## Understanding Truth Tables

This page reviews the nature of truth tables, the definitions of basic logical connectives, the rules for constructing truth tables, and the methods for using truth tables to determine whether a well-formed formula is tautologous, inconsistent, self-consistent, or contingent.

The material also discusses methods for using truth-tables to determine whether two well-formed formulas are logically equivalent, contradictory, or consistent. Finally, it covers methods for using truth-tables to determine whether an argument is valid.

A truth table lists all possible combinations of truth values. In a two-valued logic system, a single statement p has two possible truth values: truth (T) and falsehood (F). Given two statements p and q, there are four possible truth value combinations, that is, TT, TF, FT, FF. As a result, there are four rows in the truth table. With three statements, there are eight truth value combinations, ranging from TTT to FFF. In general, given n statements, there are 2 n rows (or cases) in the truth table.

 p p q p q r T T T T T T F T F T T F F T T F T F F T F F F T T F T F F F T F F F

### 3.2.1 Basic Truth Tables of the Five Connectives

Formally, the following five basic truth tables define the five connectives.

#### The Truth Table of Negation

The possible truth values of a negation are opposite to the possible truth values of the statement it negates. If p is true, then ∼p is false. If p is false, then ∼p is true.

 p ∼p T F F T

#### The Truth Table of Conjunction

A conjunction pq is true only when both of its conjuncts are true. It is false in all other three cases.

 p q p • q T T T T F F F T F F F F

#### The Truth Table of Disjunction

A disjunction pq is false only when both of its disjuncts are false. In the other three cases, the disjunction is true.

 p q p ∨ q T T T T F T F T T F F F

#### The Truth Table of Conditional

A conditional is false only when its antecedent is true but its consequent is false. This is so because p ⊃ q says that p is a sufficient condition of q. Now if p is true but q is false, then p cannot be a sufficient condition for q. Consequently, the conditional p ⊃ q would be false.

 p q p ⊃ q T T T T F F F T T F F T

#### The Truth Table of Biconditional

A biconditional p ≡ q is true only when both p and q share the same truth value. If p and q have opposite truth values, then the biconditional is false.

 p q p ≡ q T T T T F F F T F F F T

### 3.2.2 Determining the Truth Value of a Compound Statement

The truth value of a compound statement is determined by the truth values of the simple statements it contains and the basic truth tables of the five connectives. In the following example, the statements C and D are given as true, but E is given as false. To determine the truth value of the conditional, we first write down the given truth value under each letter. Afterwards, using the truth table of conjunction, we can determine the truth value of the antecedent C • D. Because both C and D are true, C • D is true. We then write down “T” under the dot “•” to indicate that C • D is true. Finally, since the antecedent C • D is true, but the consequent E is false, the conditional is false. The final truth value is written under the horseshoe “⊃”.

In the next example, the compound statement is a disjunction. The statement H is given as true, but G and K false. To figure out the truth value of the disjunction, we need to first determine the truth value of the second disjunct ∼H ⊃ K. Since H is true, ∼H is false. We write down “F” under the tilde “∼”. Next, since the antecedent ∼H is false and the consequent K is false, the conditional ∼H ⊃ K is true. So we write down “T” under the horseshoe “⊃”. In the last step, we figure out that the disjunction is true because the first disjunct G is false but the second disjunct ∼H ⊃ K is true.

In the third example, we try to determine the truth value of a biconditional statement from the given truth values that A and D are true, but M and B are false. Since M is false, but A is true, from the third row in the truth table of biconditional, we know that M ≡ A is false, and write down “F” under the triple bar “≡”. We then decide that D ⊃ B is false because D is true but B is false. Next we write down “T” under the tilde “∼” to indicate that ∼(M ≡ A) is true. Finally, we can see that the whole conjunction is false because the second conjunct is false.

### 3.2.3 Three Properties of Statement

In Propositional Logic, a statement is tautologous, self-contradictory or contingent. Which property it has is determined by its possible truth values.

#### Tautology

A statement is tautologous if it is logically true, that is, if it is logically impossible for the statement to be false. If we look at the truth table of a tautology, we would see that all its possible truth values are Ts. One of the simplest tautology is a disjunction such as D ∨ ∼D. To see all the possible truth values of D ∨ ∼D, we construct its truth table by first listing all the possible truth values of the statement D under the letter “D”. Next, we derive the truth values in the green column under the tilde “∼” from the column under the second “D”. The tilde column highlighted in green lists all the possible truth values of ∼D. Finally, from the column under the first “D” and the tilde column, we can come up with all the possible truth values of D ∨ ∼D. They are highlighted in red color, and since the red column under the wedge “∨” lists all the possible truth values of D ∨ ∼D, we put a border around it to indicate that it is the final (or main) column in the truth table. Notice that the two possible truth values in this column are Ts. Since the truth table lists all the possible truth values, it shows that it is logically impossible for D ∨ ∼D to be false. Accordingly, it is a tautology.

The next tautology K ⊃ (N ⊃ K) has two different letters: “K” and “N”. So its truth table has four (2 2 = 4) rows. To construct the table, we put down the letter “T” twice and then the letter “F” twice under the first letter from the left, the letter “K”. As a result, we have “TTFF” under the first “K” from the left. We then repeat the column for the second “K”. Under “N”, the second letter from the left, we write down one “T” and then one “F” in turn until the column is completed. This results in having “TFTF” under “N”. Next, we come up with the possible truth values for N ⊃ K since it is inside the parentheses. The column is highlighted in green. Afterwards, we derive the truth values for the first horseshoe (here highlighted in red) based on the truth values in the first K column and the second horseshoe column (i.e., the green column). Finally we put a border around the first horseshoe column to show it is the final column. We see that all the truth values in that column are all Ts. So K ⊃ (N ⊃ K) is a tautology.

A statement is self-contradictory if it is logically false, that is, if it is logically impossible for the statement to be true. After completing the truth table of the conjunction D • ∼D, we see that all the truth values in the main column under the dot are Fs. The truth table illustrates clearly that it is logically impossible for D • ∼D to be true. The conjunction G • ∼(H ⊃ G) has two distinct letters “G” and “H”, so its truth table has four rows. We write down “TTFF” under the first “G” from the left, and then repeat the values under the second “G”. Under “H”, we put down “TFTF”. We then derive the truth values under the horseshoe from the H column and the second G column. Next, all the possible truth values for ∼(H ⊃ G) are listed under the tilde (highlighted in blue). Notice they are opposite to the truth values in the horseshoe column. Afterwards, we use the first G column and the tilde column to come up with the truth values listed under the dot. Since they are all F s, G • ∼(H ⊃ G) is a self-contradiction.

#### Contingent Statements

A statement is contingent if it is neither tautologous nor self-contradictory. In other words, it is logically possible for the statement to be true and it is also logically possible for it to be false. The conditional D ⊃ ∼D is contingent because its final column contains both a T and a F. Since each row in the truth table represents one logical possibility, this shows that it is logically possible for D ⊃ ∼D to be true, as well as for it to be false. In constructing the truth table for B ≡ (∼E ⊃ B), we need to first come up with the truth values for ∼E because it is the antecedent of the conditional ∼E ⊃ B inside the parentheses. We then use the truth values under the tilde and the second “B” to derive the truth values for the horseshoe column. Next we come up with the column under the triple bar from the first B column and the horseshoe column. The final column has three Ts, representing the logical possibility of being true, and one F, representing the logical possibility of being false. Since its main column contains both T and F, B ≡ (∼E ⊃ B) is contingent. ### 3.2.4 Relations between Two Statements

By comparing all the possible truth values of two statements, we can determine which of the following logical relations exists between them: logical equivalence, contradiction, consistency and inconsistency.

#### Logical Equivalence

Two statements are logically equivalent if they necessarily have the same truth value. This means that their possible truth values listed in the two final columns are the same in each row.

To see whether the pair of statements K ⊃ H and ∼H ⊃ ∼K are logically equivalent to each other, we construct a truth table for each statement. Notice in both truth tables, the statement K has the truth value distribution TTFF, and H has the truth value distribution TFTF. This is crucial because we need to make sure that we are dealing with the same truth value distributions in each row. For instance, in the third row, K is false but H is true in both truth tables. After we complete both truth tables, we see that the two main (or final) columns are identical to each other. This shows that the two statements are logically equivalent.

It is important to be able to tell whether two English sentences are logically equivalent. To see whether these two statements

1. The stock market will fall if interest rates are raised.
2. The stock market won’t fall only if interest rates are not raised.

are logically equivalent, we first symbolize them as R ⊃ F and ∼F ⊃ ∼R. We then construct their truth tables. Since the final two columns are identical, indeed they are logically equivalent.

Two statements are contradictory to each other if they necessarily have the opposite truth values. This means that their truth values in the final columns are opposite in every row of the truth tables. After completing the truth tables for D ⊃ B and D • ∼B, we can see clearly from the two final columns that they are contradictory to each other. #### Consistency

Two statements are consistent if it is logically possible for both of them to be true. This means that there is at least one row in which the truth values in both the final columns are true.

To find out whether ∼A • ∼R and ∼(R • A) are consistent with each other, we construct their truth tables below. Notice again that we have to write down “TTFF” under both “A”, and then “TFTF” under both “R”. After both truth tables are completed, we can see that in the fourth row of the final column each statement has T as its truth value. Since each row stands for a logical possibility, this means that it is logically possible for both of them to be true. So they are consistent with each other.

If we cannot find at least one row in which the truth values in both the final columns are true, then the two statements are inconsistent. That is, it is not logically possible for both of them to be true. In other words, at least one of them must be false. Therefore, if it comes to our attention that two statements are inconsistent, then we must reject at least one of them as false. Failure to do so would mean being illogical.

In the final columns of the truth tables of M • S and ∼(M ≡ S), we do not find a row in which both statements are true. This shows that they are inconsistent with each other, and at least one of them must be false. There can be more than one logical relation between two statements. If two statements are contradictory to each other, then they would have opposite truth values in every row of the main columns. As a result, there cannot be a row in which both statements are true. This means that they must also be inconsistent to each other. However, if two statements are inconsistent, it does not follow that they must be contradictory to each other. The above pair, M • S and ∼(M ≡ S) are inconsistent, but not contradictory to each other. The last row of the two final columns shows that it is logically possible for both statements to be false.

For logically equivalent statements to be consistent with one another, they have to meet the condition that none of them is a self-contradictory statement. The final column of a self-contradictory statement contains no T. So it is not logically possible for a pair of self-contradictory statements to be consistent with each other.

### 3.2.5 Truth Tables for Arguments

A deductive argument is valid if its conclusion necessarily follows from its premises. That is, if the premises are true, then the conclusion must be true. This means that if it is logically possible for the premises to be true but the conclusion false, then the argument is invalid. Since a truth table lists all logical possibilities, we can use it to determine whether a deductive argument is valid. The whole process has three steps:

1. Symbolize the deductive argument;
2. Construct the truth table for the argument;
3. Determine the validity—look to see if there is at least one row in the truth table in which the premises are true but the conclusion false. If such a row is found, this would mean that it is logically possible for the premises to be true but the conclusion false. Accordingly, the argument is invalid. If such a row is not found, this would mean that it is not logically possible to have true premises with a false conclusion. Therefore, the argument is valid.

To decide whether the argument

 If young people don’t have good economic opportunities, there would be more gang violence. Since there is more gang violence, young people don’t have good economic opportunities. 3.2a

is valid, we first symbolize each statement to come up with the argument form. Next, we line up the three statements horizontally, separating the two premises with a single vertical line, and the premises and the conclusion with a double vertical line. Afterwards, we write down “TTFF” under “O” and “TFTF” under “V”. We then derive the truth values for ∼O. Next, we complete the column under the horseshoe and put a border around it. We then complete the column for the second premise V and the conclusion ∼O. The three columns with borders list all the possible truth values for the three statements. To determine the validity of (3.2a), we go over the three main columns row by row to see whether there is a row in which the premises are true, but the conclusion false. We find such a case in the first row. This means that it is logically possible for the premises to be true, but the conclusion false. So (3.2a) is invalid.

To determine whether Argument (3.2b) is valid, we check the three final columns row by row to see if there is a row in which the premises are true but the conclusion false. We do not find such a row. So (3.2b) is valid.

 Psychics can foretell the future only if the future has been determined. But the future has not been determined. It follows that psychics cannot foretell the future. 3.2b In the next example, there are three different letters in the argument form of (3.2c). So its truth table has eight (23 = 8) rows. To exhaust all possible truth value combinations, we write down “TTTTFFFF” under the first letter from the left, “E”. For the second letter from the left, “F”, we put down “TTFFTTFF”, and for the third, “P”, “TFTFTFTF”. For the first premise, we first come up with the truth value for F • P and write down the truth values under the dot. We then derive the truth values under the horseshoe using the first E column and the dot column. Next, we fill out the columns for ∼E and ∼P. After the truth table is completed, we go over the three main columns to see if there is at least one row with true premises but a false conclusion. We find such a case in the fifth and the seventh rows. So the argument is invalid.

 Public education will improve only if funding for education is increased and parents are more involved in the education process. Since public education is not improving, we can conclude that there is not enough parental involvement in the education process. 3.2c Notice the next argument (3.2d) has three premises. After symbolization, its argument form contains three different letters. So its truth table has eight rows. After completing the truth table, we check each row of the four final columns, looking for rows with true premises but a false conclusion. We do not find any. So (3.2d) is valid.

 If more money is spent on building prisons, then less money would go to education. But kids would not be well-educated if less money goes to education. We have spent more money building prisons. As a result, kids would not be well-educated. 3.2d

##### Exercise #1

Determine the truth-values of the following symbolized statements. Let A, B, and C be true; G,H, and K, false; M and N, unknown truth-value. You need to show how you determine the truth-value step by step.

1. A • ∼G
2. ∼A ∨ G
3. ∼(A ⊃ G)
4. ∼G ≡ (B ⊃ K)
5. (A ⊃ ∼C) ∨ C
6. B • (∼H ⊃ A)
7. M ⊃ ∼G
8. (M ∨ ∼A) ∨ H
9. (N • ∼N) ≡ ∼K
10. ∼(M ∨ ∼M) ⊃ N