Truth table proofs
Truth tables are used to show all possible values that a given logical expression might take.
Definition of AND
The following gives the definition of the logical AND operation.
|A||B||A ∧ B|
Proof of one of De Morgan's laws
A truth table can also be used to prove a logical identity. The following proves De Morgan's law that ¬ (A ∧ B) is equivalent to (¬ A) ∨ (¬ B). Notice that the truth values for these (the last two columns in the table) are always the same.
|A||B||A ∧ B||¬ (A ∧ B)||(¬ A) ∨ (¬ B)|
Disproof of a logical fallacy
Truth tables can also be used to disprove a logical fallacy. For example, one fallacy is to assume the converse of an implication holds. If you know that (A → B) is true, and you know that B is true, the fallacy would be to then conclude that A must also be true. Note that the statement - "if an animal is a mammal then it is also a vertebrate" - is true. If we have an animal that is a vertebrate (for example, a dog), it would be a fallacy to now conclude that the animal must also be a mammal.
Consider this truth table.
|A||B||A → B (which is equivalent to ¬ A ∨ B)|