This study will look at the incidence of gastric and/or duodenal ulcers in patients with osteoarthritis or rheumatoid arthritis in four different treatment groups (a) MK0782 21 mg plus enteric-coated aspirin, (b) MK0782 42 mg plus enteric-coated aspirin, (c) celecoxib 200 mg plus enteric-coated aspirin, and enteric-coated aspiring alone.