Salt is the newest of the three alchemical principles, and the only one whose introduction can be dated and attributed.
Two became three
For several centuries the doctrine held that metals were compounded of two principles, a sulphur and a mercury, a theory that came into Latin Europe from the Arabic tradition through the translations of the twelfth and thirteenth centuries. Paracelsus, in the sixteenth century, added Salt as a third, and the resulting set is the tria prima.
What the addition did
The two-principle theory was about metals. Adding a third principle, standing for body, fixity and what remains after burning, turned the scheme into a general account of all substances including living ones, which is what Paracelsus wanted it for. His interests were medical, and a theory that could describe a body and a mineral in the same terms was the point of the exercise.
That extension is the reason the tria prima appears in medical and pharmaceutical writing of the following century and the older pair largely does not.
Not table salt
As with the other two, the principle is not the everyday substance of the same name. Alchemical writers distinguished the philosophical salt from any particular salt on the bench, and the ash left after combustion is the usual illustration rather than the definition. Reading the principle as sodium chloride makes the sources incoherent.
Sources The Secrets of AlchemyLawrence M. Principe with Paracelsus: An Introduction to Philosophical Medicine in the Era of the RenaissanceWalter Pagel