Crítica, Revista Hispanoamericana de Filosofía, Volume 31, number 93, diciembre 1999
Tablas semánticas y metalógica (El caso de la lógica de segundo orden)
[Semantic Tableaux and Meta-Logic]
Ángel Nepomuceno Fernández
Departamento de Filosofía y Lógica y Filosofía de la Ciencia
Universidad de Sevilla

Abstract: Beth's method of semantic tableaux has been utilized in first order logic to obtain some results. From a metalogical point of view, a known result can be proved: completeness of first order logic. Can we extend this method to second order logic? What about completeness? Given a second order formal language, for every formula a prenex form can be defined so that some of them have a first order formula as its matrix, then the mentioned method could be applied to study several classes of formulae, particularly ∑¦, ∏¦and Zykov's inverses. In order to do that some modifications of such method must be introduced, so that we take known rules for first order semantic trees, by applying them, modifying the treatment of "∃" and defining specific rules for some second order formulae, then we can achieve some results relative to such classes similar to that obtained by means of abstract model theory. On the other hand, a restricted completeness (to certain class of sentences) could be settled.

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