strict interpretation
"Correct in the premises" is used when referring to something that is accurate or true within a particular setting or situation. "On the premises" typically refers to something physically present or located within a specific place or property.
Stated premises are explicitly mentioned in an argument or discussion, while unstated premises are assumed to be true but are not explicitly stated. Both stated and unstated premises play a role in supporting the conclusion of an argument. Unstated premises are often implied and can be inferred by understanding the context of the discussion.
It is "onto the premises" when referring to moving something onto a property or location. "Into" is used to describe movement inside an area.
on the premises
It does not match your Premises. That is how we can use Premises.
"NOTICE TO QUIT - A request from a landlord to his tenant, to quit the premises lessed, and to give possession of the same to him, the landlord, at a time therein mentioned." you can get more info for the site itself http://www.lectlaw.com/def2/n076.htm
You can say either (or 'leasehold premises'). Best avoid 'demised premises' when used in a lay sense however, for the reason mentioned in the question and also because it is an unnecessarily technical term for other than legal reasons. (Solicitor)
In a Lease agreement for a Building being constructed, it may say:'said premises and the building and other improvements thereon, upon completion of construction, are hereinafter called the "demised premises".'Etymology and usage: 'Premises' is from Medieval Latin praemissa, a plural word, meaning the "things mentioned before" and was used in legal documents. In Middle English, premises was defined as being "that property, collectively, which is specified in the beginning of a legal document and which is conveyed, as by grant, …". In due course premises was also legally extended to mean "a house or building with its grounds or appurtenances," a usage which was recorded before 1730.In a general sense today, premises means a piece of land together with the building(s) on it, or even just a building with all its parts. Therefore the word 'premises' is always plural when used in this way.
I think you mean premises. Premises has several meanings. For example, you could say,"Sandra was removed from the premises," which means that Sandra was asked to leave the property where she had previously been. Premises can also means a set of assumptions or suppositions, as in "The premises underlying the theory of evolution include the ideas that all life is interrelated and that all species are descended from a common ancestor." If you mean promises, then you are referring to a commitment to do something.
Business premises is correct, just as you have it.
on
never and not at all.