Question

unit 2.7 deductive reasoning mastery check
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.
answer:

Explanation:

1. The Law of Detachment states that if \(p

ightarrow q\) is a true conditional statement and \(p\) is true, then \(q\) is true. Here, \(p = \) "I drive over the speed limit" and \(q=\) "I will get a ticket". Since \(p\) (I drove over the speed limit) is given as true, by the Law of Detachment, \(q\) (I got a ticket) is valid.

1. The Law of Syllogism states that if \(p

ightarrow q\) and \(q
ightarrow r\), then \(p
ightarrow r\). Let \(p=\) "Amanda goes to the restaurant", \(q = \) "she orders a hamburger", \(r=\) "she will get fries". We have \(p
ightarrow q\) and \(q
ightarrow r\), so by the Law of Syllogism \(p
ightarrow r\) (If Amanda goes to the restaurant, then she will get fries) is valid.

1. For the Law of Syllogism, we need two conditionals where the conclusion of one is the hypothesis of the other. Here, the first conditional is \(p

ightarrow q\) (\(p=\) "it snows", \(q=\) "I will not go to school") and the second is \(p
ightarrow r\) (\(r=\) "I will wear my snow - boots"). There is no transitive relation (\(q
ightarrow r\) is not formed from the given conditionals in the correct way for the Law of Syllogism), so the conclusion is invalid.

Answer:

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