In this article some new results are presented about planar quadrangle quality measures. On the one hand, a modi cation of the triangle quality measure based on the Frobenius norm is proposed, in order to comply with the case where the reference element is no longer equilateral, but a right isosceles triangle. Provided this modi cation, a natural use of the Frobenius norm for quadrangular eleme...

a. ‖x‖ ≥ 0 for all x ∈ IRn, and ‖x‖ = 0 if and only if x is the zero vector. b. ‖αx‖ = |α|‖x‖. c. ‖x+ y‖ ≤ ‖x‖+ ‖y‖ [called the “triangle inequality”]. While this may seem a bit esoteric, consider that if x ∈ IR1, i.e. if x is a real number, the absolute value function, | · |, satisfies all of the above properties, and therefore constitutes as a norm. Another familiar norm would be the Euclidea...

of the origin in C ' , and vice versa. But clearly, not every convex compact neighbourhood of the origin in C ~' belongs to a lub norm. The principal problem to be considered here is the geometric characterizat ion of those convex bodies in C ~' which are associated with lub norms. The question arises because of the fact tha t m a n y of the properties of vector norms fall quite easily and natu...

The label semantics linguistic representation framework is introduced as an alternative approach to computing and modelling with words, based on the concept of appropriateness rather than graded truth. A possible worlds interpretation of label semantics is then proposed. In this model it is shown that certain assumptions about the underlying possible worlds can result in appropriateness measure...

As it was shown in [7], Monoidal t-norm based logic (MTL) is a logic of left-continuous t-norms. In other words, this means that MTL enjoys the standard completeness theorem. In this paper we present a different proof of this theorem. In fact, we prove even more since we show that MTL is complete w.r.t. the class of standard MTL-algebras with finite congruence lattices or equivalently with fini...

A dialogue game based approach to the problem of providing a deeper semantic foundation for t-norm based fuzzy logics is explored. In particular, various versions, extensions and alternatives to Robin Giles’s dialogue and betting game for Lukasiewicz logic are re-visited and put in the context of other foundational research in logic. It emerges that dialogue games cover a wide range of topics r...

It is known that the monoidal t-norm based logic (MTL) and many of its schematic extensions are decidable. The usual way how to prove decidability of some schematic extension of MTL is to show that the corresponding class of algebras of truth values has the finite model property (FMP) or the finite embeddability property (FEP). However this method does not work for the extensions whose correspo...

In this paper we prove the strong standard completeness of interval-valued monoidal t-norm based logic (IVMTL) and some of its extensions. For other extensions we show that they are not strong standard complete. We also give a local deduction theorem for IVMTL and other extensions of interval-valued monoidal logic. Similar results are obtained for interval-valued fuzzy logics expanded with Baaz...