Łukasiewicz logic

Key takeaways

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

Reference excerpt

In mathematics and philosophy, Łukasiewicz logic ( WUUK-ə-SHEV-itch, Polish: [wukaˈɕɛvitʂ]) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued modal logic; it was later generalized to n-valued (for all finite integers n) as well as infinitely-many-valued (ℵ0-valued) variants, both propositional and first order. The ℵ0-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the Łukasiewicz–Tarski logic. It belongs to the classes of t-norm fuzzy logics and substructural logics. Łukasiewicz logic was motivated by Aristotle's suggestion that bivalent logic was not applicable to future contingents, e.g. the statement "There will be a sea battle tomorrow". In other words, statements about the future were neither true nor false, but an intermediate value could be assigned to them, to represent their possibility of becoming true in the future. This article presents the Łukasiewicz(–Tarski) logic in its full generality, i.e. as an infinite-valued logic. For an elementary introduction to the three-valued instantiation Ł3, see three-valued logic.

Language The propositional connectives of Łukasiewicz logic are → {\displaystyle \rightarrow } ("implication"), and the constant ⊥ {\displaystyle \bot } ("false"). Additional connectives can be defined in terms of these:

¬ A = d e f A → ⊥ A ∨ B = d e f ( A → B ) → B A ∧ B = d e f ¬ ( ¬ A ∨ ¬ B ) A ↔ B = d e f ( A → B ) ∧ ( B → A ) ⊤ = d e f ⊥ → ⊥ {\displaystyle {\begin{aligned}\neg A&=_{def}A\rightarrow \bot \\A\vee B&=_{def}(A\rightarrow B)\rightarrow B\\A\wedge B&=_{def}\neg (\neg A\vee \neg B)\\A\leftrightarrow B&=_{def}(A\rightarrow B)\wedge (B\rightarrow A)\\\top &=_{def}\bot \rightarrow \bot \end{aligned}}}

The ∨ {\displaystyle \vee } and ∧ {\displaystyle \wedge } connectives are called weak disjunction and conjunction, because they are non-classical, as the law of excluded middle does not hold for them. In the context of substructural logics, they are called additive connectives. They also correspond to lattice min/max connectives. In terms of substructural logics, there are also strong or multiplicative disjunction and conjunction connectives, although these are not part of Łukasiewicz's original presentation:

A ⊕ B = d e f ¬ A → B A ⊗ B = d e f ¬ ( A → ¬ B ) {\displaystyle {\begin{aligned}A\oplus B&=_{def}\neg A\rightarrow B\\A\otimes B&=_{def}\neg (A\rightarrow \neg B)\end{aligned}}}

There are also defined modal operators, using the Tarskian Möglichkeit:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Łukasiewicz logic

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

In research: Łukasiewicz logic appears in science 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 Łukasiewicz 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: Łukasiewicz logic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Fuzzy logic, Many-valued logic, so understanding it makes those chapters shorter.
In everyday life: Look for Łukasiewicz 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 Łukasiewicz logic in 20 minutes

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

Frequently asked questions

What is Łukasiewicz logic in simple terms?

In mathematics and philosophy, Łukasiewicz logic ( WUUK-ə-SHEV-itch, Polish: [wukaˈɕɛvitʂ]) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued modal logic; it was later generalized to n-valued (for all finite integers n) a…

Why does Łukasiewicz logic matter?

Because it connects several science 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 Łukasiewicz 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 Łukasiewicz logic.