Conditional Statements

This is VERY IMPORTANT!

• We will show the failure of the fairness criteria by showing the first part of the conditional is true and the second part is false.

• For example, consider the conditional statement:

“If you buy a lottery ticket then you will win a million dollars.”

• To show this is not true, somebody has to buy a lottery ticket and not win a million dollars.

Proving a Conditional is False

In the example “If you buy a lottery ticket, then you will win a million dollars.”

• The first part “you buy a lottery ticket” is called the antecedent.

• The second part “you will win a million dollars” is called the consequent.

To show a conditional statement is false, we must show the antecedent is true and the consequent is false.

In any other case, the conditional statement is logically true.

Logic Puzzle

Consider the 4 boxes below as cards lying on a table.

You can only see one side of the card but it is given that on one side there is a number and on the other side is a letter.

A 3 D 4

Consider the conditional

If there is a “D” on one side, then there is a “3” on the other side.

Which cards must be flipped over to reveal the other side so as to verify that the conditional is actually true?

Logic Puzzle

Consider the 4 boxes below as cards lying on a table.

You can only see one side of the card but it is given that on one side there is a number and on the other side is a letter.

A 3 D 4

The given conditional was

If there is a “D” on one side, then there is a “3” on the other side.

The answer is you must flip over the “D” and the “4” cards.

(1) You have to check if the “D” card has a “3” on the other sideAnd (2) you also have to check if the “4” card has a “D” on the other sideBecause both of those possibilities would prove the conditional to be false.

Another Logical Question

Now consider the following situation:

Suppose you are the owner of a bar on South Beach. The police pull up outside and tell you there was a report of under-age drinking in your establishment.

You tell the police: “If there is anyone inside drinking, then they are of legal drinking age.”

Suppose the boxes below are again cards but this time represent the 4 people in the bar. On one side of the card is person’s age and on the other side of the card is the type of refreshment that person is drinking.

Which of these cards must the police flip over to verify that you are telling the truth?

Coke 33 Vodka 15

Logic Does Not Come Naturally

Which of these cards must the police flip over to verify that you are telling the truth?

Coke 33 Vodka 15

Now the question is very easy! Only the second two cards matter – those are the ones that must be flipped to check if the statement is false. Those are the only cases were it is possible to prove the statement is false.

In this situation the logic is easy – because it is familiar – and its possible that something in our brains is also very good at finding someone who is somehow cheating at something- but the logic is exactly the same as the previous but more abstract example. In the pure abstract case, you might have more trouble with the logic – it really doesn’t come naturally – so be very careful!

You tell the police: “If there is anyone inside drinking, then they are of legal drinking age.”

Fairness Criteria

• All Fairness Criteria can be written in the form of a conditional.

• To prove a method of voting fails one of these fairness criteria, we must find an example where the first part of the criterion (the antecedent) is satisfied and the second part (the consequent) is not satisfied.

Proving the Failure of Fairness Criteria

• For example, consider the majority criterion, which states

“If a candidate has a majority of first place votes, then that candidate should win.”

To prove that a certain method of voting (like Borda Count) can sometimes fail the majority criterion will must find an example where a candidate has a majority of first places votes and does not win the election.

• As another example, suppose we want to show the plurality method fails the Condorcet Winner Criterion, which states

– “If a candidate beats all other candidates in one-on-one contests, then that candidate should win the election.”

– To prove plurality fails this criterion we must find an example where a certain candidate beats all other candidates in one-on-one contests and does not win the election.