1. About Logic

Where is Anna?

If Anna is at work, her car is not in the driveway.
Her car is not in the driveway.

So Anna is at work.

Given we can be sure that the first premise has no exceptions, is that a valid conclusion?

It may feel convincing, and you start to rely on it. You send Anna a message asking whether she wants to meet at the coffee shop during her lunch break. But then Anna replies that she has a day off and is at the spa.

The premises are all true:

If Anna is at work, her car is indeed not in the driveway. It is parked in front of her office building.

Also her car is now not in the driveway. Her car is at the spa.

So the conclusion may have felt convincing at first, but then a counterexample emerged.

If you saw that right away, good. But more complicated conclusions exist.

Are we supposed to exclude all counterexamples when we want to draw a valid conclusion?

How should we do that?

Logic

Logic studies inference.

We can now look more closely at what distinguishes a conclusion that really follows from its premises from one for which a counterexample remains possible.

Consider:

15 is even or 15 is divisible by 3.
15 is not even.

Thus: 15 is divisible by 3.

Or:

All men are mortal.
Socrates is a man.

Thus: Socrates is mortal.

In the first examples, the conclusion follows from the premises.

Now compare:

15 is not even or 15 is divisible by 3.
15 is not even.

Thus: 15 is divisible by 3.

Or:

All men are mortal.
Socrates is mortal.

Thus: Socrates is a man.

These do not follow in the same way from the premises.

But how can we know that the first two have no counter examples, while the second pair has them?

What constrains the first two conclusions, and why can that constraint become so convincing?

What is missing in the second two cases, and why do their conclusions fail to acquire the same force?

To answer that, we need a way to make the relevant possibilities explicit enough to inspect.

This essay is not an introduction to logic. For that purpose it is much too thin.

We will construct only as much formal machinery as the examples require, keeping the resulting systems small enough that the relevant possibilities and mappings remain open to inspection.

The aim is to trace where the constraint inside these systems comes from and why that constraint can become convincing.

We will also ask what gives the results of applying such a system to a particular problem enough convincing force to become operative in inquiry.

That may also tell us whether this kind of constraint gives us any further basis for calling one conviction better than another.

There is no natural point at which an investigation like this becomes complete. More machinery could always be added. We will follow only a few paths far enough that the relevant sources of convincing force can be inspected rather than merely assumed.