13,324 research outputs found
Interval-valued algebras and fuzzy logics
In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter
Fuzzy inequational logic
We present a logic for reasoning about graded inequalities which generalizes
the ordinary inequational logic used in universal algebra. The logic deals with
atomic predicate formulas of the form of inequalities between terms and
formalizes their semantic entailment and provability in graded setting which
allows to draw partially true conclusions from partially true assumptions. We
follow the Pavelka approach and define general degrees of semantic entailment
and provability using complete residuated lattices as structures of truth
degrees. We prove the logic is Pavelka-style complete. Furthermore, we present
a logic for reasoning about graded if-then rules which is obtained as
particular case of the general result
A comparison of techniques for learning and using mathematics and a study of their relationship to logical principles
Various techniques exist for learning mathematical concepts, like experimentation and exploration, respectively using mathematics, like modelling and simulation. For a clear application of such techniques in mathematics education, there should be a clear distinction between these techniques.
A recently developed theory of fuzzy concepts can be applied to analyse the four mentioned concepts. For all four techniques one can pose the question of their relationship to deduction, induction and abduction as logical principles. An empirical study was conducted with 12-13 aged students, aiming at checking the three reasoning processes
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