This note gives two new axiomatizations of each of the modal logics Vern by extension of K with, respectively, an inference rule and an axiom schema other than Vern.

The paper contains two concepts of implication that can be called analytic in the sense of Parry [5] or Fine [2]. Contrary to the Parry's approach, these implications do not involve S4 strict implication but the classical one. This fact refers to the similar notion of so-called demodalized analytic implication of Dunn [1] and strong implication of Vanderveken, cf. [4, 6]. The features of analytic classical implications are presented in the form of two propositional logics on the pure implicational language.

This is a survey paper of approaches to the concept of logical implication. Roughly stated the main motivation of these approaches is to provide a necessary and sufficient condition for a set of propositions to logically imply a single proposition. In regard to their affinities these approaches are grouped into two: the transformational conception and the informational conception. Some approaches in each conception are philosophical and some are mathematical in character, their common assumption being that they reflect a previous intuition (...) – in some cases – already at work in classical mathematics. The first part of the paper is devoted to approaches within the transformational conception of logical implication emphasizing the contributions of Bolzano, Russell, Tarski, Quine and current model – theory. The second part of the paper discusses approaches within the informational conception of logical implication. Carnap and Bar Hillel’s extrinsic approach and Corcoran’s intrinsic approach developed in his information-theoretic logic are respectively compared and some of their philosophical underpinnings are brought to light. The paper concludes with some final remarks concerning the ontic question of the nature of logical implication and the epistemic question of the human access to this relation in both conceptions of logical implication. The article also attempts to connect the views of these representative thinkers with the current philosophical debate on the nature of logical consequence and the role of set-theoretic model theory in philosophy of logic. (shrink)

The paper concludes an earlier one on extensions of atomistic semantic frames. Three kinds of extension are considered: the adjunctive, the conjunctive, and the disjunctive one. Some theorems are proved on extending “Humean” frames, i.e. such that the elementary situations constituting their universa are separated by the maximally coherent sets of them.