5. Filling in a Truth Table

As stated earlier, the truth table is divided into two sets of columns:  (1) the left set of columns associated with the distinct atomic sentences found in the problem, and (2) the right set of columns depicting the problem itself.

While the majority of our work in completing the table comes from filling in the right set of columns, we must begin every truth table problem by filling in the left set of columns – the starting resources of our table.

The left set of columns depicts the possible combinations of TRUE and FALSE that the distinct atomic sentences may have.  We say distinct here to indicate that though an atomic sentence may appear multiple times within the same problem, we need to track what happens when that atomic sentence is considered FALSE, for example.  More broadly, the left set of columns depicts the set of inputs that we use to determine what will be the outputs of the problem.

Importantly, then, the inputs that we use as the starting resources for our problem are obvious to us from the beginning:  the goal of the truth table is to depict all possible ways in which the logical statement may turn out to be TRUE or FALSE, and we simply need to make sure that we evaluate all possible combinations of TRUE and FALSE among the distinct atomic sentences for our table to be complete.

A logical statement is composed of atomic sentences or atomic sentences and operators.  We can easily determine the set of possible truth values for an atomic sentence:  by definition, an atomic sentence may either be TRUE or FALSE.  So, a truth table for an atomic sentence is quite simple:

This table represents all the possible truth values that the logical statement ‘P’ may hold according to our system.  To be sure, logical statements can and often are much more complicated than being composed of a single atomic sentence.

The primary tool that we will use for determining how to ‘read’ a logical statement is by finding what is called the ‘governing connective’ in the logical statement.  The governing connective is the logical operator that best represents the logical statement as a whole.  We are already familiar with determining what the governing connective, just in the context of mathematics.

Consider the following logical statement:

P -> ~(Q & R)

At this point, we may be able to set up the truth table, but we are unsure of how to fill in the table because we do not yet know how ‘read’ the logical statement.  We do not know how to understand the relationship between atomic sentences and logical operators.  We know that this logical statement involves three atomic sentences (P, Q, and R), and we know that each atomic sentence is something that is capable of being TRUE or FALSE.  However, we do not know how the presence of the logical operators affects the truth value of the logical statement as a whole.

Now consider the following mathematical statement:

3 + (2 – 1)²

We are able to recognize quickly a few key aspects of this mathematical statement:  (1) there are three mathematical operators present in the statement (the addition sign, the subtraction sign, and the exponent); (2) we can only determine the right answer to this problem if we apply the mathematical operations in the right order; and (3) the final operation that we perform in order to get the answer is addition.

Mathematical operators function in very similar ways as our logical operators:  for a given set of inputs, the mathematical operator produces an output.

Consider the following example:

2 – 1 = 1

The mathematical operator is subtraction, and the numbers 2 and 1 are ‘inputted’ into the operator, producing the answer of 1.  Subtraction is a binary operator insofar as it requires two inputs before the operation can take place.  Addition, multiplication, and division are other basic mathematical operators that are also binary:  each of these operators require two inputs before the operation can take place, and output of each operation is determined by the definition of the operator.

2 + 1 = 3

Here the operator is addition, and the output of the operation is different even though the numbers inputted into the operator are the same.

The exponent is an operator that is unary:  only one input is required by the operation.

Returning to our more complicated mathematical statement, we need to figure out the order of operations to get the correct answer.  Depending on educational background, you may have learned the following mnemonic that helps us remember the order of operations:

Brackets

Exponents

Division

Multiplication

Addition

Subtraction

To solve a mathematical statement, we need to first perform operations with parts of the statement that are brackets; we then work with exponents; then, division and multiplication (these operations are on the same ‘tier’ in order); then, addition and subtraction (these operations are on the same tier in the order).

The mathematical statement we are focused on is:

3 + (2 – 1)²

Following BEDMAS, we are required to carry out the solution in the following order:

We must first perform the operation within the brackets, where the subtraction operations is present:

3 + (1)²

Then, we carry out the operation with the exponent:

3 + 1

Finally, we carry out the addition operation and get the final answer:

4

Each of these operations took a set of inputs and produced outputs, and the subsequent outputs were used as inputs for later steps in our solution.  A very similar process is going to play out when we start solving logical statements.  We need to figure out the order of operations for the logical statement, and, as we are completing the problem, the outputs from parts of our solution will be used as inputs for other parts of the solution.

Consider the following logical statement:

P & Q

First, following the practical approach to constructing a truth table, we get the following:

A mathematical statement that is similar is the following:

1 + 2

In order to get the answer, we would have to input the numbers 1 and 2 into the binary operator that is addition.  Performing the operation, our answer is 3.

When beginning to fill in the truth table, we need to recall a key point:  the starting resources of our truth table represent the total set of possible combinations of truth values for the atomic sentences present in the logical statement.

Whereas 1 and 2 are the numbers inputted into the addition operation, the truth values for P and Q on a given row are inputted into the conjunction operator.  So, while we get the complete answer to the mathematical problem through one calculation, we can only get the complete answer to our truth table by performing the operation for every row in the table.  The starting resources represent the possible truth values for the most basic components of any logical statement – atomic sentences.

Since the conjunction is a binary logical operator, its operation can only take place when two inputs are inputted into the operation.  In row 1, we are supposing that P is TRUE and Q is TRUE.  Recalling the definition of the conjunction operator, a conjunction produces a TRUE only when both inputs are TRUE.  So, the output for the conjunction on row 1 is TRUE.

We continue down the rows, subbing in the truth values for P and Q that are specified on each row.  We have completed this truth table only when we have performed four operations, which turns out to be accounting for all possible combinations of TRUES and FALSES for P and Q.  So, for rows 2-4, we must input the truth values for P and Q, and then determine what the output is after the conjunction operation is performed.  We get the following:

This is a complete truth table because every column with a logical operator is filled in.

Another Example:

Consider the following logical statement that is more complicated:

(P & Q) v Q

First, we fill out the truth table’s header row and starting resource columns:

Since there are multiple logical operators found in this statement, we must be take care to complete the truth table in the proper order.  One practical approach to determining the order of operations for a logical statement is to translate the statement into a mathematical statement.  Doing so allows us to see more easily which inputs are used where to complete the table.

Atomic sentences are like numbers, so we can assign P = 1 and Q = 2, just so we can track which number refers to which atomic sentence.

We can translate any logical operator into any mathematical operator.  Since we are concerned with tracking which inputs are used to complete which operation, we do not need to overcomplicate our mathematical statement.

Any binary logical operator may be translated into the addition operator.  Any unary operator may be translated into the exponent operator.

(P & Q) v Q

Turns into the following mathematical statement:

(1 + 2) + 2

The reason why translating the logical statement into a mathematical one is that we can easily see the proper order of operations required to get the correct answer.  We first must solve for what is inside the brackets:

1 + 2

The addition operator is binary and requires two inputs to be completed.  The inputs in this case are 1 and 2.  The result within the brackets is 3.  Returning to the whole statement, we get:

(3) + 2

Notice that in order to perform the next mathematical operation, we need to use the output of our operation carried out inside the brackets.  In other words, this addition operator is binary, and, from its perspective, the inputs are the number that is the result of the operation within the brackets, 3,  and 2.  Since no other operations are ‘at play’ within the statement, this is the final operation and provides us with the overall answer of 5.  Now let us turn to the logical statement to see the similarities that arise when completing the table.

Just as our first step in solving the mathematical statement is to perform the addition operation within the brackets, the first step we must take to solve the logical statement is to complete the truth column for the conjunction that resides within the brackets.

(P & Q)

Like addition, the conjunction operator is also binary and requires two inputs.  Whereas the numbers inside the brackets served as the inputs for the addition operator, the atomic sentences P and Q are the inputs for the conjunction operator.  We only had to perform the addition operation within the brackets once to determine the output of that addition.  For our logical statement, we need to perform the operation for every row in the column, and each row tells us what the inputs are for conjunction operation.  So, whereas we took the numbers 1 and 2 as inputs for the addition operator in our mathematical statement, we take the truth values for P and Q as inputs for the conjunction operator.  In the first row, our starting resources tell us to suppose that P is TRUE and Q is TRUE.  The conjunction operator produces an output of TRUE only when both inputs are TRUE; otherwise, the operator produces an output of FALSE.  So, the output for first row in this case is TRUE.

Now, we simply move onto the next row, noting what the inputs are and filling in the column for the conjunction operator.

What have we accomplished?  We have determined the ‘impact’ that the conjunction operator has on the logical statement for all possible combinations of TRUE and FALSE for our atomic sentences.  Importantly, we have not yet completed the truth table since there is still a column for a logical operator that needs to be filled in.  With respect to our mathematical statement, we have accomplished the equivalent of performing the first operation.

Original statement:

(1 + 2) + 2

First operation performed:

(3) + 2

Notice that our next step for solving the mathematical statement is to perform the remaining addition, where the inputs for the addition are 2 and the outcome of our first operation.  A similar pattern takes place on our truth table.  The disjunction is a binary operator, requiring two inputs to produce an output.  From the perspective of the disjunction, the two inputs are Q and the outcome of our first logical operation.  Our first operation has reduced (P & Q) down to one simple column filled with truth values, and the disjunction operation requires us to compare that simple column with the column that represents the right disjunct, Q.  We already know the column that represents Q, as it is found in our starting resources.  Recall that the disjunction operator produces a FALSE only when both inputs are FALSE; otherwise, the operator produces a TRUE.

Looking at the first row of our truth table, the truth value for the left disjunct is found in the first row of the column under the conjunction, and the truth value for the right disjunct is found in the first row of the column under the atomic sentence Q.

To finish the column for the disjunction, we simply go down the remaining rows, where each row determines what the inputs are for the operation.

With respect to our mathematical statement, we have accomplished the equivalent of performing the second operation.

Original statement:

(1 + 2) + 2

First operation performed:

(3) + 2

Second operation performed:

5

Notice that we just as we have ‘solved’ the mathematical statement after completing the second operation, we have also ‘solved’ the logical statement after filling in the column under the disjunction.  In other words, the column under the disjunction best represents the ‘final answer’ to the logical statement or best represents the logical properties of the statement overall.

The logical statements that will be evaluated in this text will eventually be more complicated, but the mechanics of processing the solution will remain the same.  We could always add more numbers and operators to our mathematical statement, but the underlying ‘rhythm’ to solving the statement is unchanged – just more steps!