A truth table is a complete map of how a logical expression evaluates for every possible situation. Truth tables can be used as an aide when designing and debugging logical conditions; they help verify logical equivalence, and test whether an argument is always true, always false, or sometimes true and sometimes false. They can also help reason what circuits do without drawing every wire.
Variables
Logical statements can be evaluated using any variables (such as A, B, or C). However, P and Q have become the universally accepted standard in propositional logic.
Why P and Q?
By convention, truth tables often use the letters P and Q to represent logical statements. Each letter stands for a statement that is either true or false. For example, P might represent the proposition “It is raining”. In propositional logic a proposition is a declarative statement or claim that can be evaluated as either objectively true or false.
P and Q represent logical variables in the same way x and y represent unknown variables in traditional mathematics. The letters themselves have no special meaning, they are simply a long-standing convention in propositional logic.
Example: OR
Logical OR will be true if at least one of the statements is true, but false only if both statements are false.
For example:
- Let P = “it is raining”
- Let Q = “it is windy”
The statement “P or Q” is true unless it is not raining AND it is not windy.
Logic Symbols
Symbols are used to streamline logical representation. For example, to say “P or Q is true” the symbol ∨ is used: P ∨ Q.
Basic logic operators:
- ∧ (AND)
- ∨ (OR)
- ¬ (NOT)
Full list of logic symbols (Wikipedia.org)
Example Truth Table
Let’s create a truth table for “P or Q” (P ∨ Q).
Define Propositions
Our first step is to define the propositions.
- Let P = “It is raining”.
- Let Q = “It is windy”.
List Combinations
Second, we list every possible combination.
- P is True, Q is True: (It is raining AND it is windy).
- P is True, Q is False: (It is raining AND it is NOT windy.
- P is False, Q is True: (It is NOT raining AND it is windy).
- P is False, Q is False: (It is NOT raining AND it is NOT windy).
Evaluate Logical expressions
Finally, we evaluate the logical expressions.
- P ∨ Q is True: if either P is true, or Q is true, or both are true.
- P ∨ Q is False: only when both P and Q are false.
Create the “P ∨ Q” Truth Table
The truth table for P ∨ Q maps out every possible combination into rows. This gives us a quick view of all the combination statements we’ve created.
For example, in the first row of the table, P is true, Q is true, and P ∨ Q is true. Since both P and Q are true, the truth value for P ∨ Q (P or Q) is true. The last column will only be false if both P and Q are false.
| P (It is raining) | Q (It is windy) | P ∨ Q (It is raining OR it is windy) |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Reading Truth Tables
When you look at a truth table, read each row as one possible combination of inputs. The column headings will explain each part of the row.
Example: P ∧ Q Truth Table
This table shows all possible combinations for the proposition P ∧ Q (P is true AND Q is true). Each row represents one possible combination of truth values. Columns show the values of each variable and the resulting expression.
| P | Q | P ∧ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Under the column headings, the first row shows three T (true) values. Let’s take this first row step by step through each of the columns.
- In the column under the heading of “P”, there is a T for true. This means P is true.
- The next column, Q, also has a T for true.
- The final column, P ∧ Q is the final truth value for that row.
One way to read this row is: “P is true, Q is true, P ∧ Q is true”. The last column, P ∧ Q, has the value true because the AND (∧) requires both P and Q to be true.
You’ll notice that the rest of the rows have a final truth value of false. That is because in the rest of the rows either P, Q, or both have a value of false. For P ∧ Q, the final column can only be true if both P and Q are true.
Summary
Truth tables give a systematic, complete view of how logical expressions behave for all input combinations. They’re foundational for understanding logic gates, Boolean algebra, and reasoning about logical statements.
- A truth table maps every possible combination of truth values for a logical expression.
- They help with designing and debugging logic, checking logical equivalence, and analyzing whether statements are always true, always false, or sometimes true.
- They’re also useful for understanding logic circuits.
Further Reading
- Logic Gates: Truth tables provide a systematic way to describe how logic gates behave for every possible combination of inputs.
- Intro to Truth Tables & Boolean Algebra and, by the same author, a logic puzzle: A Classic Logic Puzzle: Vampires, Insanity & Transylvanian Sisters
