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Conditionals and consequences
 Journal of Applied Logic
, 2007
"... Abstract. We examine the notion of conditionals and the role of conditionals in inductive logics and arguments. We identify three mistakes commonly made in the study of, or motivation for, nonclassical logics. A nonmonotonic consequence relation based on evidential probability is formulated. With r ..."
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Cited by 13 (11 self)
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Abstract. We examine the notion of conditionals and the role of conditionals in inductive logics and arguments. We identify three mistakes commonly made in the study of, or motivation for, nonclassical logics. A nonmonotonic consequence relation based on evidential probability is formulated. With respect to this acceptance relation some rules of inference of System P are unsound, and we propose refinements that hold in our framework. 1 Three mistakes Pure Mathematics is the class of all propositions of the form ‘p implies q’... And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member... [45, p.3]. Thus begins the precursor of Principia Mathematica, Russell’s Principles of Mathematics, and thus begins the sad and confusing twentieth century tale of implication.
Analyzing the Core of Categorial Grammar
, 2001
"... Even though residuation is at the core of Categorial Grammar [11], it is not always immediate to realize how standard logic systems like Multimodal Categorial Type Logics (MCTL) [17] actually embody this property. In this paper we focus on the basic system NL [12] and its extension with unary modal ..."
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Cited by 11 (4 self)
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Even though residuation is at the core of Categorial Grammar [11], it is not always immediate to realize how standard logic systems like Multimodal Categorial Type Logics (MCTL) [17] actually embody this property. In this paper we focus on the basic system NL [12] and its extension with unary modalities NL(3) [16], and we spell things out by means of Display Calculi (DC) [3, 10]. The use of structural operators in DC permits a sharp distinction between the core properties we want to impose on the logic system and the way these properties are projected into the logic operators. We will show how we can obtain Lambek residuated triple n, = and of binary operators, and how the operators 3 and 2 introduced by Moortgat in [16] are indeed their unary counterpart.
Duality in knowledge sharing
 IN 7TH INTERNATIONAL SYMPOSIUM ON ARTIFICIAL INTELLIGENCE AND MATHEMATICS, FT
, 2002
"... I propose a formalisation of knowledge sharing scenarios that aims at capturing the crucial role played by an existing duality between ontological theories one wants to merge and particular situations that need to be linked. I use diagrams in the Chu category and colimits over these diagrams to acco ..."
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Cited by 11 (9 self)
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I propose a formalisation of knowledge sharing scenarios that aims at capturing the crucial role played by an existing duality between ontological theories one wants to merge and particular situations that need to be linked. I use diagrams in the Chu category and colimits over these diagrams to account for the reliability and optimality of knowledge sharing systems. Furthermore, I show how we may obtain a deeper understanding of a system that shares knowledge between a probabilistic logic program and Bayesian belief networks by reanalysing the scenario in terms of the present approach.
Cutelimination and proofsearch for biintuitionistic logic using nested sequents
, 2008
"... We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant cal ..."
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Cited by 11 (3 self)
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We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cutelimination proof. We then present the derived calculus, and then present a proofsearch strategy which allows it to be used for automated proof search. We prove that this search strategy is terminating and complete by showing how it can be used to mimic derivations obtained from an existing calculus GBiInt for biintuitionistic logic. As far as we know, our new calculus is the first sequent calculus for biintuitionistic logic which uses no semantic additions like labels, which has a purely syntactic cutelimination proof, and which can be used naturally for backwards proofsearch.
Types as graphs: Continuations in type logical grammar
, 2005
"... Using the programminglanguage concept of CONTINUATIONS, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of insitu quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as c ..."
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Cited by 11 (8 self)
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Using the programminglanguage concept of CONTINUATIONS, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of insitu quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as continuations are the tool of choice for reasoning about evaluation order and side effects in programming languages, our system provides a principled, typelogical way to model evaluation order and side effects in natural language. We illustrate with an improved account of quantificational binding, weak crossover, whquestions, superiority, and polarity licensing.
Information Flow and Relevant Logics
 Logic, Language and Computation: The 1994 Moraga Proceedings. CSLI
, 1996
"... this paper I show that these hints and gestures are true. And perhaps truer than those that made them thought at the time ..."
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Cited by 10 (5 self)
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this paper I show that these hints and gestures are true. And perhaps truer than those that made them thought at the time
Sufficient conditions for cut elimination with complexity analysis
 Annals of Pure and Applied Logic
, 2007
"... Sufficient conditions for first order based sequent calculi to admit cut elimination by a SchütteTait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related with the calculus. The con ..."
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Cited by 9 (4 self)
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Sufficient conditions for first order based sequent calculi to admit cut elimination by a SchütteTait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related with the calculus. The conditions are general enough to be satisfied by a wide class of sequent calculi encompassing, among others, some sequent calculi presentations for the first order and the propositional versions of classical and intuitionistic logic, classical and intuitionistic modal logic S4, and classical and intuitionistic linear logic and some of its fragments. Moreover the conditions are such that there is an algorithm for checking if they are satisfied by a sequent calculus.
Galois Connections in Categorial Type Logic
, 2001
"... The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In x2 we do the basic modeltheoretic and ..."
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Cited by 9 (4 self)
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The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are orderpreserving; in this paper instead, we consider the addition of orderreversing, Galois connected operators. In x2 we do the basic modeltheoretic and prooftheoretic groundwork. In x3 we use the expressive power of the Galois connected operators to restrict the scopal possibilities of generalized quanti er expressions, and to describe a typology of polarity items.
Reasoning Under Inconsistency: The Forgotten Connective
 In Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence (IJCAI’05
, 2005
"... In many frameworks for reasoning under inconsistency, it is implicitly assumed that the formulae from the belief base are connected using a weak form of conjunction. When it is consistent, a belief base B = {ϕ1,..., ϕn}, where the ϕi are propositional formulae, is logically equivalent to the base {ϕ ..."
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Cited by 9 (3 self)
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In many frameworks for reasoning under inconsistency, it is implicitly assumed that the formulae from the belief base are connected using a weak form of conjunction. When it is consistent, a belief base B = {ϕ1,..., ϕn}, where the ϕi are propositional formulae, is logically equivalent to the base {ϕ1 ∧... ∧ ϕn}. However, when it is not consistent, both bases typically lead to different conclusions. This illustrates the fact that the comma used in base B has to be considered as an additional, genuine connective, and not as a simple conjunction. In this work we define and investigate a propositional framework with such a “comma connective”. We give it a semantics and show how it generalizes several approaches for reasoning from inconsistent beliefs. 1
An Encompassing Framework for Paraconsistent Logic Programs
 J. Applied Logic
, 2003
"... We propose a framework which extends Antitonic Logic Programs [13] to an arbitrary complete bilattice of truthvalues, where belief and doubt are explicitly represented. Inspired by Ginsberg and Fitting 's bilattice approaches, this framework allows a precise de nition of important operators fo ..."
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Cited by 8 (4 self)
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We propose a framework which extends Antitonic Logic Programs [13] to an arbitrary complete bilattice of truthvalues, where belief and doubt are explicitly represented. Inspired by Ginsberg and Fitting 's bilattice approaches, this framework allows a precise de nition of important operators found in logic programming, such as explicit and default negation. In particular, it leads to a natural semantical integration of explicit and default negation through the Coherence Principle [38], according to which explicit negation entails default negation. We then de ne Coherent Answer Sets, and the Paraconsistent Wellfounded Model semantics, generalising many paraconsistent semantics for logic programs. In particular, Paraconsistent WellFounded Semantics with eXplicit negation (WFSXp ) [3, 11]. The framework is an extension of Antitonic Logic Programs for most cases, and is general enough to capture Probabilistic Deductive Databases, Possibilistic Logic Programming, Hybrid Probabilistic Logic Programs, and Fuzzy Logic Programming. Thus, we have a powerful mathematical formalism for dealing simultaneously with default, paraconsistency, and uncertainty reasoning. Results are provided about how our semantical framework deals with inconsistent information and with its propagation by the rules of the program.