This paper contains five observations concerning the intended meaning of the intuitionistic logical constants: (1) if the explanations of this meaning are to be based on a non-decidable concept, that concept should not be that of `proof"; (2) Kreisel"s explanations using extra clauses can be significantly simplified; (3) the impredicativity of the definition of → can be easily and safely ameliorated; (4) the definition of → in terms of `proofs from premises" results in a loss of the inductive character of (...) the definitions of ∨ and ∃ and (5) the same occurs with the definition of ∀ in terms of `proofs with free variables". (shrink)

A distinction is introduced between itemized and non-itemized plural predication. It is argued that a full-fledged system of plural logic is not necessary in order to account for the validity of inferences concerning itemized collective predication. Instead, it is shown how this type of inferences can be adequately dealt with in a first-order logic system, after small modifications on the standard treatment. The proposed system, unlike plural logic, has the advantage of preserving completeness. And as a result, inferences such as (...) ‘Dick and Tony emptied the bottle, hence Tony and Dick emptied the bottle’ are shown to be first-order. (shrink)

En el presente artículo se examinan algunos de los problemas suscitados por el análisis lógico de la predicación colectiva. Su tratamiento habitual en lógica de primer orden no es del todo satisfactorio, y la búsqueda de otros modos de representación abre interesantes perspectivas. Se investiga, en particular, la introducción de predicados poligraduados, señalando algunas deficiencias e insuficiencias en la literatura reciente sobre los mismos. Argumento que la oposición colectivo/distributivo debe localizarse en la estructura lógica dei enunciado en cuestión. /// In (...) this paper I investigate the logical analysis of statements in which a predicate applies collectively to a plurality of subjects. The standard treatment of such cases in first-order logic is not completely satisfactory, and the search for alternative modes of representation opens up interesting perspectives. I concentrate on the introduction of multigrade predicates, pointing out a number of weaknesses in the current literature on the subject. I argue that the collective/distributive opposition must be located in the logical structure of the statement in question. (shrink)

In this paper I dispute the current view that intuitionistic logic is the common basis for the three main trends of constructivism in the philosophy of mathematics: intuitionism, Russian constructivism and Bishop’s constructivism. The point is that the so-called ‘Markov’s principle’, which is accepted by Russian constructivists and rejected by the other two, is expressible in intuitionistic first-order logic, and so it appears to have the status of a logical principle. The result of appending this principle to a complete intuitionistic (...) axiom system for first-order predicate logic constitutes a new logic, which could well be called ‘Markov’s logic’, and which should be regarded as the true logical system underlying Russian constructivism. (shrink)