Abstract
In recent years, the compositional distributional approach in computational linguistics has opened the way for an integration of the lexical aspects of meaning into Lambek’s type-logical grammar program. This approach is based on the observation that a sound semantics for the associative, commutative and unital Lambek calculus can be based on vector spaces by interpreting fusion as the tensor product of vector spaces. In this paper, we build on this observation and extend it to a ‘vector space semantics’ for the general Lambek calculus, based on algebras over a field K (or K-algebras), i.e. vector spaces endowed with a bilinear binary product. Such structures are well known in algebraic geometry and algebraic topology, since Lie algebras and Hopf algebras are important instances of K-algebras. Applying results and insights from duality and representation theory for the algebraic semantics of nonclassical logics, we regard K-algebras as ‘Kripke frames’ the complex algebras of which are complete residuated lattices. This perspective makes it possible to establish a systematic connection between vector space semantics and the standard Routley-Meyer semantics of (modal) substructural logics.
| Original language | English |
|---|---|
| Pages (from-to) | 853-873 |
| Number of pages | 21 |
| Journal | Journal of applied logistics |
| Volume | 7 |
| Issue number | 5 |
| Early online date | 14 Sept 2020 |
| Publication status | Published - Sept 2020 |
Bibliographical note
Funding Information:∗The research of the first and third author is supported by a NWO grant under the scope of the project “A composition calculus for vector-based semantic modelling with a localization for Dutch” (360-89-070).
Funding Information:
†The research of the second author is supported by the Young Scholars Program of Shandong University (11090089964225).
Funding Information:
‡The research of the fourth and fifth author is supported by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054, and a Delft Technology Fellowship awarded to the fourth author in 2013.
Publisher Copyright:
© 2020, College Publications. All rights reserved.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.
Funding
∗The research of the first and third author is supported by a NWO grant under the scope of the project “A composition calculus for vector-based semantic modelling with a localization for Dutch” (360-89-070). †The research of the second author is supported by the Young Scholars Program of Shandong University (11090089964225). ‡The research of the fourth and fifth author is supported by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054, and a Delft Technology Fellowship awarded to the fourth author in 2013.