20 research outputs found
Partially-Ordered Multi-Type Algebras, Display Calculi and the Category of Weakening Relations
We define partially-ordered multi-type algebras and use them as algebraic semantics for multi-type display calculi that have recently been developed for several logics, including dynamic epistemic logic [7], linear logic[10], lattice logic [11], bilattice logic [9] and semi-De Morgan logic [8]
Vector spaces as Kripke frames
In recent years, the compositional distributional approach in computational
linguistics has opened the way for an integration of the \emph{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 \emph{general} Lambek calculus, based on \emph{algebras over
a field} (or -algebras), i.e. vector spaces endowed
with a bilinear binary product. Such structures are well known in algebraic
geometry and algebraic topology, since they are important instances of Lie
algebras and Hopf algebras. Applying results and insights from duality and
representation theory for the algebraic semantics of nonclassical logics, we
regard -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
Semi De Morgan Logic Properly Displayed
In the present paper, we endow semi De Morgan logic and a family of its axiomatic extensions with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis of the variety of semi De Morgan algebras, and applies the guidelines of the multi-type methodology in the design of display calculi
d-Frames as algebraic duals of bitopological spaces
Achim Jung and Drew Moshier developed a Stone-type duality theory for bitopological spaces, amongst others, as a practical tool for solving a particular problem in the theory of stably compact spaces. By doing so they discovered that the duality of bitopological spaces and their algebraic counterparts, called d-frames, covers several of the known dualities.
In this thesis we aim to take Jung's and Moshier's work as a starting point and fill in some of the missing aspects of the theory. In particular, we investigate basic categorical properties of d-frames, we give a Vietoris construction for d-frames which generalises the corresponding known Vietoris constructions for other categories, and we investigate the connection between bispaces and a paraconsistent logic and then develop a suitable (geometric) logic for d-frames
Zero-one laws with respect to models of provability logic and two Grzegorczyk logics
It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems
Probabilistic epistemic updates on algebras
The present article contributes to the development of the mathematical theory of epistemic updates using the tools of duality theory. Here, we focus on Probabilistic Dynamic Epistemic Logic (PDEL). We dually characterize the product update construction of PDEL-models as a certain construction transforming the complex algebras associated with the given model into the complex algebra associated with the updated model. Thanks to this construction, an interpretation of the language of PDEL can be defined on algebraic models based on Heyting algebras. This justifies our proposal for the axiomatization of the intuitionistic counterpart of PDEL