Results 1  10
of
17
A Classical Linear λcalculus
, 1997
"... This paper proposes and studies a typed λcalculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natu ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
This paper proposes and studies a typed λcalculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natural deduction formulation of classical linear logic. This formulation is compared in detail to the sequent calculus formulation. In an appendix I shall also demonstrate a somewhat hidden connexion with the paradigm of control operators for functional languages which gives a new computational interpretation of Parigot's techniques.
Suszko's Thesis and dyadic semantics
 Department of Mathematics, Instituto Superior Técnico
"... A wellknown result by W\'ojcickiLindenbaum shows that any tarskian logic is manyvalued, and another result by Suszko shows how to provide 2valued semantics to these very same logics. This paper investigates the question of obtaining 2valued semantics for manyvalued logics, including paraconsis ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
A wellknown result by W\'ojcickiLindenbaum shows that any tarskian logic is manyvalued, and another result by Suszko shows how to provide 2valued semantics to these very same logics. This paper investigates the question of obtaining 2valued semantics for manyvalued logics, including paraconsistent logics, in the lines of the socalled ``Suszko's Thesis". We set up the bases for developing a general algorithmic method to transform any truthfunctional finitevalued semantics satisfying reasonable conditions into a computable quasi tabular 2valued semantics, that we call dyadic. We also discuss how ``Suszko's Thesis" relates to such a method, in the light of truthfunctionality, while at the same time we reject an endorsement of Suszko's philosophical views about the misconception of manyvalued logics.
On negation: Pure local rules
, 2003
"... This is an initial systematic study of the properties of negation from the point of view of abstract deductive systems. A unifying framework of multipleconclusion consequence relations is adopted so as to allow us to explore symmetry in exposing and matching a great number of positive contextua ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
This is an initial systematic study of the properties of negation from the point of view of abstract deductive systems. A unifying framework of multipleconclusion consequence relations is adopted so as to allow us to explore symmetry in exposing and matching a great number of positive contextual subclassical rules involving this logical constant among others, wellknown forms of proof by cases, consequentia mirabilis and reductio ad absurdum. Finer definitions of paraconsistency and the dual paracompleteness can thus be formulated, allowing for pseudoscotus and ex contradictione to be di#erentiated and for a comprehensive version of the Principle of NonTriviality to be presented. A final proposal is made to the e#ect that pure positive rules involving negation being often fallible a characterization of what most negations in the literature have in common should rather involve, in fact, a reduced set of negative rules.
Ineffable Inconsistencies
 Paraconsistency with no Frontiers
"... For any given consistent tarskian logic it is possible to find another nontrivial logic that allows for an inconsistent model yet completely coincides with the initial given logic from the point of view of their associated singleconclusion consequence relations. A paradox? This short note... ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
For any given consistent tarskian logic it is possible to find another nontrivial logic that allows for an inconsistent model yet completely coincides with the initial given logic from the point of view of their associated singleconclusion consequence relations. A paradox? This short note...
On Theorems Equivalent with Kotzig's Result on Graphs with Unique 1Factors
, 2001
"... We show that several known theorems on graphs and digraphs are equivalent. The list of equivalent theorems include Kotzig's result on graphs with unique 1factors, a lemma by Seymour and Giles, theorems on alternating cycles in edgecolored graphs, and a theorem on semicycles in digraphs. We co ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We show that several known theorems on graphs and digraphs are equivalent. The list of equivalent theorems include Kotzig's result on graphs with unique 1factors, a lemma by Seymour and Giles, theorems on alternating cycles in edgecolored graphs, and a theorem on semicycles in digraphs. We consider computational problems related to the quoted results; all these problems ask whether a given (di)graph contains a cycle satisfying certain properties which runs through p prescribed vertices. We show that all considered problems can be solved in polynomial time for p < 2 but are NPcomplete for p # 2. 1
Possibletranslations semantics (extended abstract
 CombLog’04 — Proceedings of the Workshop on Combination of Logics: theory and applications, held in Lisbon, PT
, 2004
"... This text aims at providing a bird’s eye view of possibletranslations semantics ([10, 24]), defined, developed and illustrated as a very comprehensive formalism for obtaining or for representing semantics for all sorts of logics. With that tool, a wide class of complex logics will very naturally tu ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
This text aims at providing a bird’s eye view of possibletranslations semantics ([10, 24]), defined, developed and illustrated as a very comprehensive formalism for obtaining or for representing semantics for all sorts of logics. With that tool, a wide class of complex logics will very naturally turn out to be (de)composable by way of some suitable combination of simpler logics. Several examples will be mentioned, and some related special cases of possibletranslations semantics, among which are society semantics and nondeterministic semantics, will also be surveyed. 1 Logics, translations, possibletranslations Let a logic L be a structure of the form 〈S, �〉, where S denotes its language (its set of formulas) and � ⊆ Pow(S)×Pow(S) represents its associated consequence relation (cr), somehow defined so as to embed some formal model of reasoning. Call any subset of S a theory. As usual, capital Greek letters will denote theories, and lowercase Greek will denote formulas; a sequence such as Γ, α, Γ ′ � ∆ ′ , β, ∆ should be read as asserting that Γ ∪ {α} ∪ Γ ′ � ∆ ′ ∪ {β} ∪ ∆. Morphisms between any two of the above structures will be called translations. So, given any two logics, L1=〈S1, �1 〉 and L2=〈S2, �2〉, a mapping t: S1 → S2 will constitute a translation from L1 into L2 just in case the following holds:
Towards a Classical Linear λcalculus
 PROC. OF THE TOKYO CONFERENCE ON LINEAR LOGIC
, 1996
"... This paper considers a typed calculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natural deduction formu ..."
Abstract
 Add to MetaCart
This paper considers a typed calculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natural deduction formulation of classical linear logic. I shall also demonstrate a somewhat hidden connexion with the continuationpassing paradigm which gives a new computational interpretation of Parigot's techniques and possibly a new style of continuation programming.
Tools and Techniques for Formalising Structural Proof Theory
"... Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are all cases of a proof completed in the l ..."
Abstract
 Add to MetaCart
Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are all cases of a proof completed in the literature. Two intermediate results can be used to drastically reduce the amount of effort needed in proofs of Cut admissibility: Weakening and Invertibility. Indeed, whereas there are proofs of Cut admissibility which do not use Invertibility, Weakening is almost always necessary. Use of these results simply shifts the bureaucracy, however; Weakening and Invertibility, whilst more easy to prove, are still not trivial. We give a framework under which sequent calculi can be codified and analysed, which then allows us to prove various results: for a calculus to admit Weakening and for a rule to be invertible in a calculus. For the latter, even though many calculi are investigated, the general condition is simple and easily verified. The results have been applied to G3ip, G3cp, G3c, G3s, G3LC and G4ip. Invertibility is important in another respect; that of proofsearch. Should all rules in a calculus be invertible, then terminating rootfirst proof search gives a decision procedure
Bases of admissible rules of Łukasiewicz logic
, 2009
"... We construct explicit bases of singleconclusion and multipleconclusion admissible rules of propositional Łukasiewicz logic, and we prove that every formula has an admissibly saturated approximation. We also show that Łukasiewicz logic has no finite basis of admissible rules. ..."
Abstract
 Add to MetaCart
We construct explicit bases of singleconclusion and multipleconclusion admissible rules of propositional Łukasiewicz logic, and we prove that every formula has an admissibly saturated approximation. We also show that Łukasiewicz logic has no finite basis of admissible rules.
Multivalued Semantics: Why and How
, 2008
"... According to Suszko’s Thesis, any multivalued semantics for a logical system can be replaced by an equivalent bivalent one. Moreover: bivalent semantics for families of logics can frequently be developed in a modular way. On the other hand bivalent semantics usually lacks the crucial property of an ..."
Abstract
 Add to MetaCart
According to Suszko’s Thesis, any multivalued semantics for a logical system can be replaced by an equivalent bivalent one. Moreover: bivalent semantics for families of logics can frequently be developed in a modular way. On the other hand bivalent semantics usually lacks the crucial property of analycity, a property which is guaranteed for the semantics of multivalued matrices. We show that one can get both modularity and analycity by using the semantic framework of multivalued nondeterministic matrices. We further show that for using this framework in a constructive way it is best to view “truthvalues” as information carriers, or “informationvalues”.