A Very Short Introduction to Logic

Some people in the modern world will boldly declare that we do not need logic. Logic is, after all, to them, nothing more than a convenient side effect of biology that helps us survive in a jungle or a conspiratorial system created to make sure mathematics and sciences remain inaccessible to certain groups of people. Neither of these framings of logic are true. In fact, logic can be framed in a third way - the way I will consider logic.

I consider logic to be almost below natural phenomena - describing exactly what are the boundaries of what is possible. I consider logic to be so fundamental that if other universes exist that are totally different to our own the exact same laws of logic will apply. After all, in what universe can some object be at once containing all properties of A and no properties of A?

Recently I have been studying logic via using the seminal text "How to Prove It" by Velleman. I have the most recent edition, from 2019. in the book Velleman teachs us how to think in a systematically logical way. The motivation for Velleman is to be able to prove mathematical statements, but I think for a more general reader the motivation would be to learn logic itself.

I'll give you a flavour of the system of logic, and I trust you will see the potential uses of it. Consider the following three arguments

What can we say about these arguments? Each of them has three lines, that is, two propositions and one conclusion. There are many more complex forms, but today let's just focus on this simple syllogism.

Arguments number one and three are most similar in the sense that they are valid. What does valid mean? Valid means that both propositions, if true, lead to the conclusion. We can see that if it must either snow or rain tomorrow, but it is too hot for snow, it must rain. Argument three is a more definitional argument. Notice how we have no idea what a scramjammer or a flangerumbler is; but we can still conclude that the argument is at least valid. Notice how these statements are always "true" or "correct" in the sense that if both propositions are true the conclusion must necessarily be true.

How would we attack such an argument then? We could do so by realising that both propositions may not actually be true. For example, must it really rain or snow tomorrow? Could neither happen? Is this losipolisy really a scramjammer? Could it not be something else?

This is in contrast to argument two – which is not valid. The propositions, even if both true, do not lead to the conclusion. When attacking this argument, you could try and argue that the propositions are not true, but in general when confronted with such an argument should point out that it is not valid.

I hope to turn this into a series of articles that will explain some of the techniques of logic and proofs. Keep an eye out for them!