Question

law of detachment
$p\to q$
$\frac{p}{\therefore q}$
law of syllogism
$p\to q$
$\frac{q\to r}{\therefore p\to r}$
determine whether the conclusion following the given statements is justified by the law of detachment or the law of syllogism. if neither, write invalid.
1 given: if i drive over the speed limit, then i will get a ticket.
given: i drove over the speed limit.
conclusion: i got a ticket.
answer:
2 given: if amanda goes to the restaurant, then she will order a hamburger.
given: if she orders a hamburger, then she will get fries.
conclusion: if amanda goes to the restaurant, then she will get fries.
answer:
3 given: if it snows, then i will wear my snow boots.
given: if it snows, then i will not go to school that day.
conclusion: if i do not go to school, then i will wear my snow boots.

Explanation:

1. For the first problem, we have \(p\to q\) (If I drive over the speed limit (\(p\)), then I will get a ticket (\(q\))) and \(p\) (I drove over the speed limit). According to the Law of Detachment (\(p\to q\) and \(p\) implies \(q\)), the conclusion \(q\) (I got a ticket) is valid.
2. For the second problem, we have \(p\to q\) (If Amanda goes to the restaurant (\(p\)), then she will order a hamburger (\(q\))) and \(q\to r\) (If she orders a hamburger (\(q\)), then she will get fries (\(r\))). According to the Law of Syllogism (\(p\to q\) and \(q\to r\) implies \(p\to r\)), the conclusion \(p\to r\) (If Amanda goes to the restaurant, then she will get fries) is valid.
3. For the third problem, we have \(p\to q\) (If it snows (\(p\)), then I will wear my snow boots (\(q\))) and \(p\to s\) (If it snows (\(p\)), then I will not go to school that day (\(s\))). There is no valid logical rule (neither Law of Detachment nor Law of Syllogism) to get from these two statements to \(s\to q\) (If I do not go to school, then I will wear my snow boots).

Answer:

1. Law of Detachment
2. Law of Syllogism
3. Invalid