Sequent calculi and interpolation for non-normal logics

arXiv (Cornell University), 2020

We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative and enjoy cut elimination and subformula property. Our proposal applies the multi-type methodology in the design of proper display calculi, starting from a semantic analysis which motivates syntactic translations from single-type non-normal modal logics to multi-type normal poly-modal logics.

Studia Logica, 2021

The tetravalent modal logic ($${\mathcal {TML}}$$ TML ) is one of the two logics defined by Font and Rius (J Symb Log 65(2):481–518, 2000) (the other is the normal tetravalent modal logic$${{\mathcal {TML}}}^N$$ TML N ) in connection with Monteiro’s tetravalent modal algebras. These logics are expansions of the well-known Belnap–Dunn’s four-valued logic that combine a many-valued character (tetravalence) with a modal character. In fact, $${\mathcal {TML}}$$ TML is the logic that preserves degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic $${\mathcal {TML}}$$ TML and the algebras is not so good as in $${{\mathcal {TML}}}^N$$ TML N , but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus (see Font and Rius in J Symb Log 65(2):481–518, 2000). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut-elimination property...

2018

We introduce labelled sequent calculi for quantified modal logics with non-rigid and and non-denoting terms. We prove that these calculi have the good structural properties of G3-style calculi. In particular, all rules are height-preserving invertible, weakening and contraction are height-preserving admissible and cut is admissible. Finally, we show that each calculus gives a proof-theoretic characterization of validity in the corresponding class of models.

Logica Universalis, 2007

2006

2019

We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative and enjoy cut elimination and subformula property. Our proposal applies the multi-type methodology in the design of display calculi, starting from a semantic analysis based on the translation from monotonic modal logic to normal bi-modal logic.

2013

In this thesis we consider generic tools and techniques for the proof-theoretic investigation of not necessarily normal modal logics based on minimal, intuitionistic or classical propositional logic. The underlying framework is that of ordinary symmetric or asymmetric two-sided sequent calculi without additional structural connectives, and the point of interest are the logical rules in such a system. We introduce the format of a sequent rule with context restrictions and the slightly weaker format of a shallow rule. The format of a rule with context restrictions is expressive enough to capture most normal modal logics in the S5 cube, standard systems for minimal, intuitionistic and classical propositional logic and a wide variety of non-normal modal logics. For systems given by such rules we provide sufficient criteria for cut elimination and decidability together with generic complexity results. We also explore the expressivity of such systems with the cut rule in terms of axioms in a Hilbert-style system by exhibiting a corresponding syntactically defined class of axioms along with automatic translations between axioms and rules. This enables us to show a number of limitative results concerning amongst others the modal logic S5. As a step towards a generic construction of cut free and tractable sequent calculi we then introduce the notion of cut trees as representations of rules constructed by absorbing cuts. With certain limitations this allows the automatic construction of a cut free and tractable sequent system from a finite number of rules. For cases where such a system is to be constructed by hand we introduce a graphical representation of rules with context restrictions which simplifies this process. Finally, we apply the developed tools and techniques and construct new cut free sequent systems for a number of Lewis' conditional logics extending the logic V. The systems yield purely syntactic decision procedures of optimal complexity and proofs of the Craig interpolation property for the logics at hand. 5

2021

We introduce labelled sequent calculi for quantified modal logics with definite descriptions. We prove that these calculi have the good structural properties of G3-style calculi. In particular, all rules are height-preserving invertible, weakening and contraction are height-preserving admissible and cut is syntactically admissible. Finally, we show that each calculus gives a proof-theoretic characterization of validity in the corresponding class of models.

Studia Logica

The global consequence relation of a normal modal logic Λ is formulated as a global sequent calculus which extends the local sequent theory of Λ with global sequent rules. All global sequent calculi of normal modal logics admits global cut elimination. This property is utilized to show that decidability is preserved from the local to global sequent theories of any normal modal logic over K4. The preservation of Craig interpolation property from local to global sequent theories of any normal modal logic is shown by prooftheoretic method.

arXiv (Cornell University), 2023

This paper studies nested sequents for quantified modal logics. In particular, it considers extensions of the propositional modal logics definable by the axioms D, T, B, 4, and 5 with varying, increasing, decreasing, and constant domains. Each calculus is proved to have good structural properties: weakening and contraction are height-preserving admissible and cut is (syntactically) admissible. Each calculus is shown to be equivalent to the corresponding axiomatic system and, thus, to be sound and complete. Finally, it is argued that the calculi are internali.e., each sequent has a formula interpretation-whenever the existence predicate is expressible in the language.