Sequent calculi and interpolation for non-normal modal and deonticlogics

G3-style sequent calculi for the logics in the cube of non-normal modal logics and for their deontic extensions are studied. For each calculus we prove that weakening and contraction are height-preserving admissible, and we give a syntactic proof of the admissibility of cut. This implies that the subformula property holds and that derivability can be decided by a terminating proof search whose complexity is in PSPACE. These calculi are shown to be equivalent to the axiomatic ones and, therefore, they are sound and complete with respect to neighbourhood semantics. Finally, it is given a Maehara-style proof of Craig's interpolation theorem for most of the logics considered

Tableaux clausal para lógica modal

Trabalho de Conclusão de Curso (graduação)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Ciência da Computação, 2016.Este trabalho apresenta a definição de um cálculo tableaux clausal proposto para lógicas modais, além de apresentar uma implementação do mesmo. O objetivo geral é aplicar o método refutacional tableaux utilizando a forma normal construída para resolução clausal na tentativa de gerar o máximo de informações possíveis sobre conjuntos de fórmulas na linguagem modal, sem que seja necessário a especificação de um número elevado de regras de inferência no cálculo, o que tornaria a tarefa de implementar razoavelmente mais difícil. Estas informações extraídas contêm singularidades dos conjuntos de fórmulas que podem auxiliar em decisões relacionadas aos métodos de provas aplicados aos mesmos. As provas de correção para o cálculo são apresentadas. A implementação realizada, apesar de bem sucedida, revelou imperfeições com relação ao desempenho de execução. Esforços direcionados à melhoria de tais imperfeições, bem como uma análise das informações extraídas, são deixados como trabalhos futuros.This work presents a clausal tableaux calculus for modal logics and na implementation for it. The main goal is to combine aspects of two different proof methods, the inference rules of a tableaux calculus and the normal form employed in a clausal resolution method. The goal is to generate as much information as possible about sets of formulae in modal logic, without the need to define a large number of inference rules in the calculus, which would make the job of implementing the calculus reasonably harder. The pieces of extracted information may help to characterise singularities of the set of formulae, which could then help to make decisions about what proof method to apply to such a set. The implementation is correct, but does not perform well. Efforts in the direction of a more robust implementation and the analysis of the extracted information are left as future work

Proof Search in Multi-Agent Dialogues for Modal Logic

In computer science, and also in philosophy, modal logics play an important role in various areas. They can be used to model knowledge structures among software-agents, behaviour of computer systems, or ontologies. They also provide mathematical tools to perform reasoning in these models, e.g., to extract common knowledge of agents, check whether security-relevant problems might occur when running a program, or to detect contradictions in a set of terminological definitions. Intuitionistic or constructive propositional logic can be considered as a special kind of modal logic. Constructive modal logics, as a combination of intuitionistic propositional logic and classical modal logics, describe a family of modal systems which are, compared to the classical variant, more restrictive concerning the validity of formulas. To prove validity of a statement formalized in such a logic, various reasoning procedures (also called calculi) have been investigated. There are especially many variants of sequent and tableau systems which can be used easily to find proofs by applying given syntactical rules one after another. Sometimes there are different possibilities to find a proof for the same formula within the same calculus. It also happens that a bad choice of non-invertible rule applications at the wrong time makes it impossible to finish the proof successfully, although the formula is provable. For this reason, a normalization of deductions in a calculus is desired. This restricts the possibilities to apply rules arbitrarily and emphasizes the situations in which significant, non-invertible rule applications are necessary. Such a normalization is enforced in so-called focused sequent systems. Another attempt to find a normalized calculus leads to dialogical logic, a game-theoretic reasoning technique. Usually, two players, one proponent and one opponent, argue about an assertion, expressed as a formula and stated by the proponent at the beginning of the play. The kinds of arguments, namely attacks and defences, are bound to special game rules. These are designed in such a way that the proponent has a winning strategy in the game if and only if his initial statement is a valid formula. The dialogical approach is very flexible as the game rules can be adjusted easily. Sets of rules exist to perform reasoning in many different kinds of logic, however proving soundness and completeness of dialogical calculi is complex and, if at all, often only considered very roughly in the literature. The standard two-player dialogues do not have much potential to enforce normalization like focus sequent systems. However, it turns out that introducing further proponent-players who fight against one opponent in a round-based setting leads to a normalization as described above. The flexibility of two-player games is largely preserved in multi-proponent dialogues. Other ordinary sequent systems can easily be transferred into the dialectic setting to achieve a normalization. Further, the round-based scheduling induces a method to parallelize the reasoning process. Modifying the game rules makes it possible to construct new intermediate or even more restrictive logics. In this work, dialogical systems with multiple proponents are presented for intuitionistic propositional logic and modal logics S4 and CS4. Starting with the former one, it is shown that the normalization can be transferred easily to both the latter systems. Informal game rules are introduced and, to make them concrete and unambiguous, translated into the dialogical sequent-style calculi DiaSeqI, DiaSeqS4, and DiaSeqCS4. An extra system for intuitionistic logic, which guarantees termination in proof searches, even if the target formula is not valid, is also provided. Soundness and completeness of all these presented dialogical sequent calculi is proven formally, by showing that it is always possible to translate derivations in the game-oriented approach into another sound and complete sequent system and vice versa. Thereby, a new (ordinary) multi-conclusion sequent calculus for CS4 is introduced for which adequateness is shown, too. The multi-proponent dialogical systems of this work are compared to different sequent calculi and other dialogical attempts found in literature. A comprehensive survey of such approaches is also part of this thesis