In this work we extend results from [4], [3] and [2] about propositional calculi with graded modalities to the predicative level. Our semantic is based on Kripke models with a single domain of interpretation for all the worlds. Therefore the axiomatic system will need a suitable generalization of the Barcan formula. We haven't considered semantics with world-relative domains because they don't present any new difficulties with respect to classical case. Our language will have, as in [1], constant and function symbols, (...) but they will have a rigid interpretation. In this instance the terms will be considered "rigid designators", that is, their interpretations will be the same in all possible worlds. Nevertheless we will not consider languages with identity: in this way we avoid the problems of identity in modal contexts. On the other hand, we think that in that context would arise essentially the same problems which occur in classical predicative modal logic. Finally we will consider only denumerable languages because the extension to non denumerable ones isn't conceptually problematic. (shrink)

This work intends to be a generalization and a simplification of the techniques employed in [2], by the proposal of a general strategy to prove satisfiability theorems for NLGM-s (= normal logics with graded modalities), analogously to the well known technique of the canonical models by Lemmon and Scott for classical modal logics.

We go on along the trend of [2] and [1], giving an axiomatization of S4 0 and proving its completeness and compactness with respect to the usual reflexive and transitive Kripke models. To reach this results, we use techniques from [1], with suitable adaptations to our specific case.

We go on along the trend of [2] and [1], giving an axiomatization of S4⁰ and proving its completeness and compactness with respect to the usual reflexive and transitive Kripke models. To reach this results, we use techniques from [1], with suitable adaptations to our specific case.

We study a modal system ¯T, that extends the classical (prepositional) modal system T and whose language is provided with modal operators M inn (nN) to be interpreted, in the usual kripkean semantics, as there are more than n accessible worlds such that.... We find reasonable axioms for ¯T and we prove for it completeness, compactness and decidability theorems.

We study a modal system $\overline{T}$, that extends the classical modal system T and whose language is provided with modal operators $M_{n}$ to be interpreted, in the usual kripkean semantics, as "there are more than n accessible worlds such that...". We find reasonable axioms for $\overline{T}$ and we prove for it completeness, compactness and decidability theorems.