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Giuseppe Greco

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I&C Journal 2022 Journal Article

Non-normal modal logics and conditional logics: Semantic analysis and proof theory

  • Jinsheng Chen
  • Giuseppe Greco
  • Alessandra Palmigiano
  • Apostolos Tzimoulis

We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal applies the multi-type methodology in the design of proper display calculi, starting from a semantic analysis which motivates syntactic translations from single-type non-normal modal logics to multi-type normal poly-modal logics.

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FLAP Journal 2020 Journal Article

Vector Spaces as Kripke Frames.

  • Giuseppe Greco
  • Fei Liang
  • Michael Moortgat
  • Alessandra Palmigiano
  • Apostolos Tzimoulis

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.

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