Invalid Statements

Invalid Statements#

Not everything is true. Even if something is written as a proper logic statement, that doesn’t mean it is true. Proving something is invalid is much easier that proving it is true. You only need one example that breaks the rules.

The statement “all even numbers are prime” is false. The number 6 is even, but is it not prime. It is the created from \(2 * 3 = 6\).

Invalid Arguments#

A logic argument is invalid if the premises can be true and the conclusion can be false. When doing proofs, we were proving that if the premise is true then the conclusion had to be true as well.

The following argument is invalid.

\[A, A \implies B \therefore \neg B\]

We can look at the whole truth table for this expression. We only care about rows where the premise are both true.

A	B	Premise 1 (\(A\))	Premise 2 (\(A \implies B\))	Conclusion (\(\neg B\))	Notes
True	True	True	True	False	Premises true but conclusion is false
True	False	True	False	True	A premise is false
False	True	False	True	False	A premise is false
False	False	False	True	True	A premise is false

This argument is invalid because when A is true and B is true, the premises are both true but the conclusion is false. This row means the argument cannot be proven. There is a situation where it does not hold.

To show something is invalid you need to find a setting of the variables that makes all the premises true. That setting of variables also needs to make the conclusion false. You only need one example to show something is invalid. It is entirely possible that in some situations that conclusion is true. There just needs to be one example that doesn’t work for a statement to be valid.

Remember, when a statement is valid, it is always valid.

Example 1#

Is it valid or invalid that \(A \vee B \therefore A \wedge B\)?

This is invalid.

Let A=True and B=False
The premise \(A \vee B\) is true because true \(\vee\) false = true.
The conclusion is false because true \(\wedge\) false = false.

There exists a setting of variables that makes the premise true and the conclusion false. That means the argument is invalid.

Notice that with A=True and B=True, the conclusion is true. It is possible for the argument to work sometimes. It just doesn’t work all the time.

Example 2#

Is it valid or invalid that \(A \implies B, B \therefore A\)?