Question

For each of these arguments determine whether the argument is correct or incorrect and explain why.

a) Everyone enrolled in the university has lived in a dormitory. Mia has never lived in a dormitory. Therefore, Mia is not enrolled in the university.

b) A convertible car is fun to drive. Isaac’s car is not a convertible. Therefore, Isaac’s car is not fun to drive.

c) Quincy likes all action movies. Quincy likes the movie Eight Men Out. Therefore, Eight Men Out is an action movie.

d) All lobstermen set at least a dozen traps. Hamilton is a lobsterman. Therefore, Hamilton sets at least a dozen traps.

Short answer

a) The argument is correct. Reason: Universal instantiation and Modus tollens.

b) The argument is incorrect. Reason: The fallacy of denying the hypothesis.

c) The argument is incorrect. Reason: The fallacy of affirming the conclusion.

d) The argument is correct. Reason: Universal instantiation and Modus ponens.

Step-by-step solution

Rules of inference

1. Universal instantiation: This rule of inference is used to conclude that \(P(a)\)is true, where ais a particular member of the domain, with the given premise\(\forall xP(x)\).

2. Modus Ponens: \((P(a) \wedge (P(a) \to Q(a)) \to Q(a)\)

3. Modus Tollens: \((\neg Q(a) \wedge (P(a) \to Q(a)) \to \neg P(a)\)

Part (a)

Let “x is enrolled in the university.” denoted as\(P(x)\).

Let “x has lived in a dormitory.” denoted as\(Q(x)\),where x belongs to the domain.

So, the given argument can be written as:

1.\(\forall x(P(x) \to Q(x))\)

2. \(\neg Q(a)\) where arepresents Mia.

3. \(\neg P(a)\)

The first two are the premises and the third statement is the conclusion.

Step Reason

1.\(\forall x(P(x) \to Q(x))\) Premise

2.\(P(a) \to Q(a)\) Universal instantiation from (1)

3.\(\neg Q(a)\) Premise

4.\((\neg Q(a) \wedge (P(a) \to Q(a)) \to \neg P(a)\) Modus Tollens from (2) and (3)

Therefore,\(\neg P(a)\)is true and hence the given argument is correct.

Part (b)

Let “x is a convertible car.” denoted as \(P(x)\).

Let “x is fun to drive.” denoted as \(Q(x)\),where x belongs to the domain (All cars).

Given argument can be written as:

1. \(\forall x(P(x) \to Q(x))\)

2. \(\neg P(a)\)where arepresents Isaac’s car.

3. \(\neg Q(a)\)

The first two are the premises and the third statement is the conclusion.

Universal instantiation of premise (1) can be given as \(P(a) \to Q(a)\)but there is no law of inference such that \(((P(a) \to Q(a)) \wedge \neg P(a)) \to \neg Q(a)\)is true since \(\neg Q(a)\)can be false but \(\neg P(a)\) can still be true.

Therefore, it is a case of the fallacy of denying the hypothesis.

Part (c)

Let \(P(x)\) be the proposition that “Quincy likes x.”

Let \(Q(x)\) be the proposition that “x is an action movie.”, where x belongs to the domain (all movies).

Given argument can be written as:

1. \(\forall x(Q(x) \to P(x))\)

2. \(P(a)\) where arepresents the movie Eight Men Out.

3. \(Q(a)\)

The first two are the premises and the third statement is the conclusion.

Universal instantiation of premise (1) can be given as \(Q(a) \to P(a)\)but there is no law of inference such that \(((Q(a) \to P(a)) \wedge P(a)) \to Q(a)\)is true since \(Q(a)\)can be false but \(P(a)\)can still be true.

Therefore, it is a case of the fallacy of affirming the conclusion.

Part (d)

Let\(P(x)\)be the proposition that “x set at least a dozen traps.”

Let\(Q(x)\)be the proposition that “x is a lobsterman.”, where x belongs to the domain.

Given argument can be written as:

4.\(\forall x(Q(x) \to P(x))\)

5.\(Q(a)\)where arepresents Hamilton.

6.\(P(a)\)

The first two are the premises and the third statement is the conclusion.

Step Reason

5.\(\forall x(Q(x) \to P(x))\) Premise

6.\(Q(a) \to P(a)\) Universal instantiation from (1)

7.\(Q(a)\) Premise

8.\((Q(a) \wedge (Q(a) \to P(a)) \to P(a)\) Modus Ponens from (2) and (3)

Therefore,\(P(a)\)is true and hence the given argument is correct.