how can i know if a student was in the campus? edit: using first order logic(logic)

Question

i don't understand this at all. how can i deduct if a student was at the campus?

EDIT: i tried to convert all of the following into a first order logic in a closed domain (and some closed world assumptions) to better understand the subject, i.e: if we know that Helen is a professor, then only helen is a professor can be inferred from the statement).

sorry for the change, but it is very important for me to do it like this so i can learn it correctly and methodically. i also tried to put the sentences into first order logic like i asked here

i know that:

1. every professor counsels at least one student

$\forall x,y(C(x,y) \land \forall x,y -> (x=p \land y=(\exists S(s)))$

2. Helen is a professor

$\exists x(P(x) \land x-> x='Helen')$

3. every student has a counselor, who is a professor

$\forall x,y(H(x,y) \land \forall x,y(x=s \land y=c) \land c -> p)$

4. counseling meeting occur at the campus

$\exists x,y(occurs(x,y) \land \forall x,y ((x=m \land i) \land y = u)$

5. Leno is a student

$\exists x(S(x) \land x -> x= s)$

6. every counselor meets with all of his counselees(the ones he counsels)

$\forall x,y(M(x,y) \land x=c \land M(x,y) -> y=h )$ (a counselor meets with all of his counselees, and if he meets someone it must be his counselee)

where:

S(x) - x is a student

P(x) - x is a professor

C(x,y) - x counsels y

M(x,y) - x meets with y

O(x,y) - x occurs at y

H(x,y) - x has y

p - a professor

s - a student

c - a counselor

m - a meeting

h - a counselee

i - a counseling

u - a campus

i don't understand, how can i conclude if helen was at the campus or not?

please help me with that and explain so i can learn and apply what you do on about a dozen more i've got. they had not taught us this well

edit: to make more sense, at least for me, i am trying to "formalize" the given sentences using first order logic to make it simpler, i am hoping the formalization would be correct.

Answer 1

Let $P$ be the set of all professors, and $S$ the set of all students. You can then interperet the statements as follows:

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(1) For all $p\in P$, there exists an $s\in S$ such that $p$ counsels $s$. We can term this statement $C$ for 'counsels' and write it symbolically as...

$$\forall p\in P.\exists s\in S:C(p,s)$$

(2) There exists a $p\in P$ such that $p=$Helen. Symbollically...

$$\exists p\in P:p=\text{Helen}$$

or, more succinctly...

$$\text{Helen}\in P$$

(3) For all $s\in S$, there exists a $p\in P$ such that $p$ counsels $s$. Symbollically... $$\forall s\in S.\exists p\in P:C(p,s)$$

(4) Counseling meetings occurs on campus. We can denote 'counseling occurs on campus' with $L$ for 'location'.

(5) There exists an $s\in S$ such that $s=$Leno. Symbollically...

$$\exists s\in S:s=\text{Leno}$$

or, more succinctly...

$$\text{Leno}\in S$$

(6) For all $p\in P$, there exists a subset $S_p\subseteq S$ such that for all $s\in S_p$, $p$ meets with $S$. We may denote '$p$ meets with $s$' by $M$ for 'meeting. Symbollically...

$$\forall p\in P.\exists S_p\subseteq S:(\forall s\in S_p.M(p,s))$$

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Let $K(p,s)\equiv\text{$p$ and $s$ are on campus}$.

We want to show that there is some $s$ such that $K(\text{Helen},s)$ is true.

It is clear that $C(p,s)\land L\land M(p,s)\implies K(p,s)$ - if $p$ counsels $s$ and all counseling meetings are on campus, and $p$ meets with $s$, then, at some point, $p$ and $s$ must both be on campus. From (2), we know that Helen is a professor and from (1) we know that all professors counsel at least one student. The conjunction of (1) and (2) then implies that there is a student $s$, such that $C(\text{Helen},s)$. Likewise, from (6), we know that $M(\text{Helen},s)$ must be true. Finally, $C(\text{Helen},s)\land L\land M(\text{Helen},s)\implies K(\text{Helen},s)$ ($L$ is always true). Therefore, at some point Helen must be on campus.

Symbolic:

$$Q:=\text{Helen}\in P\land(\forall p\in P.\exists s\in S:C(p,s))\implies \exists s\in S:C(\text{Helen},s)$$

$$U:=\text{Helen}\in P\land(\forall p\in P.\exists S_p\subseteq S:(\forall s\in S_p.M(p,s)))\implies \exists S_{\text{Helen}}\subseteq S:(\forall s\in S_{\text{Helen}}:M(\text{Helen},s))$$

$$R:=Q\land U\implies \exists s\in S:(C(\text{Helen},s)\land M(\text{Helen},s))$$

$$L\equiv\top$$

$$R\land L\implies \exists s\in S:K(\text{Helen},s)$$

(I would appreciate if someone could double check this)

Answer 2

Hint

From "every professor counsels at least one student" and "Helen is a professor" we get :

"Helen counsels at least one student".

From it and "every counselor meets with his counselees" we get :

"Helen meets its student".

From it and "counseling meeting occur at the campus" we get :

"Helen meets its student at the campus".

This implies that both Helen and the student counseled (?) by Helen are in the campus.

In order to formalize it, you have to do some more work...