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Decidability and Finite Model Property of Substructural Logics
 In The Tbilisi Symposium on Logic, Language and Computation
, 1998
"... 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 l ..."
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Cited by 8 (2 self)
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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.
On the Undecidability of Some Subclassical Firstorder Logics
"... A general criterion for the undecidabily of subclassical 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 (rstorder) C and C are intro ..."
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Cited by 1 (1 self)
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A general criterion for the undecidabily of subclassical 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 (rstorder) C and C are introduced and shown to enjoy cutelimination. 1
SUBSTRUCTURAL LOGICS ∗
"... A b s t r a c t. The present paper is concerned with the cut eliminability for some sequent systems of noncommutative substructural logics, i.e. substructural logics without exchange rule. Sequent systems of several extensions of noncommutative logics FL and LBB ′ I, which is sometimes called T → − ..."
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A b s t r a c t. The present paper is concerned with the cut eliminability for some sequent systems of noncommutative substructural logics, i.e. substructural logics without exchange rule. Sequent systems of several extensions of noncommutative logics FL and LBB ′ I, which is sometimes called T → − W, will be introduced. Then, the cut elimination theorem and the decision problem for them will be discussed in comparison with their commutative extensions. 1.
Francesco Belardinelli
"... Algebraic aspects of cut elimination Abstract. We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the comple ..."
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Algebraic aspects of cut elimination Abstract. We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasicompletion of these Gentzen structures. It is shown that the quasicompletion is a generalization of the MacNeille completion. Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and OkadaTerui [17].
On the Role of Implication in Formal Logic
"... 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 no ..."
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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 higherorder 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, cutelimination...
On the Role of Implication in Formal Logic
, 1998
"... 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 no ..."
Abstract
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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 higherorder 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, cutelimination...