Learn the four rules ~, v, •, >
1. ~ means the opposite of what follows the ~
2. FvF is the only false case of v, all others T.
3. T•T is the only true case of •, all others F.
4. T>F is the only false case of >, all others T.

Write the expression across the paper.
Under the K's put TTTTFFFF
Under the P's put TTFFTTFF
Under the R's put TFTFTFTF

 [K • (P v ~ R)] • [K > (R • ~ P)] 
  T    T     T      T    T     T
  T    T     F      T    F     T
  T    F     T      T    T     F
  T    F     F      T    F     F
  F    T     T      F    T     T
  F    T     F      F    F     T
  F    F     T      F    T     F
  F    F     F      F    F     F

Under the ~'s, put F if ~ is before a T
and F if ~ is before a T

 [K • (P v ~ R)] • [K > (R • ~ P)] 
  T    T   F T      T    T   F T
  T    T   T F      T    F   F T
  T    F   F T      T    T   T F
  T    F   T F      T    F   T F
  F    T   F T      F    T   F T
  F    T   T F      F    F   F T
  F    F   F T      F    T   T F
  F    F   T F      F    F   T F

Erase the T's and F's in the columns you just use
to get the last columns you put in.

 [K • (P v ~ R)] • [K > (R • ~ P)] 
  T    T   F        T    T   F 
  T    T   T        T    F   F 
  T    F   F        T    T   T 
  T    F   T        T    F   T 
  F    T   F        F    T   F 
  F    T   T        F    F   F 
  F    F   F        F    T   T 
  F    F   T        F    F   T 

Staying within the first innermost parentheses (),
under the v, put T everywhere except where v is
between two F's. This is the only time we put F. 
[You will notice that this his rule is the exact 
opposite of what we will put under •].

 [K • (P v ~ R)] • [K > (R • ~ P)] 
  T    T T F        T    T   F 
  T    T T T        T    F   F 
  T    F F F        T    T   T 
  T    F T T        T    F   T 
  F    T T F        F    T   F 
  F    T T T        F    F   F 
  F    F F F        F    T   T 
  F    F T T        F    F   T  

Erase the two columns of T's and F's that we used
to get the last column we made:

 [K • (P v ~ R)] • [K > (R • ~ P)] 
  T      T          T    T   F 
  T      T          T    F   F 
  T      F          T    T   T 
  T      T          T    F   T 
  F      T          F    T   F 
  F      T          F    F   F 
  F      F          F    T   T 
  F      T          F    F   T 

Staying within the other innermost parentheses, ()
under the •, put F everywhere except where • is
between two T's. This is the only time we put T. 
[This rule is the exact opposite of what we put 
under v].

 [K • (P v ~ R)] • [K > (R • ~ P)] 
  T      T          T    T F F 
  T      T          T    F F F 
  T      F          T    T T T 
  T      T          T    F F T 
  F      T          F    T F F 
  F      T          F    F F F 
  F      F          F    T T T 
  F      T          F    F F T 

Erase the two columns of T's and F's that we used
to get the last column we made:

 [K • (P v ~ R)] • [K > (R • ~ P)] 
  T      T          T      F  
  T      T          T      F  
  T      F          T      T  
  T      T          T      F  
  F      T          F      F  
  F      T          F      F  
  F      F          F      T  
  F      T          F      F  

Staying within the first innermost brackets, [],
under the •, as before, put F everywhere except 
where • is between two T's. This is the only 
time we put T. 

 [K • (P v ~ R)] • [K > (R • ~ P)] 
  T T    T          T      F  
  T T    T          T      F  
  T F    F          T      T  
  T T    T          T      F  
  F F    T          F      F  
  F F    T          F      F  
  F F    F          F      T  
  F F    T          F      F 

Erase the two columns of T's and F's that we used
to get the last column we made:

 [K • (P v ~ R)] • [K > (R • ~ P)] 
    T               T      F  
    T               T      F  
    F               T      T  
    T               T      F  
    F               F      F  
    F               F      F  
    F               F      T  
    F               F      F

Staying within the other innermost brackets, [],
under the >, put T everywhere except where > has 
T on the left and F on the right. This is the 
only time we put T. 

 [K • (P v ~ R)] • [K > (R • ~ P)] 
    T               T F    F  
    T               T F    F  
    F               T T    T  
    T               T F    F  
    F               F T    F  
    F               F T    F  
    F               F T    T  
    F               F T    F 

Erase the two columns of T's and F's that we used
to get the last column we made:

 [K • (P v ~ R)] • [K > (R • ~ P)] 
    T                 F      
    T                 F      
    F                 T      
    T                 F      
    F                 T      
    F                 T      
    F                 T      
    F                 T     

Now that we have finished all the parentheses
and brackets, we are dow to just 2 columns of T's 
and F's. Under the final • outside all parentheses
and brackets we put F's for everything but T•T.
There are none, so we put F's for everything.

 [K • (P v ~ R)] • [K > (R • ~ P)] 
    T            F    F      
    T            F    F      
    F            F    T      
    T            F    F      
    F            F    T      
    F            F    T      
    F            F    T      
    F            F    T

We erase the two columns of T's and F's that we used
to get the last column we made:

 [K • (P v ~ R)] • [K > (R • ~ P)] 
                 F          
                 F          
                 F          
                 F          
                 F          
                 F          
                 F          
                 F    

Since all the values are F, the statement is a 
logical contradiction, which means it is always
false.

[In other problems, when all the values are T, the 
statement is a logical tautology, or always true.]

[In other problems when some are T and some are F,
the statement is a contingency, or sometimes true
and sometimes false.]

Edwin