Exercise 1.13.5: Determine and prove whether an argument in English is valid or invalid. Prove whether...

Exercise 1.13.5: Determine and prove whether an argument in English is valid or invalid.

Prove whether each argument is valid or invalid. First find the form of the argument by defining predicates and expressing the hypotheses and the conclusion using the predicates. If the argument is valid, then use the rules of inference to prove that the form is valid. If the argument is invalid, give values for the predicates you defined for a small domain that demonstrate the argument is invalid.

QUESTION A AND B ALREADY SOLVED. PLEASE solve part C, D, E using same method, and give some explantion for your answer. THANKS!

The domain for each problem is the set of students in a class.

(a)

Every student on the honor roll received an A.
No student who got a detention received an A.
No student who got a detention is on the honor roll.

H(x): x is on the honor roll
A(x): x received an A.
D(x): x got a detention.

∀x (H(x) → A(x))
¬∃x (D(x) ∧ A(x))
∴ ¬∃x (D(x) ∧ H(x))

Valid.

1.	∀x (H(x) → A(x))	Hypothesis
2.	c is an arbitrary element	Element definition
3.	H(c) → A(c)	Universal instantiation, 1, 2
4.	¬∃x (D(x) ∧ A(x))	Hypothesis
5.	∀x ¬(D(x) ∧ A(x))	De Morgan's law, 4
6.	¬(D(c) ∧ A(c))	Universal instantiation, 2, 5
7.	¬D(c) ∨ ¬A(c)	De Morgan's law, 6
8.	¬A(c) ∨ ¬D(c)	Commutative law, 7
9.	¬H(c) ∨ A(c)	Conditional identity, 3
10.	A(c) ∨ ¬H(c)	Commutative law, 9
11.	¬D(c) ∨ ¬H(c)	Resolution, 8, 10
12.	¬(D(c) ∧ H(c))	De Morgan's law, 11
13.	∀x ¬(D(x) ∧ H(x))	Universal generalization, 2, 12
14.	¬∃x (D(x) ∧ H(x))	De Morgan's law, 13

(b)

No student who got an A missed class.
No student who got a detention received an A.
No student who got a detention missed class.

M(x): x missed class
A(x): x received an A.
D(x): x got a detention.

¬∃x (A(x) ∧ M(x))
¬∃x (D(x) ∧ A(x))
∴ ¬∃x (D(x) ∧ M(x))

The argument is not valid. Consider a class that consists of a single student named Frank. If M(Frank) = D(Frank) = T and A(Frank) = F, then the hypotheses are all true and the conclusion is false. In other words, Frank got a detention, missed class, and did not get an A.

(c)

Every student who missed class got a detention.
Penelope is a student in the class.
Penelope got a detention.
Penelope missed class.

(d)

Every student who missed class got a detention.
Penelope is a student in the class.
Penelope did not miss class.
Penelope did not get a detention.

(e)

Every student who missed class or got a detention did not get an A.
Penelope is a student in the class.
Penelope got an A.
Penelope did not get a detention.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Using a truth table determine whether the argument form is valid or invalid p ∧ q...

Using a truth table determine whether the argument form is valid or invalid p ∧ q →∼ r p∨∼q ∼q→p ∴∼ r

Determine whether the following categorical syllogisms are valid or invalid. Categorical syllogism 1 No M are...

Determine whether the following categorical syllogisms are valid or invalid. Categorical syllogism 1 No M are P. No M are S No S are P. Categorical syllogism 2 Some M are not P. Some M are S. Some S are not P. Categorical syllogism 3 Some M are not P. Some M are not S. Some S are P. Categorical syllogism 4 All M are P. All S are M. All S are P. Categorical syllogism 6 No M are...

Determine each of the following arguments’ forms to be valid or invalid. You may use the...

Determine each of the following arguments’ forms to be valid or invalid. You may use the Venn Diagram proof method, the rules/fallacies method, or any other method, of your choice. 1. Some athletes are not baseball players and some baseball players are not basketball players. Therefore, some athletes are not basketball players.2. All creationists are fundamentalists because all fundamentalists are religious people and all creationists are religious people. 3. As no conservationists are litterers, no environmentalists are litterers, because all...

Introduction to logic: Translate each argument using the letters provided and prove the argument valid using...

Introduction to logic: Translate each argument using the letters provided and prove the argument valid using all eight rules of implication. Sam will finish his taxes and Donna pay her property taxes or Sam will finish his taxes and Henry will go to the DMV. If Sam finishes his taxes, then his errands will be done and he will be stress-free for a time. Therefore, Sam will finish his taxes and he will be stress-free for a time. (S, D,...

7. Prove that the folowing is a valid argument, explaining each rule of inference used to...

7. Prove that the folowing is a valid argument, explaining each rule of inference used to arrive at the conclusion. “I take the bus or I walk. If I walk I get tired. I do not get tired. Therefore I take the bus.” “All lions are fierce. Some lions do not drink coffee. Therefore, some fierce creatures to not drink coffee."

Are the criticisms of absorption cost systems are valid or invalid and why?

Please use any of the methods to prove whether each of the following arguments is valid...

Please use any of the methods to prove whether each of the following arguments is valid or invalid. For each problem, please identify the method that you have decided to employ and make sure to show your work. 1. It is obvious that nuclear energy is needed. Nuclear energy is needed if and only if solar energy cannot be harnessed. And it is also true both that solar energy can be harnessed only if funds to do so are available,...

The argument in the following scenario was invalid. Discuss the logical fallacies that apply to the...

The argument in the following scenario was invalid. Discuss the logical fallacies that apply to the reasoning and why they matter in light of the scenario You are engaged in an intense discussion with your friend, Bill, who works in the IT department at your university. Bill complains that many students are using P2P (peer-to-peer) file-sharing applications on the university’s network to download excessive amounts of unauthorized copyrighted material. He also claims that the most effective solution to this problem...

Explain the difference between a valid argument and a sound argument, and give an example of...

Explain the difference between a valid argument and a sound argument, and give an example of each.

Which of the following is a valid C statement? Explain what each valid statement does. For invalid statements, explain the problem.

Which of the following is a valid C statement? Explain what each valid statement does. For invalid statements, explain the problem. a. int items [5]; b. int items []; c. int items = {3, 7,2 }; d. int items [] = {3, 7, 2}; e. int items [3] = {3, 7,2};

Subjects