##

**12 July 2017**

**05:00PM - 07:00PM**

##### Room: **Catalana**

##### Chairs: **Brunella Gerla, Diego Valota and Pietro Codara**

## Recent trends in many-valued logic and fuzziness

**Abstract - ** We study commutative idempotent semirings in general, and some examples in particular. We show that the class Red of semiring reducts of MV-algebras, although axiomatized by a first order theory, is not axiomatized by a geometric theory (in the topos-theoretic sense) or a universal-existential first order theory. Then we perform comparisons between the class Red, the class of all semirings, and some so-called exotic semirings.

**Abstract - ** Fuzzy relational compositions based on fuzzy quantifiers naturally do not preserve all the properties that are preserved for "standard" fuzzy relational compositions and, in many cases, the property is preserved only in a weaker form. For example, the associativity, that is preserved in the standard case derived from the universal and the existential quantifiers, generally does not hold for the case of compositions based on fuzzy quantifiers. However, is it the case that only the standard quantifiers lead to the preservation of such properties? Without any restriction on the shape of the fuzzy relations, the answer is positive.

**Abstract - ** A structural description of absorbent-continuous group-like commutative residuated lattices over complete, order-dense chains will be presented. The theorem is sharp, no further generalization is possible. Group-like commutative residuated lattices will be characterized as Abelian lattice-ordered groups deprived of their cancellative property only. The so- called partial-lexicographic product constructions (two of them) will be introduced, which construct group-like commutative residuated lattices. As a side-effect, it gives rise to the so-called involutive ordinal sum construction, which constructs group-like commutative residuated lattices from a family of group-like commutative residuated lattices. Via two decomposition theorems, corresponding to the partial-lexicographic product constructions, it will be shown that any order-dense group-like commutative residuated chain, which has only a finite number of idempotents can be built by iterating finitely many times the partial-lexicographic product constructions using solely totally ordered Abelian groups, as building blocks. The result extends the famous structural description of totally ordered Abelian groups by Hahn, to order-dense group-like commutative residuated chains with finitely many idempotents.

**Abstract - ** A two-fold general approach to the theory of formal concept analysis is introduced by considering intuitionistic fuzzy sets valued on a residuated lattice as underlying structure for the construction.

**Abstract - ** In this paper we study Moisil logic, a many-valued system based on the idea of nuancing. We prove a completeness theorem for graded deduction and make a first attempt towards a game semantics for this logic.

**Abstract - ** We provide a standard completeness proof which uniformly applies to a large class of axiomatic extensions of Involutive Monoidal T-norm Logic (IMTL). In particular, we identify sufficient conditions on the proof calculi which ensure density elimination and then standard completeness. Our argument contrasts with all previous approaches for involutive logics which are logic-specific.

**Abstract - ** In a previous paper, it was shown that the (minimal) modal logic \${$\backslash$}MLn\$ with fuzzy accessibility relations over the finite-valued \{{$\backslash$}L\}ukasiewicz logic {$\backslash$}L\$\_n\$ and a corresponding multi-modal logic \${$\backslash$}mMLn\$ (with a modality \${$\backslash$}Box\_a\$ for each value \$a\$ in the \$n\$-valued {$\backslash$}L\$\_n\$-chain) had the same expressive power when the language is extended with truth-constants. In this paper we partially extend these results partially when replacing the underlying logic {$\backslash$}L\$\_n\$ by the infinite-valued \{{$\backslash$}L\}ukasiewicz logic (with rational truth constants in the language). We prove that the (standard) tautologies of the modal logic \${$\backslash$}ML\$ (resp. \${$\backslash$}mML\$) are in fact the common tautologies of all the logics \${$\backslash$}MLn\$ (resp. all the logics \${$\backslash$}mMLn\$) when letting \$n\$ vary over \${$\backslash$}mathbb\{N\}\$. This fact opens the door to show an alternative proof of the finite model property for these logics and hence their decidability.

**Abstract - ** We give a [0,1]-functional representation of the finitely generated free algebras in the variety generated by Chang's MV-algebra C, and in the variety generated by the left continuous t-norm arising as Jenei's rotation JPi of the product t-norm. We generalise the construction of JPi from C by building a family T\_n of involutive t-norm algebras such that the MV-algebras in the variety generated by T\_n form the variety generated by S\_n\^{}{$\backslash$}omega and L\_\{n+1\}.