Consequences and conjectures in preordered sets

A model for “crisp reasoning” with fuzzy sets

“… Since for all P in F (min ) is \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$C_{\min}(P) \in F(\min)$\end{document}, it has sense to consider although it is only \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$P \subset C_{\min}(P)$\end{document}. It should be noticed that to define C . in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$F({\bf \cdot})$\end{document} and not in [0, 1] X is a realistic constraint 6. Usually, consequence operators are applied to all subsets \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$P \in [0, 1]^X$\end{document}, allowing P to contain contradictory elements with which it cannot be accepted that P , and C .…”

“…It should be noticed that to define C . in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$F({\bf \cdot})$\end{document} and not in [0, 1] X is a realistic constraint 6. Usually, consequence operators are applied to all subsets \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$P \in [0, 1]^X$\end{document}, allowing P to contain contradictory elements with which it cannot be accepted that P , and C .…”

“…and Sp . mappings from \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$F({\bf \cdot})$\end{document} into [0, 1] X , that can be seen6,7 as the operators of hypotheses and speculations , respectively, defined by and, in particular, Of course, it follows that showing that Hyp min and Sp min are the biggest among these operators .…”

On Conjectures in t-Norm Based Fuzzy Logics

“…The paper deals with models of ordinary reasoning as defined by Trillas et al [2,11,8], based on the notions of conjecture and hypothesis [13]. These models try to capture the main properties of some of the basic types of ordinary reasoning: deduction, induction, abduction and speculative reasoning.…”

“…If this set is finite, to obtain its consequences is usually reduced to look for the consequences of the conjunction (or meet in algebraic terms) of all the premises. This is done for instance in [11], in the setting of preordered sets, where for a given set of premises the consequence operator C ∧ provides as consequences all those elements greater or equal (with respect to a given preorder) than the conjunction of its premises. Nevertheless, there are different ways of defining a consequence operator, even in that general setting.…”