"An Introduction To Logic by The Philosophers' Web Magazine."
Introduction To Logic
An Introduction To Logic - Part 1
By Francis Moorcroft
Logic is the study of argument and reasoning, the study of why certain conclusions
flow from given premises. This preliminary definition is, however, a title 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 flowing 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 recognize 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 (4above), 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 recognize 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 recognized 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:
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
Further reading
Introduction to Logic, (many editions).
Francis Moorcroft
Department of Philosophy
University of Hull.