TESTING FOR VALIDITY WITH TRUTH VALUE ANALYSIS

We can use Truth Value Analysis to test an argument for validity.  Remember, one way to test an argument for validity to to determine whether the premises are consistent with the negation of the conclusion.  This was the method we used to show that a consistency checker can be used to test validity.  Truth Value Analysis enables us to construct our own consistency checker.

Since the only cases that are of any interest when testing for validity are those in which the conclusion is false (why is that?  click here for the answer) we begin by assuming that the conclusion is false.  To do so we assign truth values to the simple statements occurring in the conclusion so as to make the whole come out false.  then using those same assignments we attempt to make all of the premises true.  If we can succeed in making the premises true after having made the conclusion false, then the argument is non-valid.  If we cannot make the premises true after having made the conclusion false, that is, if making the conclusion false requires at least one premise to be false, then the argument is valid.

Consider the following argument:

Q,    R    S,    ~P  ▼ ~R           ~Q  ▼  ~S

Since a disjunction is false only when each of the disjuncts is false, the only case in which the conclusion of this argument comes out false is that in which both Q and S are true.  So we assume Q and S to be true.

 P → Q R → S ~ P ▼ ~ R ~ Q ▼ ~ S T T ⊥ T ⊥ T ⊥

This assumption immediately reveals that both of the premises are true, since conditionals with true consequents are true.  We know that P  É Q and   R  É  S are true without needing to know the values of P and R.

 P → Q R → S ~ P ▼ ~ R ~ Q ▼ ~ S T T ⊥ T ⊥ T T T ⊥

If is it possible for us to assign truth and falsity to P and R so as to make the third premise,  ~P  ▼ ~R, come out true, then we will have shown that the argument is non-valid, for we will have shown that the premises are indeed consistent with the negation of the conclusion.  Since the premise is a disjunction, if either P or R is false, the premise resolves to true.

 P → Q R → S ~ P ▼ ~ R ~ Q ▼ ~ S T T ⊥ T ⊥ T T T ⊥ T T T ⊥ ⊥ T T T ⊥

Contrast the previous argument with:

Q,    Q   R,   R    S,    S  T             T

Since the only case in which a conditional is false is that in which the antecedent is true and the consequent false, so we assume P to be true and T to be false.  Having made that assumption, though, the only way we can make the first premise true is by making Q true.  But once we make Q true, we have to make R true to make the second premise true.  Now we are forced to make S true in order to make the third premise true.  But once we make S true, the fourth premise MUST be false, since we already made T false.  Accordingly, this argument is VALID because the truth of the premises is inconsistent with the falsity of the conclusion.

 P → Q Q → R R → S S → T P → T T T ⊥ T ⊥ ⊥ T T T ⊥ T T T T ⊥ T T T T ⊥ ⊥ ⊥