Simplified Kripke Semantics for K45-Like Gödel Modal Logics and Its Axiomatic Extensions

Abstract

In this paper we provide a simplified, possibilistic semantics for the logics K45(G), i.e. a many-valued counterpart of the classical modal logic K45 over the [0, 1]-valued Gödel fuzzy logic $$\mathbf{G}$$ G . More precisely, we characterize K45(G) as the set of valid formulae of the class of possibilistic Gödel frames $$\langle W, \pi \rangle $$ ⟨ W , π ⟩ , where W is a non-empty set of worlds and $$\pi : W \mathop {\rightarrow }[0,1]$$ π : W → [ 0 , 1 ] is a possibility distribution on W. We provide decidability results as well. Moreover, we show that all the results also apply to the extension of K45(G) with the axiom (D), provided that we restrict ourselves to normalised Gödel Kripke frames, i.e. frames $$\langle W, \pi \rangle $$ ⟨ W , π ⟩ where $$\pi $$ π satisfies the normalisation condition $$\sup _{w \in W} \pi (w) = 1$$ sup w ∈ W π ( w ) = 1 .

Reasoning with belief functions over Belnap–Dunn logic

“…The concrete semantics of the modality can be defined in two ways. First, using a modal language with Kripke semantics where the measure is defined on the set of states as done by, e.g., Gärdenfors (1975); Delgrande and Renne (2015); Delgrande et al (2019) for qualitative probabilities, by Dautović et al (2021) for the quantitative ones, and by Rodriguez et al (2022) for the possibility and necessity measures. Second, employing a two-layered formalism (cf.…”

Paraconsistent Gödel Modal Logic

“…For example, K45 and KD45 Gödel logics can be used to formalise possibilistic reasoning since they are complete w.r.t. normalised and, respectively, non-normalised possibilistic frames [35].…”