Cut-Simulation and Impredicativity

We investigate cut-elimination and cut-simulation in impredicative (higher-order) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic -- in our case a sequent calculus for classical type theory -- is like adding cut. The phenomenon equally applies to prominent axioms like Boolean- and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are built-in instead of being treated axiomatically.Comment: 21 page

Two loop detection mechanisms: a comparison

In order to compare two loop detection mechanisms we describe two calculi for theorem proving in intuitionistic propositional logic. We call them both MJ Hist, and distinguish between them by description as `Swiss' or `Scottish'. These calculi combine in different ways the ideas on focused proof search of Herbelin and Dyckhoff & Pinto with the work of Heuerding emphet al on loop detection. The Scottish calculus detects loops earlier than the Swiss calculus but at the expense of modest extra storage in the history. A comparison of the two approaches is then given, both on a theoretic and on an implementational level

Grafting Hypersequents onto Nested Sequents

We introduce a new Gentzen-style framework of grafted hypersequents that combines the formalism of nested sequents with that of hypersequents. To illustrate the potential of the framework, we present novel calculi for the modal logics $\mathsf{K5}$ and $\mathsf{KD5}$, as well as for extensions of the modal logics $\mathsf{K}$ and $\mathsf{KD}$ with the axiom for shift reflexivity. The latter of these extensions is also known as $\mathsf{SDL}^+$ in the context of deontic logic. All our calculi enjoy syntactic cut elimination and can be used in backwards proof search procedures of optimal complexity. The tableaufication of the calculi for $\mathsf{K5}$ and $\mathsf{KD5}$ yields simplified prefixed tableau calculi for these logic reminiscent of the simplified tableau system for $\mathsf{S5}$, which might be of independent interest

Cut-free Calculi and Relational Semantics for Temporal STIT Logics

We present cut-free labelled sequent calculi for a central formalism in logics of agency: STIT logics with temporal operators. These include sequent systems for Ldm , Tstit and Xstit. All calculi presented possess essential structural properties such as contraction- and cut-admissibility. The labelled calculi G3Ldm and G3Tstit are shown sound and complete relative to irreflexive temporal frames. Additionally, we extend current results by showing that also Xstit can be characterized through relational frames, omitting the use of BT+AC frames

Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi

We have recently presented a general method of proving the fundamental logical properties of Craig and Lyndon Interpolation (IPs) by induction on derivations in a wide class of internal sequent calculi, including sequents, hypersequents, and nested sequents. Here we adapt the method to a more general external formalism of labelled sequents and provide sufficient criteria on the Kripke-frame characterization of a logic that guarantee the IPs. In particular, we show that classes of frames definable by quantifier-free Horn formulas correspond to logics with the IPs. These criteria capture the modal cube and the infinite family of transitive Geach logics

Generic Modal Cut Elimination Applied to Conditional Logics

We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity were explicitly stated as open in the literature

Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents

We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants