# Logic

The basic part of logic in mathematics is quite
simple. It is all built around simple *binary* statements or *propositions*.
These statements are ones that can either be true *or *false. Here’s an
example of a proposition:

My name is Michael

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This statement can either be true or false. My name
can’t ‘sort of’ be Michael, it either is or it is not. Here’s an example of a
statement which *isn’t* a binary statement:

How many names do I have?

This statement is a question. It can’t be true or
false, it needs a *quantitative* answer (i.e. the number of names I
have). Propositions will generally make a statement describing or predicting
something. Questions or statements that require an answer can’t be
propositions.

There are five major *connectors* that you can use
in logic to write *logic sentences*.

### OR connector

The ‘OR’ connector means the same as the English word. For instance, consider this statement:

This afternoon I will play tennis OR I will read (or both)

Here I have used the OR connector to tell the reader
that *one* of the two events is going to happen this afternoon, or both.
If I used ‘t’ to describe tennis, and ‘r’ to describe reading, then I could
rewrite this statement:

_{}

The _{} symbol is the ‘OR’ symbol – it
means at least *one *of the two things on either side of it will happen.
Remember the ‘or both’ part – it’s commonly forgotten.

### AND connector

This also comes straight from what ‘and’ means in the English language. Take this sentence for example, a slight modification of the last one:

This afternoon I will play tennis AND I will read

This tells the reader that *both* events are going
to happen this afternoon, me playing tennis but also me reading. The
mathematical symbol for ‘AND’ is _{}:

_{}

### IMPLIES connector

The word imply is a bit more complicated than ‘and’ and ‘or’, but not by much. It is used in everyday language, for instance something like:

Jason goes to primary school – this implies he is less than 15 years old

It’s a bit trickier to analyse whether an imply statement is true or false. There are three possibilities:

·
The first part of the statement is true, and the second part is
true as well. In that case the *implication* is *true*. This would
be the case if Jason went to primary school, and he was under 15 years old.

·
The first part of the statement is true, but the second part is
false. In that case the implication is *false*. This would be the case
if Jason went to primary school but was 30 years old for some reason.

·
The first part of the statement is false. In this case, the
implication is *always* true. This is sometimes confusing for people.
When the first part of the statement is false, the whole implication really has
no meaning whatsoever, it’s not really true *or* false. Whether the
second part of the statement is true can only be judged when the first part is
true – the second part *leads on* from the first part. So the convention
is that when the first part is false, we just say that the whole meaningless
implication is *true*.

Say Jason doesn’t go to primary school. Does this tell us anything for sure about whether he’s less than 15 years old? Not really. He could be 2 years old – too young for primary school but still under 15 years old. Alternatively, he could be 40 years old and working in a factory. Because we don’t have any way of judging whether the statement’s true or false, we give it the benefit of the doubt and just say it’s true.

Often the implication connector is talked about in terms of an ‘if then’ statement, like this:

*If* Jason goes to
primary school *then* he is a kid

The mathematical symbol for IMPLIES is an arrow with
two lines – _{}.
If we use ‘PS’ for primary school and ‘K’ for kid under 15 years old, the
mathematical way to write the last statement is:

_{}

### EQUIVALENCE connector

This is when two separate statements depend on each other. If something happens in one statement then something has to happen in the other statement. For instance:

I will play tennis this afternoon if I read a book, and I’ll read a book if I play tennis.

This is probably the most complicated one to analyse. Here are the four situations that can happen:

·
I play tennis, but I don’t read a book. In this case the
proposition is *false*, because it says that if I play tennis I’ll read a
book, which I didn’t.

·
I play tennis, and I read a book. This makes the proposition *true*.

·
I don’t play tennis, but read a book. This makes the proposition
*false*. The statement says that I *will* play tennis if I read a
book. I’ve read a book, but I haven’t played tennis – the statement is false.

·
I don’t play tennis, and I don’t read. This makes the
proposition *true*. The statement says that I’ll only play tennis if I read
a book, and that I’ll only read if I play tennis. Since I didn’t read, I
didn’t play tennis and vice versa. So not doing either agrees with the
statement and makes it true.

This connector is often called the ‘if and only if’ statement. You can rephrase the statement this way:

I will play tennis this afternoon ** if and only if**
I read a book, and I will read a book

**I play tennis this afternoon.**

__if and only if__I find this form easier to work with. The mathematical
symbol for EQUIVALENCE is a double headed arrow – _{}. We can rewrite the statement
mathematically like this:

_{}

### NOT connector

This is when you take something and make it the
opposite of what it was before. So if it was true, you make it false. If it
was false, you make it true. The logic symbol for this NOT connector is ‘**~**’.
So if I start with a statement like:

I will play tennis this afternoon

I can convert this into a mathematical statement just like this:

_{}

If I put a NOT symbol in front of this, I reverse the statement:

_{}

This reverses the statement so it is now:

I will not play tennis this afternoon