In propositional logic, a truth table serves two purposes: defining logical connectives and testing validity of arguments.

Testing Validity with a Truth Table

We can constructing a truth table that includes all possible combinations of basic propositions. By examining the truth table, we can determine if there is any row where the premises are all true, but the conclusion is false.

This can be view as a form of model checking.

\begin{prooftree} \noLine \AXC{L R} \UIC{L} \UIC{ R} \end{prooftree}

\begin{array}{ll|l|l|l} L & R & L \vee R & L & \neg R \ \hline \text { T } & \text { T } & \text { T } & \text { T } & \text { F } \ \text { T } & \text { F } & \text { T } & \text { T } & \text { T } \ \text { F } & \text { T } & \text { T } & \text { F } & \text { F } \ \text { F } & \text { F } & \text { F } & \text { F } & \text { T } \end{array}

Testing validity using a truth table has some limitations. First, a large number of basic propositions can lead to combinatorial explosion of rows, making truth table impractical. Second, truth tables are no longer sufficient for testing validity in predicate logic, and other methods must be employed.