I am having difficulties understanding when to interpret a sentence as a conditional statement and when to interpret it as a conjunction.
For example in the following problem:
Show that the premises “A student in this class has not read the book,”
and “Everyone in this class passed the first exam” imply the
conclusion “Someone who passed the first exam has not read the book.”
Solution:
Let C(x) be “x is in this class,” B(x) be “x has read the
book,” and P(x) be “x passed the first exam.” The premises are ∃x(C(x)∧￢B
(x)) and ∀x(C(x) → P(x)). The conclusion is ∃x(P(x)∧￢B(x)). These steps
can be used to establish the conclusion from the premises.
The second premise - ∀x(C(x) → P(x)) I interpreted as ∀x(C(x) ∧ P(x)). In English I read it as "For all 'x', x is a student in this class AND 'x' has passed the exam". Why is it not valid to interpret the sentence this way?
If you say
$\forall x (C(x) \land P(x))$
then you say that everything in the domain is a student in the course and passed the exam.
However, presumably there are things in the domain other than students in the course. For example, that book that the student didn't read, or the exam that they took. So if they are in the domain, you would get that the book passed the exam, or that the exam is a sudent in the course ... which is not what you want. What you want is that if something is a student in the course, then they passed the exam, i.e. you want the conditional.
Of course, it is possible that the domain is simply all the students in the course, but then why would you need a predicate $C(x)$? If the domain was all students in the course, you could simply use $\exists x \neg B(x)$ for the first premise, and $\forall x P(x)$ for the second.
So, either way, $\forall (C(x) \land P(x))$ is not what you want.
Hope that helps!