Incompleteness Theorems: The Logical Necessity of Inconsistency

Incompleteness theorems prove that there are logically undecidable propositions, i.e., that there are propositions that are neither provable nor disprovable in certain classes of theories.

Incompleteness of Principia Mathematica was proved informally using proof by contradiction in a stratified metatheory by Gödel [1931] with restrictive conditions. Then Rosser [1936] informally proved incompleteness using proof by contradiction in a stratified metatheory assuming consistency of Principia Mathematica.

However, information on the modern Internet is pervasively inconsistent and restricting reasoning to use only stratified metatheories is impractical. Consequently Direct Logic has been developed which is inconsistency tolerant and does without stratified metatheories. And incompleteness has been formally self-proved for every theory of Direct Logic without requiring the hypothesis of consistency. Moreover, because incompleteness is self-proved, logically necessary inconsistency immediately follows.



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Relevance logic

Relevance logics represent an approach to logic that is:

  1. Well understood, with a reasonably long history;
  2. Posess good proof and model theories;
  3. Are paraconsistent (i.e., tolerate inconsistency) without needing artificial techniques such as stratification or reflection.

How do the aims and achievements of direct logic compare and relate to those of relevance logic?

Relevance Logic versus Direct Logic

Direct Logic is a minimal fix to classical mathematical logic and statistical probability (fuzzy) inference that meets the requirements of modern computer science by addressing the following issues: inconsistency tolerance, contrapositive inference bug, and direct argumentation.

For example, in classical logic, the contrapositve holds for inference. The same issue affects probabilistic (fuzzy) inference. Also, in the Tarskian framework of classical mathematical logic, a theory cannot directly express argumentation.

Relevance Logic arose from attempts to axiomatise the notion that an implication P Q should be regarded to hold only if the hypothesis P is “relevant” to the conclusion Q. According to [Routley 1979], “The abandonment of disjunctive syllogism is indeed the characteristic feature of the relevant logic solution to the implicational paradoxes.” Since Direct Logic incorporates disjunctive syllogism ((Φ∨Ψ), Â¬Φ Ψ) and does not support disjunction introduction (Ψ Φ∨Ψ), it is not a Relevance Logic.

Direct Logic makes the following contributions over Relevance Logic:

•   Boolean Equivalences

•   Splitting (including Splitting by Negation)

•   Two-way Deduction Theorem (Natural Deduction)

•   Self-refutation

•   Incompleteness Theorem self-provable

•   Direct argumentation

•   Logical necessity of inconsistency

For more information please see Common sense for concurrency and inconsistency tolerance using Direct LogicTM ArXiv 0812.4852