The aim of this paper is to enrich the minimal logic of propositions of Meaning and Speech Acts and to analyze the logical form of modal propositions according to which it is necessary, possible or contingent that things are so and so. Unlike truth functions, modalities enrich the structure of constituents of propositions. We will analyze in the new modal logic the structure of constituents and the truth conditions of modal propositions. We will in particular generalize the notion of tautology so as to apply it to modal as well as truth functional tautologies. We will also generalize the relation of strong implication and study how modal propositions are relacted by strict, strong and analytic implication.
Strong implication is the finest non-classical relation of implication between propositions.
It is finite, decidable and cognitively realized. Whenever a proposition strongly implies another, it is not posible to express that proposition without eo ipso knowing that it implies the other proposition. We will state the laws of propositional identity of the new modal logic and show that it avoids paradoxes of other logics. We will also formulate the laws of minimal consistency and paraconsistency that exist in that logic thanks to strong implication.