THE EFFECT OF THE KOREAN NUMBER SYSTEM ON YOUNG CHILDREN'S COUNTING: A NATURAL EXPERIMENT IN NUMERICAL BILINGUALISM

In Korea, two number systems are in simultaneous use. One is somewhat irregular (like the English language number system), and the other regular (similar to the Chinese number system). This natural experiment allows a direct test of the hypothesis that structural regularity of the number system facilitates the acquisition of counting. Korean children initially suffer from their numerical ‘bilingualism’: they find it difficult to learn two counting systems at the same time. Yet, after this initial difficulty, they soon master the generative rules which are transparent in the regular system. The cognitive advantage provided by the regular number system then combines with other factors, like intensive teaching and parental pressure, to produce school achievement in mathematics superior to US children's.     

Formalization of functionally complete propositional calculus with the functor of implication as the only primitive term

The most difficult problem that Leniewski came across in constructing his system of the foundations of mathematics was the problem of defining definitions, as he used to put it. He solved it to his satisfaction only when he had completed the formalization of his protothetic and ontology. By formalization of a deductive system one ought to understand in this context the statement, as precise and unambiguous as possible, of the conditions an expression has to satisfy if it is added to the system as a new thesis. Now, some protothetical theses, and some ontological ones, included in the respective systems, happen to be definitions. In the present essay I employ Leniewski's method of terminological explanations for the purpose of formalizing ukasiewicz's system of implicational calculus of propositions, which system, without having recourse to quantification, I first extended some time ago into a functionally complete system. This I achieved by allowing for a rule of implicational definitions, which enabled me to define any propositionforming functor for any finite number of propositional arguments.To the memory of Jerzy Supecki     

On a problem of P(α, δ, π) concerning generalized Alexandroff s cube

Universality of generalized Alexandroff's cube plays essential role in theory of absolute retracts for the category of , -closure spaces. Alexandroff's cube. is an , -closure space generated by the family of all complete filters. in a lattice of all subsets of a set of power .Condition P(, , ) says that is a closure space of all , -filters in the lattice ( ), .Assuming that P (, , ) holds, in the paper [2], there are given sufficient conditions saying when an , -closure space is an absolute retract for the category of , -closure spaces (see Theorems 2.1 and 3.4 in [2]).It seems that, under assumption that P (, , ) holds, it will be possible to givean uniform characterization of absolute retracts for the category of , -closure-spaces.Except Lemma 3.1 from [1], there is no information when the condition P (, , ) holds or when it does not hold.The main result of this paper says, that there are examples of cardinal numbers, , , such that P (, , ) is not satisfied.Namely it is proved, using elementary properties of Lebesgue measure on the real line, that the condition P (, ₁, 2 ) is not satisfied.Moreover it is shown that fulfillment of the condition is essential assumption in, Theorems 2.1 and 3.4 from [1] i.e. it cannot be eliminated.     

On decidable consequence operators

The main theorem says that a consequence operator is an effective part of the consequence operator for the classical prepositional calculus iff it is a consequence operator for a logic satisfying the compactness theorem, and in which every finitely axiomatizable theory is decidable.     

Book reviews

Science as Salvation: a Modern Myth and its Meaning, Mary Midgley, 1994. London, Routledge x +256pp., Hb 04 15062713, £35; Pb 04 15107733, £8.99

Philosophical Naturalism, David Papineau, 1993 Oxford, Basil Blackwell xii +219pp., Hb 0631189025, £40; Pb 0631189033, £14.99

F. H. Bradley, Writings on Logic and Metaphysics, James W. Allard & Guy Stock (Eds), 1994. Oxford, Clarendon Press xv+357pp, Hb 0–198–24445–2, £40.00; Pb 0–198–24438‐X, £14.95

Invariance and Heuristics: Essays in Honour of Heinz Post, Steven French & Harmke Kamminga (Eds), 1993 Boston Studies in the Philosophy of Science, Vol. 148 Kluwer Academic Publishers, Dordrecht

Beyond Reason: Essays on the Philosophy of Paul Feyerabend, GONZALO MUNÉVAR (Ed.), 1991. Dordrecht, Kluwer Academic Publishers xxi + 535pp., hb, ISBN 0–7923–1272–4, £104.20

World Changes: Thomas Kuhn and the Nature of Science, Paul Horwich (Ed.), 1993. Cambridge, MA, Bradford Books/MIT Press vi + 356pp., pb, ISBN 0262581388, £14.95

Realism Rescued: How Scientific Progress is Possible, Jerold L. Aronson, Rom Harré & Eileen Cornell Way, 1994 London, Duckworth vii +213pp., Hb 0715624768, £30.00

Scientific Nihilism: On the Loss and Recovery of Physical Explanation, Daniel Athearn, 1994. State University of New York Press, Albany ix + 387pp., Hb ISBN 0–7914–1807–3, £52

Theism, Atheism, and Big Bang Cosmology, William Lane Craig & Quentin Smith, 1993. Oxford, Clarendon Press x +342pp., Hb 0198263481, £35; Pb 019826383X, £13.95     

Finite structural axiomatization of every finite-valued propositional calculus

In [2] A. Wroski proved that there is a strongly finite consequence C which is not finitely based i.e. for every consequence C ⁺ determined by a finite set of standard rules C C ⁺. In this paper it will be proved that for every strongly finite consequence C there is a consequence C ⁺ determined by a finite set of structural rules such that C(Ø)=C ⁺(Ø) and = (where , are consequences obtained by adding to the rules of C, C ⁺ respectively the rule of substitution). Moreover it will be shown that under certain assumptions C=C ⁺.