1. About Logic

Logic studies inference.

We can begin by comparing conclusions that seem to follow from their premises with conclusions that do not.

Consider:

If the alarm light is on, the machine has stopped.
The alarm light is on.

Thus: The machine has stopped.

Or:

All men are mortal.
Socrates is a man.

Thus: Socrates is mortal.

If we take the first sentence as an exceptionless condition, the conclusion follows from the two premises.

Now compare:

If the alarm light is on, the machine has stopped.
The alarm light is off.

Thus: The machine has not stopped.

Or:

All men are mortal.
Socrates is mortal.

Thus: Socrates is a man.

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

What constrains the first two conclusions while the premises remain in place, 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?

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

We will try to construct formal systems that let us inspect that difference rather than merely relying on our initial sense that one inference follows and the other does not.

The goal is to trace some basic sources of logical constraint and to examine how those sources can acquire convincing force.

We will not attempt to reconstruct the convincing force of modern first-order logic by analysing all the mathematical machinery on which it depends.

That investigation would be worthwhile, but it would hinder what we actually want to achieve here now.

First-order logic is already a mature formal system. Its language, semantics, proof methods, and metatheory have been developed together with mathematics and with other parts of logic. Much of the convincing force available to a trained user is carried by structures whose own justification and history would have to be reconstructed in turn.

Here we will instead construct only as much formal machinery as each example requires, and keep the resulting systems small enough that the relevant possibilities and mappings remain open to inspection.

There is no natural point at which an investigation like this becomes complete. More machinery could always be added.

Here we will follow only a few paths far enough that their constraint can be inspected rather than merely assumed.