THE LOGIC NOTES

Truth table Glossary

Definition

A truth table is simply a way of representing a truth function in the form of a look-up table giving the value of the function for each combination of inputs.

Comments

Each truth-functional connective has its truth table. Moreover, every formula of propositional logic expresses a truth function, so every formula has a truth table as well.

Truth tables of any arity can generally be written as columns of values. However, for binary operations including the standard binary connectives we usually write them in a square matrix format with rows corresponding to values of the left subformula and columns corresponding to values of the right one.

Examples

  1. Conjunction has the truth table:
    AND 0 1
    0 0 0
    1 0 1
    The top right entry in the table, for instance, gives the value of A AND B in the case where A is false and B is true.
  2. The formula p IFF (q IFF NOTr) has the truth table:
    p q r p IFF (q IFF NOTr)
    0 0 0 1
    0 0 1 0
    0 1 0 0
    0 1 1 1
    1 0 0 0
    1 0 1 1
    1 1 0 1
    1 1 1 0
    The fifth row of this table, for instance, gives the value of the formula in the case where p is true and both q and r are false. From the table, we can note that this formula is true iff the number of true inputs is even.

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