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The ProofTheory and Semantics of Intuitionistic Modal Logic
, 1994
"... Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpret ..."
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Cited by 102 (0 self)
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Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are selfjustifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their
Strongly Analytic Tableaux for Normal Modal Logics
, 1994
"... A strong analytic tableau calculus is presentend for the most common normal modal logics. The method combines the advantages of both sequentlike tableaux and prefixed tableaux. Proper rules are used, instead of complex closure operations for the accessibility relation, while non determinism and cu ..."
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Cited by 48 (13 self)
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A strong analytic tableau calculus is presentend for the most common normal modal logics. The method combines the advantages of both sequentlike tableaux and prefixed tableaux. Proper rules are used, instead of complex closure operations for the accessibility relation, while non determinism and cut rules, used by sequentlike tableaux, are totally eliminated. A strong completeness theorem without cut is also given for symmetric and euclidean logics. The system gains the same modularity of Hilbertstyle formulations, where the addition or deletion of rules is the way to change logic. Since each rule has to consider only adjacent possible worlds, the calculus also gains efficiency. Moreover, the rules satisfy the strong Church Rosser property and can thus be fully parallelized. Termination properties and a general algorithm are devised. The propositional modal logics thus treated are K, D, T, KB, K4, K5, K45, KDB, D4, KD5, KD45, B, S4, S5, OM, OB, OK4, OS4, OM + , OB + , OK4 + ,...
Modality in Dialogue: Planning, Pragmatics and Computation
, 1998
"... Natural language generation (NLG) is first and foremost a reasoning task. In this reasoning, a system plans a communicative act that will signal key facts about the domain to the hearer. In generating action descriptions, this reasoning draws on characterizations both of the causal properties of the ..."
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Cited by 36 (9 self)
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Natural language generation (NLG) is first and foremost a reasoning task. In this reasoning, a system plans a communicative act that will signal key facts about the domain to the hearer. In generating action descriptions, this reasoning draws on characterizations both of the causal properties of the domain and the states of knowledge of the participants in the conversation. This dissertation shows how such characterizations can be specified declaratively and accessed efficiently in NLG. The heart of this dissertation is a study of logical statements about knowledge and action in modal logic. By investigating the prooftheory of modal logic from a logic programming point of view, I show how many kinds of modal statements can be seen as straightforward instructions for computationally manageable search, just as Prolog clauses can. These modal statements provide sufficient expressive resources for an NLG system to represent the effects of actions in the world or to model an addressee whose knowledge in some respects exceeds and in other respects falls short of its own. To illustrate the use of such statements, I describe how the SPUD sentence planner exploits a modal knowledge base to
Labelled Tableaux for MultiModal Logics
 Theorem Proving with Analytic
, 1995
"... this paper we present a tableaulike proof system for multimodal logics based on D'Agostino and Mondadori's classical refutation system KE [DM94]. The proposed system, that we call KEM , works for the logics S5A and S5P(n) which have been devised by Mayer and van der Hoek [MvH92] for formalizing th ..."
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Cited by 17 (9 self)
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this paper we present a tableaulike proof system for multimodal logics based on D'Agostino and Mondadori's classical refutation system KE [DM94]. The proposed system, that we call KEM , works for the logics S5A and S5P(n) which have been devised by Mayer and van der Hoek [MvH92] for formalizing the notions of actuality and preference. We shall also show how KEM works with the normal modal logics K45, D45, and S5 which are frequently used as bases for epistemic operators  knowledge, belief (see, for example [Hoe93, Wan90]), and we shall briefly sketch how to combine knowledge and belief in a multiagent setting through KEM modularity
FirstOrder Intensional Logic
 Annals of Pure and Applied Logic
, 2003
"... Firstorder modal logic is very much under current development, with many di#erent semantics proposed. The use of rigid objects goes back to Saul Kripke. More recently several semantics based on counterparts have been examined, in a development that goes back to David Lewis. ..."
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Cited by 9 (2 self)
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Firstorder modal logic is very much under current development, with many di#erent semantics proposed. The use of rigid objects goes back to Saul Kripke. More recently several semantics based on counterparts have been examined, in a development that goes back to David Lewis.
A ProofPlanning Framework with explicit Abstractions based on Indexed Formulas
 Electronic Notes in Theoretical Computer Science
, 2001
"... A major motivation of proofplanning is to bridge the gap between highlevel, cognitively adequate reasoning for specific domains, and calculuslevel reasoning to ensure soundness. For high reasoning levels the cognitive adequacy of representation and reasoning techniques is a major issue, while ..."
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Cited by 6 (5 self)
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A major motivation of proofplanning is to bridge the gap between highlevel, cognitively adequate reasoning for specific domains, and calculuslevel reasoning to ensure soundness. For high reasoning levels the cognitive adequacy of representation and reasoning techniques is a major issue, while for lower reasoning levels the adequacy wrt. the modelled domain is important. Furthermore, proof construction is an engineering task and there is a need to support the design and application of proofsearch engineering methods. To this end we present a framework to explicitly support di#erent reasoning levels. To structure reasoning levels the framework allows for an explicit representation of abstractions and proofsearch refinement techniques. In order to ensure soundness within a reasoning level, we use techniques developed in the context of matrix characterisation relying on the notion of indexed formulas. Furthermore, we introduce a uniform concept for contextual reasoning, and sketch basic tacticals for the definition of tactics to organise the overall proofsearch inside and across di#erent reasoning levels.
Interpolation for first order S5
 Journal of Symbolic Logic
"... web page: comet.lehman.cuny.edu/fitting ..."
Prefixed Tableaus and Nested Sequents
, 2010
"... Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tabl ..."
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Cited by 5 (1 self)
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Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational variants of each other, roughly in the same way that tableaus and Gentzen sequent calculi are notational variants. This immediately gives rise to new modal nested sequent systems which may be of independent interest. We discuss some of these, including those for some justification logics that include standard modal operators.
KED: A Deontic Theorem Prover
 on Legal Application of Logic Programming, ICLPâ€™94
, 1994
"... Deontic logic (DL) is increasingly recognized as an indispensable tool in such application areas as formal representation of legal knowledge and reasoning, formal specification of computer systems and formal analysis of database integrity constraints. Despite this acknowledgement, there have been fe ..."
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Cited by 2 (1 self)
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Deontic logic (DL) is increasingly recognized as an indispensable tool in such application areas as formal representation of legal knowledge and reasoning, formal specification of computer systems and formal analysis of database integrity constraints. Despite this acknowledgement, there have been few attempts to provide computationally tractable
Firstorder MultiModal Deduction
"... This report aims to help provide such links by providing a set of extremely general results about firstorder multimodal deduction in terms of analytic tableaux and a prefix representation of possible worlds. We first provide sound and complete ground tableau and sequent inference systems, extendin ..."
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Cited by 1 (1 self)
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This report aims to help provide such links by providing a set of extremely general results about firstorder multimodal deduction in terms of analytic tableaux and a prefix representation of possible worlds. We first provide sound and complete ground tableau and sequent inference systems, extending and refining those presented in [Fitting and Mendelsohn, 1998] to the multimodal case. Then we show how to apply general prooftheoretic techniques to derive an equivalent calculus where Herbrand terms streamline proof search [Lincoln and Shankar, 1994]. Finally, we derive a lifted multimodal sequent inference system, which uses unification (or constraintsatisfaction) to resolve the values of variables, in the style of [Voronkov, 1996]. From one point of view, this report can be regarded as the multimodal generalization of the results presented for linear logic and firstorder modal logic in [Lincoln and Shankar, 1994, Fitting, 1996, Fitting and Mendelsohn, 1998]; alternatively, it can be seen as recasting into a modal setting the results of [Stone, 1999b], which investigates firstorder intuitionistic logic along similar lines. Formal modal logic goes back eighty years [Lewis, 1918, Lewis and Langford, 1932]. Yet according to McCarthy [McCarthy, 1997], for example, the modal logic literature still does not offer a formalism with the intensional expressive powerincluding fresh modalities defined ad hoc,and means to describe knowing what by concise and easily manipulated formulasthat is needed for knowledge representation in Artificial Intelligence. Moreover, typical results from the modal logic literature do not support the design of specialized modal inference mechanisms to solve particular knowledge representation tasks. The approach adopted here is a response t...