Assume some proved things cannot be disproved

Sunday July 16, 2023

Given a valid proof in a system of assumptions and inference rules, there is no sure way to know that there isn't also a contradictory proof, except by assumption outside the system.

Some systems allow a proof of every statement. For example, take a system where anything follows from a contradiction, as is common, and also assume a contradiction.

Systems that prove everything are not honest. In such systems, you can prove that the system is consistent. You can prove the law of non-contradiction, and that the system contains no contradictions. Seeing such a proof does not establish that an opposite proof doesn't exist. The situation is akin to the futile exercise of trying to determine whether a person is a liar by asking them.

The situation is also bad even if attempting to show for just one proved statement that there isn't a proof of its negation. Socrates presents argument 1, demonstrating conclusion A. Socrates then presents argument 2, demonstrating conclusion B, which is that there is no argument for the negation of A. Socrates then presents argument 3, demonstrating conclusion C, which is that there is no argument for the negation of B. And so on, forever, for every initial conclusion A.

A proof of consistency cannot be convincing, whether it be for a system overall or even for a single statement in the system (that for a specific proof, there isn't a proof for the opposite conclusion).

Stepping up a level, to a meta-system where the consistency of the original system can be proved, does not solve the problem, because the meta-system hasn't itself been shown to be consistent.

Gödel's second incompleteness theorem is a special case of this phenomenon. It shows that for formal systems with certain properties, they can't prove that they are consistent.

Introducing the law of non-contradiction as an axiom, or assuming that the system contains no contradictions, does not solve the problem. Such a claim made within the system can always coexist with its negation, if it is present.

If a system proves both a statement and its negation, there is no ambiguity. If a system proves a statement but its negation is not proved, it is necessary to assume that there is no such proof.