The consequence relation preserving logical information

Information is contained in statements and «flows» from their structure and meaning of expressions they contain. The information that flows only from the meaning of logical constants and logical structure of statements we will call logical information. In this paper we present a formal explication of this notion which is proper for sentences being Boolean combination of atomic sentences. 1 Therefore we limit ourselves to analyzing logical information flowing only from the meaning of truth-value connectives and logical structure of sentences connected with these connectives

Mereological sets of distributive classes

We will present an elementary theory in which we can speak of mereological sets composed of distributive classes. Besides the concept of a distributive class and the membership relation, it will possess the notion of a mereological set and the relation of being a mereological part. In this theory we will interpret Morse’s elementary set theory (cf. Morse [11]). We will show that our theory has a model, if only Morse’s theory has one

Simplified Kripke style semantics for some very weak modal logics

In the present paper we examine very weak modal logics C1, D1, E1, S0.5◦, S0.5◦+(D), S0.5 and some of their versions which are closed under replacement of tautological equivalents (rte-versions). We give semantics for these logics, formulated by means of Kripke style models of the form , where w is a «distinguished» world, A is a set of worlds which are alternatives to w, and V is a valuation which for formulae and worlds assigns the truth-vales such that: (i) for all formulae and all worlds, V preserves classical conditions for truth-value operators; (ii) for the world w and any formula ϕ, V(⬜ϕ,w) = 1 iff ∀x∈A V(ϕ,x) = 1; (iii) for other worlds formula ⬜ϕ has an arbitrary value. Moreover, for rte-versions of considered logics we must add the following condition: (iv) V(⬜χ,w) = V(⬜χ[ϕ/ψ],w), if ϕ and ψ are tautological equivalent. Finally, for C1, D1and E1 we must add queer models of the form in which: (i) holds and (ii') V(⬜ϕ,w) = 0, for any formula ϕ. We prove that considered logics are determined by some classes of above models

Mereological sets of distributive classes

We will present an elementary theory in which we can speak of mereological sets composed of distributive classes. Besides the concept of a distributive class and the membership relation, it will possess the notion of a mereological set and the relation of being a mereological part. In this theory we will interpret Morse’s elementary set theory (cf. Morse [11]). We will show that our theory has a model, if only Morse’s theory has one

On Theses Without Iterated Modalities of Modal Logics Between C1 and S5. Part 1

This is the first, out of two papers, in which we identify all logics between C1 and S5 having the same theses without iterated modalities. All these logics canbe divided into certain groups. Each such group depends only on which of thefollowing formulas are theses of all logics from this group: (N), (T), (D), ⌜(T)∨ ☐q⌝,and for any n 0 a formula ⌜(T) ∨ (altn)⌝, where (T) has not the atom ‘q’, and(T) and (altn) have no common atom. We generalize Pollack’s result from [12],where he proved that all modal logics between S1 and S5 have the same theseswhich does not involve iterated modalities (i.e., the same first-degree theses).Zadanie „ Wdrożenie platformy Open Journal System dla czasopisma „ Bulletin of the Section of Logic” finansowane w ramach umowy 948/P-DUN/2016 ze środków Ministra Nauki i Szkolnictwa Wyższego przeznaczonych na działalność upowszechniającą naukę