Question

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

a)     All students in this class understand logic. Xavier is a student in this class. Therefore, Xavier understands logic.

b)    Every computer science major takes discrete mathematics. Natasha is taking discrete mathematics. Therefore, Natasha is a computer science major.

c)     Everyone who eats granola every day is healthy. Linda is not healthy. Therefore, Linda does not eat granola every day.

Answer 1

a) Lets assume:

P(x) = "x is a student in this class"

Q(x) = "x understands logic"

We can rewrite the given sentence:

1. $\forall x(P(x)\to Q(x))$ premise
2. P(Xavier) premise
3. $P(Xavier) \to Q(Xavier)$ universal installation from 1
4. Q(Xavier) modus ponens from 2 and 3

Step 4 means that "Xavier" understand logic" and this corresponds with the given sentence thus the argument is correct.

b) Lets assume:

P(x) = "x is a computer science major"

Q(x) = "x takes discrete mathematics"

We can rewrite the given sentence:

1. $\forall x(P(x)\to Q(x))$ premise
2. P(Natasha) premise
3. $P(Natasha) \to Q(Natasha)$ universal installation from 1

There is no rule for inference that allows us to conclude P(Natasha) which means "Natasha is a computer science major." and thus the argument is incorrect.

c) Lets assume:

P(x) = "x eats granola every day"

Q(x) = "x is healthy"

We can rewrite the given sentence:

1. $\forall x(P(x)\to Q(x))$ premise
2. $\neg Q(Linda)$ premise
3. $P(Linda) \to Q(Linda)$ universal installation from 1
4. $\neg P(Linda)$ modus tollens from 2 and 3

Step 4 means that "Linda does not eat granola every day." and this corresponds with the given sentence thus the argument is correct.

a) correct

b) incorrect

c) correct