It should be something close to the following (bulkfied, but caveat emptor - I haven't compiled this so there might be the odd typo.  It also won't scale if there are thousands of specialities associated with each contact):

trigger setupPCP on Contact_Speciality__c (after insert, after update, after delete) {
  // get the contacts
  Set<Id> contactids=new Set<Id>();
  Map<Id, Contact_Speciality__c> specMap;

  // figure out which map is populated
  if (Trigger.isInsert || Trigger.isUpdate)
  {
     specRecs=Trigger.newMap();
  }  
  else
  {
    specRecs=Trigger.oldMap();
  }

  // get the contact ids from the speciaity records
  for (Contact_Speciality__c spec : specMap.values)
  {
    contactIds.add(spec.Contact__c);
  }

  // now retrieve the contacts and all of their associated specialities, so that we can figure out the correct value for the contact
  List<Contact> toUpdate=new List<Contact>();
  for (Contact cont : [select id, PCP__c, (select id, Speciality__c from Contact_Specialities__r) from Contact where id in :contactIds])
  {
    // iterate the specialities looking for speciality values that require the PCP flag to be set on the contact
    boolean checkPCP=false;
    for (Contact_Speciality__c cSpec : cont.Contact_Specialities__r)
    {
       if (cSpec.Speciality__c=='Family Medicine' || cSpec.Speciality__c=='General Medicine' | cspec.Speciality__c=='Family_Practice__c')
       {
           checkPCP=true;
           break;       // no need to continue with the inner loop as we know we need to check the PCP box on the contact
       }
    }

    // is the result different to the contact record?
    if (checkPCP!=cont.PCP__c)
    {
       cont.PCP__c=checkPCP;
       toUpdate.add(cont);
    }
    
  }

  if (!toUpdate.isEmpty())
  {
    update toUpdate;
  }
}