Common argument patterns

Some argument types are so common they have their own names. Learning to recognise these patterns will help you recognise valid arguments.

This section introduces four common argument patterns, and some simple variations on them.

All of the argument patterns in this section are valid.

• Modus ponens
• Modus tollens
• Disjunctive syllogism
• Hypothetical syllogism
• Some notes on Conditionals and generalisations

​Modus ponens

Consider the following argument:

P1) If Rover is a dog, then Rover is a mammal.
P2) Rover is a dog.
                           
C) Rover is a mammal.

It has the following pattern:

If p then q
p
Therefore q

The letters ‘p’, ‘q’, ‘r’ etc. are a traditional way of representing statements. Any statement can be inserted in the place of ‘p’ and ‘q’, and the resulting argument will be valid. Learning to recognise some  common argument forms makes it much easier to identify valid (and invalid) arguments.

Here’s another argument with the same pattern.

P1) If Winston Peters is a Cabinet Minister, then New Zealand First must have entered into a coalition agreement with either the National Party or the Labour Party.
P2) Winston Peters in a Cabinet Minister.
                                                            
C) New Zealand First must have entered into a coalition agreement with either the National Party or the Labour Party.

This argument has longer statements, but the basic pattern is the same. The pattern is identified as Modus ponens, so it must be valid.

Modus tollens

Modus tollens is another very common valid argument form. It has the following pattern:

If p then q
Not q
Therefore not p

The first premise claims that if p occurs, then q must also occur. The second premise points out that q hasn’t occurred. So it has to follow that p hasn’t occurred either. Why? Because if p had occurred, then q would also have occurred. And we know q hasn’t occurred.

It may be easier to see with an example.

P1) If Hillary Clinton has won the last election, then she would be president.
P2) Hillary Clinton is not president.
                                              
C) Hillary Clinton did not win the last election.

It’s clear that it is not possible for this conclusion to be false while these premises are true. This is a valid argument.

Try this one. Remember, you’re looking to see whether the pattern of Modus tollens applies.

Try another one:

Disjunctive syllogism

A ‘disjunction’ is a complex statement where two statements are joined with an ‘or’ (or another word serving the same role). A disjunctive syllogism is a valid argument with the following form:

p or q
Not p
Therefore q

This argument form is valid because the initial premise dictates that one of the two options must hold, and the second premise asserts that one does not hold. It follows that the other must hold.

Here is an example.

P1) Chess is the most challenging board game or Monopoly is the most challenging board game.
P2) Chess is not the most challenging board game.
                                                               
C) Monopoly is the most challenging board game.

It does not matter whether it is what is before the ‘or’ or what is after the ‘or’ which is denied. But it must be denied.

You can practice applying the pattern in the following questions.

A disjunctive syllogism is often expressed using an ‘either… or… construction. For instance:

P1) Either there will be a recession, or house prices will continue to rise.
P2) House prices will not continue to rise.
                                             
C) There will be a recession.

Sometimes a disjunctive syllogism uses ‘either’ along with ‘or’, and sometimes it doesn’t. It doesn’t change the force of ‘or’ either way. ‘Either’ is generally used rhetorically, to emphasise the contrast between the two options. There is more on ‘either’ in the next section on Structural fallacies.

Hypothetical syllogism

A hypothetical syllogism creates a ‘chain’ of conditional claims. So long as the links of the chain occur in the right way, where each leads to the next, the intermediate links can be omitted. Here’s an example:

P1) If housing prices continue to rise, then rents will continue to rise.
P2) If rents continue to rise, then rental accommodation will become unaffordable for the working poor.
                                                                                
C) If housing prices continue to rise, then rental accommodation will become unaffordable for the working poor.

It has the general form:

If p then q
If q then r
Therefore if p then r

When checking an argument form, the order of the premises is irrelevant. That is because validity treats the premises as a collection of claims. You are welcome to change the order of the premises if it makes it easier for you.

Conditionals and generalisations

Conditionals

Several of the basic argument patterns above use conditional claims.

A conditional is an ‘if… then…’ statement. The ‘if…’ part of the statement is called the ‘antecedent‘, and then ‘then…’ part is called the ‘consequent‘.

Conditionals are sometimes expressed in a different order. The ‘antecedent’ is the ‘if…’ clause no matter what order the parts are presented in.

So, the English sentence ‘If Borka is a goose, then Borka is a bird’ means the same thing as ‘Borka is a bird if she’s a goose’.

The same conditional can be expressed in more ways than this. One version of a conditional that people find especially tricky is ‘only if’. Suppose there is a sign in the university carpark which says: “Staff permit holders only”. This means “You can only park here if you are a staff permit holder”. Think about what this involves. It does not mean that if you are a staff permit holder you must park there. It doesn’t forbid staff permit holders from parking in other places. What it means if that if you are not a staff permit holder, you must not park there. That is equivalent to “If you park here, then you are (must be) a staff permit holder”.

What this shows is that the order of antecedent and consequent in a conditional statement is very important, and they cannot simply be reversed. To see this, let’s return to the goose example.

It is true that “if Borka is a goose, then Borka is a bird”. However, it is not the case that “If Borka is a bird then she is a goose”. Some birds are not geese. So, “Borka is a bird only if she is a goose” is false. The ‘only if’ claim which is equivalent to “If Borka is a goose then she is a bird” is “Borka is a goose only if she’s is a bird”.

Conditional statements can also be expressed using ‘unless’. “If Borka is a goose then she’s a bird” is equivalent to “Borka is not a goose unless she’s a bird”. We often use ‘unless’ in contrast to a ‘not’ claim. So, someone might say “I won’t babysit for you unless you pay me”, which means the same as “If I babysit for you then you are (must be) paying me”.

Conditionals and generalisations

There is also an important relationship between conditionals and generalisations. The reason why ‘if Borka is a goose, then Borka is a bird’ is true, is because the generalisation ‘All geese are birds’ is true.

Any hard generalisation can be expressed as a conditional. ‘All geese are birds’ can be expressed as ‘if something is a goose, then it is a bird’. This means that the basic argument patterns which use conditionals all have forms which use generalisations instead.

All As are Bs
x is an A
Therefore x is a B

is a variation on Modus ponens, using a generalisation in the place of a conditional.

Here is an example of a hypothetical syllogism, using generalisations instead of conditionals:

P1) All squares are quadrilaterals.
P2) All quadrilaterals are polygons.
                                         
C) All squares are polygons.

Any argument with this form will be valid.

Here are some for you to try: