Joachim Lambek : the interplay of mathematics, logic, and linguistics / / Claudia Casadio and Philip J. Scott (editors)

• Pubblicazione: Cham, Switzerland : , : Springer, , [2021]; ©2021
• Descrizione fisica: 1 online resource (453 pages) : illustrations
• Disciplina: 511.3
• Soggetto topico: Mathematical linguistics; Logic, Symbolic and mathematical
• Persona (resp. second.): ScottPhilip J.; CasadioC (Claudia)
• Nota di contenuto: Intro -- Preface -- Contents -- List of Contributors -- Introduction -- I. A brief biography -- II. Academic Work -- III. The papers in this volume -- References -- Lambek's Syntactic Calculus and Noncommutative Variants of Linear Logic: Laws and Proof-Nets -- 1 Formulations of Lambek's Syntactic Calculus in the framework of noncommutative variants of Linear Logic -- 1.1 Formulation of LC in the multiplicative fragment of Noncommutative Intuitionistic Linear Logic -- 1.2 Formulation of LC in the multiplicative fragments of Noncommutative Classical Linear Logic and Cyclic Linear Logic -- 2 Proof-Nets for Lambek's Syntactic Calculus -- 2.1 Cyclic multiplicative proof-nets -- 2.2 Proof-nets for LC -- 3 Geometrical formulation of laws of LC through Proof-Nets for LC -- 3.1 Geometrical representation of Monotonicity laws -- 3.2 Geometrical Representation of Application Laws, Expansion Laws and Type-raising Laws -- 3.3 Geometrical representation of Composition Laws, Geach Laws and Switching Laws -- 4 New laws emerged from proof-nets for LC and their linguistic applications -- 4.1 Laws of Composition -- 4.2 Laws related to Geach rules -- 4.3 Swiching laws -- 5 Conclusions -- References -- Sheaf Representations and Duality in Logic -- Preface -- 1 Gelfand duality -- 2 Grothendieck duality for commutative rings -- 3 Lambek-Moerdijk sheaf representation for toposes -- 3.1 Lambek's modified sheaf representation for toposes -- 4 Local sheaf representation for toposes -- 5 Stone duality for Boolean algebras -- 6 Stone duality for Boolean pretoposes -- 7 Sheaf representation for pretoposes -- 8 Logical schemes -- References -- On the naturalness of Mal'tsev categories -- Introduction -- Acknowledgements -- 1 Mal'tsev operations -- 2 Mal'tsev categories -- 2.1 Definition and examples -- 2.2 Yoneda embedding for internal structures -- 3 Characterizations. ; 3.1 Unital characterization -- 3.2 Centralization of equivalence relations -- 3.3 Groupoid characterization -- 3.4 Base-change characterization -- 4 Stiffly and naturally Mal'tsev categories -- 5 Regular Mal'tsev categories -- 6 Regular Mal'tsev categories and the calculus of relations -- 7 Baer sums in Mal'tsev categories -- References -- Extensions of Lambek Calculi -- 1 Introduction -- 2 Residuation -- 3 Sequent systems for L and NL -- 4 Substructural logics -- 5 Linear logics -- 6 Final comments -- References -- Categories with Families: Unityped, Simply Typed, and Dependently Typed -- 1 Introduction -- 2 Dependent type theory and categories with families -- 2.1 Martin-L¨of type theory -- 2.2 Categories with families -- 3 Unityped cwfs -- 3.1 Plain ucwfs -- 3.2 Contextual ucwfs -- 3.3 - -ucwfs -- 4 Simply-typed cwfs -- 4.1 Plain scwfs -- 4.2 Finite products as structure -- 4.3 Finite products as property -- 4.4 Adding function types -- 5 Dependently typed categories with families -- 5.1 Plain cwfs -- 5.2 Extensional identity types, -types, and finite limits -- 5.3 -types and locally cartesian closed categories -- References -- The Mathematics of Text Structure -- 1 Introduction -- 2 Background: DisCoCat -- 2.1 diagrams -- 2.2 From grammar to wirings -- 2.3 Internal wirings of meanings -- 2.4 Models of meaning -- 2.5 Comparing meanings -- 3 Features and flaws of DisCoCat -- 4 Composing sentences: meet DisCoCirc -- 4.1 Naive composition of sentences for DisCoCat -- 4.2 Sentences as I/O-processes -- 4.3 The use of states -- 4.4 DisCoCat from DisCoCirc -- 4.5 Individual and subgroup meanings -- 4.6 Example -- 4.7 Other cognitive modes -- 5 Logic and language -- 5.1 Dynamic epistemic logic from language -- 5.2 Linear and non-linear -- 6 Concrete models -- 6.1 Sketch of a concrete model -- 6.2 Computing text meaning -- 6.3 Comparing texts. ; 7 Physical embodiment -- References -- Aspects of Categorical Recursion Theory -- 1 Introduction -- 1.1 On Lambek's Questions -- 1.2 General Notation and Background -- 2 What is a computable function? -- 2.1 The primitive recursive functions -- 2.2 The computable functions -- 2.3 Some Newer Models of Computability -- 3 Lambek's Categorical Proof Theory -- 3.1 A brief history -- 3.2 Internal Languages and free categories -- 3.3 CCCs and the Curry-Howard-Lambek correspondence -- 3.4 Elementary toposes and HAH -- 4 What are computable functions in categories? -- 4.1 Natural Numbers Objects and Prim -- 4.2 Representability -- 4.3 Going beyond the primitive recursive functions: free CCCs -- 4.4 Some properties of the free topos -- 4.5 C-monoids and Untyped Lambda Calculi -- 4.6 Plotkin's characterization of Kleene's μ-recursion -- 5 Abstract Computability -- 5.1 Categories of Partial Maps -- 5.2 Turing Categories -- 5.3 Computable maps and PCAs -- 6 Realizability -- 6.1 Kleene Realizability -- 6.2 Realizability Toposes -- 6.3 PCAs and Toposes -- 7 Other Directions -- 7.1 Traced Categories -- 7.2 Typed PCAs -- 7.3 Computation at higher types -- 7.4 Higher-order computation in toposes -- 7.5 Complexity Theory -- Conclusion -- References -- Morphisms of Rings -- Introduction -- 1 Double categories -- 2 Companions and conjoints -- 3 Matrix-valued homomorphisms -- 4 The graded double category of rings -- 5 Adjoint bimodules -- References -- Pomset Logic The other approach to noncommutativity in logic -- 1 Presentation -- 2 A glimpse of pomset logic -- 3 Structured sequents as dicographs of formulas -- 3.1 Looking for structured sequents -- 3.2 Directed cographs or dicographs -- 3.3 Dicograph inclusion and (un)folding -- 3.4 Folding and unfolding pomset logic sequents -- 3.5 A sequent calculus attempt with sp pomset of formulas -- 4 Proof nets. ; 4.1 Handsome pomset proof nets -- 4.2 Cut and cut-elimination -- 4.3 From sequent calculus and rewrite proofs to dicog-PN -- 5 Denotational semantics of pomset logic within coherence spaces -- 5.1 Coherence spaces -- 5.2 A sound and faithful interpretation of proof nets in coherence spaces -- 6 Sequentialisation with pomset sequents or dicographs sequents -- 7 Pomset logic in deep inference style -- 7.1 Standard multiplicative linear logic as cograph rewriting -- 7.2 Deep pomset is SBV -- 7.3 Cut-elimination in Deep Pomset and in SBV -- 8 Grammatical use -- 8.1 Proof nets with links -- 8.2 Grammars with partial proof nets -- 9 Conclusion and perspective -- References -- Pregroup Grammars, Their Syntax and Semantics -- 1 Introduction -- 2 Pregroup Grammars -- 2.1 Mathematical Definition -- 2.2 Linguistic Applications -- 3 Set Theoretic and Vector Space Semantics -- 3.1 Ambiguous Set Theoretic Semantics -- 3.2 Vector Space Semantics -- 3.3 Direct Product of Vector Spaces -- 3.4 Two Attempts -- 4 Data-Driven Vector Space Semantics -- 5 What is Truth? -- 6 Conclusion -- References -- The Sequent Calculus of Skew Monoidal Categories -- 1 Introduction -- 2 Skew Monoidal Categories -- 3 A Skew Monoidal Sequent Calculus -- 4 An Equational Theory on Cut-Free Derivations -- 5 A Focused Subsystem of Canonical Derivations -- 6 Comparison with Bourke and Lack -- 7 Conclusion and Future Work -- References -- Appendix A Obituary and two expository articles -- A.1 Obituary (and a short biography) of J. Lambek -- Two Expository Articles of J. Lambek -- A.2 Quaternions in Physics -- A.3 Pregroup Grammars.
• Titolo autorizzato: Joachim Lambek
• ISBN: 3-030-66545-3
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910484293003321
• Lo trovi qui: Univ. Federico II
• Serie: Outstanding Contributions to Logic