Logic is the study of argument and reasoning, the study of why certain conclusions follow from given premises. This preliminary definition is, however, a little too broad as it stands: everyday arguments may contain some emotional or rhetorical force (consider the arguments advanced by lovers and politicians!) and such arguments are not our concern here. Instead we will be concentrating on just one aspect of arguments – validity.
A valid argument is defined to be one in which given that the premises of the argument are true then the conclusion must be true or has to be true or – equivalently – if the premises are true then the conclusion cannot be false,. This definition may seem abstract and so some examples may help.
Consider the following argument:
- All iron is magnetic.
- My garden furniture is made of iron.
- Therefore, my garden furniture is magnetic.
If the two premises of this argument (labelled 1 and 2 above) are true – which they are – then the conclusion (labelled 3 above) must also be true; there is no way that the premises could be true and the conclusion false. But consider this argument:
- Some metals are magnetic.
- My cooking pans are made of metal.
- Therefore, my cooking pans are magnetic.
Here we have an example of an invalid argument: one where the premises are true but the conclusion is false. 4 is true and so is 5 – as my pans are made of aluminium they are made of metal – but the conclusion, 6, is false.
I hope that these examples are clear in suggesting the difference between a valid argument and an invalid one. After all, if we could not recognise the difference between valid and invalid arguments then we wouldn’t be able to start doing logic at all. What must be reinforced at this point, though, is that it is arguments that are valid or invalid but propositions – those items that are the premises and conclusions of arguments – that are true or false.
So far we have considered only a couple of particular arguments, and logicians are not interested in whether iron is or is not magnetic, which other metals are magnetic or what my garden furniture is made of. Instead, logicians are interested in the general forms of arguments and in the general forms of propositions. If we return to 1, the first premise of the first argument above, we can see that it shares something in common with:
All cats are felines.
All Germans are Europeans.
All people who understand logic are highly intelligent individuals.
All of these examples have a common form: there is a subject term of the proposition – iron, cats, Germans, persons who understand logic – and a predicate term where something is said to be a property of the subject – that it is magnetic, or a feline, a European or a highly intelligent individual. So all of these examples have a common form:
All S are P
where S is the subject term and P is the predicate term. If we look again at the first premise of the second argument (4 above), we can see that it has something in common with:
Some cats are ginger.
Some Germans are speakers of French.
Some logicians are not very good writers.
I hope that you are able to recognise the subject terms and the predicate terms in these examples.
But these two forms of propositions are not the only ones that there can be: logicians have traditionally recognised four forms of what they have called categorical propositions. These are:
All S are P
No S are P
Some S are P
Some S are not P
It is common to label these A, E, I and 0, respectively.
Again, there is a common form to the categorical propositions. They all have the form:
(Quantifier) S (copula) P
where the quantifier says how many of the subject term – all, none or some comes under the predicate term; the copula is a word which joins the subject and predicate terms together and the predicate term may be applied to, or denied of, the subject.
A syllogism ( is an argument with two premises and a conclusion, each of which is a categorical proposition. Put there is a further restriction: a syllogism contains three terms, a subject term, a predicate term and a middle term, where the middle term occurs only in the two premises and not in the conclusion. An example of a syllogism is:
All M are P
All S are M
Therefore, all S are P.
If this feels a little abstract then you may like an instance of this form of argument:
All humans are mortal.
All Greeks are humans.
Therefore, all Greeks are mortals.
How many syllogisms are there? Each of the two premises and the conclusion is a categorical proposition and there are four forms of categorical propositions. The mood of a syllogism is a list of the forms of the categorical propositions in the premises and conclusion: for example, the last example has the mood AAA. So are there only 64 (= 4 x 4 x 4) forms of syllogism? The answer is ‘no’ there is more to a syllogism than the form of the categorical propositions in the premises and conclusion. Compare the syllogism above with:
All dogs have four legs.
All cats have four legs.
Therefore, all cats are dogs.
This syllogism is clearly invalid but yet seems to be of the same form as that given earlier – they are both of mood AAA. But if we replace ‘cats’, ‘dogs’ and ‘four legs’ with S, P and M we can see that this syllogism has the form:
All P are M
All S are M
Therefore, All S are P.
The first syllogism had a first premise of the form ‘All M are P’ while the second had a first premise of the form ‘All P are M’ . That is, the order in which the middle term occurs is different – and important.
A complete specification of a syllogism can be given by its mood and figure, where the figure describes the position of the middle term in each of the premises. There are four possible moods for the syllogism:
|1st premise||M P||P M||M P||P M|
|2nd premise||S M||S M||MS||M S|
|Conclusion||S P||S P||S P||S P|
|figure i||figure ii||figure iii||figure iv|
Giving the mood and figure of a syllogism is sufficient to specify that syllogism. The above two examples are AAA figure i and AAA figure ii, respectively. As there are 4 figures and each can be of 64 possible moods then there are 256 (= 4 x 64) possible syllogisms.
The systematic study of the syllogism began with Aristotle, and occupied logicians for around 2000 years. The majority verdict is that there are 24 forms of valid syllogism – although this is contended. Various ways of deciding whether a syllogism is valid were developed: memorising the valid forms; learning rules that specified them; various methods of diagrams. These are given in any good textbook on the subject. Then a new era in logic began in 1872…