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     

Three-element nonfinitely axiomatizable matrices

There are exactly two nonfinitely axiomatizable algebraic matrices with one binary connective o such thatx(yz) is a tautology of . This answers a question asked by W. Rautenberg in [2], P. Wojtylak in [8] and W. Dziobiak in [1]. Since every 2-element matrix can be finitely axiomatized ([3]), the matrices presented here are of the smallest possible size and in some sense are the simplest possible.Presented byWolfgang Rautenberg     

Finite axiomatization for some intermediate logics

LetN. be the set of all natural numbers (except zero), and letD _n ^* = {k ∈N ∶k|n} ∪ {0} wherek¦n if and only ifn=k.x f or somex∈N. Then, an ordered setD _n ^* = 〈D _n ^* , ? _n , wherex? _ny iffx¦y for anyx, y∈D _n ^* , can easily be seen to be a pseudo-boolean algebra. In [5], V.A. Jankov has proved that the class of algebras {D _n ^* ∶n∈B}, whereB =,{k ∈N∶ ? \(\mathop \exists \limits_{n \in N} \) (n > 1 ≧n ² k)is finitely axiomatizable. The present paper aims at showing that the class of all algebras {D _n ^* ∶n∈B} is also finitely axiomatizable. First, we prove that an intermediate logic defined as follows: $$LD = Cn(INT \cup \{ p_3 \vee [p_3 \to (p_1 \to p_2 ) \vee (p_2 \to p_1 )]\} )$$ finitely approximatizable. Then, defining, after Kripke, a model as a non-empty ordered setH = 〈K, ?〉, and making use of the set of formulas true in this model, we show that any finite strongly compact pseudo-boolean algebra ? is identical with. the set of formulas true in the Kripke modelH _B = 〈P(?), ?〉 (whereP(?) stands for the family of all prime filters in the algebra ?). Furthermore, the concept of a structure of divisors is defined, and the structure is shown to beH D _n ^* = 〈P (D _n ^* ), ?〉for anyn∈N. Finally, it is proved that for any strongly compact pseudo-boolean algebraU satisfying the axiomp ₃∨ [p ₃→(p₁→p₂)∨(p₂→p₁)] there is a structure of divisorsD _* ⁿ such that it is possible to define a strong homomorphism froomiH D _n ^* ontoH D _U . Exploiting, among others, this property, it turns out to be relatively easy to show that \(LD = \mathop \cap \limits_{n \in N} E(\mathfrak{D}_n^* )\) .     

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 ⁺.     

Strong completeness with respect to finite kripke models

We prove that each intermediate or normal modal logic is strongly complete with respect to a class of finite Kripke frames iff it is tabular, i.e. the respective variety of pseudo-Boolean or modal algebras, corresponding to it, is generated by a finite algebra. The author wishes to thank the Editors for calling his attention to the fact that the result of this paper concerning the intermediate logics was announced earlier by A. Wroński at the conference “Logical calculi”, Wrocław, October 5–7, 1975, though without proof. Wroński's result has not been published.     

On finite approximability of ψ-intermediate logics

The aim of this note is to show (Theorem 1.6) that in each of the cases: = {, }, or {, , }, or {, , } there are uncountably many -intermediate logics which are not finitely approximable. This result together with the results known in literature allow us to conclude (Theorem 2.2) that for each : either all -intermediate logics are finitely approximate or there are uncountably many of them which lack the property.     

TRAINING CONSERVATION OF QUANTITY (LIQUIDS) IN WEST AFRICAN (BAOULE) CHILDREN

A developmental ‘lag’ is found in the Conservation of Quantity (Liquids) in a sample of 73 rural, schooled Baoulé children (Ivory Coast, West Africa) aged 7 to 14 years. This study was designed to investigate if this ‘lag’ could be reduced or bridged through training and whether it could be attributed to differences in ‘performance’ or in ‘competence’. A sub-sample of 28 children aged 7 to 9 years was divided into two matched groups, one being trained for Conservation of Liquids and the other for Class Inclusion (Lavallée and Dasen in press). Statistically significant training effects were observed, which generalized to other concrete operational concepts and remained stable over at least one month. Since no ‘actualization’ or very fast learning occurred, the ‘lag’ was attributed to differences in competence; the training was sufficient to bridge the developmental lag.