The ExecuteSQL() function can calculate this (Note this is NOT the script step of nearly the same name). A new way to count unique values in FileMaker 12
You could set up a conditional value list of your dates using your existing relationship as the means to limit the value list to just the dates for this participant. Then ValueCount ( ValueListItems ( Get ( FileName ) ; "YourValueListNameHere" ) ) will return the number of unique date values for your participant.
You could also modify your data model from:
Appointments::_fkParticipantID = Dates::_fkParticipantID AND
Appointments::Date = Dates::Date
where you limit a given record in Dates to a single record for any unique combination of Date and ParticipantID. Then you can count the related records in Dates. The way you'd use this is to enable "allow creation..." for Dates and use this script every time you create a new appointment record:
Set Field [Dates::Date ; Appointments::Date ]
This step creates a new record if one does not already exist, but produces no net change if a related record already exists so your duplicates never appear in the Dates table like they do in Appointments.
Since I am new to filemaker I chose the second option with the conditional list. That looked the easiest among the options. It worked fine! Thank you so much for your quick reply!