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.


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