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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”, let $B(x)$ be “$x$ has read the book”, and let $P(x)$ be “$x$ passed the first exam.” The premises are $\exists x\,(C(x) \land \neg B(x))$ and $\forall x\,(C(x) \to P(x))$.

How do I know, judging by the wording, when a premise will be $p \land q$ as opposed to $p \to q$? So in this example, how do I know “A student in this class has not read the book” cannot be $\exists x\,(C(x) \to \neg B(x))$, to say that for some student, if he is in this class, then he has not read the book. I am often wrong in my steps of simplification due to my incorrect interpretation.

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    $\begingroup$ "∃x(C(x) → ¬B(x))" might be translated as "there is a student who if in this class has not read the book" which might be satisfied by finding a student not in the class (though perhaps has read the book). That is is not what the question says with "A student in this class has not read the book". $\endgroup$ – Henry Oct 14 '14 at 22:52
  • $\begingroup$ @Henry Extend this comment to an answer? This seems to answer the question. $\endgroup$ – Pål GD Oct 14 '14 at 22:52
  • $\begingroup$ @Pål GD: Feel free to do so if you wish $\endgroup$ – Henry Oct 14 '14 at 22:57
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The premises are $\exists x(C(x)\wedge\neg B(x))$ and $\forall x(C(x)\to P(x))$.

How do I know, judging by the wording, when a premise will be $p\wedge q$ as opposed to $p\to q$ ?

The key is whether the quantifier is existential or universal.

Take statements of the form "There is an entity $x$ in the category $Q$ that has a property $P$". The statements are true when an example can be found that is both in the category and having the property. It's not implied that it possesses the property because it's in the category nor vice versa. It's just that at least one exists that is jointly both.

So: $\exists x\, (Q(x)\wedge P(x))$

Take statements of the form "Every/Each/All entities $x$ in the category $Q$ possess a property $P$". These statements are true if being in the category means that any such entity possess the property, while the converse is not necessarily so. This is an implication.

Thus: $\forall x\; (Q(x)\to P(x))$, or equivalently: $\forall x\; (\neg Q(x)\vee P(x))$ ("all entities either are not in this category, or have this property.")

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