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Following K. Segerberg [22] , D. Kozen [15] and V. Pratt [19] , who have been introduced dynamic propositional logic and dynamic algebras, dynamic propositional Łukasiewicz logic DPŁ (dynamic n-valued propositional Łukasiewicz logic D P Ł n ) and dynamic MV-algebras (dynamic M V n -algebras) are introduced and theories of the logic DPŁ ( D P Ł n ) and dynamic MV-algebras ( M V n -algebras) are developed. Dynamic MV-algebras (dynamic M V n -algebras) are algebraic counterparts of the logic DPŁ ( D P Ł n ), that in turn represent two-sorted algebras that combine the varieties of MV-algebras ( M V n -algebras) ( M , ⊕ , ⊙ , ∼ , 0 , 1 ) and regular algebras ( R , ∪ , ; , ⁎ ) into a single finitely axiomatized variety ( M , R , ◇ ) resembling R-module with “scalar” multiplication ◇. Kripke semantics is developed for dynamic propositional Łukasiewicz logic (dynamic n-valued propositional Łukasiewicz logic D P Ł n ).

This paper proposes a generalization of the Kripke semantics of intuitionistic logicIL appropriate forintuitionistic Łukasiewicz logicIŁL —a logic in the intersection betweenIL and (classical) Łukasiewicz logic. This generalised Kripke semantics is based on theposet sumconstruction, used in Bova and Montagna (Theoret Comput Sci 410(12):1143–1158, 2009). to show the decidability (and PSPACE completeness) of the quasiequational theory of commutative, integral and bounded GBL algebras. The main idea is that$$w \Vdash \psi $$w⊩ψ—which forILis a relation between worldswand formulas$$\psi $$ψ, and can be seen as a function taking values in the booleans$$(w \Vdash \psi ) \in {{\mathbb {B}}}$$(w⊩ψ)∈B—becomes a function taking values in the unit interval$$(w \Vdash \psi ) \in [0,1]$$(w⊩ψ)∈[0,1]. An appropriate monotonicity restriction (which we callsloping functions) needs to be put on such functions in order to ensure soundness and completeness of the semantics.

We introduce a relational semantics based on poset products, and provide sufficient conditions guaranteeing its soundness and completeness for various substructural logics. We also demonstrate that our relational semantics unifies and generalizes two semantics already appearing in the literature: Aguzzoli, Bianchi, and Marra's temporal flow semantics for H\'ajek's basic logic, and Lewis-Smith, Oliva, and Robinson's semantics for intuitionistic Lukasiewicz logic. As a consequence of our general theory, we recover the soundness and completeness results of these prior studies in a uniform fashion, and extend them to infinitely-many other substructural logics.

It is introduced an immune dynamicn-valued Łukasiewicz logic$$ID{\L }_n$$IDŁnon the base ofn-valued Łukasiewicz logic$${\L }_n$$Łnand corresponding to it immune dynamic$$MV_n$$MVn-algebra ($$IDL_n$$IDLn-algebra),$$1< n < \omega $$1

Decidability of the validity problem is established for a family of many-valued modal logics, notably Godel modal logics, where propositional connectives are evaluated according to the order of values in a complete sublattice of the real unit interval [ 0 , 1 ] , and box and diamond modalities are evaluated as infima and suprema over (many-valued) Kripke frames. If the sublattice is infinite and the language is sufficiently expressive, then the standard semantics for such a logic lacks the finite model property. It is shown here, however, that, given certain regularity conditions, the finite model property holds for a new semantics for the logic, providing a basis for establishing decidability and PSPACE-completeness. Similar results are also established for S5 logics that coincide with one-variable fragments of first-order many-valued logics. In particular, a first proof is given of the decidability and co-NP-completeness of validity in the one-variable fragment of first-order Godel logic.

The Belnap–Dunn logic (also known as First Degree Entailment, or FDE) is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of view of Abstract Algebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies.

In the literature, little attention has been paid to the development of solvers for systems of mathematical fuzzy logic, and in particular, there are few works concerned with infinitely-valued logics. In this paper it is presented mNiBLoS (a modal Nice BL-Logics Solver): a modular SMT-based solver complete with respect to a wide family of continuous t-norm based fuzzy modal logics (both with finite and infinite universes), restricting the modal structures to the finite ones. At the propositional level, the solver works with some of the best known infinitely-valued fuzzy logics (including BL, Łukasiewicz, Gödel and product logics), and with all the continuous t-norm based logics that can be finitely expressed in terms of the previous ones; concerning the modal expansion, mNiBLoS imposes no boundary on the cardinality of the modal structures considered. The solver allows to test 1-satisfiability of equations, tautologicity and logical consequence problems. The logical language supported extends the usual one of fuzzy modal logics with rational constants and the Monteiro-Baaz Δ operator. The code of mNiBLoS is of free distribution and can be found in the web page of the author. © 2016 Elsevier Inc., This work was supported by the CSIC Intramural project 201450E045:“Logal: Lógica y algoritmos”, the joint project of Austrian Science Fund (FWF) I1897-N25 and Czech Science Foundation (GACR) 15-34650L, the institutional support RVO:67985807 and the MINECO project EdeTRI:TIN2012-39348-C02-01. The author wishes to thank the anonymous reviewers for their useful commentaries.

The contribution of this paper lies with providing a systematically specified and intuitive interpretation pattern and delineating a class of relational structures (frames) and models providing a natural interpretation of logical operators on an underlying propositional calculus of Positive Lattice Logic (the logic of bounded lattices) and subsequently proving a generic completeness theorem for the related class of logics, sometimes collectively referred to as (non-distributive) Generalized Galois Logics (GGL’s).

We translate unconstrained and constrained input/output logics as introduced by Makinson and van der Torre to modal logics, using adaptive logics for the constrained case. The resulting reformulation has some additional benefits. First, we obtain a proof-theoretic (dynamic) characterization of input/output logics. Second, we demonstrate that our framework naturally gives rise to useful variants and allows to express important notions that go beyond the expressive means of input/output logics, such as violations and sanctions.

The logic BN4 was defined by R.T. Brady in 1982. It can be considered as the 4-valued logic of the relevant conditional. E4 is a variant of BN4 that can be considered as the 4-valued logic of (relevant) entailment. The aim of this paper is to define reduced general Routley-Meyer semantics for BN4 and E4. It is proved that BN4 and E4 are strongly sound and complete w.r.t. their respective semantics.