What does a truth table show?
Method: A truth table lists every possible truth-value assignment and calculates the compound statement for each row. It is a completeness check: no case is hidden or skipped.
Build all four truth cases
- Name the two statements without changing their meaning.
- Choose the connective used between them.
- Read all four rows, including the cases that feel unlikely.
- Translate a false row back into ordinary language.
Read implication without causal assumptions
For A implies B, the only false row is A true and B false. A promise such as 'If the alarm is armed, the light is on' fails only when the alarm is armed while the light is off.
What a truth table cannot establish
Use false rows as counterexample targets
The most useful row is often the one that makes the compound statement false. It describes the exact counterexample a rule must prevent. For equivalence, inspect both mismatched rows. For exclusive OR, inspect the both-true row as well as both-false. Naming those cases turns a symbolic table into a practical rule test.
Labels may contain complete statements such as the alarm is armed and the warning light is on. Keeping those sentences beside the Boolean values makes a false result readable without translating A and B from memory.
Boundary of the result
Material implication evaluates the stated relationship. It does not prove that A causes B.
Continue with the method
Use the conditional logic guide and the reasoning test and the probability calculator for two independent events to move from a quick result to a complete reasoning process.