How does logic work?

Today I ran across what is likely the most elemental example of logic ever devised. Aristotle is credited with its earliest forms and I am therefore sure that it is taught to every first year philosophy student on the planet; but that does not take away its simplistic beauty.

It is a syllogism.

Much like the best scientific theories are simple, the best examples of logic are simple. So here we go.

A syllogism is an example of using ==deductive reasoning== to reach a conclusion.

Deductive reasoning starts from the general to produce claims about the specific. Inductive reasoning starts from specific observations to derive a general claim.

The syllogistic form is two or more premises that, if true, must mean the conclusion is true. For example:

All men are human
Jon is a man
Therefore, Jon is human

As with all simple things, it is only simple because the hard work has already been done by other people. There are actually two complicated things going on in this tiny piece of prose that make it a valid example of logic.

The two premises have a function: the major premise is the premise “All men are human”. It is major because it states the general claim about all men. The second premise “Jon is a man” is the minor premise. It is minor because it states the specific case, about Jon in particular, and how it relates to the major premise. In order to logically relate these two things, the minor premise must be shown to be related to the major premise. The reason for that follows.

The second thing that is going is called “distribution of the middle term”. The middle term is the thing that joins the major and minor premise. It’s easy to pick out because it appears (in some form) in both the major and minor premise. In this case, the middle term is man/men.

Distribution of the term “man” means that the major premise makes a claim about all members of that category. In this case, “All men are human”. And since the minor premise has been linked to the major premise via the middle term, logic dictates that the minor premise takes on all the characteristics that have been claimed for the major premise.

If all those criteria are present, you have a logical argument. A major premise makes a statement about all members of population (distributed) and the minor statement claims the specific case is included in the population of the major premise (middle term), so therefore the conclusion must be true.

That’s a nice text book answer, but let’s look at some examples when things go wrong.

What happens if the major premise does not distribute the middle term?

Some men are human
Jon is a man
Therefore, Jon is human

==This is a logic error==

Since the middle term is not distributed (ie, only ‘some’ men are human, not ‘all’ men) it becomes obvious the conclusion is now suspect. Jon may or may not be human in this case.

What happens if the middle term is omitted?

All men are human
Jon is tall
Therefore, Jon is human

==This is a logic error==

Again, it becomes very obvious how the conclusion becomes invalid if the rules of logic are not followed. The lack of a middle term means the two premises are not tied together in any way, so the required relationship fails.

My final example deals with the distinction between logic and truth. It is entirely possible to construct a valid logical argument that is not true. Using the same model, it may look like this:

All puppies can fly
Fido is a puppy
Therefore, Fido can fly

==This is a factual error==

This is a perfectly logical statement. It follows all of the rules. I stated a major premise, I ensured the existence of a middle term between the major and minor premise and I distributed it properly therefore my conclusion must be correct. Sadly, it’s patently wrong, but truth is not the province of logic so the truth of the argument does nothing to invalidate its perfect logic.

In real life, things are rarely so obvious. But the tendency to equate logic with truth sometimes allows us to accept, or arrive at, incorrect conclusions because of coherent logical arguments even if they contain untrue premises. I can think of examples in social circles where conclusions are drawn about people based on the community groups they belong to. However, sometimes framing those arguments in a syllogism can highlight either the logical fallacies of them, or the untrue premises the conclusion relies upon.

Title image from xkcd