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Published in:
01-01-2012 | Erratum
Erratum to: n-Fold implicative basic logic is Gödel logic
Published in: Soft Computing | Issue 1/2012
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Excerpt
Unfortunately, there are some errors in the online published article.
1.
The proof of Proposition 5 is wrong. Equation (10) can be utilized only if the power of an element x is n, not if it is n − 1 as is done in the proof. Moreover, an n + 1 element Lukasiewicz chain is an explicit example that Proposition 5 does not hold: {1} is an n-fold implicative filter in that algebra, however, the Lukasiewicz product is not idempotent.
2.
Corollary 6, Theorem 8 and Theorem 9 are based on Proposition 5, therefore also they do not hold. Since Gödel logic trivially satisfies axiom (12), we can only say that Gödel logic is an example of n-fold implicative basic logic, however, n-fold implicative basic logic is not, in general, Gödel logic, contrary to the claim in Theorem 9.
3.
The inaccuracy of Proposition 5 also implies that the problem of the relation of n-fold implicative basic logic and n-fold positive implicative basic logic remains open, contrary to what is alleged in the Conclusion.