Logic Part 2: Modus ponens, modus tollens and Chuck Norris

In Logic Part 1 I wrote about the use of basic Aristotelian syllogisms and their fallacies. I am now going to talk about propositional hypothetical logic. Sounds scary, right? Don’t worry! As with the basic syllogisms, it’s also pretty easy!
Propositional hypothetical logic

This is it…

If P, then Q

There you have it! If P, then Q, easy-peasy! ‘If P, then Q’ is called a hypothetical proposition.

A hypothetical syllogism is simply a local syllogism (like in Part 1) but where the first premise is a hypothetical proposition. For example,

Major premise: If P, then Q

Minor premise: P is true

Conclusion: Therefore, Q is true

By confirming P we can then also confirm Q. Here is the same hypothetical sllogism but with a Chuck Norris example,
If you get into a fight with Chuck Norris you will die

Alison got into a fight with Chuck Norris

Therefore, Alison died!

If you wanted to map this out on a Venn diagram it would look something like this…

The antecedent and the consequent

Now when you have a hypothetical proposition (If P, then Q) you have to know the correct terminology for the two parts. The first part (If P) is called the antecedent and the second (then Q) is called the consequent.

There are two types of hypothetical syllogisms: modus ponens and modus tollens.
Modus ponens

Modus ponens when the antecedent is affirmed to prove the conclusion true (the Chuck Norris example above is a case of modus ponens). Here is a reminder,

If P, then Q

P is true

Therefore, Q is true

Modus tollens

Modus tollens is when the consequent is denied thus inferring that the antecedent is false. For example,

If P, then Q

Not Q

Therefore, not P

We know that if P is true then Q would be true (becuase the minor premise is if P, then Q). From this, we can infer that if Q isn’t true it means that neither is P.
If you get into a fight with Chuck Norris you will die

Alison is not dead

Therefore, Alison did not get into a fight with Chuck Norris

With modus tollens the minor premise denies Q which allows us to infer that P is not true.

Modus ponens and modus tollens are the two true forms of hypothetical logic. As with the basic logical syllogisms the hypothetical syllogisms also have fallacies that provide false conclusions.
Denying the antecedent

The first logical fallacy is denying the antecedent,

If P, then Q

Not P

Therefore, not Q

Denying P does not allow us to infer anything about Q (even if the syllogism works )! Here is another example of denying the antecedent,

If you punch a police officer you will get arrested

John did not punch a police officer

Therefore, John did not get arrested

In reality, John could have got arrested for a number of reasons (even if he didn’t punch a police officer). It is because of this that when you deny the antecedent in a hypothetical syllogism your conclusion is unsound (quick reminder: ‘unsound’ means that it is not necessarily true; John may or may not have punched a police officer).
Affirming the consequent

The second fallacy of hypothetical logic is when you affirm the consequent,

If P, then Q

Q is true

Therefore, P is true

This is wrong because affirming Q does not allow us to infer anything about P. Hopefully this flamingo themed example will help,

If you have a pet flamingo named George then you are an awesome person

Fred is an awesome person

Therefore, Fred has a per flamingo called George

Whilst it is possible that Fred is an awesome person because he has a pet flamingo named George he could also be awesome for other reasons. Perhaps he used to train with Chuck Norris? Whatever the case, we cannot soundly conclude that he has a pet flamingo named George.

From the Venn diagram we can see that people who have a pet flamingo called Gertrude are never awesome people.

We’ve now covered basic logical syllogisms, logical fallacies, enthymemes, hypothetical propositional logic and hypothetical fallacies. More to follow in Logic Part III.