I have the following English question
There exists one school student.
And I would like to translate it to ALCQO or First order logic.
I have written the following:
∃SchoolStudent
or in First Order Logic:
∃x(SchoolStudent(x))
However, I know from mathematics and the theory that ∃ is translated as "at least one" or "some". Thus, I am wondering if the following two options are correct:
approach 1
¬∃x(SchoolStudent(x))
approach 2
(≥1student.SchoolStudent)⊓(≤1student.SchoolStudent)
∃SchoolStudent
is meaningless in ALCQO. The only constructor available in ALCQO using existential restrictions are of the form
∃r.C
which is interpreted as
{d ∈ ΔI | there is an e ∈ ΔI with (d, e) ∈ rI and e ∈ CI}
In first order logic ∃r.C translates to πx(∃r.C) = ∃y.r(x, y) ∧ πy(C).
To define a school with only 1 student you will have to have the concepts School and Student and a role which can be used to associate a student with a school, say hasStudent
.
School ≡ ∃hasStudent.Student ⊓ ≤1hasStudent.Student
This means School here is the set of schools that have only a single student.
But we can also define School as anything that has more than 1 student and then define a specific individual of the school as a school with only 1 student.
School ≡ ∃hasStudent.Student
School(x)
≤1hasStudent.Student(x)