The ŁII and ŁII1/2 logics were introduced by Godo, Esteva and Montagna in [4] and further developed in my work [2]. These types of logic unite many other known propositional and predicate logics, including the three mainly investigated ones (Godel, Product and Łukasiewicz logic).
The aim of this paper is to show a tight connection between the ŁII logic and the product involutive logic. This logic was introduced by Esteva, Godo, Hájek and Navara in their paper [3].
We will see that all the connectives of the ŁII logic are definable from the connectives of this logic. In addition we show that the ŁII logic is an schernatic extension of this logic by a single axiom. We also make some simplification of the axiomatic system of this logic.
We propose a generalization of simple coalition games in the context of games with fuzzy coalitions. Mimicking the correspondence of simple games with non-constant monotone formulas of classical logic, we introduce simple Łukasiewicz games using monotone formulas of Łukasiewicz logic, one of the most prominent fuzzy logics. We study the core solution on the class of simple Łukasiewicz games and show that cores of such games are determined by finitely-many linear constraints only. The non-emptiness of core is completely characterized in terms of balanced systems and by the presence of strong veto players.