Basic structural fallacies

A fallacy is a bad argument. These fallacies are similar to the valid argument patterns above, and are frequently confused with them. These argument forms, however, are invalid. It is a good idea to practise recognising the difference: you will need to pay close attention to the patterns.

This section looks at three common structural fallacies:

• The fallacy of affirming the consequent
• The fallacy of denying the antecedent
• Disjunctive fallacies

The fallacy of affirming the consequent

The fallacy of affirming the consequent is an invalid argument which is often mistaken for Modus ponens or Modus tollens. It has the following form.

If p then q
q
Therefore p

​In this fallacy, the consequent of the first premise is affirmed in the second premise. Such an argument isn’t valid. The first premise claims that when p occurs, q must also occur. But it does not claim that the occurrence of q guarantees the occurrence of p.

The problem is easy to see with a couple of examples.

P1) If Bernie Sanders is the president of the US, then Hillary Clinton is not president of the US.
P2) Hillary Clinton is not the president of the US.
                                                         
C) Bernie Sanders is the president of the US.

The first premise is true. It is true that ‘If Bernie Sanders is the president of the US then Hillary Clinton is not the president of the US’. There can only be one president of the US at a time, and if it is Sanders, then it isn’t Clinton. The second premise is also true. Hillary Clinton is not the president of the US. But it certainly doesn’t follow from that that Bernie Sanders is the president.

Here’s another example.

P1) If this shape is a square then its sides are equal in length.
P2) This shape’s sides are equal in length.
                                        
C) This shape is a square.

It is true that any square will have sides equal in length. But it is perfectly possible to have a shape with equal sides which is not a square. (An equilateral triangle, for instance.) So this conclusion does not follow from the premises.

It is relatively easy to see the problem with these examples. Sometimes examples which affirm the consequent are harder to spot. It helps to look at the form of the argument carefully. It often helps to work out the form using letters instead of statements. This prevents you from being distracted by what you know (or believe) is true.

Remember, the fallacy of affirming the consequent is an invalid argument form.

Can you correctly identify the argument form?

The fallacy of denying the antecedent

The fallacy of denying the antecedent is another invalid argument which is often mistaken for Modus ponens or Modus tollens. It has the following form.

If p then q
Not p
Therefore not q

Here an example of the argument: you should be able to see that this one is invalid.

P1) If it’s wrong to eat meat then it’s wrong to eat human beings.
P2) It’s not wrong to eat meat.
                                             
C) It’s not wrong to eat human beings.

It must be true that if it’s wrong to eat meat then it’s wrong to eat humans. But it does not follow from it being permissible to eat meat that it’s permissible to eat people. There can be compelling reasons to not eat people even if it turns out that eating other sorts of meat is okay.

When assessing these it’s important not to get distracted by what you know or believe. Consider this example:

P1) If Hillary Clinton is president then the president is a woman.
P2) Hillary Clinton is not president.
                                               
C) The president is not a woman.

The conclusion does not follow from the premises. It is possible to imagine a world in which it was true that Hillary Clinton was not president but where some other woman was president. That is a world where the premises are true and the conclusion is false. Therefore the argument is not valid.

Disjunctive fallacies

A disjunctive syllogism has the following form. It is valid.

p or q
Not p
Therefore q

This form, however, is not valid:

p or q
p
Therefore not q

There are, however, situations in which we would be willing to accept arguments of the second form. Here’s an example.

P1) Donald Trump or Joe Biden will win the next presidential election.
P2) Joe Biden will win the next presidential election.
                                                                  
C) Donald Trump will not win the next presidential election.

You may well look at this argument and say that it is valid: if the premises are both true, then the conclusion has to be true also.

The reason for this is that you know there can only be one winner of the next election. It is not possible for both Trump and Biden to win. So, if Biden wins, Trump cannot. Here, the real meaning of P1 is ‘Donald Trump or Joe Biden will win the next election, but not both’.

Sometimes in English we use the word ‘or’ to mean ‘or, and both are possible’, and sometimes we use or to mean ‘or, but not both’. The first of these is called an ‘inclusive or’, and the second is called an ‘exclusive or’.

The form given for a disjunctive syllogism above is valid on either interpretation of ‘or’. But this form

p or q
p
Therefore not q

is only valid if the ‘or’ is exclusive. That is, it can only be valid if it is not possible to have both p and q. This is something which can only be determined by considering the meaning of the claim made. It cannot be determined by looking at the form of the argument.

When you try to assess this sort of argument, think to yourself ‘Is it possible for both p and q to occur?’. If it is possible, then the argument isn’t valid. If it’s impossible, then the argument is valid. However, it may help to add ‘but not both’ to the disjunctive premise.

​Sometimes people claim that an ‘either… or…’ construction is used to show that both options cannot hold: that is, an ‘either… or…’ construction indicates an exclusive ‘or’. It might be useful if we used ‘either… or…’ in this way in English, but we don’t. If I said to you ‘Bring either beer or wine to the party; I don’t mind’, and you turned up with both, you would be rightfully put out if I then said ‘You brought both. You can’t come in. I said to bring either one or the other.’ The presence of the word ‘either’ does not tell you whether an exclusive ‘or’ or an inclusive ‘or’ is being used. You must use your common sense for that. A good rule of thumb is if you’re not sure, suppose the ‘or’ is inclusive.