Substructural logic

Key takeaways

Substructural 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 Substructural logic to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Substructural logic from memory before moving on to harder problems.

Reference excerpt

In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity. Two of the more significant substructural logics are relevance logic and linear logic.

Examples In a sequent calculus, one writes each line of a proof as

Γ ⊢ Σ {\displaystyle \Gamma \vdash \Sigma } . Here the structural rules are rules for rewriting the LHS of the sequent, denoted Γ, initially conceived of as a finite string (sequence) of propositions. The standard interpretation of this string is as conjunction: we expect to read

A , B ⊢ C {\displaystyle {\mathcal {A}},{\mathcal {B}}\vdash {\mathcal {C}}}

as the sequent notation for

(A and B) implies C. Here we are taking the RHS Σ to be a single proposition C (which is the intuitionistic style of sequent); but everything applies equally to the general case, since all the manipulations are taking place to the left of the turnstile symbol ⊢ {\displaystyle \vdash } . Since conjunction is a commutative and associative operation, the formal setting-up of sequent theory normally includes structural rules for rewriting the sequent Γ accordingly—for example for deducing

B , A ⊢ C {\displaystyle {\mathcal {B}},{\mathcal {A}}\vdash {\mathcal {C}}}

from

A , B ⊢ C {\displaystyle {\mathcal {A}},{\mathcal {B}}\vdash {\mathcal {C}}} . There are further structural rules corresponding to the idempotent and monotonic properties of conjunction: from

Γ , A , A , Δ ⊢ C {\displaystyle \Gamma ,{\mathcal {A}},{\mathcal {A}},\Delta \vdash {\mathcal {C}}}

we can deduce

Γ , A , Δ ⊢ C {\displaystyle \Gamma ,{\mathcal {A}},\Delta \vdash {\mathcal {C}}} . Also from

Γ , A , Δ ⊢ C {\displaystyle \Gamma ,{\mathcal {A}},\Delta \vdash {\mathcal {C}}}

one can deduce, for any B,

Γ , A , B , Δ ⊢ C {\displaystyle \Gamma ,{\mathcal {A}},{\mathcal {B}},\Delta \vdash {\mathcal {C}}} . Linear logic, in which duplicated hypotheses 'count' differently from single occurrences, leaves out both of these rules, while relevant (or relevance) logics merely leaves out the latter rule, on the ground that B is clearly irrelevant to the conclusion. The above are basic examples of structural rules. It is not that these rules are contentious, when applied in conventional propositional calculus. They occur naturally in proof theory, and were first noticed there (before receiving a name).

Premise composition There are numerous ways to compose premises (and in the multiple-conclusion case, conclusions as well). One way is to collect them into a set. But since e.g. {a,a} = {a} we have contraction for free if premises are sets. We also have associativity and permutation (or commutativity) for free as well, among other properties. In substructural logics, typically premises are not composed into sets, but rather they are composed into more fine-grained structures, such as trees or multisets (sets that distinguish multiple occurrences of elements) or sequences of formulae. For example, in linear logic, since contraction fails, the premises must be composed in something at least as fine-grained as multisets.

History Substructural logic is a relatively young field. The first conference on the topic was held in October 1990 in Tübingen, as "Logics with Restricted Structural Rules". During the conference, Kosta Došen proposed the term "substructural logics", which is now in use today.

See also Substructural type system Residuated lattice

References

Paoli, Francesco (2002). Substructural Logics: A Primer. Trends in Logic. Vol. 13. Dordrecht: Springer Science & Business Media. doi:10.1007/978-94-017-3179-9. ISBN 978-90-481-6014-3. Restall, Greg (2000). An Introduction to Substructural Logics. London and New York: Routledge. ISBN 0-415-21533-1.

Further reading Galatos, Nikolaos; Jipsen, Peter; Kowalski, Tomasz; Ono, Hiroakira (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and Practical Reasoning. Vol. 2. Amsterdam: Elsevier. ISBN 978-0-444-52141-5.

External links Media related to Substructural logic at Wikimedia Commons Restall, Greg. "Substructural logics". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. ISSN 1095-5054. OCLC 429049174.

Worked examples

Example 1 — a first encounter with Substructural logic

Start with the simplest possible case. Write down what Substructural 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 Substructural 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 Substructural 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 Substructural logic

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

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

Frequently asked questions

What is Substructural logic in simple terms?

In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity. Two of the more significant substructural logics are relevance logic and linear logic.

Why does Substructural 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 Substructural 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 Substructural logic.