I have this sentence: "All teachers and students are here." Also I have these predicates: t(x) - x are teachers, s(x) - x are students, h(x) - x are here. In my textbook the sentence is written in a predicate language this way: $\forall x((s(x)∨t(x))\to h(x)$ .
The question is: Why there is OR symbol instead of AND symbol?
On
This is a classic. The one I always use is 'All fruits and vegetable are nutritious'
So yes, given the 'and', you might be inclined to symbolize "All teachers and students are here" as:
$$\forall x ((T(x) \land S(x)) \to H(x))$$
but that would mean that "Anyone who is both a teacher and a student is here"
So, what's going on? Why doesn't the 'and' in English give you the logical $\land$?
It's because "All teachers and students are here" is really short for "All teachers are here and all students are here".
Indeed, we could translate this sentence as:
$$\forall x (T(x) \to H(x)) \land \forall x (S(x) \to H(x)) $$
which is equivalent to:
$$\forall x ((T(x) \to H(x) \land (S(x) \to H(x)) $$
And now, here is the crucial equivalence that swithces the $\land$ into an $\lor$:
$$(P \lor Q) \to R \Leftrightarrow (P \to R) \land (Q \to R)$$
So with that, we can symbolize the sentence also as:
$$\forall x ((T(x) \lor S(x)) \to H(x))$$
and if we translate that back to English, we get:
"Anyone who is a teacher or a student is here"
which is, if you think about it, exactly what we want!
An "and" here would mean "for every $x$, if $x$ is a teacher and $x$ is a student, then $x$ is here". In English the "and" is intended as a union, which is expressed in logic as "or".