As part of a general research programme into the expressive power of different generalisations of... more As part of a general research programme into the expressive power of different generalisations of the sequent framework we investigate hypersequent calculi given by rules of the newly introduced format of hypersequent rules with context restrictions. The introduced rule format is used to prove uniform syntactic cut elimination, decidability and complexity results. We also introduce transformations between hypersequent rules of this format and Hilbert axioms, entailing a result about the limits of such rules. As case studies, we apply our methods to several modal logics and obtain e.g. a complexity-optimal decision procedure for the logic S5 and new calculi for the logic K4.2 as well as combinations of modal logics in the form of simply dependent bimodal logics.
Starting with the deontic principles in Mīmāṃsā texts we introduce a new deontic logic. We use ge... more Starting with the deontic principles in Mīmāṃsā texts we introduce a new deontic logic. We use general proof-theoretic methods to obtain a cut-free sequent calculus for this logic, resulting in decidability, complexity results and neighbourhood semantics. The latter is used to analyse a well known example of conflicting obligations from the Vedas.
The Mīmāṃsā school of Indian philosophy elaborated complex ways of interpreting the prescriptive ... more The Mīmāṃsā school of Indian philosophy elaborated complex ways of interpreting the prescriptive portions of the Vedic Sacred Texts. The present article is the result of the collaboration of a group of scholars of logic, computer science, Western philosophy and Indian philosophy and aims at the individuation and analysis of the deontic system which is applied, but never explicitly discussed in Mīmāṃsā texts. The article outlines the basic distinction among three sorts of principles ---hermeneutic, linguistic and deontic--- proposes a first formalisation of the deontic ones and uses it to reason about a well-known example of seemingly conflicting statements, namely the prescription to undertake the malefic Śyena sacrifice and the prohibition to perform any harm.
We introduce a new Gentzen-style framework of grafted hypersequents that combines the formalism o... more 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 K5 and KD5, as well as for extensions of the modal logics K and KD with the axiom for shift reflexivity. The latter of these extensions is also known as 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.
We introduce transformations between hypersequent rules with context restrictions and Hilbert axi... more We introduce transformations between hypersequent rules with context restrictions and Hilbert axioms extending classical (and intuitionistic) propositional logic and vice versa. The introduced rules are used to prove uniform cut elimination, decidability and complexity results as well as finite axiomatisations for many modal logics given by simple frame properties. Our work subsumes many logic-tailored results and allows for new results. As a case study we apply our methods to the logic of uniform deontic frames.
Which modal logics can be 'naturally' captured by a sequent system? Clearly, this question hinges... more Which modal logics can be 'naturally' captured by a sequent system? Clearly, this question hinges on what one believes to be natural, i.e. which format of sequent rules one is willing to accept. This paper studies the relationship between the format of sequent rules and the corresponding syntactical shape of axioms in an equivalent Hilbert-system. We identify three different such formats, the most general of which captures most logics in the S5-cube. The format is based on restricting the context in rule premises and the correspondence is established by translating axioms into rules of our format and vice versa. As an application we show that there is no set of sequent rules of this format which is sound and cut-free complete for S5 and for which cut elimination can be shown by the standard permutation-of-rules argument.
In this thesis we consider generic tools and techniques for the proof-theoretic investigation of ... more 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
We consider a general format for sequent rules for not necessarily normal modal logics based on c... more We consider a general format for sequent rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic VA.
We present unlabelled cut-free sequent calculi for Lewis' conditional logic V and extensions, in ... more We present unlabelled cut-free sequent calculi for Lewis' conditional logic V and extensions, in both the languages with the entrenchment connective and the strong conditional. The calculi give rise to Pspace-decision procedures, also in the language with the weak conditional. Furthermore, they are used to prove the Craig interpolation property for all the logics under consideration, and yield a Pspace-decision procedure for a recently considered hybrid version of V.
We present a graphical representation of sequents and sequent rules, which aids in the discovery ... more We present a graphical representation of sequents and sequent rules, which aids in the discovery of cut-free sequent systems for non-iterative modal logics permitting backwards proof search in polynomial space. The technique is used to construct sequent systems for conditional logic V and KLM rational logic R.
Motivated by the fact that nearly all conditional logics are axiomatised by so-called shallow axi... more Motivated by the fact that nearly all conditional logics are axiomatised by so-called shallow axioms (axioms with modal nesting depth ≤ 1) we investigate sequent calculi and cut elimination for modal logics of this type. We first provide a generic translation of shallow axioms to (one-sided, unlabelled) sequent rules. The resulting system is complete if we admit pseudo-analytic cut, i.e. cuts on modalised propositional combinations of subformulas, leading to a generic (but sub-optimal) decision procedure. In a next step, we show that, for finite sets of axioms, only a small number of cuts is needed between any two applications of modal rules. More precisely, completeness still holds if we restrict to cuts that form a tree of logarithmic height between any two modal rules. In other words, we obtain a small (Pspace-computable) representation of an extended rule set for which cut elimination holds. In particular, this entails Pspace decidability of the underlying logic if contraction is also admissible. This leads to (tight) Pspace bounds for various conditional logics.
As part of a general research programme into the expressive power of different generalisations of... more As part of a general research programme into the expressive power of different generalisations of the sequent framework we investigate hypersequent calculi given by rules of the newly introduced format of hypersequent rules with context restrictions. The introduced rule format is used to prove uniform syntactic cut elimination, decidability and complexity results. We also introduce transformations between hypersequent rules of this format and Hilbert axioms, entailing a result about the limits of such rules. As case studies, we apply our methods to several modal logics and obtain e.g. a complexity-optimal decision procedure for the logic S5 and new calculi for the logic K4.2 as well as combinations of modal logics in the form of simply dependent bimodal logics.
Starting with the deontic principles in Mīmāṃsā texts we introduce a new deontic logic. We use ge... more Starting with the deontic principles in Mīmāṃsā texts we introduce a new deontic logic. We use general proof-theoretic methods to obtain a cut-free sequent calculus for this logic, resulting in decidability, complexity results and neighbourhood semantics. The latter is used to analyse a well known example of conflicting obligations from the Vedas.
The Mīmāṃsā school of Indian philosophy elaborated complex ways of interpreting the prescriptive ... more The Mīmāṃsā school of Indian philosophy elaborated complex ways of interpreting the prescriptive portions of the Vedic Sacred Texts. The present article is the result of the collaboration of a group of scholars of logic, computer science, Western philosophy and Indian philosophy and aims at the individuation and analysis of the deontic system which is applied, but never explicitly discussed in Mīmāṃsā texts. The article outlines the basic distinction among three sorts of principles ---hermeneutic, linguistic and deontic--- proposes a first formalisation of the deontic ones and uses it to reason about a well-known example of seemingly conflicting statements, namely the prescription to undertake the malefic Śyena sacrifice and the prohibition to perform any harm.
We introduce a new Gentzen-style framework of grafted hypersequents that combines the formalism o... more 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 K5 and KD5, as well as for extensions of the modal logics K and KD with the axiom for shift reflexivity. The latter of these extensions is also known as 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.
We introduce transformations between hypersequent rules with context restrictions and Hilbert axi... more We introduce transformations between hypersequent rules with context restrictions and Hilbert axioms extending classical (and intuitionistic) propositional logic and vice versa. The introduced rules are used to prove uniform cut elimination, decidability and complexity results as well as finite axiomatisations for many modal logics given by simple frame properties. Our work subsumes many logic-tailored results and allows for new results. As a case study we apply our methods to the logic of uniform deontic frames.
Which modal logics can be 'naturally' captured by a sequent system? Clearly, this question hinges... more Which modal logics can be 'naturally' captured by a sequent system? Clearly, this question hinges on what one believes to be natural, i.e. which format of sequent rules one is willing to accept. This paper studies the relationship between the format of sequent rules and the corresponding syntactical shape of axioms in an equivalent Hilbert-system. We identify three different such formats, the most general of which captures most logics in the S5-cube. The format is based on restricting the context in rule premises and the correspondence is established by translating axioms into rules of our format and vice versa. As an application we show that there is no set of sequent rules of this format which is sound and cut-free complete for S5 and for which cut elimination can be shown by the standard permutation-of-rules argument.
In this thesis we consider generic tools and techniques for the proof-theoretic investigation of ... more 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
We consider a general format for sequent rules for not necessarily normal modal logics based on c... more We consider a general format for sequent rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic VA.
We present unlabelled cut-free sequent calculi for Lewis' conditional logic V and extensions, in ... more We present unlabelled cut-free sequent calculi for Lewis' conditional logic V and extensions, in both the languages with the entrenchment connective and the strong conditional. The calculi give rise to Pspace-decision procedures, also in the language with the weak conditional. Furthermore, they are used to prove the Craig interpolation property for all the logics under consideration, and yield a Pspace-decision procedure for a recently considered hybrid version of V.
We present a graphical representation of sequents and sequent rules, which aids in the discovery ... more We present a graphical representation of sequents and sequent rules, which aids in the discovery of cut-free sequent systems for non-iterative modal logics permitting backwards proof search in polynomial space. The technique is used to construct sequent systems for conditional logic V and KLM rational logic R.
Motivated by the fact that nearly all conditional logics are axiomatised by so-called shallow axi... more Motivated by the fact that nearly all conditional logics are axiomatised by so-called shallow axioms (axioms with modal nesting depth ≤ 1) we investigate sequent calculi and cut elimination for modal logics of this type. We first provide a generic translation of shallow axioms to (one-sided, unlabelled) sequent rules. The resulting system is complete if we admit pseudo-analytic cut, i.e. cuts on modalised propositional combinations of subformulas, leading to a generic (but sub-optimal) decision procedure. In a next step, we show that, for finite sets of axioms, only a small number of cuts is needed between any two applications of modal rules. More precisely, completeness still holds if we restrict to cuts that form a tree of logarithmic height between any two modal rules. In other words, we obtain a small (Pspace-computable) representation of an extended rule set for which cut elimination holds. In particular, this entails Pspace decidability of the underlying logic if contraction is also admissible. This leads to (tight) Pspace bounds for various conditional logics.
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Papers by Björn Lellmann