this paper, we will give a short survey of results on decision problems and the finite model property of substructural logics. The paper is far from a complete list of these results, since a lot of results have been obtained already in some restricted classes of substructural logics, like relevant logics, and therefore it is impossible to cover all of them. ( As for surveys of decision problems and the finite model property of relevant logics, see e.g. [1, 2, 7]. Also, see [16] for a survey of decision problems of logics related to linear logic. ) Our aim of the present paper is to try to compare results from different classes of substructural logics with each other and discuss them as a whole, in order to get a perspective of them.

A general criterion for the undecidabily of sub-classical rstorder logics and important fragments thereof is established. It is applied, among others, to Urquart's (original version of) C and the closely related logic C . In addition, hypersequent systems for (rst-order) C and C are introduced and shown to enjoy cut-elimination. 1

Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #-calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a "classical" version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higher-order BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic. 1991 AMS (MOS) Classification: 03B40, 03F05, 03B20 Key words: Implication, negation, combinatory logic, lambda calculus, comprehension principle, normalization, cut-elimination...

Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #-calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a "classical" version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higher-order BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic. 1991 AMS (MOS) Classification: 03B40, 03F05, 03B20 Key words: Implication, negation, combinatory logic, lambda calculus, comprehension principle, normalization, cut-elimination...