A natural language predicate is a parameterised statement about the world. For example the following predicate has parameters S#, SNAME, STATUS and CITY:
There exists a supplier identified by supplier number [S#] with name [SNAME] having status [STATUS] and located in city [CITY]
Even though this predicate is expressed in natural language, the parameters are intended to be assigned (mathematical) values to give a natural language proposition.
Natural language predicates provide the basis for data in a relational database to be interpreted as information. The terms "data" and "information" are not interchangeable, they don't mean the same thing. Data means encoded values. Information means knowledge about the world.
If a natural language predicate is "named" with a predicate symbol then it can be associated with a
formula in the predicate calculus.
For example, if we use the predicate symbol supplier for the above natural language predicate then
its corresponding formula is:
supplier(S#, SNAME, STATUS, CITY)
Note that the parameters of the natural language predicate have become variables in the predicate calculus.
If the attribute names of a given tuple match the parameter names in a given predicate, then the tuple can represent an assignment of values to the parameters of the predicate to give a proposition.
For example the above predicate can be instantiated with the tuple
{ (S#, S1), (SNAME, Smith), (STATUS, 20), (CITY, London) }
to give the proposition
There exists a supplier identified by supplier number S1 with name Smith having status 20 and located in city London
Given that in natural language we can be explicit when we quantify it seems a dangerous idea to assume it's happening implicitly.
Consider the statement: "The present King of France is bald".
What is its negation?
In logic there's a big difference between the wff p(x) and the wff (exists x p(x)).
One might treat the expression as "[The-present-King-of-France] is bald" where [The-present-King-of-France] is regarded as a symbol which is an unbound variable. In that case the negation is "[The-present-King-of-France] is not bald".
One uses natural language to define predicates on world situations. For those predicates to be meaningful some assumptions are necessary about what world situations are possible. e.g. use of the definite article might imply existence or uniqueness of something. A natural language statement about the world might be unambiguous for our needs (i.e. when we're talking about a database which has a narrow purpose and scope) but nevertheless becomes ambiguous as we throw hypothetical odd-ball situations at it.
Note that we can't put presuppositions such as the existence of something into predicates as conditional statements. Consider:
If there is a supplier identified by 'S1' then that supplier is located in a city named [CITY]
This isn't a predicate because it doesn't uniquely determine an extension in every world situation.