It seems to contain a number of objects. That's why it would be right to say ' the premises are...'.
It is the truth because I say it is.
on the premises
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)
You will have to make a logical inference for the premises, and a logical deduction for the conclusion.
syllogism
on the premises
it is these premises as the nopun is considered a plural noun according to the Oxford Dictionary.
Dogs on premises.
It is the truth because I say it is.
on the premises
It does not match your Premises. That is how we can use Premises.
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)
the part that you rent
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.
The M and C Contract defines "Premises" as the location where the services will be performed or the goods delivered as specified in the agreement. The specifics of what constitutes the Premises should be detailed in the contract to avoid any confusion or disputes.
Business premises is correct, just as you have it.
on