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A Judgmental Reconstruction of Modal Logic
 Mathematical Structures in Computer Science
, 1999
"... this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deductio ..."
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this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic [FM97] and find that it is already contained in modal logic, using the decomposition of the lax modality fl A as
Coercive Subtyping in Type Theory
 Proc. of CSL'96, the 1996 Annual Conference of the European Association for Computer Science Logic, Utrecht. LNCS 1258
, 1996
"... We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as MartinLof's type theory [NPS90] and the type theory UTT [Luo94]. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical frame ..."
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Cited by 27 (14 self)
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We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as MartinLof's type theory [NPS90] and the type theory UTT [Luo94]. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; in particular, subsumption and coercion are combined in such a way that the meaning of an object being in a supertype is given by coercive definition rules for the definitional equality. It is shown that this provides a conceptually simple and uniform framework to understand subtyping and coercion relations in type theories with sophisticated type structures such as inductive types and universes. The use of coercive subtyping in formal development and in reasoning about subsets of objects is discussed in the context of computerassisted formal reasoning. 1 Introduction A type in type theory is often intuitively thought of as a set. For example, types in MartinLof's type theory [ML84, NPS90...
Intensionality, Extensionality, and Proof Irrelevance in Modal Type Theory
 Pages 221–230 of: Symposium on Logic in Computer Science
, 2001
"... We develop a uniform type theory that integrates intensionality, extensionality, and proof irrelevance as judgmental concepts. Any object may be treated intensionally (subject only to #conversion), extensionally (subject also to ##conversion), or as irrelevant (equal to any other object at the sam ..."
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We develop a uniform type theory that integrates intensionality, extensionality, and proof irrelevance as judgmental concepts. Any object may be treated intensionally (subject only to #conversion), extensionally (subject also to ##conversion), or as irrelevant (equal to any other object at the same type), depending on where it occurs. Modal restrictions developed in prior work for simple types are generalized and employed to guarantee consistency between these views of objects. Potential applications are in logical frameworks, functional programming, and the foundations of firstorder modal logics.
ON ENDS AND MEANS. CONSTRUCTIVE TYPE THEORY AS A GUIDE FOR MODELING IN THEORY OF MIND AND ACTION
"... RÉSUMÉ – Fins et moyens. La théorie constructive des types considérée comme guide pour former des modèles en théorie de l’esprit et de l’action. Cet article se propose de clarifier la relation formelle entre « fin » et « moyen » en termes de théorie constructive des types. Selon la tradition philoso ..."
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RÉSUMÉ – Fins et moyens. La théorie constructive des types considérée comme guide pour former des modèles en théorie de l’esprit et de l’action. Cet article se propose de clarifier la relation formelle entre « fin » et « moyen » en termes de théorie constructive des types. Selon la tradition philosophique, deux conceptions s’opposent: l’une, machiavellienne, qui considère « fin » et « moyen » comme étant dans une certaine mesure indépendants, l’autre, kantienne, qui envisage cette relation comme analytique. La première semble inacceptable d’un point de vue aussi bien moral que logique, mais la seconde se heurte également à des difficultés. Celles ci peuvent, selon nous, être résolues grâce à la théorie constructive des types si l´on considère la fin d’une action comme le type des moyens aptes à réaliser cette fin. MOTS CLÉS – Action considérée comme un type, Analyticité, Moyen canonique, Relation fin/moyen, Savoir pratique (savoir faire), Théorie constructive des types SUMMARY – The following discussion aims at clarifying the formal relation between end and means in terms of constructive type theory. Philosophical tradition offers two opposed conceptions: a Machiavellian, on the one hand, dealing with end and means as to some extent independent items, and a Kantian, on the other, which regards the relation as analytical. The first seems to be both morally and logically unacceptable, but also the second one faces difficulties. I propose to resolve these difficulties along the lines of constructive type theory by considering the end of an action as the type of means which are apt to realise this end.
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"... Note that the second premise of the ⊃L rule is an unfocused sequent. From a practical point of view it is important to continue with the focusing steps in the first premise before attempting to prove the second premise, because the decomposition of B may ultimately fail when an atomic proposition is ..."
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Note that the second premise of the ⊃L rule is an unfocused sequent. From a practical point of view it is important to continue with the focusing steps in the first premise before attempting to prove the second premise, because the decomposition of B may ultimately fail when an atomic proposition is reached. Such a failure would render the possibly difficult proof of A useless. It is possible to extend the definition of L + to include conjunction and ⊤ and remove the left focus rules for conjunction. In some situations this would clearly lead to shorter proofs, but the present version appears to have less disjunctive nondeterminism. 3 Initial Sequents. There is a slight, but important asymmetry in the initial sequents: we require that we have focused on the left proposition. init ∆; P ≫·; P Since this is the only rule which can be applied when the left focus formula is atomic, a proof attempt fails in a situation where ∆; P ≫·; Q for P � = Q.