Conjunction vs Implication from a linguistic perspective.

Question

I am translating some English sentences to FOL and sometimes the use of conjunction and implication confuses me, so I am trying to come up with a rule. I have given some examples:

1. Every student doing homework is smart.

   ∀x. (student(x) ^ homework(x) → smart(x))

my rule:

doing is not a verb, hence: ^

is a verb, hence: →

2. Every student does some homework.

   ∀x. (student(x) → ∃y (homework(y)))

does a verb, hence: →

3. If all students do their homework, then all teachers are happy.

   (∀x. (student(x) → homework(x)) → (∀y. (teachers(y) → happy(y))

do a verb, hence: →

are a verb, hence: →

does my conclusion make sense?

Answer 1

The rules for using $\to$ vs. $\land$ are quite straightforward: With $\forall$, use $\to$, with $\exists$, use $\land$. This is independent of whether the predicate involved is expressed in natural language as a verb (but what makes you think "do(ing)" is not a verb? Certainly "to do" is a verb; the fact that we can form the progressive form "doing" shows this clearly), a noun or an adjective. So, no, your conclusions do not make sense unfortunately.

It is important for you to recognize the distinction between the restriction and the nuclear scope of a quantification. Roughly, the restriction is the set of individuals you talk about and the nuclear scope is what you say about these individuals. In "Every student doing homework is smart", the restriction is the students doing homework and the nuclear scope is "smart"; in "Every student does some homework", the restriction is "student" and the nuclear scope is "there is some homework they do"; in 3. the restriction of the first universal quantifier is "students" and the nuclear scope is "do their homework", of the second universal quantifier restriction and nuclear scope are "teachers" and "happy", respectively.

1. Every      | student doing homework | is smart
   quantifier | restriction            | nucleus

2. Every      | student     | does some homework
   quantifier | restriction | nucleus

3. If [all      | students | do their homework ], then [all      | teachers | are happy].
      [quant. 1 | restr. 1 | nucl. 1           ]       [quant. 2 | restr. 2 | nucl. 2  ]

When saying that $\forall$ goes with $\to$ and $\exists$ goes with $\land$, then this always refers to the connectives that goes between the restriction and the nuclear scope. You can easily see that this rule applies to all of your examples and more without inconsistencies:

1. Every      ( student doing homework --> is smart)
   quantifier (      restriction       --> nucleus )

2. Every      (   student   --> does some homework)
   quantifier ( restriction -->     nucleus       )

4. Some       ( students doing homework ^ are smart )
   quantifier (      restriction        ^  nucleus  )

5. Some       ( student     ^ does some homework )
   quantifier ( restriction ^      nucleus       )

What enters additionally in 1. is a further modification of the restriction: The restriction of the universal quantification is not limited only to the student predicate, but also the homework predicate. In general, predictions that add further details about the restriction (be it expressed as a progressive verb phrase like "doing their homework", as a relative clause like "who does their homework", or as something different like "such that they do their homework) are always treated as conjoined ($\land$-ed). What you say is that the smart predicate applies to all individuals under the restriction that they are students and doing their homework; someone who is a student but doesn't do their homework is not necessarily smart. This is why it intuitively makes sense to use $\land$: We add further conditions about the set of individuals that is being predicated over. The same would go in the nuclear scope of the quantification: If we were to say "Every student doing their homework is a smart person", we would also have a $\land$ between the predicates "smart" and "person".

1. Every      ( student ^ doing homework --> is smart )
   quantifier (      restriction         -->  nucleus )

6. Every      ( student ^ doing homework --> is smart ^ is a person)
   quantifier (      restriction         -->       nucleus         )        

But this is independent of whether we are dealing with a universal or an existential quantifier: You would get the same if you were to say "There is a student doing their homework who is smart". In that case, you would have a $\land$ between the student and the homework as well, just that because we now have "There is" ($\exists$), the connective between the restriction "student doing homework" and the nuclear scope "smart" is a $\land$ as well, whereas it is a $\to$ in the case of a universal quantification.

4. There is a ( student ^ doing homework ^ is smart)
   quantifier (      restriction         ^ nucleus )

6. Every      ( student ^ doing homework --> is smart ^ is a person)
   quantifier (       restriction        -->       nucleus         )  

So the rules are:

• Between the restriction and the nuclear scope of a quantification (between the two parts sepearated by |), use $\to$ for $\forall$ and $\land$ for $\exists$.
• Within the restriction or scope of a quantifier (within a part enclosed by |'s), use $\land$ to conjoin several predictions about the set of individuals talked about.

This makes the connectives used in your formulas completely predictable, and is independent of linguistic categories such as "verb" or "adverbial modification" or whatever.