Question 915332
Question 915332
<pre>

There was a slight mistake before.  The argument is valid. Here is the corrected version.


I do truth tables different, but it's the same way. I begin
by rewriting the argument with T's and F's under the S's and K's:

ORIGINAL TRUTH TABLE

S &#8835; (K &#8744; &#8764; S) / K &#8835; S // S &#8801; K
-------------------------------
T &#8835; (T &#8744; &#8764; T) / T &#8835; T // T &#8801; T
T &#8835; (F &#8744; &#8764; T) / F &#8835; T // T &#8801; F
F &#8835; (T &#8744; &#8764; F) / T &#8835; F // F &#8801; T
F &#8835; (F &#8744; &#8764; F) / F &#8835; F // F &#8801; F

Then I place the truth values under the symbols &#8764;, &#8835; and &#8801;

S &#8835; (K &#8744; &#8764; S) / K &#8835; S // S &#8801; K
-------------------------------
T &#8835; (T &#8744; F T) / T T T // T T T
T &#8835; (F &#8744; F T) / F T T // T F F
F &#8835; (T &#8744; T F) / T F F // F F T
F &#8835; (F &#8744; T F) / F T F // F T F

Then I erase all the used truth values:

S &#8835; (K &#8744; &#8764; S) / K &#8835; S // S &#8801; K
-------------------------------
T &#8835; (T &#8744; F  ) /   T   //   T 
T &#8835; (F &#8744; F  ) /   T   //   F 
F &#8835; (T &#8744; T  ) /   F   //   F 
F &#8835; (F &#8744; T  ) /   T   //   T 

Place truth values under symbol &#8744;:

S &#8835; (K &#8744; &#8764; S) / K &#8835; S // S &#8801; K
-------------------------------
T &#8835; (T T F  ) /   T   //   T 
T &#8835; (F F F  ) /   T   //   F 
F &#8835; (T T T  ) /   F   //   F 
F &#8835; (F T T  ) /   T   //   T

Erase used truth values:

S &#8835; (K &#8744; &#8764; S) / K &#8835; S // S &#8801; K
-------------------------------
T &#8835;    T      /   T   //   T 
T &#8835;    F      /   T   //   F 
F &#8835;    T      /   F   //   F 
F &#8835;    T      /   T   //   T 

Place truth values under &#8835; 

S &#8835; (K &#8744; &#8764; S) / K &#8835; S // S &#8801; K
-------------------------------
T T    T      /   T   //   T 
T F    F      /   T   //   F 
F T    T      /   F   //   F 
F T    T      /   T   //   T 



Erase used truth values:

S &#8835; (K &#8744; &#8764; S) / K &#8835; S // S &#8801; K
-------------------------------
  T           /   T   //   T 
  F           /   T   //   F 
  T           /   F   //   F 
  T           /   T   //   T

Place truth values under /, which is the same same 
as &#8743; (and, conjunction)

S &#8835; (K &#8744; &#8764; S) / K &#8835; S // S &#8801; K
-------------------------------
  T           T   T   //   T 
  F           F   T   //   F 
  T           F   F   //   F 
  T           T   T   //   T


Erase the used truth values


S &#8835; (K &#8744; &#8764; S) / K &#8835; S // S &#8801; K
-------------------------------
              T            T 
              F            F 
              F            F 
              T            T

Place truth values under the // which is the same as &#8835;


 S &#8835; (K &#8744; &#8764; S) / K &#8835; S // S &#8801; K
-------------------------------
               T        T   T
               F        T   F
               F        T   F
               T        T   T

Valid.  There are only T's under the //

Edwin</pre>