scholarly journals Connecting Sequent Calculi with Lorenzen-Style Dialogue Games

2021 ◽  
pp. 115-141
Author(s):  
Christian G. Fermüller
Keyword(s):  
Intuitionistic Logic ◽  
Linear Logic ◽  
Substructural Logics ◽  
Point Of View ◽  
Sequent Calculi ◽  
Dialogue Games ◽  
Structured Information ◽  
Logical Connectives ◽  
Pluralistic Approach ◽  
Intuitionistic Linear Logic

Abstract Lorenzen has introduced his dialogical approach to the foundations of logic in the late 1950s to justify intuitionistic logic with respect to first principles about constructive reasoning. In the decades that have passed since, Lorenzen-style dialogue games turned out to be an inspiration for a more pluralistic approach to logical reasoning that covers a wide array of nonclassical logics. In particular, the close connection between (single-sided) sequent calculi and dialogue games is an invitation to look at substructural logics from a dialogical point of view. Focusing on intuitionistic linear logic, we illustrate that intuitions about resource-conscious reasoning are well served by translating sequent calculi into Lorenzen-style dialogue games. We suggest that these dialogue games may be understood as games of information extraction, where a sequent corresponds to the claim that a certain information package can be systematically extracted from a given bundle of such packages of logically structured information. As we will indicate, this opens the field for exploring new logical connectives arising by consideration of further forms of storing and structuring information.

scholarly journals A uniform framework for substructural logics with modalities

10.29007/93qg ◽  
2018 ◽  
Author(s):  
Bjoern Lellmann ◽  
Carlos Olarte ◽  
Elaine Pimentel
Keyword(s):  
General Framework ◽  
Linear Logic ◽  
Local System ◽  
Substructural Logics ◽  
Point Of View ◽  
Theoretical Point ◽  
Logical Framework ◽  
Modal Operators ◽  
Nested Sequents ◽  
Logical Rules

It is well known that context dependent logical rules can be problematic both to implement and reason about. This is one of the factors driving the quest for better behaved, i.e., local, logical systems. In this work we investigate such a local system for linear logic (LL) based on linear nested sequents (LNS). Relying on that system, we propose a general framework for modularly describing systems combining, coherently, substructural behaviors inherited from LL with simply dependent multimodalities. This class of systems includes linear, elementary, affine, bounded and subexponential linear logics and extensions of multiplicative additive linear logic (MALL) with normal modalities, as well as general combinations of them. The resulting LNS systems can be adequately encoded into (plain) linear logic, supporting the idea that LL is, in fact, a “universal framework” for the specification of logical systems. From the theoretical point of view, we give a uniform presentation of LL featuring different axioms for its modal operators. From the practical point of view, our results lead to a generic way of constructing theorem provers for different logics, all of them based on the same grounds. This opens the possibility of using the same logical framework for reasoning about all such logical systems.

scholarly journals The Girard Translation Extended with Recursion

1995 ◽  
Vol 2 (13) ◽  
Author(s):  
Torben Braüner
Keyword(s):  
Intuitionistic Logic ◽  
Linear Logic ◽  
Full Version ◽  
Intuitionistic Linear Logic

<p>This paper extends Curry-Howard interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion. The resulting term languages, the rec-calculus and the linear rec-calculus respectively, are given sound<br />categorical interpretations. The embedding of proofs of IL into proofs of ILL given by the Girard Translation is extended with the rules for recursion, such that an embedding of terms of the rec-calculus into terms of the linear rec-calculus is induced via the extended Curry-Howard isomorphisms. This embedding is shown to be sound with respect to the categorical interpretations.</p><p><br />Full version of paper to appear in Proceedings of CSL '94, LNCS 933, 1995.

Substructural fuzzy logics

2007 ◽  
Vol 72 (3) ◽  
pp. 834-864 ◽  
Author(s):  
George Metcalfe ◽  
Franco Montagna
Keyword(s):  
Linear Logic ◽  
Substructural Logics ◽  
Unit Interval ◽  
Residuated Lattices ◽  
Cut Elimination ◽  
Algebraic Semantics ◽  
Fuzzy Logics ◽  
The Real ◽  
Gentzen Systems ◽  
Intuitionistic Linear Logic

AbstractSubstructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) V ((B → A)∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MIX and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1].

The Logic of Bunched Implications

1999 ◽  
Vol 5 (2) ◽  
pp. 215-244 ◽  
Author(s):  
Peter W. O'Hearn ◽  
David J. Pym
Keyword(s):  
Intuitionistic Logic ◽  
Linear Logic ◽  
Structural Rules ◽  
Intuitionistic Linear Logic

AbstractWe introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers and which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels.

scholarly journals Hybrid linear logic, revisited

2019 ◽  
Vol 29 (8) ◽  
pp. 1151-1176
Author(s):  
KAUSTUV CHAUDHURI ◽  
JOËLLE DESPEYROUX ◽  
CARLOS OLARTE ◽  
ELAINE PIMENTEL
Keyword(s):  
Fixed Points ◽  
Linear Logic ◽  
Transition System ◽  
Point Of View ◽  
Theoretical Point ◽  
Monoidal Structure ◽  
Computational Tree Logic ◽  
Intuitionistic Linear Logic

HyLL (Hybrid Linear Logic) is an extension of intuitionistic linear logic (ILL) that has been used as a framework for specifying systems that exhibit certain modalities. In HyLL, truth judgements are labelled by worlds (having a monoidal structure) and hybrid connectives (at and ↓) relate worlds with formulas. We start this work by showing that HyLL's axioms and rules can be adequately encoded in linear logic (LL), so that one focused step in LL will correspond to a step of derivation in HyLL. This shows that any proof in HyLL can be exactly mimicked by a LL focused derivation. Another extension of LL that has extensively been used for specifying systems with modalities is Subexponential Linear Logic (SELL). In SELL, the LL exponentials (!, ?) are decorated with labels representing locations, and a pre-order on such labels defines the provability relation. We propose an encoding of HyLL into SELL⋒ (SELL plus quantification over locations) that gives better insights about the meaning of worlds in HyLL. More precisely, we identify worlds as locations, and show that a flat subexponential structure is sufficient for representing any world structure in HyLL. This shows that HyLL's monoidal structure is not reflected in LL derivations, hence not increasing the expressiveness of LL, from a proof theoretical point of view. We conclude by proposing the notion of fixed points in multiplicative additive HyLL (μHyMALL), which can be encoded into multiplicative additive linear logic with fixed points (μMALL). As an application, we propose encodings of Computational Tree Logic (CTL) into both μMALL and μHyMALL. In the former, states are represented as atoms in the linear context, hence reflecting a more operational view of CTL connectives. In the latter, worlds represent states of the transition system, thus exhibiting a pleasant similarity with the semantics of CTL.

The undecidability of second order linear logic without exponentials

1996 ◽  
Vol 61 (2) ◽  
pp. 541-548 ◽  
Author(s):  
Yves Lafont
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Linear Logic ◽  
Classical Case ◽  
Second Order ◽  
Order Linear ◽  
Phase Semantics ◽  
Counter Machines ◽  
Intuitionistic Linear Logic

AbstractRecently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicative-additive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicative-additive fragment of second order classical linear logic is also undecidable, using an encoding of two-counter machines originally due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics.

scholarly journals Relations and Non-commutative Linear Logic

1991 ◽  
Vol 20 (372) ◽  
Author(s):  
Carolyn Brown ◽  
Doug Gurr
Keyword(s):  
Intuitionistic Logic ◽  
Arbitrary Number ◽  
Linear Logic ◽  
Expressive Power ◽  
Cut Elimination ◽  
Natural Class ◽  
Structural Rules ◽  
Restricted Sense ◽  
Intuitionistic Linear Logic ◽  
Natural Way

<p>Linear logic differs from intuitionistic logic primarily in the absence of the structural rules of weakening and contraction. Weakening allows us to prove a proposition in the context of irrelevant or unused premises, while contraction allows us to use a premise an arbitrary number of times. Linear logic has been called a ''resource-conscious'' logic, since the premises of a sequent must appear exactly as many times as they are used.</p><p>In this paper, we address this ''experimental nature'' by presenting a non-commutative intuitionistic linear logic with several attractive properties. Our logic discards even the limited commutativityof Yetter's logic in which cyclic permutations of formulae are permitted. It arises in a natural way from the system of intuitionistic linear logic presented by Girard and Lafont, and we prove a cut elimination theorem. In linear logic, the rules of weakening and contraction are recovered in a restricted sense by the introduction of the exponential modality(!). This recaptures the expressive power of intuitionistic logic. In our logic the modality ! recovers the non-commutative analogues of these structural rules. However, the most appealing property of our logic is that it is both sound and complete with respect to interpretation in a natural class of models which we call relational quantales.</p>

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