First-order Logic Formalisation Help I am having trouble with formalising the following two sentences:
(a) A person who is not registered as a student is not entitled to take the Philosophy exam.
(b) Ant is not entitled to take part in any activity in the School
For (a) my attempt is ∀x(Px → ∀y(Sy → ¬Rxy) → ¬Exp)
Formalisation Key:
Px: x is a person
Sx: x is a student
Rxy: x is registered as y
Exy: x is entitled to take y
p: is the Philosophy exam
Does this make sense? Would it be alright to leave the universal quantifiers separated in this way?
For (b) my attempt is ∀x(Ax ∧ Ixs → ¬Tax).
Key:
Ax: x is an activity
Ixy: x is in y
Txy: x is entitled to take part in y
s: is 'the school'
Would it be right to formalise 'the school' as an individual constant 's' (i.e. that 'the school' is referring to some specific school in a domain and this is its name) ??. Thanks.
 A: 
(a) A person who is not registered as a student is not entitled to
take the Philosophy exam.

Letting the universe of discourse be the set of persons, the simplest formalisation is $$\forall x \;(Ex\to Sx).$$ A literal and self-contained formalisation is $$∀x \;(Px\to¬Sx\to ¬Ex)$$ (note that this is read as $∀x \;(Px\to(¬Sx\to ¬Ex)),$ which is equivalent to $$∀x \;((Px\land ¬Sx)\to ¬Ex)$$ as well as $$∀x \;((Px\land Ex)\to Sx).$$
Your attempt doesn't look correct:

*

*

¬Expp: is the Philosophy exam

Why is the symbolisation key interpreting/restricting a variable?


*

Rxy: x is registered as y

What does it mean to say that a variable is registered as another
variable?


*

$∀x(Px → ∀y(Sy → ¬Rxy) → ¬Exp)$

contains an unquantified variable, $p,$ so technically is not a
sentence/proposition.

(b) Ant is not entitled to take part in any activity in the School

Letting the universe of discourse be the set of activities, this can be formalised as $$∀x\;(Sx\to¬Ax).$$ If you don't wish to specify the universe of discourse, then: $$∀x\;(Cx\to Sx\to¬Ax),$$ which can be simplified as in the previous example.
    (aCtivity, School, Ant)

For (b) my attempt is $∀x(Ax ∧ Ixs → ¬Tax).$ Ax: x is an
activity Ixy: x is in y Txy: x is entitled to take part in
y s: is 'the school'

You accidentally omitted the last line: “a: (is) Ant.” In any case, this attempt has the same issues as above. In particular, none of ‘Philosophy exam’, ‘school’ and ‘Ant’ needs to be a variable.
