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Using Typed Lambda Calculus to Implement Formal Systems on a Machine
 Journal of Automated Reasoning
, 1992
"... this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper [24] describes the basic features of the LF. In this paper we are going to provide a broader illustration of its a ..."
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Cited by 83 (14 self)
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this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper [24] describes the basic features of the LF. In this paper we are going to provide a broader illustration of its applicability and discuss to what extent it is successful. The analysis (of the formal presentation) of a system carried out through encoding often illuminates the system itself. This paper will also deal with this phenomenon.
Transformation Methods in LDS
 In Logic, Language and Reasoning. An Essay in Honor of Dov Gabbay
, 1997
"... this paper we shall, instead, use a fragment of this family of logics as a casestudy to illustrate a set of methods originating in the LDS program. In particular, we aim to illuminate the following aspects: (I) By virtue of the extra power of labels and labelling algebras, traditional proof systems ..."
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Cited by 3 (3 self)
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this paper we shall, instead, use a fragment of this family of logics as a casestudy to illustrate a set of methods originating in the LDS program. In particular, we aim to illuminate the following aspects: (I) By virtue of the extra power of labels and labelling algebras, traditional proof systems can be transformed so as to become applicable over a much wider territory whilst retaining a uniform structure. Different logics can be obtained by defining different labelling algebras, which therefore act as "parameters", and the transition from one logic to another can be captured as a parameterchanging process which leaves the structure of deductions unchanged
REPRESENTABLE IDEMPOTENT COMMUTATIVE RESIDUATED LATTICES
"... Abstract. It is proved that the variety of representable idempotent commutative residuated lattices is locally finite. The ngenerated subdirectly irreducible algebras in this variety are shown to have at most 3n+1 elements each. A constructive characterization of the subdirectly irreducible algebra ..."
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Cited by 2 (1 self)
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Abstract. It is proved that the variety of representable idempotent commutative residuated lattices is locally finite. The ngenerated subdirectly irreducible algebras in this variety are shown to have at most 3n+1 elements each. A constructive characterization of the subdirectly irreducible algebras is provided, with some applications. The main result implies that every finitely based extension of positive relevance logic containing the mingle and GĂ¶delDummett axioms has a solvable deducibility problem. 1.
Equality In Linear Logic
, 1996
"... reference is [Ros]). Quantales were introduced by Mulvey ([Mul]) as an algebraic tool for studying representations of noncommutative C algebras. Informally, a quantale is a complete lattice Q equipped with a product distributive over arbitrary sup's. The importance of quantales for Linear Logi ..."
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reference is [Ros]). Quantales were introduced by Mulvey ([Mul]) as an algebraic tool for studying representations of noncommutative C algebras. Informally, a quantale is a complete lattice Q equipped with a product distributive over arbitrary sup's. The importance of quantales for Linear Logic is revealed in Yetter's work ([Yet]), who proved that semantics of classical linear logic is given by a class of quantales, named Girard quantales, which coincides with Girard's phase semantics. An analogous result is obtained for a sort of noncommutative linear logic, as well as intuitionistic linear logic without negation, which suggest that the utilisation of the theory of quantales (or even weaker structures, such that *autonomous posets) might be fruitful in studying the semantic of several variants of linear logic. As usual, we denote the order in a lattice by , while W and V denote the operatio
unknown title
, 803
"... Logics preserving degrees of truth from varieties of residuated lattices ..."
A CATEGORY EQUIVALENCE FOR ODD SUGIHARA MONOIDS AND ITS APPLICATIONS
"... Abstract. An odd Sugihara monoid is a residuated distributive latticeordered commutative idempotent monoid with an orderreversing involution that fixes the monoid identity. The main theorem of this paper establishes a category equivalence between odd Sugihara monoids and relative Stone algebras. In ..."
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Abstract. An odd Sugihara monoid is a residuated distributive latticeordered commutative idempotent monoid with an orderreversing involution that fixes the monoid identity. The main theorem of this paper establishes a category equivalence between odd Sugihara monoids and relative Stone algebras. In combination with known results, it swiftly determines which varieties of odd Sugihara monoids are [strongly] amalgamable and which have the strong [or weak] epimorphismsurjectivity property. In particular, the full variety is shown to have all of these properties. The results extend, with slight modification, to the case where the algebras are bounded. Logical applications include immediate answers to some questions about projective and finite Beth definability and interpolation in the uninormbased logic IUML, its boundless fragment and all of their extensions. 1.