document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1188876>Question 1188876</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the finite universe method to prove that the following argument is invalid:<BR>\n" );
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document.write( "1.	(x)Ax⊃(∃x)Bx	 <BR>\n" );
document.write( "2.	(∃x)Ax           	/ (∃x)Bx </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849291 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=849291\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to use the finite universe method to demonstrate the invalidity of the argument:\r<BR>\n" );
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document.write( "1. Define a Finite Universe:\r<BR>\n" );
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document.write( "Let the universe consist of just one object, which we'll call \"a.\"  So, our universe U = {a}.\r<BR>\n" );
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document.write( "2. Interpret the Premises and Conclusion:\r<BR>\n" );
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document.write( "Premise 1: (x)Ax ⊃ (∃x)Bx becomes (Aa ⊃ Ba). This means \"If everything has property A, then there is something that has property B.\"<BR>\n" );
document.write( "Premise 2: (∃x)Ax becomes Aa. This means \"There is something that has property A\" (In our universe, this means 'a' has property A).<BR>\n" );
document.write( "Conclusion: (∃x)Bx becomes Ba. This means \"There is something that has property B\" (In our universe, this means 'a' has property B).<BR>\n" );
document.write( "3. Assign Truth Values:\r<BR>\n" );
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document.write( "We need to assign truth values (True or False) to the properties A and B for the object 'a' in such a way that the premises are true, but the conclusion is false.\r<BR>\n" );
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document.write( "Let Aa be True. ('a' has property A).<BR>\n" );
document.write( "Let Ba be False. ('a' does not have property B).<BR>\n" );
document.write( "4. Evaluate the Premises:\r<BR>\n" );
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document.write( "Premise 1: (Aa ⊃ Ba) becomes (True ⊃ False), which is False.<BR>\n" );
document.write( "Since premise 1 is false, the argument is invalid.\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "We have created a scenario where 'a' has property A (Aa is True) and 'a' does not have property B (Ba is False).  This makes Premise 2 (∃x)Ax true. However, the conclusion (∃x)Bx is false.\r<BR>\n" );
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document.write( "The crucial point is Premise 1 (x)Ax ⊃ (∃x)Bx. This premise becomes false in our model. Since we could make the premises true while the conclusion is false, the argument is invalid. </SPAN>\n" );
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