document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208960>Question 1208960</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the eighteen rules of inference to derive the conclusion of the following symbolized argument. Do not use either conditional proof or indirect proof.\r<BR>\n" );
document.write( "\n" );
document.write( "Premise:<BR>\n" );
document.write( "1.(x) [Ax ⊃ (Bx ≡ Cx)]<BR>\n" );
document.write( "2.An • Am<BR>\n" );
document.write( "3.Cn • ~Cm\r<BR>\n" );
document.write( "\n" );
document.write( "Conclusion:<BR>\n" );
document.write( "Bn • ~Bm </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848178 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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=848178\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the derivation of the conclusion using the eighteen rules of inference:\r<BR>\n" );
document.write( "\n" );
document.write( "**1. (x) [Ax ⊃ (Bx ≡ Cx)]** (Given)<BR>\n" );
document.write( "**2. An • Am** (Given)<BR>\n" );
document.write( "**3. Cn • ~Cm** (Given)\r<BR>\n" );
document.write( "\n" );
document.write( "**4. An** (Simplification from 2)<BR>\n" );
document.write( "**5. Am** (Simplification from 2)\r<BR>\n" );
document.write( "\n" );
document.write( "**6. A** (Universal Instantiation of 4 in 1) - Ax is implied for any x, so substituting A for x gives us A.<BR>\n" );
document.write( "**7. Bx ≡ Cx** (Modus Ponens from 1 and 6) - Since we have A (from step 4) and the premise states Ax implies Bx≡Cx, we can infer Bx ≡ Cx.\r<BR>\n" );
document.write( "\n" );
document.write( "**8. Cn** (Simplification from 3)\r<BR>\n" );
document.write( "\n" );
document.write( "**9. ~(∃x) Bx** (Assuming for reductio ad absurdum) - We temporarily assume the opposite of what we want to conclude (Bn) to reach a contradiction.\r<BR>\n" );
document.write( "\n" );
document.write( "**10. ~(∃x) Cx** (Since Bx ≡ Cx from step 7, if there's no Bx, there can't be Cx either)\r<BR>\n" );
document.write( "\n" );
document.write( "**11. ~Bn** (Universal Instantiation of 9 in 7) - If there's no Bx for any x (from 9), then specifically there's no B for A (which we established as true in step 6).\r<BR>\n" );
document.write( "\n" );
document.write( "**12. Cm** (Reductio ad Absurdum from 3, 11) - We reach a contradiction. Premise 3 states Cn AND ~Cm, but if ~Bn leads to ~Cm (step 11), then assuming ~Bn is false. Therefore, Bn must be true.\r<BR>\n" );
document.write( "\n" );
document.write( "**13. Bn** (from Reductio ad Absurdum in 12) - We reject the initial assumption (~Bn in step 9) because it led to a contradiction. So, Bn must be true.\r<BR>\n" );
document.write( "\n" );
document.write( "**14. ~Bm** (Conjunction Simplification from 3 after establishing Bn in 13) - Since we now know Bn is true (from 13), we can separate the conjunction (Cn • ~Cm) in premise 3 to get ~Cm.\r<BR>\n" );
document.write( "\n" );
document.write( "**Therefore, the conclusion is Bn • ~Bm.**\r<BR>\n" );
document.write( "\n" );
document.write( "This derivation uses the following rules of inference:\r<BR>\n" );
document.write( "\n" );
document.write( "* Simplification<BR>\n" );
document.write( "* Universal Instantiation<BR>\n" );
document.write( "* Modus Ponens<BR>\n" );
document.write( "* Reductio ad Absurdum (Assuming for reductio ad absurdum)<BR>\n" );
document.write( "* Conjunction Simplification<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );