Solution for Section 1.3 Question 1b: Invalid argument approach

1b)  Let c represent "Bill is a cheater",  and let s represent "Bill sits in the back row." Now write each of the premises and the conclusion in symbolic form.

The first premise (p₁) is "If Bill is a cheater, then Bill sits in the back row." This can be written symbolically as  c s.
The second premise (p₂) is "Bill sits in the back row." This can be written symbolically as  s.
The conclusion (q) is "Therefore Bill is a cheater." This can be written symbolically as  c.

Remember that an argument is written as a conjunction of the premises implies the conclusion. So this argument can be represented as
[(c s) L (s)] c.

To determine if it is possible for an argument to be invalid, we are looking for truth values for the variables which make the conclusion false but all of the premises true.

For the conclusion to be FALSE, we need c to be FALSE.

For the premises to be TRUE we need:

• c s to be true (1);
• s to be true (2).

Can you find a set of truth values for c and s, which make the conclusion false but the premises true?
Since premise (2) has the least number of variables, start with that premise.

For premise (2) to be true we need s to be TRUE.
For premise (1) to be true, and knowing that s is true,  c can be TRUE or FALSE.

We have now found a set of truth values which make the conclusion false and the premises true, namely s true and c false. Therefore the argument is NOT valid.

Back to Section 1.3