Классификация пропозициональных логик.

Аннотация

In this article Smirnov's problem about the finding of the unified foundation for the clssification of implicational logics is discussed. The author formulated and solved the problem of the appropriate extension of an implicational fragment $H_\rightarrow$ of inuitionistic propositional logic to an implicational fragment $TV_\rightarrow$ of classical propositional logic. As a result the logical constructions in the form of Boolean lattices of implicational logics are obtained. The application of the modus ponens and substitution rules to these consructions permits to get the whole classes (including ifinite ones) of new logics. The transition from implicational logics to full logics are considered. The article contains the following sections: 1. Introduction; 2. Lattice of implicational logics $L(H_\rightarrow)$; 3. Lattice of implicational logics $L(TV_\rightarrow)$: classical versions of BCI, BCK and $R_\rightarrow$ logics; 4. Maximal lattice $L(TV_\rightarrow)$: $RM_\rightarrow$, and $L_{\varepsilon\rightarrow}$ logics:; 5. Lattice of implicational logics $L(S5_\rightarrow)$; 6. Full propositional logics and basic principles of classification.