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181
A theory of indirection via approximation
 IN POPL
, 2010
"... Building semantic models that account for various kinds of indirect reference has traditionally been a difficult problem. Indirect reference can appear in many guises, such as heap pointers, higherorder functions, object references, and sharedmemory mutexes. We give a general method to construct m ..."
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Cited by 18 (9 self)
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Building semantic models that account for various kinds of indirect reference has traditionally been a difficult problem. Indirect reference can appear in many guises, such as heap pointers, higherorder functions, object references, and sharedmemory mutexes. We give a general method to construct models containing indirect reference by presenting a “theory of indirection”. Our method can be applied in a wide variety of settings and uses only simple, elementary mathematics. In addition to various forms of indirect reference, the resulting models support powerful features such as impredicative quantification and equirecursion; moreover they are compatible with the kind of powerful substructural accounting required to model (higherorder) separation logic. In contrast to previous work, our model is easy to apply to new settings and has a simple axiomatization, which is complete in the sense that all models of it are isomorphic. Our proofs are machinechecked in Coq.
Multiple conclusions
 In 12th International Congress on Logic, Methodology and Philosophy of Science
, 2005
"... Abstract: I argue for the following four theses. (1) Denial is not to be analysed as the assertion of a negation. (2) Given the concepts of assertion and denial, we have the resources to analyse logical consequence as relating arguments with multiple premises and multiple conclusions. Gentzen’s mult ..."
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Cited by 17 (2 self)
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Abstract: I argue for the following four theses. (1) Denial is not to be analysed as the assertion of a negation. (2) Given the concepts of assertion and denial, we have the resources to analyse logical consequence as relating arguments with multiple premises and multiple conclusions. Gentzen’s multiple conclusion calculus can be understood in a straightforward, motivated, nonquestionbegging way. (3) If a broadly antirealist or inferentialist justification of a logical system works, it works just as well for classical logic as it does for intuitionistic logic. The special case for an antirealist justification of intuitionistic logic over and above a justification of classical logic relies on an unjustified assumption about the shape of proofs. Finally, (4) this picture of logical consequence provides a relatively neutral shared vocabulary which can help us understand and adjudicate debates between proponents of classical and nonclassical logics. Our topic is the notion of logical consequence: the link between premises and conclusions, the glue that holds together deductively valid argument. How can we understand this relation between premises and conclusions? It seems that any account begs questions. Painting with very broad brushtrokes, we can sketch the landscape
The stories of logic and information
 In Handbook of the Philosophy of Information, P. Adriaans and
, 2008
"... ..."
A Gentzen System for Reasoning with ContraryToDuty Obligations: A Preliminary Study
 Proc. ∆eon’02
, 2002
"... In this paper we present a Gentzen system for reasoning with contraryto duty obligations. The intuition behind the system is that a contraryto duty is a special kind of normative exception. The logical machinery to formalize this idea is taken from substructural logics and it is based on the defin ..."
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Cited by 17 (10 self)
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In this paper we present a Gentzen system for reasoning with contraryto duty obligations. The intuition behind the system is that a contraryto duty is a special kind of normative exception. The logical machinery to formalize this idea is taken from substructural logics and it is based on the definition of a new nonclassical connective capturing the notion of reparational obligation. Then the system is tested against wellknown contrarytoduty paradoxes.
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 16 (7 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.
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 operato ..."
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Cited by 16 (6 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.
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 15 (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.
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 15 (4 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.
Lambek Calculus with Nonlogical Axioms
 Language and Grammar, Studies in Mathematical Linguistics and Natural Language
, 2002
"... We study Nonassociative Lambek Calculus and Associative Lambek Calculus enriched with nitely many nonlogical axioms. We prove that the nonassociative systems are decidable in polynomial time and generate contextfree languages. In [1] it has been shown that nite axiomatic extensions of Associa ..."
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Cited by 14 (10 self)
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We study Nonassociative Lambek Calculus and Associative Lambek Calculus enriched with nitely many nonlogical axioms. We prove that the nonassociative systems are decidable in polynomial time and generate contextfree languages. In [1] it has been shown that nite axiomatic extensions of Associative Lambek Calculus generate all recursively enumerable languages; here we give a new proof of this fact. We also obtain similar results for systems with permutation and n ary operations.
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 13 (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.