document.write( "Question 1061347: i need help for the following ,,,need to construct a formal proof for the following valid arguments..\r
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Algebra.Com's Answer #676155 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "~w → ~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b\r\n" );
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document.write( "Indirect proof by abbreviated truth table.  Assume the \r\n" );
document.write( "conclusion a→b is false, yet all the premises are true.\r\n" );
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document.write( "So we put F under the main connective of the conclusion,\r\n" );
document.write( "and T's under the main connectives of all the premises:\r\n" );
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document.write( "~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  \r\n" );
document.write( "  T           T      T    T           F   \r\n" );
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document.write( "The only way the conclusion can be false is for a to be true\r\n" );
document.write( "and b to be false.\r\n" );
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document.write( "So we begin by putting T's under all the a's, and F's \r\n" );
document.write( "under all both b's:\r\n" );
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document.write( "~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  \r\n" );
document.write( "  T T         T      TF   T T        TFF  \r\n" );
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document.write( "Now look at the 1st premise ~w→~a.  Since a has T under it,\r\n" );
document.write( "the ~ before it must have F under it:\r\n" );
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document.write( "~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  \r\n" );
document.write( "  TFT         T      TF   T T        TFF\r\n" );
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document.write( "Therefore the ~ before the w must have F under it to\r\n" );
document.write( "make the premise true.  And that means the w must have\r\n" );
document.write( "T under it, and therefore both w's must have T's under them:\r\n" );
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document.write( "~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  \r\n" );
document.write( "FTTFT     T   T      TF   T T        TFF\r\n" );
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document.write( "Now we look at the 3rd premise s→b, since b has F under it,\r\n" );
document.write( "s must have F under it so that that premise is true. So\r\n" );
document.write( "we put F under all the s's:\r\n" );
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document.write( "~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  \r\n" );
document.write( "FTTFT     T   TF    FTF   T T        TFF\r\n" );
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document.write( "Now we look back at the 2nd premise ~(w٨p)vs. Since s has \r\n" );
document.write( "F under it, the ~ must have T under it to make that premise \r\n" );
document.write( "true, so we put T under the ~ of that premise.  \r\n" );
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document.write( "~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  \r\n" );
document.write( "FTTFT   T T   TF    FTF   T T        TFF \r\n" );
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document.write( "Therefore we must put F under the ٨ of that 2nd premise, and\r\n" );
document.write( "therefore F under all the p's\r\n" );
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document.write( "~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  \r\n" );
document.write( "FTTFT   T TFF TF    FTF   T T  F     TFF\r\n" );
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document.write( "Now we look at the 4th premise: ~(a٨~p)\r\n" );
document.write( "Since p has F under it, the ~ before it must have T under it\r\n" );
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document.write( "~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  \r\n" );
document.write( "FTTFT   T TFF TF    FTF   T T TF     TFF\r\n" );
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document.write( "But since the first ~ has a T under it, the ٨ must have an F\r\n" );
document.write( "under it, but it CANNOT, because it has T's on both sides of it.\r\n" );
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document.write( "Thus we have reached a contradiction.  \r\n" );
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document.write( "We put X under the ٨ to show that this is where we have\r\n" );
document.write( "reached our contradiction:\r\n" );
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document.write( "~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  \r\n" );
document.write( "FTTFT   T TFF TF    FTF   T TXTF     TFF\r\n" );
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document.write( "Therefore it is impossible for the conclusion to be false \r\n" );
document.write( "if all the premises are true.\r\n" );
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document.write( "Therefore we have proved the argument indirectly, by\r\n" );
document.write( "abbreviated truth table.  If your teacher wanted you to\r\n" );
document.write( "use a different method, tell me in the thank-you note\r\n" );
document.write( "form below and I'll get back to you by email.  No\r\n" );
document.write( "charge ever.  I do this for fun.\r\n" );
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document.write( "Edwin
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