Introduction to Logic:

People often incorrectly say things like "that's illogical", or "the only logical thing to do". Of course, as part of common parlance, we all know exactly what they mean, but in the world of logic, their statements are erroneous. For instance the man who supports Manchester United one week, then switches allegiance to Manchester City the next, is fickle, but he may not be illogical. Before we look at how to use logic, we need to ask one important question:

What is Logic?

It is perhaps a slight miscomprehension to link logic totally with what is true and what is false. Although notions of truth are involved, logic is more concerned ultimately with the validity of arguments, and the consistency of statements. Logic is a tool we can use to test whether a particular argument is valid or not.

Arguments:

Arguments normally consist of a set of statements, known as premises, and a conclusion. The nature of an argument is such that if the premises are true, then the conclusion that follows from them must also be true. An often quoted example is:

1) All men are mortal.
2) Socrates is a man.
3) Therefore Socrates is mortal.

It is clear to see that if statement 1) is true and statement 2) is true, then there is no way that 3) can be false. We therefore have a valid argument. Now, we must not get the idea of a valid argument confused with the idea of a good argument. Let's take the following example:

1) All men can fly like Superman and have six legs.
2) Socrates is a man.
3) Therefore Socrates can fly like Superman and has six legs.

Now this is a totally insane argument, and no one in their right mind would accept it as being true. However, it is logically valid because if the premises (statements 1 and 2) are true, then the conclusion is also true. The fundamental root of logical validity lies in the concept of logical possibility.

Logical Possibility:

If something is logical, then it does not necessarily mean that it is true in the world, or even probable. To help gain a concept of logical possibility, it is helpful to consider the idea of "possible worlds".

Let us imagine that there are very many, perhaps an infinite number of, "worlds". With this system, we can imagine that things are different in each. One of these worlds is the "actual world" - i.e., the one in which we live - but all the others are our own creation. Now, for something to be logically possible, all we need is for it to be able to happen in one of these worlds we created. Now, take the argument I quoted earlier:

1) All men can fly like Superman and have six legs.
2) Socrates is a man.
3) Therefore Socrates can fly like Superman and has six legs.

Now in our possible worlds, we can create one where men do in fact fly like Superman and have six legs. It is not the actual world of course, but that does not matter here. The possible worlds are a way of rejecting any set of statements of an argument that contain a contradiction. If an argument has a contradiction within it, then it cannot be possible in any possible world. Here is an example to demonstrate:

1) All lemons are yellow all over.
2) This is a lemon.
3) Therefore this is blue all over.

Clearly, there is a contradiction here. There is no possible world that we can create where we have a lemon that is both yellow all over and blue all over. This is therefore a logical impossibility. We can use ideas of logical possibility to define validity:

A set of statements forms a valid argument if there is no possible world in which the premises are true, and the conclusion false.

It follows from this that any conclusion can follow a contradiction and it still be a valid argument. This is because there is no possible world where the premises are true and the conclusion false since the premises can never be true anyway.

Necessary Truth:

A necessary truth, also known as a tautology, is something that is true in all possible worlds. For instance, mathematical laws are necessarily true. We cannot conjure up an imaginary world where two plus two does not equal four. The number it equals in a world might not be called "four", but it will be the same thing, just as water is the same as H2O. An example of a necessary statement outside the realms of mathematics would be something along the lines of:

"My shoes are black all over, or they are not black all over."

It is clear that there is no possible situation where this is not true.

If a necessary truth should be the conclusion of an argument, then that argument is always valid because the conclusion can never be false in any possible world. For instance:

1) My shoes are black all over.
2) My shoes are brown all over.
3) Therefore two plus two equals four.

Here there is an obvious contradiction in the premises, yet the conclusion is a necessary truth, so the argument is valid. A rule of logic is that anything entails a tautology.

Conversely, it is another rule that anything can follow a contradiction, as this is another situation where it can never be the case that the premises are true and the conclusion false.

Wilfred Hodges, "Logic"

Thanks to tdent for pointing out that I neglected to mention that anything can follow a contradiction.