Relevance logic

Key takeaways

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

Reference excerpt

Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, but not universally, called relevant logic by British and, especially, Australian logicians, and relevance logic by American logicians. Relevance logic aims to capture aspects of implication that are ignored by the "material implication" operator in classical truth-functional logic, namely the notion of relevance between antecedent and conditional of a true implication. This idea is not new: C. I. Lewis was led to invent modal logic, and specifically strict implication, on the grounds that classical logic grants paradoxes of material implication such as the principle that a falsehood implies any proposition. Hence "if I'm a donkey, then two and two is four" is true when translated as a material implication, yet it seems intuitively false since a true implication must tie the antecedent and consequent together by some notion of relevance. And whether or not the speaker is a donkey seems in no way relevant to whether two and two is four. In terms of a syntactical constraint for a propositional calculus, it is necessary, but not sufficient, that premises and conclusion share atomic formulae (formulae that do not contain any logical connectives). In a predicate calculus, relevance requires sharing of variables and constants between premises and conclusion. This can be ensured (along with stronger conditions) by, e.g., placing certain restrictions on the rules of a natural deduction system. In particular, a Fitch-style natural deduction can be adapted to accommodate relevance by introducing tags at the end of each line of an application of an inference indicating the premises relevant to the conclusion of the inference. Gentzen-style sequent calculi can be modified by removing the weakening rules that allow for the introduction of arbitrary formulae on the right or left side of the sequents. A notable feature of relevance logics is that they are paraconsistent logics: the existence of a contradiction will not necessarily cause an "explosion." This follows from the fact that a conditional with a contradictory antecedent that does not share any propositional or predicate letters with the consequent cannot be true (or derivable).

History Relevance logic was proposed in 1928 by Soviet philosopher Ivan E. Orlov (1886 – circa 1936) in his strictly mathematical paper "The Logic of Compatibility of Propositions" published in Matematicheskii Sbornik. The basic idea of relevant implication appears in medieval logic, and some pioneering work was done by Ackermann, Moh, and Church in the 1950s. Drawing on them, Nuel Belnap and Alan Ross Anderson (with others) wrote the magnum opus of the subject, Entailment: The Logic of Relevance and Necessity in the 1970s (the second volume being published in the nineties). They focused on both systems of entailment and systems of relevance, where implications of the former kinds are supposed to be both relevant and necessary.

Axioms The early developments in relevance logic focused on the stronger systems. The development of the Routley–Meyer semantics brought out a range of weaker logics. The weakest of these logics is the relevance logic B. It is axiomatized with the following axioms and rules.

A → A {\displaystyle A\to A}

A ∧ B → A {\displaystyle A\land B\to A}

A ∧ B → B {\displaystyle A\land B\to B}

( A → B ) ∧ ( A → C ) → ( A → B ∧ C ) {\displaystyle (A\to B)\land (A\to C)\to (A\to B\land C)}

A → A ∨ B {\displaystyle A\to A\lor B}

B → A ∨ B {\displaystyle B\to A\lor B}

( A → C ) ∧ ( B → C ) → ( A ∨ B → C ) {\displaystyle (A\to C)\land (B\to C)\to (A\lor B\to C)}

A ∧ ( B ∨ C ) → ( A ∧ B ) ∨ ( A ∧ C ) {\displaystyle A\land (B\lor C)\to (A\land B)\lor (A\land C)}

¬ ¬ A → A {\displaystyle \lnot \lnot A\to A}

The rules are the following.

A , A → B ⊢ B {\displaystyle A,A\to B\vdash B}

A , B ⊢ A ∧ B {\displaystyle A,B\vdash A\land B}

A → B ⊢ ( C → A ) → ( C → B ) {\displaystyle A\to B\vdash (C\to A)\to (C\to B)}

A → B ⊢ ( B → C ) → ( A → C ) {\displaystyle A\to B\vdash (B\to C)\to (A\to C)}

A → ¬ B ⊢ B → ¬ A {\displaystyle A\to \lnot B\vdash B\to \lnot A}

Stronger logics can be obtained by adding any of the following axioms.

( A → B ) → ( ¬ B → ¬ A ) {\displaystyle (A\to B)\to (\lnot B\to \lnot A)}

( A → B ) ∧ ( B → C ) → ( A → C ) {\displaystyle (A\to B)\land (B\to C)\to (A\to C)}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Relevance logic

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

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

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

Frequently asked questions

What is Relevance logic in simple terms?

Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics.

Why does Relevance 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 Relevance 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 Relevance logic.