## Foundations of Mathematics## Mathematical and Formal Logic |

## What are truth values and truth functions? |

As seen above, when discussing propositional calculus, a proposition is any declarative sentence that is either true (T) or false (F). Mathematicians refer to T or F as the *truth value* of the statement.

The combinations of such statements are known as *truth functions,* with their true values determined from the overall true values of their contents. Truth-functional analysis includes the following logical operators:

*Negation*—The negation of a statement is false if the original statement is true, and true if the original statement is false; it refers to “it is not the case that” or simply “not” in natural language.

*Conjunction*—The conjunction of two statements is true only if both are true and false in all other instances; it refers to “and” in natural language.

*Alteration*—Alteration (or disjunction) of two statements is false only if both are false and true in all other instances; it refers to “or” (and “either … or”) in natural language.

*Conditional*—Conditional (or implication) is false only if the antecedent is true and the consequent is false, and is true in all other instances; it refers to “if … then” or “implies” in natural language.

*Biconditional*—Biconditional (double implication or bi-implication) is true only if the two statements have the same value, either true or false; it refers to “if and only if…” in natural language.