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The Relevance of Semantic Subtyping
 In IEEE Symposium on Logic in Computer Science (LICS
, 2002
"... We compare Meyer and Routley's minimal relevant logic B+ with the recent semanticsbased approach to subtyping introduced by Frisch, Castagna and Benzaken in the definition of a type system with intersection and union. We show that  for the functional core of the system  such notion of subty ..."
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Cited by 53 (9 self)
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We compare Meyer and Routley's minimal relevant logic B+ with the recent semanticsbased approach to subtyping introduced by Frisch, Castagna and Benzaken in the definition of a type system with intersection and union. We show that  for the functional core of the system  such notion of subtyping, which is defined in purely settheoretical terms, coincides with the relevant entailment of the logic B+ . 1
Diamonds are a Philosopher's Best Friends. The Knowability Paradox and Modal Epistemic Relevance Logic (Extended Abstract)
 Journal of Philosophical Logic
, 2002
"... Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the ..."
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Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution o#ered is in the spirit of the constructivist attitude usually maintained by defenders of the antirealist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.
The Semantics of Entailment Omega
, 2002
"... This paper discusses the relation between the minimal positive relevant logic B+ and intersection and union type theories. There is a marvellous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the twea ..."
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Cited by 4 (2 self)
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This paper discusses the relation between the minimal positive relevant logic B+ and intersection and union type theories. There is a marvellous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking B intersect T of B+ , which saves implication and conjunction but drops disjunction. The filter models of the lambdacalculus (and its intimate partner Combinatory Logic CL) of the first author and her coauthors then become theory models of these calculi. (The logician's Theory is the algebraist's Filter.) The coincidence extends to a dual interpretation of key particles  the subtype translates to provable >, type intersection to conjunction, function space > to implication and whole domain omega to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper and to be expected that type union U should correspond to the logical disjunction \/ of B+ . But the simulation of functional application by a fusion (or modus ponens product) operation o on theories leaves the key Bubbling lemma of work on ITD unprovable for the \/prime theories now appropriate for the modelling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of lambda and CL.
Intersection Type Systems and Logics Related to the Meyer–Routley System B +
, 2003
"... Abstract: Some, but not all, closed terms of the lambda calculus have types; these types are exactly the theorems of intuitionistic implicational logic. An extension of these simple (→) types to intersection (or →∧) types allows all closed lambda terms to have types. The corresponding → ∧ logic, rel ..."
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Abstract: Some, but not all, closed terms of the lambda calculus have types; these types are exactly the theorems of intuitionistic implicational logic. An extension of these simple (→) types to intersection (or →∧) types allows all closed lambda terms to have types. The corresponding → ∧ logic, related to the Meyer–Routley minimal logic B + (without ∨), is weaker than the → ∧ fragment of intuitionistic logic. In this paper we provide an introduction to the above work and also determine the →∧ logics that correspond to certain interesting subsystems of the full →∧ type theory. 1 Simple Typed Lambda Calculus In standard mathematical notation “f: α → β ” stands for “f is a function from α into β. ” If we interpret “: ” as “∈ ” we have the rule: f: α → β t: α f(t) : β This is one of the formation rules of typed lambda calculus, except that there we write ft instead of f(t). In λcalculus, λx.M represents the function f such that fx = M. This makes the following rule a natural one: [x: α] M: β λx.M: α → β We now set up the λterms and their types more formally.
Toward Isomorphism of Intersection and Union Types ∗ Dedicated to Corrado Böhm on the occasion of his 90th Birthday
"... This paper investigates type isomorphism in a λcalculus with intersection and union types. It is known that in λcalculus, the isomorphism between two types is realised by a pair of terms inverse one each other. Notably, invertible terms are linear terms of a particular shape, called finite heredit ..."
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This paper investigates type isomorphism in a λcalculus with intersection and union types. It is known that in λcalculus, the isomorphism between two types is realised by a pair of terms inverse one each other. Notably, invertible terms are linear terms of a particular shape, called finite hereditary permutators. Typing properties of finite hereditary permutators are then studied in a relevant type inference system with intersection and union types for linear terms. In particular, an isomorphism preserving reduction between types is defined. Reduction of types is confluent and terminating, and induces a notion of normal form of types. The properties of normal types are a crucial step toward the complete characterisation of type isomorphism. The main results of this paper are, on one hand, the fact that two types with the same normal form are isomorphic, on the other hand, the characterisation of the isomorphism between types in normal form, modulo isomorphism of arrow types. 1
Preciseness of Subtyping on Intersection and Union Types?
"... Abstract. The notion of subtyping has gained an important role both in theoretical and applicative domains: in lambda and concurrent calculi as well as in programming languages. The soundness and the completeness, together referred to as the preciseness of subtyping, can be considered from two dif ..."
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Abstract. The notion of subtyping has gained an important role both in theoretical and applicative domains: in lambda and concurrent calculi as well as in programming languages. The soundness and the completeness, together referred to as the preciseness of subtyping, can be considered from two different points of view: denotational and operational. The former preciseness is based on the denotation of a type which is a mathematical object that describes the meaning of the type in accordance with the denotations of other expressions from the language. The latter preciseness has been recently developed with respect to type safety, i.e. the safe replacement of a term of a smaller type when a term of a bigger type is expected. We propose a technique for formalising and proving operational preciseness of the subtyping relation in the setting of a concurrent lambda calculus with intersection and union types. The key feature is the link between typings and the operational semantics. We then prove soundness and completeness getting that the subtyping relation of this calculus enjoys both denotational and operational preciseness. 1
Recent Work in Relevant Logic Mark Jago Forthcoming in Analysis. Draft of April 2013.
"... Relevant logics are a group of logics which attempt to block irrelevant conclusions being drawn from a set of premises. The following inferences are all valid in classical logic, where A and B are any sentences whatsoever: • from A, one may infer B → A, B → B and B ∨¬B; ..."
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Relevant logics are a group of logics which attempt to block irrelevant conclusions being drawn from a set of premises. The following inferences are all valid in classical logic, where A and B are any sentences whatsoever: • from A, one may infer B → A, B → B and B ∨¬B;
⋆ Component—Universal Logic = Rigid Logic + Flexible Logic = Formal Logic + Dialectical Logic
"... www.unilog.org www.unilog.cn ..."
On Isomorphism of “Functional” Intersection and Union Types∗
"... Type isomorphism is useful for retrieving library components, since a function in a library can have a type different from, but isomorphic to, the one expected by the user. Moreover type isomorphism gives for free the coercion required to include the function in the user program with the correct typ ..."
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Type isomorphism is useful for retrieving library components, since a function in a library can have a type different from, but isomorphic to, the one expected by the user. Moreover type isomorphism gives for free the coercion required to include the function in the user program with the correct type. The present paper faces the problem of type isomorphism in a system with intersection and union types. In the presence of intersection and union, isomorphism is not a congruence and cannot be characterised in an equational way. A characterisation can still be given, quite complicated by the interference between functional and non functional types. This drawback is faced in the paper by interpreting each atomic type as the set of functions mapping any argument into the interpretation of the type itself. This choice has been suggested by the initial projection of Scott’s inverse limit λmodel. The main result of this paper is a condition assuring type isomorphism, based on an isomorphism preserving reduction. 1