~L\r<BR>\n" );
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document.write( "**Proof:**\r<BR>\n" );
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document.write( "6.  ~P ∨ ~M (1, Implication)<BR>\n" );
document.write( "7.  ~C ∨ M (2, Implication)<BR>\n" );
document.write( "8.  ~P ⊃ ~E (4, Simplification)<BR>\n" );
document.write( "9.  ~E ⊃ ~C (4, Simplification)<BR>\n" );
document.write( "10. P ∨ ~P (5, Copy)<BR>\n" );
document.write( "11. P (10, Tautology)<BR>\n" );
document.write( "12. ~M (6, 11, Modus Ponens)<BR>\n" );
document.write( "13. ~C (7, 12, Modus Tollens)<BR>\n" );
document.write( "14. ~L (3, 13, Disjunctive Syllogism)\r<BR>\n" );
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document.write( "**Therefore, ~L is proven.**\r<BR>\n" );
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document.write( "**3) Argument:**\r<BR>\n" );
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document.write( "1.  O ⊃ (Q • N)<BR>\n" );
document.write( "2.  (N ∨ E) ⊃ S / O ⊃ S\r<BR>\n" );
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document.write( "**Proof:**\r<BR>\n" );
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document.write( "3.  ~O ∨ (Q • N) (1, Implication)<BR>\n" );
document.write( "4.  N ∨ E ⊃ S (2, Copy)<BR>\n" );
document.write( "5.  ~O ∨ Q (3, Simplification)<BR>\n" );
document.write( "6.  ~O ∨ N (3, Simplification)<BR>\n" );
document.write( "7.  N ∨ E (6, Addition)<BR>\n" );
document.write( "8.  S (4, 7, Modus Ponens)<BR>\n" );
document.write( "9.  ~O ∨ S (6, 8, Hypothetical Syllogism)<BR>\n" );
document.write( "10. O ⊃ S (9, Implication)\r<BR>\n" );
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document.write( "**Therefore, O ⊃ S is proven.**<BR>\n" );
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