Aristotle's Logic

Before Aristotle, people argued well or badly. After him, there was a way to say why an argument was bad that did not depend on knowing whether its subject matter was true. That is the whole achievement, and it is easy to underrate because we have absorbed it so completely.

The move is this: replace the terms of an argument with letters, and look at what is left. What is left is the form, and the form alone determines whether the conclusion follows.

The letters are the invention

Take an argument about Socrates and mortality. Now take one about triremes and seaworthiness. They are about nothing in common — but write them schematically and they are the same object:

All A are B
All B are C
∴ All A are C

Every instance of this pattern is truth-preserving: if the premises are true, the conclusion cannot fail to be. You do not need to know what A, B and C are, and that is exactly the point. Aristotle's schematic letters — as far as we know, the first use of variables in the history of argument — turned "is this a good argument?" from a question about the world into a question about a pattern.

He catalogued the patterns. A syllogism has two premises and a conclusion, three terms, and each proposition is one of four types: all A are B, no A are B, some A are B, some A are not B. Arrange these systematically and you get a finite space of possible forms — most invalid, a handful valid — which he sorted into figures and proved the valid ones from a small base. It is the first formal system, and it was carried out before anyone had a notation for it.

Validity and truth come apart

Here is the distinction that the whole apparatus exists to enforce, and the one that most arguments in the wild still trip over.

Validity is a property of the form: the conclusion follows from the premises. Truth is a property of the individual propositions. An argument can be valid with false premises and a false conclusion. It can also be invalid while every proposition in it happens to be true.

The bottom-left cell is where real damage happens. An argument whose premises are all true and whose conclusion is true, but whose form is invalid, looks like knowledge and is not. It supplies no support for its conclusion; it merely sits next to it.

All experts agree the system is secure
The system has never been breached
∴ The system is secure

Form:  All A are B · C has not been D · ∴ C is B
Verdict: invalid — the conclusion is asserted, not derived.
         Both premises can be true and it proves nothing.

What his logic could not do

The system dominated for over two thousand years, and Kant, writing in the 1780s, thought it was finished — complete, with nothing left to add. He was wrong in a specific and interesting way.

Syllogistic handles one level of generality at a time. It struggles with inferences involving relations, and it cannot express a sentence with nested quantifiers at all: every student has an advisor and some advisor advises every student have different meanings and different consequences, but syllogistic has no way to write the difference down. Two millennia of logicians noticed the gap and patched around it without solving it.

Frege's Begriffsschrift, in 1879, replaced the whole apparatus: quantifiers, variables ranging over objects, and functions applied to arguments. That is the logic underneath every proof assistant and type system in use today. Aristotle's system was not so much corrected as absorbed — every valid syllogism remains valid, as a small special case of something much larger.

Why it still matters at a keyboard

Every static type check is a formal validity check with the content stripped out. The compiler does not know what your UserID means; it knows the shape of the composition, and it rejects the ones that cannot go through. When it passes your program, it is telling you exactly what a valid syllogism tells you: if your premises hold — your types describe what you think they describe — the conclusion follows. It is not telling you the premises are true. Sound types, wrong model of the domain, and you get a correct derivation of the wrong thing.

The habit transfers to reading arguments too, and it is the cheapest reasoning tool there is. Strip the terms, look at the shape, and ask whether the shape could carry you from true premises to a false conclusion. Most persuasive-sounding bad arguments fail this immediately, and they fail it for reasons that have nothing to do with the topic being argued — which is precisely what Aristotle was pointing at.