# Formalizing sentences into first order logic

in regards to how can i know if a student was in the campus? edit: using first order logic(logic), which dealt with inferring, i am asking here assistance with formalizing the sentences with the following emphasis:

• closed world emphasis(What is not currently known to be true, is false)

• domain closure(There are no more domain elements than those named by constant symbols)

did i do the following correctly?

1. every professor counsels at least one student

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

1. Helen is a professor

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

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

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

1. counseling meeting occur at the campus

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

1. Leno is a student

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

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

new attempt: $$\forall x,y((P(x) \land S(y) \land C(x,y) -> M(x,y))$$

$$\forall x,y(M(x,y) \land x=c \land M(x,y) -> y=h )$$ (e 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

would appreciate your input and corrections. some where quite complicated for me.

thank you very much for helping

(note, as mentioned, those 2 questions are different in their subject. here the subject is converting the sentences into first order logic under the given assumptions(closed world and domain closure) whereas the other was about inferring)

• You confused the explanations for closed world and closed domain, it's the other way round. – lemontree May 3 at 10:03
• yeah sorry i was so confused and tired. at least it's funny. editing it now. except that could you take a look at the first order logic conversion i've tried to do? – Avariable May 3 at 11:17

## 2 Answers

First of all some general comments:

1) You are erroneously conflating constant names with variables.
Constant names like $$h$$ or $$l$$ are used to refer to particular individuals (like Helen or Leno). Their meaning is fixed and given by the model's interpretation function, which is why they belong to the non-logical symbols (= the symbols that are specific to a particular predicate logic language and whose meaning needs to be defined) and you have to provide a translation key from them.
Variables, on the other hand, as used with quantifiers, usually named $$x, y, z, ...$$ belong to the logical symbols (= the symbols that are common to all predicate logic languages, and whose meaning is not determined by an interpretation function). Variables are called variables because other than constants, their meaning (= the object they refer to) is not fixed (as the meaning of the name $$h$$ or $$l$$ would be), but varies depending on what the chosen assignment function maps them to. The meaning of a variable is given by the local context, such as a universal or existential quantification; for these reasons, you do not provide a translation key for variables.
The only particular, specified, fixed individuals you need refer to in this assignment are Helen and Leno, hence $$h$$ and $$l$$ are the only constant sybmols you need to use, and the only ones you have to provide a translation key for. All other occurences of terms, such as the quantification over all professors or some students, are variables, which have no place in the translation key.

2) The name of constant names or variables does not say anything about what they refer to.
The naming of constant names is arbitrary: As a constant name for Helen you could choose $$leno$$ and as a constant name for Leno you could choose $$helen$$, or $$cheesecake$$ or whatever you want. The name does not say anything about its meaning. The only thing that determines what $$helen$$ or $$leno$$ refer to is the interpretation you specify in the translation key.
Likewise, the naming of variables is arbitrary: You could call a variable named $$x$$ $$y$$ instead or $$u$$ or $$blah$$ or whatever you want; their interpreation does not depend on the name given. The only thing that determines what a variable refers to is the assignment function.
If often make sense to choose mnemnoic names, to make it easier to see what your symbols refer to, e.g. to choose $$h$$ or $$helen$$ as a constant name for Helen, or (less commonly done) choose mnemonic names for variables, e.g. $$p$$ if with $$p$$ you quantify over the set of professors. But such a name choice alone does not have any consequences on their interpreation.
This means that in a formula like $$\forall p \exists s (C(p,s))$$, the variables $$p$$ and $$s$$ are not restricted to any domain: $$p$$ and $$s$$ are just variables that could stand for anything, so $$\forall p \exists s (C(p,s))$$ means "Everyone counsels someone". If you want a restriction, you need to make that explicit: "Every professor counsels some student" has to be formalize as $$\forall p(P(p) \to \exists s (S(s) \land C(p,s))$$, or equivalently $$\forall x(P(x) \to \exists y (S(y) \land C(x,y))$$, since the name of the variables doesn't matter.

This is the key for the non-logical symbols that I would choose:

Predicates (= properties of and relations between individuals):

• $$P(x)$$ = x is a professor
• $$S(x)$$ = x is a student
• $$C(x,y)$$ = x counsels y
• $$M(x,y)$$ = x meets with y
• $$A(x)$$ = x is at the campus

Constant symbols (= names that refer to particular individuals):

• $$h$$ = Helen
• $$l$$ = Leno

Function symbols: none

1. Every professor counsels at least one student

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

$$\forall xy(C(x,y)) \land ...)$$ means "Everyone counsels everyone." This is wrong because a) every profesor counsels only at least one student, not all of them and b) the domains are not restricted to the professors and students.
I don't know what you mean by the rest of your formula behind $$\land$$, it's not even syntactically correct, i.e. simply not a predicate logic formula at all.
$$\forall x,y -> (...)$$ is not syntactically correct: $$\to$$ expects a formula on its left-hand side, but there is not any formula $$\forall x,y$$ quantifies over.
$$y=(\exists S(x)))$$ is not syntactically correct either. $$=$$ expects a term to its right-hand side, but $$\exists S(s)$$ is not a term. It is not even a formula, because the variable behind the quantifier is mising.

Instead, the sentence is

For every $$x$$ which is a professor, there exists some $$y$$ such that $$y$$ is a student and $$x$$ counsels $$y$$

which is translated as

$$\forall x(P(x) \to \exists y(S(y) \land C(x,y))$$

1. Helen is a professor

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

Again, this is not a formula, because again, $$\to$$ expects a formula to its left-hand side, but $$x$$ is not a formula, but a term.

I suppose you wanted to say

There is a professor, and this professor is Helen

$$\exists x(P(x) \land x = h)$$

which could be abbreviated to just

Helen is a professor

$$P(h)$$

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

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

Your formula says "Everyone has everyone and everyone is identical to s (which you want to be some constant name that refers to some student?) and everyone is identical to c (again, a constant name that refers to some undspecified counselor)" and the rest behind the $$\land$$ is, again, not even a proper formula, because neither $$c$$ nor $$p$$ are formulas, so you can't $$\to$$ them. This formula doesn't make sense at all.

Hint: The sentence can be paraphrased as

For every $$x$$ who is a student, there is a $$y$$ such that $$y$$ is a professor and counsels them.

This is very similar to 1. Try to correct your formula along the lines of this explanation.

1. counseling meeting occur at the campus

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

This one is a bit tricky, but let me first translate what your formula says:

"There are x and y such that x occurs at y, and all x are identical to m (a constant for a particular but in your translation key not further specified meeting), and i??? (not a formula) and all y are identical to u (a constant which refers to a particular but unspecified campus)." I don't even what you want to say with this.

Instead, one could paraphrase "All meetings occur at the campus" as

For all pairs of x and y where x has a meeting with y, x is at the campus and y is at the campus

which, according to the translation key above, would be formalized as

$$\forall x,y(M(x,y) \to A(x) \land A(y))$$

1. Leno is a student

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

This statement is nowhere needed to derive the desired inference, but the problems are the same as before and the correct solution completely analogous to 2. Try correcting this one according to the previous explanations.

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

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

No, this is not what your formula says. Your formula says "Everyone meets with everyone (you didn't restrict the x to the counselors and the y to the students they counsel) and all x are identical to a particular counselor c and if x meets y, then y is identical to Helen."

Instead, your formlua should be

For all $$x$$ and $$y$$, if $$x$$ is a professor and $$y$$ is a student and $$x$$ counsels $$y$$, then $$x$$ meets $$y$$.

Try to find the formalization for this one, and feel free to update your post with your revised version for feedback.

• my new attempt for 6: $\forall x,y((P(x) \land S(y) \land C(x,y) -> M(x,y))$ – Avariable May 5 at 14:50
• one additional small question if i may, because we are assuming a closed world and domain closure, is it correct to say it can be inferred from that "database" that Halen consulted Leno? (because of those two assumptions) – Avariable May 5 at 14:54
• 6.: correct Re. your second question: Yes, if we know she consulted at least one student and the only students whose existence we know of is Leno, then under that assumption we can infer that Helen consulted Leno. However, this is not needed to derive the inference that Helen is at the campus; it suffices to know that she counsels some student (hence has conuseling meetings at the campus) without having to mention in the argument who that student is. – lemontree May 5 at 19:21
• thank you so much! tried to give you a lot of times upvotes, but i don't have enough reputation to do so :( – Avariable May 6 at 21:20
1. for all p, (P(p) implies exists s with (S(s) and C(p,s))
2. P(Helen)