Over the last twenty years, in all of these neighbouring fields, modal systems have been developed that we call multi-dimensional. (Our definition of multi ...

In this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic.

In this paper, we introduce a general technology, calledtaming, for finding well-behaved versions of well-investigated logics. Further, we state completeness, decidability, definability and interpolation results for a multimodal logic, calledarrow logic, with additional operators such as thedifference operator, andgraded modalities. Finally, we give a completeness proof for a strong version of arrow logic.

We expand first order models with a tolerance relation on thedomain. Intuitively, two elements stand in this relation if they arecognitively close for the agent who holds the model. This simplenotion turns out to be very powerful. It leads to a semanticcharacterization of the guarded fragment of Andréka, van Benthemand Németi, and highlights the strong analogies between modallogic and this fragment. Viewing the resulting logic – tolerance logic– dynamically it is a resource-conscious information processingalternative to classical first order logic. The (...) differences areindicated by several examples. (shrink)