Conservative Extension Research Articles

On the conservative extensions of semantical systems: A contribution to the problem of analyticity

The attempt to distinguish within an empirical theory between a factual and a conventional component is an important task. It bears especially on the following methodological issues. 'Cognitive status' of theories: The dispute between conventionalism and its critics concerns just where this distinction should be drawn. But equally, if the conventional part of a theory determines the reference of some of its expressions, the question whether theoretical assertions of empirical science include non-conventional statements of determinate truth-value cannot be settled until an exact characterization of the con­ ventional part is given. Inter-theory relations: It is clear that logical relations between theories­ inconsistency, equivalence, reducibility, etc. - depend critically on intra­ theoretic relations. These are determined for each theory in a way which depends on the choice of conventional component. Evidential support: It is to be expected that an adequate measure of support furnished to theory by evidence should assign the highest value to statements belonging to the conventional part. It follows that the measure appropriate to the language of a given theory depends on how its factual and conventional components are distinguished. The problem of distinguishing the factual and conventional component was solved by R. Carnap for finite sets of postulates as follows [4], [5, pp. 958-66]. Let n be the conjunction of theoretical postulates and correspondence postulates and let the descriptive constants of n be distinguished into O-terms and T-terms. Let Rn be the existential closure of the formula obtained from n by proper simultaneous substitution of predicate variables for T-terms. Then the factual part of n is identified with the consequence class of Rn and its conventional part with the con­ sequence class of Rn;:) n. It is understood in addition that a partial inter­ pretation is effected for T-terms by restricting their proper interpre­ tations to those that establish the truth of Rn;:) n given prior interpretations of O-terms. Carnap made the following expository observations:

Ordinal theory in a conservative extension of predicate calculus.

Ordinal theory in a conservative extension of predicate calculus.

Clinical Report on Fractures of the Femur

During the fifteen years, 1955-69, two hundred and twenty-four patients with fractures of the femur were treated in our orthopaedic clinic.Femoral head fracture was seen in five patients and the slipping of the capital femoral epiphysis was seen in four patients. Femoral neck fracture was seen in ninety-eight patients (internal neck fracture-51 cases, external neck fracture-47 cases). Sixty-six of this fracture were treated operatively with Smith-Petersen nail and McLaughlin nailplate. Femoral head was resected in two cases and arthroplasties were performed on only four cases. There remained twenty-one conservatively treated femoral neck fractures. Femoral shaft fracture was seen also in ninety-eight patients. Fourty-two of this fracture were treated by intramedullary nailing. In sixteen cases the nailing was done by a closed techinique, without exposing the fracture site, and using an image intensifier. But the fractures of the femoral shaft of children were mainly treated by conservative extension. The fracture of femoral condyle was seen in eighteen patients and they were commonly treated operatively with blade plates and screws. Only one case of the slipping of the lower femoral epiphysis was found in our series.

TEACHING LOGIC

In this paper the author emphasizes the value of problems and problem solving in the teaching of logic. Afterwards, special attention is given to Natural Deduction. Also, this paper poses some reflections relative to the presentation of the notions of univocal connective and conservative extension in a teaching situation. Some brief remarks on Analytic Tableaux and logical terminology may also prove useful pedagogically.

The substitution method

This paper deals with Hilbert's substitution method for eliminating bound variables from first order proofs. With a first order system S framed in the ε-calculus [2] the problem is to associate a system S' without bound variables and an effective procedure for transforming derivations in S into derivations in S′. The transform of a formula A derived in S is to be an “ε-substitution instance” of A, i.e. it is obtained by replacing terms εxB(x) in A by terms of S′. In general the choice of these terms will depend on the particular derivation of A, and not on A alone. Cf. [4]. The present formulation sharpens Hilbert's original statement of the problem, i.e. that the transform of A should be finitistically verifiable, by making explicit the methods of verification used, namely those formalized in S′; on the other hand, it generalizes Hilbert's formulation since S′ need not be restricted to finitist systems.The bound variable elimination procedure can always be taken to be primitive recursive in (the Gödel number of) the derivation of A. Constructions which transcend primitive recursion can simply be built into S′.In this paper we show that if S′ is taken to be a second order system with constants for functionals, then the existence of suitable ε-substitution instances can be expressed by the solvability of certain functional equations in S′. We deal with two cases here. If S is number theory without induction, i.e. essentially predicate calculus with identity, then we can solve the equations in question by taking for S′ the free variable part S* of S with an added rule of definition of functionals by cases (recursive definition on finite ordinals), which is a conservative extension of S*.