SOLUTION: i need help for the following ,,,need to construct a formal proof for the following valid arguments.. ~ w → ~ a ~ ( w ٨p ) V s S →b ~ (a ٨~ p) ____

~w → ~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b

Indirect proof by abbreviated truth table.  Assume the 
conclusion a→b is false, yet all the premises are true.

So we put F under the main connective of the conclusion,
and T's under the main connectives of all the premises:

~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  
  T           T      T    T           F   

The only way the conclusion can be false is for a to be true
and b to be false.

So we begin by putting T's under all the a's, and F's 
under all both b's:

~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  
  T T         T      TF   T T        TFF  

Now look at the 1st premise ~w→~a.  Since a has T under it,
the ~ before it must have F under it:

~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  
  TFT         T      TF   T T        TFF

Therefore the ~ before the w must have F under it to
make the premise true.  And that means the w must have
T under it, and therefore both w's must have T's under them:

~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  
FTTFT     T   T      TF   T T        TFF

Now we look at the 3rd premise s→b, since b has F under it,
s must have F under it so that that premise is true. So
we put F under all the s's:

~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  
FTTFT     T   TF    FTF   T T        TFF

Now we look back at the 2nd premise ~(w٨p)vs. Since s has 
F under it, the ~ must have T under it to make that premise 
true, so we put T under the ~ of that premise.  

~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  
FTTFT   T T   TF    FTF   T T        TFF 

Therefore we must put F under the ٨ of that 2nd premise, and
therefore F under all the p's

~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  
FTTFT   T TFF TF    FTF   T T  F     TFF

Now we look at the 4th premise: ~(a٨~p)
Since p has F under it, the ~ before it must have T under it

~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  
FTTFT   T TFF TF    FTF   T T TF     TFF

But since the first ~ has a T under it, the ٨ must have an F
under it, but it CANNOT, because it has T's on both sides of it.

Thus we have reached a contradiction.  

We put X under the ٨ to show that this is where we have
reached our contradiction:

~w→~a / ~(w٨p)vs  / s→b / ~(a٨~p) // a→b  
FTTFT   T TFF TF    FTF   T TXTF     TFF


Therefore it is impossible for the conclusion to be false 
if all the premises are true.

Therefore we have proved the argument indirectly, by
abbreviated truth table.  If your teacher wanted you to
use a different method, tell me in the thank-you note
form below and I'll get back to you by email.  No
charge ever.  I do this for fun.

Edwin