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42
PartialGaggles Applied to Logics with Restricted Structural Rules
 In Peter SchroederHeister and Kosta Dosen, editors, Substructural Logics
, 1991
"... Law of Residuation (in their jth place) when f and g are contrapositives (with respect to their jth place) and S(f; a 1 ; : : : ; a j ; : : : ; a n ; b) iff S(g; a 1 ; : : : ; b; : : : ; a n ; a j ). (5) Two operators f , g 2 OP are relatives when they satisfy the Abstract Law of Residuation in ..."
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Cited by 40 (1 self)
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Law of Residuation (in their jth place) when f and g are contrapositives (with respect to their jth place) and S(f; a 1 ; : : : ; a j ; : : : ; a n ; b) iff S(g; a 1 ; : : : ; b; : : : ; a n ; a j ). (5) Two operators f , g 2 OP are relatives when they satisfy the Abstract Law of Residuation in some position. (6) The family of operations OP is founded when there is a distinguished operator f 2 OP (the head) such that any other operator g 2 OP is a relative of f . Definition. A partialgaggle is a tonoid T = (X; ; OP), in which OP is a founded family. As examples, consider a p.o. residuated groupoid, with OP chosen to be any of the following families of operations (ffi is the head of the families of which it is a member): fffig, fffi; /g, fffi; !g, fffi; /;!g, f/g, f!g. Note that f!;/g does not formally fall under our definition since the trace of one is not directly the contrapositive of the trace of the other, even though the trace of each is a contrapositive of the trace of f...
Substructural Logics on Display
, 1998
"... Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculu ..."
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Cited by 38 (16 self)
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Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponentialfree linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a "cyclic" counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic BiLambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bilinear, Birelevant, BiBCK and Biintuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and som...
First Order Linear Logic in Symmetric Monoidal Closed Categories
, 1991
"... There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encodin ..."
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Cited by 11 (0 self)
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There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encoding of logics, and the second a uniform treatment of their model theory. This thesis forms a case study for the work on model theory. The models of many first and higher order logics can be represented as fibred or indexed categories with certain extra structure, and this has been suggested as a general paradigm. The aim of the thesis is to test the strength and flexibility of this paradigm by studying the specific case of Girard's linear logic. It should be noted that the exact form of this logic in the first order case is not entirely certain, and the system treated here is significantly different to that considered by Girard.
Automated Generation of Analytic Calculi for Logics with Linearity
 Proceedings of CSL’04, vol. 3210 LNCS
, 2004
"... Abstract. We show how to automatically generate analytic hypersequent calculi for a large class of logics containing the linearity axiom (lin) (A ⊃ B) ∨ (B ⊃ A) starting from existing (singleconclusion) cutfree sequent calculi for the corresponding logics without (lin). As a corollary, we define ..."
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Cited by 8 (4 self)
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Abstract. We show how to automatically generate analytic hypersequent calculi for a large class of logics containing the linearity axiom (lin) (A ⊃ B) ∨ (B ⊃ A) starting from existing (singleconclusion) cutfree sequent calculi for the corresponding logics without (lin). As a corollary, we define an analytic calculus for Strict Monoidal Tnorm based Logic SMTL. 1
Duality for LatticeOrdered Algebras and for Normal Algebraizable Logics
 Studia Logica
, 1997
"... Part I of this paper is developed in the tradition of Stonetype dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compactopens of their dual St ..."
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Cited by 8 (4 self)
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Part I of this paper is developed in the tradition of Stonetype dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compactopens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality. In part II, we consider latticeordered algebras (lattices with additional operators) , extending the J'onsson and Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the J'onssonTarski additive operators. Representation of `algebras is extended to full duality. In part III we discuss applications in logic...
Tnorm based logics with ncontraction
 Neural Network World
"... Abstract: We consider two families of fuzzy propositional logics obtained by extending MTL and IMTL with the ncontraction axiom, for n ≥ 2. These logics – called CnMTL and CnIMTL – range from Gödel and classical logic (when n = 2) to MTL and IMTL (when n tends to infinity), respectively. We inves ..."
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Cited by 6 (5 self)
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Abstract: We consider two families of fuzzy propositional logics obtained by extending MTL and IMTL with the ncontraction axiom, for n ≥ 2. These logics – called CnMTL and CnIMTL – range from Gödel and classical logic (when n = 2) to MTL and IMTL (when n tends to infinity), respectively. We investigate tnorm based semantics and proof theory for CnMTL and CnIMTL. We show standard completeness and suitable analytic hypersequent calculi for them. 1.
On Urquhart's C Logic
 In International Symposium on Multiple Valued Logic (ISMVL’2000
, 2000
"... In this paper we investigate the basic manyvalued logics introduced by Urquhart in [15] and [16], here referred to as C and Cnew , respectively. We define a cutfree hypersequent calculus for Cnew and show the following results: (1) C and Cnew are distinct versions of G odel logic without contracti ..."
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Cited by 5 (2 self)
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In this paper we investigate the basic manyvalued logics introduced by Urquhart in [15] and [16], here referred to as C and Cnew , respectively. We define a cutfree hypersequent calculus for Cnew and show the following results: (1) C and Cnew are distinct versions of G odel logic without contraction. (2) Cnew is decidable. (3) In Cnew the family of axioms ((A k ! C) ^ (B k ! C)) ! ((A _ B) k ! C), with k 2, is in fact redundant. 1 Introduction The logic C was introduced by Urquhart in the chapter devoted to manyvalued logic of the Handbook of Philosophical Logic [15]. C turns out to be a basic manyvalued logic being contained in the most important formalizations of fuzzy logic [7], namely infinitevalued Godel, L/ukasiewicz and product logic (see [3]). In [9, 10] C was shown to be a particular Godel logic without contraction. A cutfree calculus for C was defined in [3]. This calculus uses hypersequents that are a natural generalization of Gentzen sequents. Due to sema...
Completions of Algebras and Completeness of Modal and Substructural Logics
 of Advances in Modal Logic
, 2003
"... this paper. So, we start our paper by giving a brief account of the MacNeille completion of partially ordered sets and of Boolean algebras, which is de ned here by using cuts (see x 35 Example F in [33]) ..."
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Cited by 5 (1 self)
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this paper. So, we start our paper by giving a brief account of the MacNeille completion of partially ordered sets and of Boolean algebras, which is de ned here by using cuts (see x 35 Example F in [33])
On the Structure of Hoops
 Algebra Universalis
, 1998
"... A hoop is a naturally ordered pocrim (i.e., a partially ordered commutative residuated integral monoid). We list some basic properties of hoops, describe in detail the structure of subdirectly irreducible hoops, and establish that the class of hoops, which is a variety, is generated, as a quasiv ..."
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Cited by 4 (0 self)
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A hoop is a naturally ordered pocrim (i.e., a partially ordered commutative residuated integral monoid). We list some basic properties of hoops, describe in detail the structure of subdirectly irreducible hoops, and establish that the class of hoops, which is a variety, is generated, as a quasivariety, by its finite members. Introduction Residuated structures arise in many areas of mathematics, and are particularly common among algebras associated with logical systems. The essential ingredients are a partial order , a binary operation of say multiplication \Delta that respects the partial order, and a binary (left) residuation operation ! characterized by c \Delta a b if and only if c a ! b. In the logical context these represent a partial ordering of an algebra of truth values, (intensional) conjunction and implication, respectively. If the partial order is a semilattice order, and the multiplication the semilattice operation, we obtain the Brouwerian semilattices  the mo...