TRUTH VALUE ANALYSIS

Sometimes we can know the truth value of a compound statement without knowing the truth values of each component simple statement. We know that a conditional with a false antecedent is true, so, if ‘P’ is false, then P (Q ▼ (R S)) is TRUE, no matter what the truth values of ‘Q,’ ‘R,’ and ‘S’ happen to be!   Similarly, since a conjunction is false if any of its conjuncts happens to be false, P (Q (R S)) is FALSE no matter what the truth values of the others.

The following rules allow you to resolve the truth value of a compound statement based on partial knowledge of the truth values of the components and without needing to construct a full truth table (and since truth tables get very large, very fast, this is good news).

n
n

n Rules for Truth Value Analysis

• nA conjunction with one or more false conjuncts is false.
• nA disjunction with one or more true disjuncts is true.
• nA conditional with a false antecedent or a true consequent is true.
• nA biconditional with a true component has the same truth value as the other component.
• nA biconditional with a false component has a truth value opposite the other component.

Consider the following formula:

{[(P ● Q) ▼ R]   (S ● ~T)}

Suppose that we know that P is false, R is true and T is true.  We can use the looping method to short cut the construction of a full 32 row truth table.  First, plug in the known values:

 {[(P ● Q) ▼ R] → (S ● ~ T)} ⊥ T T

Since T is true, we know immediately that ~T is false, so we plug in that value.  I call this resolving the truth value of the compound.

 {[(P ● Q) ▼ R] → (S ● ~ T)} ⊥ T T ⊥

Next, since we have 2 conjunctions, each with a false conjunct, we know that each conjunction resolves to false.  Note that we do not need to know the value of each conjunct if one is known to be false--the entire conjunction is false no matte what the truth value of the other happens to be.

 {[(P ● Q) ▼ R] → (S ● ~ T)} ⊥ ? T ? T ⊥ ? ⊥ ⊥

Next we need to resolve the truth value of the disjunction which is the antecedent of the conditional. Since one side of the disjunction is true, the entire disjunction is true, even though the other side is false.

 {[(P ● Q) ▼ R] → (S ● ~ T)} ⊥ ? T ? T ⊥ ⊥ T ⊥

Finally, we resolve the truth value of the entire statement based on the truth values of the antecedent and consequent.  A conditional with a true antecedent and a false consequent is false.

 {[(P ● Q) ▼ R] → (S ● ~ T)} ⊥ T T ⊥ ⊥ T ⊥ ⊥

This truth value analysis allows us to draw a conclusion as to the truth of our original statement based on partial knowledge of the truth values of the simple statements without having to construct an entire 32 row truth table and without knowing the truth values of 2 of the statement letters.

Test yourself  by taking the U-Test quiz (this is an ungraded, self-help exercise) on Truth Value Analysis.