CONSTRUCTING TRUTH TABLES

A truth table is a complete list of all the possible combinations of truth and falsity for a set of simple statements, showing the affect of each combination on the truth value of a compound having those simple statements as components.  Each combination of truth values constitutes one row of a truth table and the number of rows in a truth table is 2n where n equals the number of simple statements.  In order to determine the truth value of a compound, examine the column under the dominant operator for that compound

There is a completely mechanical procedure for constructing a truth table for a sentence. Three skills are necessary in order to do so:

• Producing an initial list of all permutations of truth and falsity assigned to the sentence letters in the sentence.
• Determining which columns in a truth table can be filled in at a given stage.
• Determining what truth values to fill into a given column.
To see how these skills work, let's form a truth table for the sentence:

P → (Q ▼ R)

(STEP 1) This sentence contains three statement letters: P, Q, and R. So first we must list all permutations of truth and falsity assigned to P, Q, and R. There will be 8 (= 23) such permutations:

 P Q R T T T T T ⊥ T ⊥ T T ⊥ ⊥ ⊥ T T ⊥ T ⊥ ⊥ ⊥ T ⊥ ⊥ ⊥

(STEP 2) We now want to start constructing the truth table. We write our target sentence next to the sentence letters. Our eventual goal is to fill in a series of columns of truth values, one under each sentence letter and connective in the target sentence.  I have put numbers above the columns for easy reference.  The first thing we notice is that Column 4 is the column for the dominant operator in our statement, so that is the last column to fill in.

 1 2 3 4 5 6 7 8 P Q R P → (Q ▼ R) T T T T T ⊥ T ⊥ T T ⊥ ⊥ ⊥ T T ⊥ T ⊥ ⊥ ⊥ T ⊥ ⊥ ⊥

(STEP 3) We now want to construct a truth table for our particular sentence. The first step is to assign truth values to each sentence letter in our sentence--values that come directly from the guide columns.  You may, once you are comfortable with constructing truth tables, wish to dispense with this step (the information is already in the guide columns) to make reading the table easier.  Completing this step we get:

 1 2 3 4 5 6 7 8 P Q R P → (Q ▼ R) T T T T T T T T ⊥ T T ⊥ T ⊥ T T ⊥ T T ⊥ ⊥ T ⊥ ⊥ ⊥ T T ⊥ T T ⊥ T ⊥ ⊥ T ⊥ ⊥ ⊥ T ⊥ ⊥ T ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

(STEP 4) To fill in the rest of the columns, we work from connectives of smaller scope to connectives of larger scope. We start with connectives of smallest scope. Since the disjunction of Q with R is a component of the main conditional, we need to construct that column first.  To do so we simply go back to our basic understanding of the rules for disjunction:  if at least one disjunct is true, then the disjunction is true.  We determine the truth value of each disjunct by examining its column (so we look at columns 6 and 8).  Rows 1, 2, 3, 5, 6, and 7 all have at least one true disjunct, so we get:

 1 2 3 4 5 6 7 8 P Q R P → (Q ▼ R) T T T T T T T T T ⊥ T T T ⊥ T ⊥ T T ⊥ T T T ⊥ ⊥ T ⊥ ⊥ ⊥ ⊥ T T ⊥ T T T ⊥ T ⊥ ⊥ T T ⊥ ⊥ ⊥ T ⊥ ⊥ T T ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

(STEP 5) Next we'll do the main conditional whose antecedent is P and whose consequent is the disjunction of Q with R.  To fill in the column, we take the truth value listed under P in Column  4  as the first input and the truth value under (Q ▼ R) in Column 7 as the second input, and then apply to these two inputs the rule for the conditional:

 1 2 3 4 5 6 7 8 P Q R P → (Q ▼ R) T T T T T T T T T T ⊥ T T T T ⊥ T ⊥ T T T ⊥ T T T ⊥ ⊥ T ⊥ ⊥ ⊥ ⊥ ⊥ T T ⊥ T T T T ⊥ T ⊥ ⊥ T T T ⊥ ⊥ ⊥ T ⊥ T ⊥ T T ⊥ ⊥ ⊥ ⊥ T ⊥ ⊥ ⊥

We discover that our original statement is CONTINGENT, that is, sometimes it comes out true, sometimes it comes out false (Row 4).

Move on to a more complicated example.