SOLUTION: Use ordinary truth tables to answer the following problem. Given the argument: B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B, this argument is: In

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Question 927200: Use ordinary truth tables to answer the following problem.

Given the argument: B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B, this argument is:

Invalid; fails in 3rd line.

Invalid; fails in 2nd line.

Invalid; fails in 1st line.

Invalid; fails in 4th line.

Valid.
I have tried to break up the question and use the T & F in the lines but keep messing up on my placement. Can someone help me so that I can understand what I am doing wrong. Thanks!
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
I use the method of truth tables where you begin
with T's and F's under only the letters, and end up 
with T's and F's under only the symbols.

Put TTTTFFFF under B
Put TTFFTTFF under M
Put TFTFTFTF under K

B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B
----------------------------------
T   T   T      T     T      T    T 
T   T   T      F     F      T    T
T   F   T      T     T      F    T
T   F   T      F     F      F    T
F   T   F      T     T      T    F
F   T   F      F     F      T    F
F   F   F      T     T      F    F
F   F   F      F     F      F    F

Do B ∨ M, which is T unless both sides false.

B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B
----------------------------------
T T T   T      T     T      T    T 
T T T   T      F     F      T    T
T T F   T      T     T      F    T
T T F   T      F     F      F    T
F T T   F      T     T      T    F
F T T   F      F     F      T    F
F F F   F      T     T      F    F
F F F   F      F     F      F    F

Now erase the columns under the first B and M columns:

B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B
----------------------------------
  T     T      T     T      T    T 
  T     T      F     F      T    T
  T     T      T     T      F    T
  T     T      F     F      F    T
  T     F      T     T      T    F
  T     F      F     F      T    F
  F     F      T     T      F    F
  F     F      F     F      F    F

Now just left of the first K column, do ~ K 
by putting the opposite of what's under K under ~.

B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B
----------------------------------
  T     T    F T     T      T    T 
  T     T    T F     F      T    T
  T     T    F T     T      F    T
  T     T    T F     F      F    T
  T     F    F T     T      T    F
  T     F    T F     F      T    F
  F     F    F T     T      F    F
  F     F    T F     F      F    F

Erase the column under K

B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B
----------------------------------
  T     T    F       T      T    T 
  T     T    T       F      T    T
  T     T    F       T      F    T
  T     T    T       F      F    T
  T     F    F       T      T    F
  T     F    T       F      T    F
  F     F    F       T      F    F
  F     F    T       F      F    F

Do B ∨ ∼ K, which is T unless both sides false.

B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B
----------------------------------
  T     T T  F       T      T    T 
  T     T T  T       F      T    T
  T     T T  F       T      F    T
  T     T T  T       F      F    T
  T     F F  F       T      T    F
  T     F T  T       F      T    F
  F     F F  F       T      F    F
  F     F T  T       F      F    F

Erase the columns we just used to put a
column under the ∨

B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B
----------------------------------
  T       T          T      T    T 
  T       T          F      T    T
  T       T          T      F    T
  T       T          F      F    T
  T       F          T      T    F
  T       T          F      T    F
  F       F          T      F    F
  F       T          F      F    F


Now just left of the second M column, do ~ M 
by putting the opposite of what's under M under ~.

B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B
----------------------------------
  T       T          T    F T    T 
  T       T          F    F T    T
  T       T          T    T F    T
  T       T          F    T F    T
  T       F          T    F T    F
  T       T          F    F T    F
  F       F          T    T F    F
  F       T          F    T F    F

Erase the column under the second M

B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B
----------------------------------
  T       T          T    F      T 
  T       T          F    F      T
  T       T          T    T      T
  T       T          F    T      T
  T       F          T    F      F
  T       T          F    F      F
  F       F          T    T      F
  F       T          F    T      F

Do K ∨ ~ M, which is T unless both sides false.
Blace result under the ∨.

B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B
----------------------------------
  T       T          T T  F      T 
  T       T          F F  F      T
  T       T          T T  T      T
  T       T          F T  T      T
  T       F          T T  F      F
  T       T          F F  F      F
  F       F          T T  T      F
  F       T          F T  T      F

Now erase the columns under the first K and ~ columns:

B ∨ M / B ∨ ∼ K // (K ∨ ∼ M) ⊃ B
----------------------------------
  T       T            T         T 
  T       T            F         T
  T       T            T         T
  T       T            T         T
  T       F            T         F
  T       T            F         F
  F       F            T         F
  F       T            T         F

Since I am not sure what / and // mean, I
can go no further. I do know what ⊃ means
but I can't go any further until I learn 
what / and // stand for.  If you'll tell
me what they stand for (conjunction?)
(biconditional?) in the thank-you note
below I'll be glad to finish.  

Edwin
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