We study the problem of finding a basis for all rules admissible in the intuitionistic propositional logic IPC. The main result is Theorem 3.1 which gives a basis consisting of all rules in semi-reduced form satisfying certain specific additional requirements. Using developed technique we also find a basis for rules admissible in the logic of excluded middle law KC.

ABSTRACT We1 study unification of formulas in modal logics and consider logics which are equivalent w.r.t. unification of formulas. A criteria is given for equivalence w.r.t. unification via existence or persistent formulas. A complete syntactic description of all formulas which are non-unifiable in wide classes of modal logics is given. Passive inference rules are considered, it is shown that in any modal logic over D4 there is a finite basis for passive rules.

Let equation image denote the inverse limit of all finite cyclic groups. Let F, G and H be abelian groups with H ≤ G. Let FβH denote the abelian group , where +βis defined by +β = + β — β) for a certain β : F → G linear mod H meaning that β = 0 and β + β — β ∈ H for all a, b in F. In this paper we show that the following hold: The (...) additive group of any nonstandard model ℤ* of the ring ℤ is isomorphic to βH for a certain β : ℤ*+/H → equation image linear mod H. equation image is isomorphic to βH for some β : equation image/H →ℚ linear mod H, though equation image is not the additive group of any model of Th and the exact sequence H → equation image → equation image/H is not splitting. (shrink)

In [5] Phillips proved that one can obtain the additive group of any nonstandard model *ℤ of the ring ℤ of integers by using a linear mod 1 function h : F ℚ, where F is the α-dimensional vector space over ℚ when α is the cardinality of *ℤ. In this connection it arises the question whether there are linear mod 1 functions which are neither addition nor quasi-linear. We prove that this is the case.

We study quasi-characterizing inference rules (this notion was introduced into consideration by A. Citkin (1977). The main result of our paper is a complete description of all self-admissible quasi-characterizing inference rules. It is shown that a quasi-characterizing rule is self-admissible iff the frame of the algebra generating this rule is not rigid. We also prove that self-admissible rules are always admissible in canonical, in a sense, logics S4 or IPC regarding the type of algebra generating rules.