Following a suggestion of Feys, we use “rigorous implication” as a translation of Ackermann's strenge Implikation ([1]). Interest in Ackermann's system stems in part from the fact that it formalizes the properties of a strong, natural sort of implication which provably avoids standard implicational paradoxes, and which is consequently a good candidate for a formalization of entailment (considered as a narrower relation than that of strict implication). Our present purpose will not be to defend this suggestion, but rather to present (...) some information about rigorous implication. In particular, we show first that the structure of modalities (in the sense of Parry [4]) in Ackermann's system is identical with the structure of modalities in Lewis's S4, and secondly that (Ackermann's apparent conjecture to the contrary notwithstanding) it is possible to define modalities with the help of rigorous implication. (shrink)

In our previous paper [5], we have studied Kripke-type semantics for propositional logics without the contraction rule. In this paper, we will extend our argument to predicate logics without the structure rules. Similarly to the propositional case, we can not carry out Henkin's construction in the predicate case. Besides, there exists a difficulty that the rules of inference () and () are not always valid in our semantics. So, we have to introduce a notion of normal models.