In some systems there are no truth tables, but rather just formal axioms (e.g. strings of symbols from a set { ~, →, (, ), variables p1, p2, p3, ... } and formula-formation rules (rules about how to make more symbol strings from previous strings by use of e.g. substitution and modus ponens). the result of such a calculus will be another formula (i.e. a well-formed symbol string). Eventually, however, if one wants to use the calculus to study notions of validity and truth, one must add axioms that define the behavior of the symbols called "the truth values" {T, F} ( or {1, 0}, etc.) relative to the other symbols.