"It is also important to point out that “1st generation” Joshua Pettit had two sons and two daughters according to the letter. None of the names of his children were given except for Increase Pettit. Some have speculated that the other son may have been named Joshua II born in 1734 who relocated to New Jersey, Virginia and South Carolina in that order. There are also some who claim the other son was a man named Zadok Pettit."
This confusion with some people naming the other son of 1st generation Joshua as Zadok may come from the relationship of Joshua Pettit (1751-1820) having a son named Zadock (1785-1834). This Zadok was a g-grandson of 1st generation Joshua which goes like this: Joshua (1701) - Increase (1726) - Joshua (1751) - Zadock (1785). To confuse the matter even more Zadock (1785) had brothers; Increase and Joshua. Good grief!!!
I have never found any other reference to another Zadock during this time period on Long Island.