Hi Tom. There are many factors that can go into this decision, including the time of year you visit and your family's interests. I will start by assuming you want to 1) visit all four theme parks, 2) would attend during reasonable time of year for crowds and 3) you are interested in experiences most attractions. Assuming this, I recommend a minimum of five days and, preferably, six.
Based upon my experience, if you are open to spending all day at a park, Animal Kingdom and Hollywood Studios can typically be completed in a day (albeit a full one!). As for Magic Kingdom and Epcot, these parks are so large and have so many neat attractions, it takes many folks a second visit. If you purchase the park hopper feature, you can plan to visit for 1/2 day at each park on a return visit. Otherwise, a sixth day may be needed.
Additional days may be required should you want to mix in a water park or schedule a "break" day sometime during the vacation. Hope this helps!