SOLUTION: INSTRUCTIONS: Use indirect truth tables to answer the following problems. Given the argument: Premises: (K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P •

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Question 975439: INSTRUCTIONS: Use indirect truth tables to answer the following problems.
Given the argument:
Premises: (K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C
This argument is:

Cogent.

Sound.

Valid.

Uncogent.

Invalid.
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 

Start the indirect truth table by putting an F under the ⊃ in the conclusion.
We are assuming that the conclusion is false. If we can now show that this makes
one of the premises false, then we will have shown that the assumption that the
conclusion was false was a bad assumption.  Thus the conclusion will be true,
and the argument valid.

(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
                                                                    F

The only way that can be false is for (A • J) to be true and C false, so put a T
under the • and F under the C

(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
                                                               T    F F
No you can put F's under all the C's

(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
      F                                                        T    F F

Since C s false, ~C is true so put a T under the ~ of ~C

(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
     TF                                                        T    F F

The only way (A • J) can be true is for both A and J to be true, so put
T's under all the A's and J's:

(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
     TF               T            T                         T T T  F F

Since A is true, the only way A ⊃ (P • R) can be true is for (P • R) to be 
true, so put a T under the • of (P • R)
 
(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
     TF               T            T       T                 T T T  F F

The only way (P • R) can be true is for both P and R to be true, so put
T's under all the P's and R's:

(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
     TF       T   T   T         T   T    T T T               T T T  F F

Since J is true, the only way J ⊃ (K • P) can be true is for (K • P) to be 
true, so put a T under the • of (K • P)

(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
     TF       T   T   T       T T   T    T T T               T T T  F F

The only way (K • P) can be true is for K to be true, so put T under
all the K's

(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
 T   TF       T   T   T     T T T   T    T T T               T T T  F F

Since P, R are both true put a T under the • of (P • R)

(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
 T   TF       T T T   T     T T T   T    T T T               T T T  F F

Since (P • R) is true, ~(P • R) id false, so put an F under the ~

(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
 T   TF     F T T T   T     T T T   T    T T T               T T T  F F

(K • ∼C) is true becatuse K and ~C are true, so put a T under the • of
(K • ∼C).

(K • ∼C) ⊃ ∼(P • R)/ J ⊃ (K • P)/ A ⊃ (P • R) Conclusion: (A • J) ⊃ C 
 T T TF     F T T T   T     T T T   T    T T T               T T T  F F

We have reached a contradiction because (K • ∼C) ⊃ ∼(P • R) is given
as a premise yet (K • ∼C) is true and ∼(P • R) is false.

Therefore since the assumption that the conclusion is false leads to a
false premise, then the argument is valid.

Edwin
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