An Ackermann constant is a formula of sentential logic built up from the sentential constant t by closing under connectives. It is known that there are only finitely many non-equivalent Ackermann constants in the relevant logic R. In this paper it is shown that the most natural systems close to R but weaker than it-in particular the non-distributive system LR and the modalised system NR-allow infinitely many Ackermann constants to be distinguished. The argument in each case proceeds by construction of (...) an algebraic model, infinite in the case of LR and of arbitrary finite size in the case of NR. The search for these models was aided by the computer program MaGIC (Matrix Generator for Implication Connectives) developed by the author at the Australian National University. (shrink)

In this paper, we introduce connectification operators for intuitionistic and classical linear algebras corresponding to linear logic and to some of its extensions withn-contraction. In particular,n-contraction (n2) is a version of the contraction rule, wheren+1 occurrences of a formula may be contracted ton occurrences. Since cut cannot be eliminated from the systems withn-contraction considered most of the standard proof-theoretic techniques to investigate meta-properties of those systems are useless. However, by means of connectification we establish the disjunction property for both intuitionistic (...) and classical affine linear logics withn-contraction. (shrink)

In this paper, we consider multiplicative-additive fragments of affine propositional classical linear logic extended with n-contraction. To be specific, n-contraction (n 2) is a version of the contraction rule where (n+ 1) occurrences of a formula may be contracted to n occurrences. We show that expansions of the linear models for (n + 1)- valued ukasiewicz logic are models for the multiplicative-additive classical linear logic, its affine version and their extensions with n-contraction. We prove the finite axiomatizability for the classes (...) of finite models, as well as for the class of infinite linear models based on the set of rational numbers in the interval [0, 1]. The axiomatizations obtained in a Gentzen-style formulation are equivalent to finite and infinite-valued ukasiewicz logics. (shrink)