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12
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 ..."
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Cited by 92 (16 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.
Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL
, 2006
"... Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem ..."
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Cited by 20 (6 self)
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Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.
Multiplicative Conjunction as an Extensional Conjunction
 Journal of the IGPL
, 1997
"... We show that the rule that allows the inference of A from A\Omega B is admissible in many of the basic multiplicative (intensional) systems. By adding this rule to these systems we get, therefore, conservative extensions in which the tensor behaves as classical conjunction. Among the systems obtain ..."
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Cited by 4 (4 self)
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We show that the rule that allows the inference of A from A\Omega B is admissible in many of the basic multiplicative (intensional) systems. By adding this rule to these systems we get, therefore, conservative extensions in which the tensor behaves as classical conjunction. Among the systems obtained in this way the one derived from RMIm (= multiplicative linear logic together with contraction and its converse) has a particular interest. We show that this system has a simple infinitevalued semantics, relative to which it is strongly complete, and a nice cutfree Gentzentype formulation which employs hypersequents (= finite sequences of ordinary sequents). Moreover: classical logic has a simple, strong translation into this logic. This translation uses definable connectives and preserves the consequence relation of classical logic (not just the set of theorems). Similar results, but with a 3valued semantics, obtain if instead of RMIm we use RMm (the purely multiplicative fragment ...
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
Analysis of two fragments with negation and without implication of the logic of residuated lattices
, 2004
"... The logic of (commutative integral bounded) residuated lattices is known under different names in the literature: monoidal logic [9], intuitionistic logic without contraction [1], HBCK [13], etc. It is usually given, up to definitional equivalence, in the language 〈∨,∧, ∗,¬,→, 0, 1〉. It was shown i ..."
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Cited by 3 (2 self)
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The logic of (commutative integral bounded) residuated lattices is known under different names in the literature: monoidal logic [9], intuitionistic logic without contraction [1], HBCK [13], etc. It is usually given, up to definitional equivalence, in the language 〈∨,∧, ∗,¬,→, 0, 1〉. It was shown in [1] that this logic, denoted there by IPC∗\c, is algebraizable (the variety of residuated lattices is its equivalent algebraic semantics) and that is the external deductive system associated to the Gentzen system determined by the sequent calculus FLew [10, 12]. This logic is decidable [5]. The 〈→〉fragment and the 〈∗,→〉fragment of this logic (i.e., the BCKlogic and the POCRIMlogic) are also algebraizable and decidable [5]. This paper contains a summary of the results obtained in [6] about the 〈∨, ∗,¬, 0, 1〉fragment and the 〈∨,∧, ∗,¬, 0, 1〉fragment of the logic of residuated lattices. As regards the algebraic aspects of this study, we introduce the notion of pseudocomplementation with respect to the monoidal operation ∗ (see [3] in the case of ∧). We then define two new classes of algebras: the class of commutative integral
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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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 GödelDummett axioms has a solvable deducibility problem. 1.
Full Nonassociative Lambek Calculus with Distribution: Models and Grammars
"... We study Nonassociative Lambek Calculus with additives ∧,∨, satisfying the distributive law (Full Nonassociative Lambek Calculus with Distribution DFNL). We prove that formal grammars based on DFNL, also with assumptions, generate contextfree languages. The proof uses prooftheoretic tools (interp ..."
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We study Nonassociative Lambek Calculus with additives ∧,∨, satisfying the distributive law (Full Nonassociative Lambek Calculus with Distribution DFNL). We prove that formal grammars based on DFNL, also with assumptions, generate contextfree languages. The proof uses prooftheoretic tools (interpolation) and a construction of a finite model, employed in [13] in the proof of Strong Finite Model Property of DFNL. We obtain analogous results for different variants of DFNL, e.g. BFNL, which admits negation ¬ such that ∧,∨, ¬ satisfy the laws of boolean algebra, and HFNL whose underlying lattice is a Heyting algebra. Our proof also yields Finite Embeddability Property for boolean and Heyting algebras, supplied with an additional residuation structure. 1
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.
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"... local deduction theorem and interpolation for substructural logics over FL. Dedicated to the memory of Willem Johannes Blok Abstract. Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural l ..."
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local deduction theorem and interpolation for substructural logics over FL. Dedicated to the memory of Willem Johannes Blok Abstract. Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.