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12
An intuitionistic theory of types
"... An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongl ..."
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An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis.
Logics of Formal Inconsistency
 Handbook of Philosophical Logic
"... 1.1 Contradictoriness and inconsistency, consistency and noncontradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory ..."
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Cited by 45 (19 self)
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1.1 Contradictoriness and inconsistency, consistency and noncontradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory
A Taxonomy of Csystems
, 2002
"... The logics of formal inconsistency (LFIs) are paraconsistent logics which permit us to internalize the concepts of consistency or inconsistency inside our object language, introducing new operators to talk about them, and allowing us, in principle, to logically separate the notions of contradictorin ..."
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Cited by 41 (15 self)
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The logics of formal inconsistency (LFIs) are paraconsistent logics which permit us to internalize the concepts of consistency or inconsistency inside our object language, introducing new operators to talk about them, and allowing us, in principle, to logically separate the notions of contradictoriness and of inconsistency. We present the formal definitions of these logics in the context of General Abstract Logics, argue that they in fact represent the majority of all paraconsistent logics existing up to this point, if not the most exceptional ones, and we single out a subclass of them called Csystems, as the LFIs that are built over the positive basis of some given consistent logic. Given precise characterizations of some received logical principles, we point out that the gist of paraconsistent logic lies in the Principle of Explosion, rather than in the Principle of NonContradiction, and we also sharply distinguish these two from the Principle of NonTriviality, considering the next various weaker formulations of explosion, and investigating their interrelations. Subsequently, we present the syntactical formulations of some of the main Csystems based on classical logic, showing how several wellknown logics in the literature can be recast as such a kind of Csystems, and carefully study their properties and shortcomings, showing for instance how they can be used to faithfully
On the computational content of the axiom of choice
 The Journal of Symbolic Logic
, 1998
"... We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the rea ..."
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Cited by 34 (1 self)
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We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We alsoshowhow to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift.
An encoding of partial algebras as total algebras
 Information Processing Letters
"... We introduce a semantic encoding of partial algebras as total algebras through a Horn axiomatization of the existence equality relation interpreted as an algebraic operation. We show that this novel encoding enjoys several important properties that make it a good tool for the execution of partial al ..."
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Cited by 3 (2 self)
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We introduce a semantic encoding of partial algebras as total algebras through a Horn axiomatization of the existence equality relation interpreted as an algebraic operation. We show that this novel encoding enjoys several important properties that make it a good tool for the execution of partial algebraic specifications through means specific to ordinary algebraic reasoning, such as term rewriting.
Implicit Programming and the Logic of Constructible Duality
"... ABSTRACT We present an investigation of duality in the traditional logical manner. We extend Nelson's symmetrization of intuitionistic logic, constructible falsity, to a selfdual logic constructible duality. We develop a selfdual model by considering an interval of worlds in an intuitionistic Kr ..."
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ABSTRACT We present an investigation of duality in the traditional logical manner. We extend Nelson's symmetrization of intuitionistic logic, constructible falsity, to a selfdual logic constructible duality. We develop a selfdual model by considering an interval of worlds in an intuitionistic Kripke model. The duality arises through how we judge truth and falsity. Truth is judged forward in the Kripke model, as in intuitionistic logic, while falsity is judged backwards. We develop a selfdual algebra such that every point in the algebra is representable by some formula in the logic. This algebra arises as an instantiation of a Heyting algebra into several categorical constructions. In particular, we show that this algebra is an instantiation of the Chu construction applied to a Heyting algebra, the second Dialectica construction applied to a Heyting algebra, and as an algebra for the study of recursion and corecursion. Thus the algebra provides a common base for these constructions, and suggests itself as an important part of any constructive logical treatment of duality. Implicit programming is suggested as a new paradigm for computing with constructible duality as its formal system. We show that all the operators that have computable least fixed points are definable explicitly and all operators with computable optimal fixed points are definable implicitly within constructible duality. Implicit programming adds a novel definitional mechanism that allows functions to be defined implicitly. This new programming feature is especially useful for programming with corecursively defined datatypes such as circular lists. iii DEDICATION To my cousin Jordan Lackey (19631995) whose courage with AIDS was an inspiration. iv
Interpolation via translations
"... A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties betwe ..."
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A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a KolmogorovGentzenGödel style translation, and a new translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a KiriyamaOno style translation with features of a KolmogorovGentzenGödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics. AMS Classification: 03C40, 03F03, 03B22
New dimensions on translations between logics
"... Abstract. After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: conservative translations, transfers and contextual translations. Though independent, such approaches are here compared and assess ..."
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Abstract. After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: conservative translations, transfers and contextual translations. Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another.
c ○ 2009 Birkhäuser Verlag Basel/Switzerland What is a Logic Translation? In memoriam
"... Abstract. We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both prooftheoretic and modeltheoretic entailment. We show how logic translations induce notions of logical e ..."
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Abstract. We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both prooftheoretic and modeltheoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading to an explanation of paradoxes that have been described in the literature. Connectives and quantifiers, although not present in the definition of logic and logic translation, can be recovered by their abstract properties and are preserved and reflected by translations under suitable conditions. 1.
Callbyvalue isn’t dual to callbyname, callbyname is dual to callbyvalue!
, 2004
"... Gentzen’s sequent calculus for classical logic shows great symmetry: for example, the rule introducing ∧ on the left of a sequent is mirror symmetric to the introduction rule for the dual operator ∨ on the right of a sequent. A consequence of this casual observation is that when Γ ⊢ ∆ is a theorem o ..."
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Gentzen’s sequent calculus for classical logic shows great symmetry: for example, the rule introducing ∧ on the left of a sequent is mirror symmetric to the introduction rule for the dual operator ∨ on the right of a sequent. A consequence of this casual observation is that when Γ ⊢ ∆ is a theorem over operators {∨,∧,¬}, then so is ∆◦ ⊢ Γ◦, where Σ◦ reverses the order of formulas in Σ, and exchanges each instance of A ∧ B (C ∨ D) in a formula of Σ with B ∨ A (D ∧ C). This symmetry of rules means that the duality of theorems is also a duality of proof structures.