Logical Reasoning
How to Build a truth table
Use a truth table to work out when a compound statement is true, or to check whether an argument is valid.
Before you start
- Know the three connectives: negation (~), 'and' (∧) and 'or' (∨) and their symbols.
- Understand that a simple statement is either true or false.
- Be able to list every true/false combination systematically (2 statements → 4 rows).
When to use it
Use a truth table to work out when a compound statement is true, or to check whether an argument is valid.
The steps
- List the simple statements (p, q) as columns and write every combination of true and false.
- Add a column for each connective in the compound statement.
- Fill in the negation, “and”, or “or” columns using their rules.
- Work outward to the final compound column.
- Read the final column to answer the question.
Worked example
Build a truth table for ~p ∨ q and state in which cases the statement is true.
- List the simple statements p and q. With two statements there are four combinations: (T, T), (T, F), (F, T), (F, F).
- Add a column for the negation ~p and a column for the compound ~p ∨ q.
- Fill ~p using the negation rule: when p = T, ~p = F; when p = F, ~p = T, giving ~p = F, F, T, T.
- Work outward to ~p ∨ q, which is true when at least one of ~p or q is true: row 1 F ∨ T = T, row 2 F ∨ F = F, row 3 T ∨ T = T, row 4 T ∨ F = T.
- Read the final column ~p ∨ q = T, F, T, T: it is true in every case except when p = T and q = F.
A second example, with a twist
This time you build columns for two compound statements and compare their final columns to test whether they are logically equivalent. Use a truth table to determine whether ~(p ∧ q) and ~p ∨ ~q are logically equivalent.
- List p and q in the usual four rows: (T, T), (T, F), (F, T), (F, F).
- Add columns for p ∧ q, then ~(p ∧ q), then ~p, ~q and finally ~p ∨ ~q.
- Fill p ∧ q (true only when both are true) = T, F, F, F; fill ~p = F, F, T, T and ~q = F, T, F, T.
- Work outward: ~(p ∧ q) = F, T, T, T; and ~p ∨ ~q gives row 1 F ∨ F = F, row 2 F ∨ T = T, row 3 T ∨ F = T, row 4 T ∨ T = T, so ~p ∨ ~q = F, T, T, T.
- Read the two final columns: both are F, T, T, T, so they are identical.
Practise this in a KBAT problem
Frequently asked questions
How many rows does my truth table need?
It depends on how many simple statements you have. For two statements like p and q you need 2² = 4 rows, one for each true/false combination.
Three statements need 2³ = 8 rows. Always list the combinations in a fixed order so you never miss one or repeat a row.
What is the difference between the 'and' and 'or' columns?
The 'and' column (p ∧ q) is true only when both p and q are true; in every other row it is false. The 'or' column (p ∨ q) is true when at least one of them is true, and false only when both are false.
Mixing up these two rules is the most common mistake.
Do I really need a separate column for every connective?
Yes. Build one working column for each connective, the negation first, then 'and' or 'or', then the final compound.
Doing it step by step keeps long statements manageable and lets you check each stage. Trying to jump straight to the final answer usually causes a sign or logic slip.