There are a variety of blood tests that will aid in the differential diagnosis of the particular type of arthritis, but no tests that will specifically identify the type.
The ESR ( Erythrocyte Sedimentation Rate) is a test performed to determine whether there is inflammation present, as will C-Reactive Protein. They may also test for Rheumatoid Factor and Anti Nuclear Antibodies, becuase there are a whole suite of disease states centred around bodies immune reponse.
@ Zeuhl re blood tests for cancer. Again, no specific tests that are exclusively diagnostic for specific types of cancer, but there are markers that are indicative, and, when combined with diagnostic imaging methods are very useful to determine the type of cancer.
If you class leukaemias as cancers, then a blood test can be exclusively diagnostic, with the data drawn from the cell counts and examination of the cell morphology.
In short, it can be quite complicated :)