Apodictic judgments are clearly provable or logically certain. Perniciously polarized societies often witness public controversies over factually provable questions. The function is then called provably secure, or just provable. Since every tautology is provable, the logic is complete. It doesn't deliver a single, coherent, scientifically sensible or provable scenario. Completeness states that all true sentences are provable. The security is provable, unbreakable, and quantifiable (in bits/second/hertz). It can mean just "not provable", leaving open whether an independent statement might be refuted. Here "logical proposition" refers to a proposition that is provable using the laws of logic. Formal mathematics is based on "provable" truth. In III.a We assume that if is provable it is implied. Together they imply that all and only validities are provable. For example, consider showing that a given proposition is "not" provable in natural deduction. A major goal in cryptography is to create cryptographic primitives with provable security. Alternatively, a number of techniques exist for efficiently generating provable primes.