To determine whether an argument is valid or not, we should prove that it is impossible that all the premises are true and the conclusion is false.

The way we do that proof is by applying one or more rules of inference on the premises and obtaining the conclusion.

Like in other exercises, it is helpful to use propositional variables or predicates instead of the proposition itself. That will keep our work as clear as possible.

Table of Contents

• 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.

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

First, we define the predicates. Notice that in this case, you cannot represent the premises using propositions.

Let:

– P(x) = “x is a student in this class”

– Q(x) = “x understand logic”

(1) ∀x (P(x) -> Q(x)) premise

(2) P(Xavier) premise

Now, we need to apply one or more rules of inference, to draw the conclusion that Xavier understands logic.

(3) P(Xavier) -> Q(Xavier) Universal instantiation using 3

(4) Q(Xavier) modus ponens using 2 and 3

The argument is valid because the conclusion follows from the truth of the premises of the argument.

Note: A simpler way to solve this exercise is to use universal modus ponens using 1 and 2.

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

Following the same approach as in the previous exercise, we get:

Let

• P(x) = “x is a computer science major”
• Q(x) = “x takes discrete mathematics”

(1) ∀x (P(x) -> Q(x)) premise

(2) Q(Natasha) premise

(3) P(Natasha) -> Q(Natasha) Universal instantiation using (1)

This argument is invalid. We cannot conclude that P(Natasha) is true from the 2 and 3.

It is a fallacy of affirming the conclusion.

Looking at this statement in another way, we can understand that someone that takes a discrete mathematics course, does not necessarily need to be a computer science major. Therefore, we cannot conclude that everyone that takes discrete mathematics is a computer science major.

c) All parrots like fruit. My pet bird is not a parrot. Therefore, my pet bird does not like fruit.

This exercise is similar to the previous one. It is a fallacy.

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

Let

• P(x) = “x eats granola every day”
• Q(x) = “x is healthy”

(1) ∀x (P(x) -> Q(x)) premise

(2) ¬Q(Linda) premise

(4) ¬P(Linda) universal modus tollens using 1 and 2

This argument is valid because the conclusion follows from the truth of the premises of the argument.

Related posts:

• What rule of inference is used in each of these arguments?
• Use rules of inference to show that the hypotheses “Randy works hard,” “If Randy works hard, then he is a dull boy,” and “If Randy is a dull boy, then he will not get the job” imply the conclusion “Randy will not get the job.”
• What rules of inference are used in this famous argument? “All men are mortal. Socrates is a man. Therefore, Socrates is mortal.” 
• For each of these collections of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. 
• For each of these arguments, explain which rules of inference are used for each step