Rules and Notation

This is Unit Five of Artificial Intelligence: rules and notation. The unit has two halves that turn out to teach the same thing. The first builds systems that reason from written rules. The second teaches how to read formal notation. Both are about where meaning actually lives. An early error entry sets the whole thing up. Replace every label in a rule system with a serial number and every derivation runs identically. The words mean nothing to the machine — the meaning lives entirely in whoever reads the output. That has a consequence the unit takes seriously. These systems produce flawless, auditable proofs, and a flawless proof from a wrong rule is a confident wrong answer with a perfect trace attached. The final case is built around exactly that. The notation half then applies the same idea to reading. A formula is compressed language, every symbol is a name somebody chose, and the standing rule is never to let one pass unread. By the end you will be able to do five things. One: say where the meaning of a rule system's labels lives. Two: run a knowledge base both directions, and say what each produces. Three: say what soundness guarantees, and what it cannot supply. Four: say why more rules add conclusions rather than competence. Five: read a formula outside-in, as a sentence.

Two entries, and the second reframes everything after it. The first names the smallest unit these systems work with. The second demonstrates that the labels on those units carry no weight at all inside the machine.

The smallest unit.

AI 0221 The Proposition

A statement that is either true or false, with the atomic ones being those containing no smaller statements. Pulling compound claims apart into atoms is the first working skill here.

And the reframing.

AI 0222 "The Computer Knows What the Words Mean"

Swap every label for a serial number and every derivation runs identically. The labels are for people — the meaning lives entirely in whoever reads the output, and never in the machinery.

Three entries building the apparatus. What a rule states, what a collection of them amounts to, and one built by hand — with a point about rules that separates them from instructions.

What a rule states.

AI 0223 The Rule

Whenever every condition holds, the conclusion holds — written with the conclusion first. It is not an instruction with a time and place attached; it states what follows from what.

The collection.

AI 0224 The Knowledge Base

Rules together with established facts — the system's entire expertise, laid out where it can be inspected. No rule says when to run; the procedures supply that.

Building one.

AI 0225 The Plant Clinic

Choosing a vocabulary of atoms and writing the rules connecting them. Note that one rule builds on another's conclusion — rules chain, and the chaining is where the reasoning comes from.

Three entries on the first procedure. Start from what is known and fire every rule that applies, over and over, until nothing new can be added. It is thorough, and thoroughness has a specific cost.

The procedure.

AI 0226 Bottom-Up: Derive Everything

Start from the facts and repeatedly fire any rule whose whole body is established, adding conclusions as new facts. Elsewhere this pair is called forward and backward chaining.

Where it stops.

AI 0227 The Fixed Point

The point where nothing further can be derived, with everything established there being the consequences. It contains conclusions nobody asked for — thorough and undirected at once.

Run whole.

AI 0228 Bottom-Up at the Clinic

A complete run showing a diagnosis assembling from three starting facts. It also derived a good deal nobody wanted — which is what undirected means in practice.

Three entries on the other direction. Start from a question rather than the facts, and something valuable falls out as a byproduct — except that the entries insist it is not a byproduct at all.

The procedure.

AI 0229 Top-Down: Start from the Question

Work backward from a question: find a rule whose conclusion matches, then take its conditions as new goals. A goal matching nothing is a dead end, and the procedure backtracks.

Run whole.

AI 0230 Top-Down at the Clinic

One run answering a single question and leaving a tree behind it. The tree is not a leftover — it is the answer's full justification.

What that tree is.

AI 0231 The Proof Is the Audit Trail

Every answer traces through numbered rules back to established facts, and the trace can be handed to someone else and checked. Auditability at this grade is a property later machinery in this course loses.

Three entries on the promises, and they are unusually careful. The pair of properties is strong, the fence buying them is named, and there is a precise statement of what the system does at the edge of its knowledge.

The pair.

AI 0232 Sound and Complete

Both procedures are sound — everything derived really does follow — and complete. But the guarantee is about consequence rather than truth: sound reasoning from wrong premises produces wrong conclusions faithfully.

What buys it.

AI 0233 The Fence

Rule bodies allow only atoms joined by and — no alternatives, no negation, no uncertainty. Loosen that fence and the guarantees start costing dearly.

At the edge.

AI 0234 Silence at the Edge

Where the rules run out the system is silent — not saying it does not know, which would itself require a rule. Silence and awareness of ignorance are different capabilities, and this machinery has only the first.

Three entries on failure, and what unites them is that none produces an error message. Nothing warns, nothing crashes — the system continues confidently, which is what makes these worth naming.

The first failure.

AI 0235 Blind to Near-Synonyms

Labels are matched literally, so a vocabulary mismatch makes a near-synonym completely unrecognised — no similarity bridges it. The failure is invisible from inside: nothing errors and nothing warns.

The second.

AI 0236 Sound Reasoning, Wrong Knowledge

A sound procedure transmits truth and cannot create it, so a rule wrong for a case yields a confident, perfectly derived wrong answer. Auditing means locating the fault — facts, rules, or procedure.

And the third.

AI 0239 All-or-Nothing

Every atom is flatly true or flatly false, so an author must either overstate a case or say nothing about it. Repeated across a whole knowledge base, that forced choice makes the system brittle.

Two entries on why this approach hit a ceiling, and the answer is not what people assume. The reasoning was never the hard part — writing down the knowledge was, and the second entry contrasts this directly with the previous unit.

The bottleneck.

AI 0237 The Knowledge Bottleneck

These systems scaled up and stalled on authorship — every exception needs somebody to write it, every vocabulary needs maintaining. Inference was never the difficulty; acquiring the knowledge was.

And the error.

AI 0238 "More Rules Make the System Smarter"

More rules add conclusions rather than competence — each one multiplies interactions and widens the vocabulary that has to stay consistent. Contrast the previous unit, where each rule powered deletions.

One full case, and its post-mortem contains the unit's sharpest observation — that the flawlessness of the trace is precisely what makes the failure dangerous.

The capstone. Derivation, proof, post-mortem.

AI 0240 The Succulent Audit

A complete case showing exactly where a rule system's confidence comes from and exactly where it fails. The trace is impeccable, and that is what makes the wrong answer dangerous — confidence with a receipt attached.

The unit's second half, and it opens with a standing rule worth adopting permanently. Formal notation is compressed language, and every symbol in it is a name somebody chose.

The standing rule.

AI 0245 Notation Is Compressed Language

Every formula is a sentence that can be said out loud. Never let one pass unread — saying it is what turns a string of symbols into a claim you can evaluate.

What a letter is.

AI 0246 Letters Are Local Names

A name for something, hired by a nearby sentence and meaning nothing outside that discussion. A formula containing an unhired letter is not deep — it is incomplete, and the hiring sentence is worth hunting for.

A common shape.

AI 0247 The Machine with a Slot

A name followed by something in brackets names a machine and what feeds it. The brackets are a mouth rather than a multiplication sign — reading them as multiplication is a standing error.

Three entries covering the grammar this course uses. Two connectives with a word order that runs against English habit, an arrow that goes one way only, and two kinds of label that look similar and do entirely different jobs.

Two connectives.

AI 0248 And, and If

One symbol joins claims and reads as and; another points from a conclusion back to its conditions and reads as if. The conclusion-first order runs against English habit, and both grammars appear in practice.

The direction.

AI 0249 The One-Way Arrow

From established conditions the conclusion may be drawn — and from an established conclusion nothing follows about the conditions. Running that arrow backwards is among the oldest reasoning errors there is.

Two kinds of label.

AI 0250 Raised and Lowered Labels

A raised label multiplies — the base used as a factor that many times — and a lowered one counts siblings for reference. Lowered labels carry no operation at all; they are name tags.

One procedure closing the unit, and it names a distinction worth keeping. Pronouncing the symbols is transliteration. Turning them into a claim is translation — and only the second one is reading.

Reading a formula.

AI 0251 Reading Outside-In

Find the main structure, state it as a sentence frame, then descend into the parts — never left to right. Pronouncing symbols is transliteration; producing the claim is translation, and only that is reading.

Five things you can now do. You can say where a rule system's meaning lives — replace every label with a serial number and nothing changes, so the meaning is entirely in the reader. You can run a knowledge base both directions: forward derives everything derivable, and backward answers one question while leaving a checkable proof. You can say what soundness guarantees and what it cannot supply — it transmits truth faithfully and cannot create it, so a wrong rule yields a perfect wrong answer. You can say why more rules add conclusions rather than competence, and contrast that with constraints, where each rule powered a deletion. And you can read a formula outside-in as a sentence, knowing that pronouncing the symbols is transliteration and not reading.