Expanding basic fuzzy logic with truth constants for component delimiters
The paper introduces and studies expansions/extensions of Hájek's basic fuzzy logic BL based on the ordinal sum decomposition of BL-chains. Each saturated BL-chain is decomposable into an ordinal sum whose non-singleton components are isomorphic to either the standard (Łukasiewicz) MV-chain or the standard Gödel chain or the standard product chain. The author introduces first the logic \(\mathrm{BL}_{\mathrm{EP}(*)}\), whose language is BL expanded with a new constant for each idempotent element of a continuous t-norm \(*\) that is the end-point of a component in the aforementioned ordinal-sum decomposition. Book-keeping axioms express idempotency and relative order of these elements. Finite strong standard completeness is proved for each \(\mathrm{BL}_{\mathrm{EP}(*)}\) with respect to its natural algebraic semantics. The logic \(\mathrm{BL}_{\mathrm{EP}(*)}\) only captures the external structure of the ordinal sum expressing the t-norm \(*\). To be able to deal with the isomorphism types of each component, the paper introduces first the logic \(\mathrm{BL}_{\mathrm{COMP}(*)}\), adding axioms characterising each component of \(*\). A canonical \(\mathrm{BL}_{\mathrm{COMP}(*)}\)-algebra is a standard \(\mathrm{BL}_{\mathrm{COMP}(*)}\)-algebra such that each one of the new constants is interpreted by its corresponding idempotent end-point element. It is proved that \(\mathrm{BL}_{\mathrm{COMP}(*)}\) is complete with respect to its canonical algebra for each continuous t-norm \(*\) which is the ordinal sum of finitely many components. A further extension of \(\mathrm{BL}_{\mathrm{COMP}(*)}\), with axioms characterising the intervals of the ordinal-sum decomposition of \(*\), gives the logic \(\mathrm{BL}_{\mathrm{COMP}(*)}\) that is proved to enjoy completeness with respect to its canonical algebra for each continuous t-norm \(*\). The paper ends with some results of computational complexity. In particular, it is shown that the t-norm \(*\) can be chosen in such a way that the tautologousness problem for \(\mathrm{BL}_{\mathrm{EP}(*)}\) can be placed arbitrarily high in the arithmetical hierarchy and that it can even be non-arithmetical. This is achieved by coding an arbitrary subset \(S\) of the natural numbers into an ordinal sum whose components are either Łukasiewicz or product. Elements of \(S\) correspond with Łukasiewicz components, elements not in \(S\) with product ones.
propositional logic of standard algebras
scientific article; zbMATH DE number 6109486