6. Filling in a Truth Table – Another Example

Let us construct and complete a truth table for the following logical statement:

P -> (~R & Q)

First, we fill out the truth table’s header row and starting resource columns.  Note that this logical statement has three distinct atomic sentences, so our truth table must account for all possible combinations of truth values for all distinct atomic sentences.  Recall that since this this statement has three distinct atomic sentences, we need a truth table with eight rows (not including the header row) in order to capture all possible combinations:

P -> (~R & Q)

We can translate this logical statement into the following mathematical statement:

P = 1, Q = 2, R = 3

1 + (3² + 2)

(Again, the reason why we translate every binary logical operator into the addition operator is because we are primarily concerned with tracking which inputs go with which operation.  We translate the negation (~) into the square [²] because, like the negation, the exponent is a unary operator.)

First operation – the exponent:

1 + (9 + 2)

Second operation – the addition in the brackets:

1 + (11)

Third operation – the remaining addition:

12

With respect to our logical statement, then, our first operation will be processing the negation using the truth value column for R as the input.  Our second operation will be processing the conjunction using the truth value column for the negation and the truth value column for Q as inputs.  Finally, our third operation will be processing the conditional using the truth value column for P and the truth value column for the conjunction as inputs.

First operation – the negation, using R as the input:

Second operation – the conjunction, using the output of the first operation and the Q as inputs:

Third operation – the conditional, using P and the output of the conjunction as inputs:

Our truth table is now complete, and the column under the conditional operator best represents the ‘final answer’ for this logical statement.