A system of relational syllogistic incorporating full Boolean reasoning

We present a system of relational syllogistic, based on classical propositional logic, having primitives of the following form: Some A are R-related to some B; Some A are R-related to all B; All A are R-related to some B; All A are R-related to all B. Such primitives formalize sentences from natural language like `All students read some textbooks'. Here A and B denote arbitrary sets (of objects), and R denotes an arbitrary binary relation between objects. The language of the logic contains only variables denoting sets, determining the class of set terms, and variables denoting binary relations between objects, determining the class of relational terms. Both classes of terms are closed under the standard Boolean operations. The set of relational terms is also closed under taking the converse of a relation. The results of the paper are the completeness theorem with respect to the intended semantics and the computational complexity of the satisfiability problem.Comment: Available at http://link.springer.com/article/10.1007/s10849-012-9165-

Information Content and Processing

Abstract: The problems extraction knowledge from natural language text is considered. An automatization approach to extraction knowledge is proposed

Algebraic foundations for qualitative calculi and networks

A qualitative representation $\phi$ is like an ordinary representation of a relation algebra, but instead of requiring $(a; b)^\phi = a^\phi | b^\phi$, as we do for ordinary representations, we only require that $c^\phi\supseteq a^\phi | b^\phi \iff c\geq a ; b$, for each $c$ in the algebra. A constraint network is qualitatively satisfiable if its nodes can be mapped to elements of a qualitative representation, preserving the constraints. If a constraint network is satisfiable then it is clearly qualitatively satisfiable, but the converse can fail. However, for a wide range of relation algebras including the point algebra, the Allen Interval Algebra, RCC8 and many others, a network is satisfiable if and only if it is qualitatively satisfiable. Unlike ordinary composition, the weak composition arising from qualitative representations need not be associative, so we can generalise by considering network satisfaction problems over non-associative algebras. We prove that computationally, qualitative representations have many advantages over ordinary representations: whereas many finite relation algebras have only infinite representations, every finite qualitatively representable algebra has a finite qualitative representation; the representability problem for (the atom structures of) finite non-associative algebras is NP-complete; the network satisfaction problem over a finite qualitatively representable algebra is always in NP; the validity of equations over qualitative representations is co-NP-complete. On the other hand we prove that there is no finite axiomatisation of the class of qualitatively representable algebras.Comment: 22 page

Interpretaciones del método de Descartes en didáctica del álgebra. Estudio documental

This communication presents a documentary study of the of the Cartesian method in three books on the teaching and learning of algebra, its influence and use in the understanding of school algebra. Arguments were found to indicate that some interpretations used in didactics regarding the mathematical practice of Descartes reveal a conception of the Cartesian method that privileges aspects of a syntactic and methodical type, leaving aside the analysis of the representational and semantic treatment found in the work. The educational importance given to the method of analysis is related to the algebraic manipulation and the construction of a symbolic system, the reflections on the interpretive, diagrammatic, and semantic needs of the method in solving problems are meager.Esta comunicación presenta un estudio documental de las interpretaciones del método cartesiano en tres libros sobre enseñanza y aprendizaje del álgebra, su influencia y uso en la comprensión del álgebra escolar. Se encontraron argumentos para indicar que algunas interpretaciones usadas en didáctica respecto a la práctica matemática de Descartes develan una concepción del método cartesiano que privilegia aspectos de tipo sintáctico y metódico, dejando a un lado el análisis del tratamiento representacional y semántico que se encuentra en la obra. La importancia educativa que se le otorga al método de análisis está relacionada con la manipulación algebraica y la construcción de un sistema simbólico, las reflexiones sobre las necesidades interpretativas, diagramáticas y semánticas del método en la solución de problemas son exiguas

A General Semantics for Logics of Affirmation and Negation

A general framework for translating various logical systems is presented, including a set of partial unary operators of affirmation and negation. Despite its usual reading, affirmation is not redundant in any domain of values and whenever it does not behave like a full mapping. After depicting the process of partial functions, a number of logics are translated through a variety of affirmations and a unique pair of negations. This relies upon two preconditions: a deconstruction of truth-values as ordered and structured objects, unlike its mainstream presentation as a simple object; a redefinition of the Principle of Bivalence as a set of four independent properties, such that its definition does not equate with normality