Abstract: Arrow logic is a widely applicable system. Many different
notions from various disciplines like mathematics, economics, computer
science, linguistics and cognitive science can be formulated in it.
Arrow logic is designed to talk about all such objects which may be
represented in a picture by arrows. Relation algebras have played a
role in the development of arrow logic, in fact relation algebras
emerge as the modal algebras of arrow logic.
Relativization in algebraic logic started, by Leon Henkin, as a
technique for generalizing representations of algebras of logics,
while also, in some cases, ‘defusing’ undesirable properties, like
undecidability. Natural question arises here, does relativization
defuse Godel incompleteness property?
Here, we study the question: Do those decidable relativized fragments
of arrow logic get rid of (weak) Godel incompleteness property?
If you are familiar with relativizations, you would say: probably YES,
relativization is defined some times as a way of turning all the
negative properties to positive ones!
If you are familiar with universal algebras, you would say: we DON'T
KNOW, indeed the classes of algebras (the relativized ones)
corresponding to those fragments don't satisfy all the known
sufficient conditions which are due to Andreka and Nemeti (inspired
We consider the pure algebraic version of this question, and then we
give an answer for the algebraic questions using mosaics. Mosaics were
introduced by Nemeti to study decidability via infinite structures,
unlike most of the known ways to show decidability. We note that some
part of this question was posed in the most recent book in algebraic
logic [Cylindric like algebras and algebraic logic, Eds: Andreka,
Ferenczi and Nemeti, 2012].
You can watch the lecture here: