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What exactly do we mean by logic? The Oxford Compact English
Dictionary gives the definition as:
The science of reasoning, proof, thinking or
But what does logic mean to us and is that different to
will explore these questions here.
One way to think of logic is as the understanding of how ideas are
used in arguments. We often think of arguments as loud discussions
between two or more people who don't agree with each other. I am
sure that you can think of times when you have argued recently -
perhaps it was with a brother or sister over what to watch on TV or
when they should go to bed.
However, an argument can simply be putting forward ideas which
explain your point of view, not necessarily in front of someone who
isn't of the same opinion. If these ideas are not logical, then
your argument will fall flat on its face. Let us look at this in
We use reasoning in the world around us without really thinking
anything of it. For example, if I were told that David Beckham is
an Englishman and that all the English are Europeans, then I can
work out that David Beckham is a European. The steps that I have
taken to come to this final statement involve logic. On the other
hand, if I were told that Emma is in class 4G and some pupils in 4G
are right-handed, then if I conclude that Emma is right-handed, I
wouldn't be thinking very logically.
However, Emma could
right-handed couldn't she? It is very important to realise that
logic and truth are not the same thing. It may be that Emma is
indeed right-handed and I am correct, but I have not used logic to
arrive at this conclusion. Similarly, David Beckham is only
European if the two facts that I was originally told are true. So,
even though I have used logical thinking, if in fact David Beckham
isn't an Englishman then he is not necessarily European.
In coming to the conclusion that David Beckham is a European (let's
assume that we know for certain he is English so our conclusion is
right), we are really using our understanding of "sets" and
"subsets" of these. English people are a subset of the set of
Europeans. In maths, Venn diagrams are often used to show this sort
of information. Below is an example which shows how you would place
the numbers 1 to 15 in the sets of "Odd numbers" and "Prime
Venn diagrams frequently help us to make our deductions more
The Ancient Greeks were the first to really develop logic, in
particular Aristotle who lived from 384 to 322 BC. Aristotle put
forward the notion of a syllogism . This is an argument in
three parts, like the examples above. A syllogism consists of two
premises and a conclusion . The first premise must
have one thing in common with the second premise. The second
premise must have one thing in common with the first premise. The
conclusion must have one thing in common with both premises.
Aristotle's example is:
||Every Greek is a person
||Every person is mortal
||Every Greek is mortal
Aristotle believed that logic should be
investigated before any other areas of knowledge. He made a lot of
progress in the understanding of logic, but all of his analysis was
done in everyday language.
It wasn't until much later that Leibniz
took Aristotle's ideas a stage further. Leibniz (who lived between
1646 and 1716) was taught Aristotle's theories at school, but
wasn't satisfied with them. He suggested that a scientific language
needed to be developed which could be more precise than using
everyday words. Leibniz got a long way in creating symbolic logic which used formulae to
help work through deductions.
Boole refined these formulae to produce a
special form of algebra called Boolean algebra. Mathematicians can
use this to write and analyse logical ideas. Others followed in his
footsteps, for example Frege and Peano who were convinced that
maths could be reduced to logic. More recently, Bertrand Russell
and Alfred Whitehead wanted to prove this. In the process they
found that this could generate paradoxes . A paradox is an expression
that seems to contradict itself, like "this statement is false" or
"I am telling you the truth when I say I am a liar".
Even though Russell and Whitebread
encountered these problems with mathematical logic, it is used a
great deal in the world today. Boolean algebra has wide
applications in telephone switching and computer technology.
The St Andrew's website www-gap.dcs.st-and.ac.uk/~history/Mathematicians/
has more details on all of these logicians as well as hundreds of