Totuustaulukko

A totuustaulukko, often translated as a truth table, is a mathematical table used in logic to determine the truth value of a compound proposition based on the truth values of its constituent simple propositions. It systematically lists all possible combinations of truth values for the simple propositions and shows the resulting truth value of the compound proposition for each combination.

Truth tables are fundamental tools in propositional logic, a branch of formal logic that deals with propositions, which are declarative statements that are either true or false. Compound propositions are formed by combining simple propositions using logical connectives such as AND (conjunction, denoted by ∧), OR (disjunction, denoted by ∨), NOT (negation, denoted by ¬), IF...THEN (implication, denoted by →), and IF AND ONLY IF (biconditional, denoted by ↔).

The construction of a truth table involves creating columns for each simple proposition and for the compound proposition. For 'n' simple propositions, there will be 2^n rows, representing all possible truth assignments (true or false) to these simple propositions. Each row is a unique combination of these truth values. The truth value of the compound proposition in each row is then calculated by applying the definitions of the logical connectives.

Truth tables are used to analyze logical arguments, determine the validity of statements, and simplify complex logical expressions. For example, they can be used to show that two logical statements are equivalent if they have the same truth value in all rows of their combined truth table. They are also essential in computer science for designing and verifying digital circuits and in philosophy for formalizing logical reasoning.