Truth Values
Calculating the Truth Value of a Compound Proposition
The truth or falsity of a proposition is called its truth value. The truth value of a compound proposition can be calculated from the truth values of its components, using the following rules:
- For a conjunction to be true, both conjuncts must be true.
- For a disjunction to be true, at least one disjunct must be true.
- A conditional is true except when the antecedent is true and the consequent false.
- For a biconditional to be true, the two input values must be the same (either both true or both false).
- A negation has the opposite value of the negated proposition.
We can use these rules to calculate the truth value of any compound proposition, beginning with the truth values of its simple components (the sentence letters) then calculating the truth value of each connective in the order that the connectives are used to join the component propositions.
Suppose A, B, and C are all true. What is the truth value of the following compound proposition?
((A • B) ⊃ ~C)
To figure it out, first take note of the order in which connectives are used to join the component propositions. If we were constructing the above WFF according to the
rules of syntax, we would start by joining A and B with “•” to make
(A • B), and we’d prefix C with “~” to make ~C. Then we would join
(A • B) and ~C with “⊃” to make
((A • B) ⊃ ~C). We follow this same order when calculating the truth value of the compound proposition:
- (A • B) is true, because both conjuncts are true.
- ~C is false, because C is true.
- ((A • B) ⊃ ~C) is false, because the antecedent (A • B) is true and the consequent ~C is false.
The last connective to be calculated is the main connective. The truth value of the main connective is the truth value of the compound proposition as a whole. (As you may recall, the main connective represents the logical structure of the compound proposition as a whole.) In the above example, the main connective is “⊃”, so the proposition is a conditional. Since the “⊃” is false, the proposition as a whole is false.
Using numerals to represent truth values
Calculating the truth value of a compound proposition can be challenging when the proposition is very complex. To make things easier, we can write the truth values beneath each of the letters and connectives in a compound proposition, using the numeral “1” to represent true and “0” to represent false, as shown in the example below.
In many logic textbooks, truth values are represented using the letter “T” for true and “F” for false. This is merely a matter of convention, but there are advantages to using numerals “1” and “0” to represent truth values (as frequently done in computer science) rather than letters. Using letters to represent truth values can be confusing when “T” and “F” are also used as sentence letters. To avoid that problem, lowercase “t” and “f” are sometimes used instead; but then the truth values are more difficult to read, especially in
truth tables, because “t” and “f” look similar at a glance. Both problems can be avoided by using numerals to represent truth values.
Suppose A, B, and C are all true, but D is false. What is the truth value of
((A • B) ⊃ (~C ∨ D))?
Step 1. The truth values of A, B, C, and D are given, so we write them beneath the sentence letters:
| (( | A | • | B | ) | ⊃ | ( | ~ | C | ∨ | D | )) |
| 1 | | 1 | | | | | 1 | | 0 | |
Step 2. The values of the conjunction and negation can be calculated from the sentence letters, so we write those next:
| (( | A | • | B | ) | ⊃ | ( | ~ | C | ∨ | D | )) |
| 1 | 1 | 1 | | | | 0 | 1 | | 0 | |
Step 3. The value of the disjunction can now be calculated. In order for a disjunction to be true, at least one of its disjuncts must be true. But neither ~C nor D is true, so the disjunction is false:
| (( | A | • | B | ) | ⊃ | ( | ~ | C | ∨ | D | )) |
| 1 | 1 | 1 | | | | 0 | 1 | 0 | 0 | |
Step 4. Finally, the value of the conditional can be calculated. Its antecedent (the conjunction) is true, and its consequent (the disjunction) is false; so the conditional is false:
| (( | A | • | B | ) | ⊃ | ( | ~ | C | ∨ | D | )) |
| 1 | 1 | 1 | | 0 | | 0 | 1 | 0 | 0 | |
Since “⊃” is the main connective, its truth value is the same as the truth value of the proposition as a whole: the proposition is false.
Calculating Truth Values When Some Components are Unknown
It is often possible to calculate the truth value of a compound proposition even when the truth values of some components are unknown, as illustrated in the following example.
Suppose P is true, but the truth values of Q and R are unknown. What is the truth value of
~(P ∨ (Q ≡ R))?
Step 1. The truth value of P is given:
Step 2. The disjunction must be true, because at least one of the disjuncts is true:
Step 3. The negation must be false, since the negated proposition (the disjunction) is true:
Since “~” is the main connective, the proposition is false.