Avoiding Contradictions in the Paradoxes, the Halting Problem, and Diagonalization

The fundamental proposal in this article is that logical formulas of the form (f ↔ ¬f) are not contradictions, and that formulas of the form (t ↔ t) are not tautologies. Such formulas, wherever they appear in mathematics, are instead reason to conclude that f and t have a third truth value, different from true and false. These formulas are circular definitions of f and t. We can interpret the implication formula (f ↔ ¬f) as a rule, a procedure, to find the truth value of f on the left side: we just need to find the truth value of f on the right side. When we use the rules to ask if f and t are true or false, we need to keep asking if they are true or false over and over, forever. Russell’s paradox and the liar paradox have the form (f ↔ ¬f). The truth value provides a straightforward means of avoiding contradictions in these problems. One broad consequence is that the technique of proof by contradiction involving formulas of the form (f ↔ ¬f) becomes invalid. One such proof by contradiction is one form of proof that the halting problem is uncomputable. The truth value also appears in Cantor’s diagonal argument, Berry’s paradox, and the Grelling-Nelson paradox.