Question

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:
4 given: if tina goes to the beach, she will wear sunscreen.
given: tina goes to the beach.
conclusion: tina will wear sunscreen.
answer:
5 given: if you get a cold, then you eat chicken soup.
given: you ate chicken soup.
conclusion: you must have a cold.
answer:
6 given: if two angles form a linear pair, then they are supplementary.
given: if two angles are supplementary, then the sum of their measures is 180°.
conclusion: if two angles form a linear pair, then the sum of their measures is 180°.
answer:

Explanation:

1

Step1: Analyze Law of Detachment

Law of Detachment: If \(p
ightarrow q\) (if \(p\) then \(q\)) and \(p\) (hypothesis is true), then \(q\) (conclusion is true). Here \(p = \) "I drive over the speed limit", \(q=\) "I will get a ticket", and we have \(p\) (I drove over the speed limit).

2

Step1: Analyze Law of Syllogism

Law of Syllogism: 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 \(p
ightarrow r\).

3

Step1: Check logical structure

We have \(p
ightarrow q\) (if it snows (\(p\)), then wear snow - boots (\(q\))) and \(p
ightarrow r\) (if it snows (\(p\)), then do not go to school (\(r\))). The conclusion \(r
ightarrow q\) is not in the form of Law of Detachment (\(p
ightarrow q\) and \(p\)) or Law of Syllogism (\(p
ightarrow q\), \(q
ightarrow r\) then \(p
ightarrow r\)).

4

Step1: Apply Law of Detachment

Let \(p=\) "Tina goes to the beach", \(q=\) "wear sunscreen". We have \(p
ightarrow q\) and \(p\) (Tina goes to the beach).

5

Step1: Check logical form

We have \(p
ightarrow q\) (if get a cold (\(p\)), then eat chicken soup (\(q\))) and \(q\) (ate chicken soup). Law of Detachment requires \(p
ightarrow q\) and \(p\), not \(p
ightarrow q\) and \(q\).

6

Step1: Check logical form

We have \(p
ightarrow q\) (if two angles form a linear pair (\(p\)), then they are supplementary (\(q\))) and \(q
ightarrow r\) (if two angles are supplementary (\(q\)), then sum is \(180^{\circ}\) (\(r\))). But the conclusion \(p
ightarrow r\) is just repeating the transitive property in a non - syllogistic way (since \(p
ightarrow q\) and \(q
ightarrow r\) would be Law of Syllogism for \(p
ightarrow r\), but here the second given is \(q
ightarrow r\) which is already a known property, not a new conditional for syllogism in a proper logical chain creation).

Answer:

1. Law of Detachment
2. Law of Syllogism
3. Invalid
4. Law of Detachment
5. Invalid
6. Invalid