TY - JOUR
AU - Pietruszczak, Andrzej
PY - 2010/03/30
Y2 - 2024/03/04
TI - Simplified Kripke style semantics for some very weak modal logics
JF - Logic and Logical Philosophy
JA - LLP
VL - 18
IS - 3-4
SE - Articles
DO - 10.12775/LLP.2009.013
UR - https://apcz.umk.pl/LLP/article/view/LLP.2009.013
SP - 271-296
AB - 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 <w,A,V>, 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 <w,V> 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.
ER -