document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207200>Question 1207200</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please use the 18 rules of natural deduction, the 4 instantiation and generalization rules to derive the conclusions of this problem.\r<BR>\n" );
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document.write( "1. (x)(Bx ⊃ Cx)<BR>\n" );
document.write( "2. (∃x)(Ax • Bx)    /(∃x)(Ax • Cx) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847973 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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=847973\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. (x)(Bx ⊃ Cx)** <BR>\n" );
document.write( "   * Premise\r<BR>\n" );
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document.write( "**2. (∃x)(Ax • Bx)** <BR>\n" );
document.write( "   * Premise\r<BR>\n" );
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document.write( "**3. Aa • Ba** <BR>\n" );
document.write( "   * Existential Instantiation (2)\r<BR>\n" );
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document.write( "**4. Ba** <BR>\n" );
document.write( "   * Simplification (3)\r<BR>\n" );
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document.write( "**5. Ba ⊃ Ca** <BR>\n" );
document.write( "   * Universal Instantiation (1)\r<BR>\n" );
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document.write( "**6. Ca** <BR>\n" );
document.write( "   * Modus Ponens (4, 5)\r<BR>\n" );
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document.write( "**7. Aa • Ca** <BR>\n" );
document.write( "   * Conjunction (3, 6)\r<BR>\n" );
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document.write( "**8. (∃x)(Ax • Cx)** <BR>\n" );
document.write( "   * Existential Generalization (7)\r<BR>\n" );
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document.write( "**Explanation:**\r<BR>\n" );
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document.write( "1. **Existential Instantiation (2):** We introduce a new constant 'a' to represent an arbitrary object that satisfies the existential quantifier in premise 2.\r<BR>\n" );
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document.write( "2. **Simplification (3):** We extract the conjunct 'Ba' from the conjunction '(Aa • Ba)'.\r<BR>\n" );
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document.write( "3. **Universal Instantiation (1):** We instantiate the universal quantifier in premise 1 with the constant 'a'.\r<BR>\n" );
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document.write( "4. **Modus Ponens (4, 5):** We apply the rule of Modus Ponens to derive 'Ca' from 'Ba' and 'Ba ⊃ Ca'.\r<BR>\n" );
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document.write( "5. **Conjunction (2, 5):** We combine 'Aa' and 'Ca' using the rule of Conjunction.\r<BR>\n" );
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document.write( "6. **Existential Generalization (6):** We generalize the statement 'Aa • Ca' to obtain the existential quantifier '(∃x)(Ax • Cx)'.\r<BR>\n" );
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document.write( "This derivation demonstrates that the conclusion (∃x)(Ax • Cx) logically follows from the given premises within the specified 7 steps.<BR>\n" );
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