This paper proposes and studies a typed λ-calculus for classical linear logic. I shall give an explanation of a multiple-conclusion 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.

This is an initial systematic study of the properties of negation from the point of view of abstract deductive systems. A unifying framework of multiple-conclusion consequence relations is adopted so as to allow us to explore symmetry in exposing and matching a great number of positive contextual sub-classical rules involving this logical constant ---among others, well-known 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 pseudo-scotus and ex contradictione to be di#erentiated and for a comprehensive version of the Principle of Non-Triviality 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.

A well-known result by W\'ojcicki-Lindenbaum shows that any tarskian logic is many-valued, and another result by Suszko shows how to provide 2-valued semantics to these very same logics. This paper investigates the question of obtaining 2-valued semantics for many-valued logics, including paraconsistent logics, in the lines of the so-called ``Suszko's Thesis". We set up the bases for developing a general algorithmic method to transform any truth-functional finite-valued semantics satisfying reasonable conditions into a computable quasi tabular 2-valued semantics, that we call dyadic. We also discuss how ``Suszko's Thesis" relates to such a method, in the light of truth-functionality, while at the same time we reject an endorsement of Suszko's philosophical views about the misconception of many-valued logics.

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 single-conclusion consequence relations. A paradox? This short note...

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 1-factors, a lemma by Seymour and Giles, theorems on alternating cycles in edge-colored 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 NP-complete for p # 2. 1

This text aims at providing a bird’s eye view of possible-translations 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 possible-translations semantics, among which are society semantics and non-deterministic semantics, will also be surveyed. 1 Logics, translations, possible-translations 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:

This paper considers a typed -calculus for classical linear logic. I shall give an explanation of a multiple-conclusion 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 continuation-passing paradigm which gives a new computational interpretation of Parigot's techniques and possibly a new style of continuation programming.

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, G3-LC and G4ip. Invertibility is important in another respect; that of proof-search. Should all rules in a calculus be invertible, then terminating root-first proof search gives a decision procedure

We construct explicit bases of single-conclusion and multiple-conclusion 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.

According to Suszko’s Thesis, any multi-valued 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 multi-valued matrices. We show that one can get both modularity and analycity by using the semantic framework of multi-valued non-deterministic matrices. We further show that for using this framework in a constructive way it is best to view “truth-values” as information carriers, or “information-values”.