Results 1  10
of
49
Continuation semantics for the Lambek–Grishin calculus
 INFORMATION AND COMPUTATION
, 2010
"... ..."
Continuation Semantics for Symmetric Categorial Grammar
, 2007
"... Categorial grammars in the tradition of Lambek [1,2] are asymmetric: sequent statements are of the form Γ ⇒ A, where the succedent is a single formula A, the antecedent a structured configuration of formulas A1,..., An. The absence of structural context in the succedent makes the analysis of a num ..."
Abstract

Cited by 19 (6 self)
 Add to MetaCart
Categorial grammars in the tradition of Lambek [1,2] are asymmetric: sequent statements are of the form Γ ⇒ A, where the succedent is a single formula A, the antecedent a structured configuration of formulas A1,..., An. The absence of structural context in the succedent makes the analysis of a number of phenomena in natural language semantics problematic. A case in point is scope construal: the different possibilities to build an interpretation for sentences containing generalized quantifiers and related expressions. In this paper, we explore a symmetric version of categorial grammar based on work by Grishin [3]. In addition to the Lambek product, left and right division, we consider a dual family of typeforming operations: coproduct, left and right difference. Communication between the two families is established by means of structurepreserving distributivity principles. We call the resulting system LG. We present a CurryHoward interpretation for LG(/, \,,) derivations. Our starting point is Curien and Herbelin’s sequent system for λμ calculus [4] which capitalizes on the duality between logical implication (i.e. the Lambek divisions under the formulasastypes perspective) and the difference operation. Importing this system into categorial grammar requires two adaptations: we restrict to the subsystemwhere linearity conditions are in effect, and we refine the interpretation to take the leftright symmetry and absence of associativity/commutativity into account. We discuss the continuationpassingstyle (CPS) translation, comparing the callbyvalue and callbyname evaluation regimes.We showthat in the latter (but not in the former) the types ofLG are associated with appropriate denotational domains to enable a proper treatment of scope construal.
Cutfree Display Calculi for Nominal Tense Logics
 Conference on Tableaux Calculi and Related Methods (TABLEAUX
, 1998
"... . We define cutfree display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use a translation of MNTL into the minimal tense logic of inequality (MTL 6= ) which is known to be properly displayable by application of Krac ..."
Abstract

Cited by 17 (7 self)
 Add to MetaCart
. We define cutfree display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use a translation of MNTL into the minimal tense logic of inequality (MTL 6= ) which is known to be properly displayable by application of Kracht's results. The rules of the display calculus ffiMNTL for MNTL mimic those of the display calculus ffiMTL 6= for MTL 6= . Since ffiMNTL does not satisfy Belnap's condition (C8), we extend Wansing's strong normalisation theorem to get a similar theorem for any extension of ffiMNTL by addition of structural rules satisfying Belnap's conditions (C2)(C7). Finally, we show a weak Sahlqviststyle theorem for extensions of MNTL, and by Kracht's techniques, deduce that these Sahlqvist extensions of ffiMNTL also admit cutfree display calculi. 1 Introduction Background: The addition of names (also called nominals) to modal logics has been investigated recently with different motivations; see...
Cutelimination and proofsearch for biintuitionistic logic using nested sequents
, 2008
"... We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant cal ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cutelimination proof. We then present the derived calculus, and then present a proofsearch strategy which allows it to be used for automated proof search. We prove that this search strategy is terminating and complete by showing how it can be used to mimic derivations obtained from an existing calculus GBiInt for biintuitionistic logic. As far as we know, our new calculus is the first sequent calculus for biintuitionistic logic which uses no semantic additions like labels, which has a purely syntactic cutelimination proof, and which can be used naturally for backwards proofsearch.
Symmetric categorial grammar
 Journal of Philosophical Logic
, 2009
"... is lost or not), is a phenomenon which a linguistic semantics ought to explain, rather than ignore. (van Benthem 1986, p 213) The LambekGrishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of typeforming operations (product and residual left a ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
(Show Context)
is lost or not), is a phenomenon which a linguistic semantics ought to explain, rather than ignore. (van Benthem 1986, p 213) The LambekGrishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of typeforming operations (product and residual left and right division) with a dual family: coproduct, left and right difference. Interaction between these two families is provided by distributivity laws. These distributivity laws have pleasant invariance properties: stability of interpretations for the CurryHoward derivational semantics, and structurepreservation at the syntactic end. The move to symmetry thus offers novel ways of reconciling the demands of natural language form and meaning. 1 1
Analyzing the Core of Categorial Grammar
, 2001
"... Even though residuation is at the core of Categorial Grammar [11], it is not always immediate to realize how standard logic systems like Multimodal Categorial Type Logics (MCTL) [17] actually embody this property. In this paper we focus on the basic system NL [12] and its extension with unary modal ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
(Show Context)
Even though residuation is at the core of Categorial Grammar [11], it is not always immediate to realize how standard logic systems like Multimodal Categorial Type Logics (MCTL) [17] actually embody this property. In this paper we focus on the basic system NL [12] and its extension with unary modalities NL(3) [16], and we spell things out by means of Display Calculi (DC) [3, 10]. The use of structural operators in DC permits a sharp distinction between the core properties we want to impose on the logic system and the way these properties are projected into the logic operators. We will show how we can obtain Lambek residuated triple n, = and of binary operators, and how the operators 3 and 2 introduced by Moortgat in [16] are indeed their unary counterpart.
A local system for intuitionistic logic
 of Lecture Notes in Artificial Intelligence
, 2006
"... Abstract. This paper presents systems for firstorder intuitionistic logic and several of its extensions in which all the propositional rules are local, in the sense that, in applying the rules of the system, one needs only a fixed amount of information about the logical expressions involved. The ma ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
(Show Context)
Abstract. This paper presents systems for firstorder intuitionistic logic and several of its extensions in which all the propositional rules are local, in the sense that, in applying the rules of the system, one needs only a fixed amount of information about the logical expressions involved. The main source of nonlocality is the contraction rules. We show that the contraction rules can be restricted to the atomic ones, provided we employ deepinference, i.e., to allow rules to apply anywhere inside logical expressions. We further show that the use of deep inference allows for modular extensions of intuitionistic logic to Dummett’s intermediate logic LC, Gödel logic and classical logic. We present the systems in the calculus of structures, a proof theoretic formalism which supports deepinference. Cut elimination for these systems are proved indirectly by simulating the cutfree sequent systems, or the hypersequent systems in the cases of Dummett’s LC and Gödel logic, in the cut free systems in the calculus of structures.
Dual intuitionistic logic revisited
 Automated Reasoning with Analytic Tableaux and Related Methods, St
, 2000
"... Abstract. We unify the algebraic, relational and sequent methods used by various authors to investigate “dual intuitionistic logic”. We show that restricting sequents to “singletons on the left/right ” cannot capture “intuitionistic logic with dual operators”, the natural hybrid logic that arises fr ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We unify the algebraic, relational and sequent methods used by various authors to investigate “dual intuitionistic logic”. We show that restricting sequents to “singletons on the left/right ” cannot capture “intuitionistic logic with dual operators”, the natural hybrid logic that arises from intuitionistic and dualintuitionistic logic. We show that a previously reported generalised display framework does deliver the required cutfree display calculus. We also pinpoint precisely the structural rule necessary to turn this display calculus into one for classical logic. 1
Classical BI (A Logic for Reasoning about Dualising Resources)
"... We show how to extend O’Hearn and Pym’s logic of bunched implications, BI, to classical BI (CBI), in which both the additive and the multiplicative connectives behave classically. Specifically, CBI is a nonconservative extension of (propositional) Boolean BI that includes multiplicative versions of ..."
Abstract

Cited by 12 (6 self)
 Add to MetaCart
(Show Context)
We show how to extend O’Hearn and Pym’s logic of bunched implications, BI, to classical BI (CBI), in which both the additive and the multiplicative connectives behave classically. Specifically, CBI is a nonconservative extension of (propositional) Boolean BI that includes multiplicative versions of falsity, negation and disjunction. We give an algebraic semantics for CBI that leads us naturally to consider resource models of CBI in which every resource has a unique dual. We then give a cuteliminating proof system for CBI, based on Belnap’s display logic, and demonstrate soundness and completeness of this proof system with respect to our semantics.
Embedding Display Calculi into Logical Frameworks: Comparing Twelf and Isabelle
 ENTCS
, 2001
"... Logical frameworks are computer systems which allow a user to formalise mathematics using specially designed languages based upon mathematical logic and Church's theory of types. They can be used to derive programs from logical specifications, thereby guaranteeing the correctness of the resulti ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
(Show Context)
Logical frameworks are computer systems which allow a user to formalise mathematics using specially designed languages based upon mathematical logic and Church's theory of types. They can be used to derive programs from logical specifications, thereby guaranteeing the correctness of the resulting programs. They can also be used to formalise rigorous proofs about logical systems. We compare several methods of implementing the display (sequent) calculus #RA for relation algebra in the logical frameworks Isabelle and Twelf. We aim for an implementation enabling us to formalise, within the logical framework, prooftheoretic results such as the cutelimination theorem for #RA and any associated increase in proof length. We discuss issues arising from this requirement. Key words: logical frameworks, higherorder logics, proof systems for relation algebra, nonclassical logics, automated deduction, display logic. 1 Supported by an Australian Research Council Small Research Grant. 2 Sup...