Dictator Dim wants to replace counting, in particular, counting the population of his land. To keep an accurate population count, Dictator Dim has instructed his secretary to add one + symbol to a record each time a citizen is born, and one - symbol to the record whenever a citizen dies. This has been going on ever since his rule began.
Dictator Dim is not concerned so much with the actual total population of his lands, just with the net change in population since his rule began. He would be forever disgraced if the net population change were ever negative. Refusing to solve this problem by counting, Dictator Dim instructs his secretary to find a sound and complete logic for the subset of all records that would not disgrace him.
More formally, define the following truth concept WFF's = {+; -}, i.e., all strings composed of only plus and minus symbols.
A WFF x is true if every prefix of x contains at least as many + symbols as - symbols.
Find a logic for this truth concept by proving that it is sound and complete. Hint: it may help to define additional judgement form(s)