# A three-valued logic

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## Abstract

The logic used in mathematics today asserts that any statement is either true or false. It is the function of logic to establish a way of determining the truth or falsity of a given statement. However, the truth or falsity of some statements either has not been, or can not be determined in the classical logic. Thus, these statements cannot be used in logical arguments since their truth value has not been established. Thus the question arises " Could a three-valued logic be developed in which the statements mentioned above would receive a third value, say M for maybe?"

Several three-valued systems of logic have been developed. The basic truth tables used for the development in this paper were defined in 1938 by S. C. Kleene. However, Kleene adheres strictly to the classical definition of what is meant by a formula being a tautology. Thus in his development, the formula p-->p is not a tautology. The notion of a formula being a tautology has been defined in this paper in such a way that the formula p-->p is a tautology in the three-valued logic. Thus many formulas which would not be tautologies in Kleene's original system are tautologies in the system developed here.

The concepts of equivalence, substitution, and consequence are defined in the three-valued logic in the same way as they are in the classical logic. Their properties and relationship with the new notion of tautology in the three-valued logic are compared with those in the classical logic. Many of the properties which hold true in the classical logic are also true or partially true in the three-valued logic.

A topic of interest in the classical logic is that of truth functions which generate all possible functions of two arguments. Two such functions, the stroke (/) and the dagger function are defined in this paper in such a way that they generate all the functions of two arguments which contain only the usual connectives. However, they do not generate all the possible truth functions of two arguments in the three-valued logic. Thus, a truth function, the square function was defined in such a way that it would generate all possible truth functions of two arguments in the three-valued logic.