Major premise: If a country has a strong economy, the government will be popular. Minor premise: The government is not popular. Conclusion:...

This argument is valid.  The fact that it is valid, however, does not necessarily mean that it is true.  A syllogism (which this argument is) is valid when its conclusion must be true if both premises are true.  In other words, the syllogism can be valid even if the premises are false.  It is valid so long as the conclusion would be true if the premises were true.

Our first premise in this syllogism says that “if a country has a strong economy, the government will be popular.”  What this means is that the government will always be popular if the economy is strong.  Put differently, the government will never be unpopular if the economy is strong.  Our second premise is that “the government is not popular.”  Looking at the first premise, we can see that the economy cannot possibly be strong in this situation.  If the government is not popular, it is not possible that the economy could be strong.  This means that, if the two premises are true, the conclusion must also be true.  Therefore, the argument is valid.

Remember, this does not mean that this argument is true.  There could be countries where the economy is strong but the government is so repressive that it is not popular.  However, that does not affect the validity of the syllogism.  The validity of the syllogism is based only on whether the conclusion must be true if the premises are true.