Hint 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.

Fill in the following sentence:

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?

Fill in the following blanks to find truth values which make 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 _________.
For premise (1) to be true, and knowing that s is _______,  c can be ___________ or ___________.

Have you been able to find a set of truth values which make all of the premises true and make the conclusion false? What do you conclude about the validity of the argument?

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