Rules of passage

Key takeaways

Rules of passage belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Rules of passage to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Rules of passage from memory before moving on to harder problems.

Reference excerpt

In mathematical logic, the rules of passage govern how quantifiers distribute over the basic logical connectives of first-order logic. The rules of passage govern the "passage" (translation) from any formula of first-order logic to the equivalent formula in prenex normal form, and vice versa.

The rules See Quine (1982: 119, chpt. 23). Let Q and Q' denote ∀ and ∃ or vice versa. β denotes a closed formula in which x does not appear. The rules of passage then include the following sentences, whose main connective is the biconditional:

Q x [ ¬ α ( x ) ] ↔ ¬ Q ′ x [ α ( x ) ] . {\displaystyle Qx[\lnot \alpha (x)]\leftrightarrow \lnot Q'x[\alpha (x)].}

The following conditional sentences can also be taken as rules of passage:

∃ x [ α ( x ) ∧ γ ( x ) ] → ( ∃ x α ( x ) ∧ ∃ x γ ( x ) ) . {\displaystyle \exists x[\alpha (x)\land \gamma (x)]\rightarrow (\exists x\alpha (x)\land \exists x\gamma (x)).}

( ∀ x α ( x ) ∨ ∀ x γ ( x ) ) → ∀ x [ α ( x ) ∨ γ ( x ) ] . {\displaystyle (\forall x\,\alpha (x)\lor \forall x\,\gamma (x))\rightarrow \forall x\,[\alpha (x)\lor \gamma (x)].}

( ∃ x α ( x ) ∧ ∀ x γ ( x ) ) → ∃ x [ α ( x ) ∧ γ ( x ) ] . {\displaystyle (\exists x\,\alpha (x)\land \forall x\,\gamma (x))\rightarrow \exists x\,[\alpha (x)\land \gamma (x)].}

"Rules of passage" first appeared in French, in the writings of Jacques Herbrand. Quine employed the English translation of the phrase in each edition of his Methods of Logic, starting in 1950.

See also First-order logic Prenex normal form Quantifier

References Willard Quine, 1982. Methods of Logic, 4th ed. Harvard Univ. Press. Jean Van Heijenoort, 1967. From Frege to Gödel: A Source Book on Mathematical Logic. Harvard Univ. Press.

External links Stanford Encyclopedia of Philosophy: "Classical Logic—by Stewart Shapiro.

Worked examples

Example 1 — a first encounter with Rules of passage

Start with the simplest possible case. Write down what Rules of passage claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rules of passage before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rules of passage ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rules of passage

In research: Rules of passage appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Rules of passage in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Rules of passage is common in secondary-school and first-year university syllabi. It links to neighbouring topics Logic stubs, Mathematical logic, so understanding it makes those chapters shorter.
In everyday life: Look for Rules of passage outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Rules of passage in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rules of passage means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Rules of passage out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rules of passage in simple terms?

In mathematical logic, the rules of passage govern how quantifiers distribute over the basic logical connectives of first-order logic. The rules of passage govern the "passage" (translation) from any formula of first-order logic to the equivalent formula in prenex normal form, and vice versa.

Why does Rules of passage matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rules of passage?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rules of passage.