Linear logic

Key takeaways

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

Reference excerpt

Linear logic is a substructural logic proposed by French logician Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such as programming languages, game semantics, and quantum physics (because linear logic can be seen as the logic of quantum information theory), as well as linguistics, particularly because of its emphasis on resource-boundedness, duality, and interaction. Linear logic lends itself to many different presentations, explanations, and intuitions. Proof-theoretically, it derives from an analysis of classical sequent calculus in which uses of (the structural rules) contraction and weakening are carefully controlled. Operationally, this means that logical deduction is no longer merely about an ever-expanding collection of persistent "truths", but also a way of manipulating resources that cannot always be duplicated or thrown away at will. In terms of simple denotational models, linear logic may be seen as refining the interpretation of intuitionistic logic by replacing cartesian (closed) categories by symmetric monoidal (closed) categories, or the interpretation of classical logic by replacing Boolean algebras by C*-algebras.

Notation comparison This article follows Girard's notation. For readers who are familiar with different notations, the following table, compiled by Paoli 2002, compares the notations for linear-logic connectives and constants across several sources.

Connectives, duality, and polarity

Syntax

The language L {\displaystyle {\mathcal {L}}} of classical propositional linear logic can be defined recursively as follows.

If φ {\displaystyle \varphi } is an atomic formula, then φ {\displaystyle \varphi } is a formula of L {\displaystyle {\mathcal {L}}} . If φ {\displaystyle \varphi } is an atomic formula, then φ ⊥ {\displaystyle \varphi ^{\bot }} is a formula of L {\displaystyle {\mathcal {L}}} . If φ {\displaystyle \varphi } and ψ {\displaystyle \psi } are formulas of L {\displaystyle {\mathcal {L}}} , then φ ⊗ ψ {\displaystyle \varphi \otimes \psi } is too. If φ {\displaystyle \varphi } and ψ {\displaystyle \psi } are formulas of L {\displaystyle {\mathcal {L}}} , then φ {\displaystyle \varphi } ⅋ ψ {\displaystyle \psi } is too. If φ {\displaystyle \varphi } and ψ {\displaystyle \psi } are formulas of L {\displaystyle {\mathcal {L}}} , then φ & ψ {\displaystyle \varphi \ \&\ \psi } is too. If φ {\displaystyle \varphi } and ψ {\displaystyle \psi } are formulas of L {\displaystyle {\mathcal {L}}} , then φ ⊕ ψ {\displaystyle \varphi \oplus \psi } is too. If φ {\displaystyle \varphi } is a formula of L {\displaystyle {\mathcal {L}}} , then ! φ {\displaystyle !\varphi } is too. If φ {\displaystyle \varphi } is a formula of L {\displaystyle {\mathcal {L}}} , then ? φ {\displaystyle ?\varphi } is too.

1 {\displaystyle {\bf {1}}} is a formula of L {\displaystyle {\mathcal {L}}} .

⊥ {\displaystyle \bot } is a formula of L {\displaystyle {\mathcal {L}}} .

⊤ {\displaystyle \top } is a formula of L {\displaystyle {\mathcal {L}}} .

0 {\displaystyle {\bf {0}}} is a formula of L {\displaystyle {\mathcal {L}}} . Here p and p⊥ range over logical atoms. For reasons to be explained below, the connectives ⊗, ⅋, 1, and ⊥ are called multiplicatives, the connectives &, ⊕, ⊤, and 0 are called additives, and the connectives ! and ? are called exponentials. We can further employ the following terminology:

Binary connectives ⊗, ⊕, & and ⅋ are associative and commutative; 1 is the unit for ⊗, 0 is the unit for ⊕, ⊥ is the unit for ⅋ and ⊤ is the unit for &. Every proposition A in CLL has a dual A⊥, defined as follows:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Linear logic

Start with the simplest possible case. Write down what Linear logic claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In engineering, 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 Linear logic 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 Linear logic 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 Linear logic

In research: Linear logic appears in engineering 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 Linear logic 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: Linear logic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Linear logic, Logic, Non-classical logic, so understanding it makes those chapters shorter.
In everyday life: Look for Linear logic 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 Linear logic in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Linear logic 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 Linear logic out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Linear logic in simple terms?

Linear logic is a substructural logic proposed by French logician Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broad…

Why does Linear logic matter?

Because it connects several engineering 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 Linear logic?

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 Linear logic.