science

Infinite-valued logic

Infinite-valued logic is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Infinite-valued logic rather than just read about it. In short: In logic, an infinite-valued logic (or real-valued logic or infinitely-many-valued logic) is a many-valued logic in which truth values comprise a continuous range. Traditionally, in Aristotle's logic, logic other than bivalent logic was abnormal, as the law of the excluded middle precluded more than two possible values (i.e., "true" and "false") for any proposition.

Key takeaways

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

Reference excerpt

In logic, an infinite-valued logic (or real-valued logic or infinitely-many-valued logic) is a many-valued logic in which truth values comprise a continuous range. Traditionally, in Aristotle's logic, logic other than bivalent logic was abnormal, as the law of the excluded middle precluded more than two possible values (i.e., "true" and "false") for any proposition. Modern three-valued logic (trivalent logic) allows for an additional possible truth value (i.e., "undecided") and is an example of finite-valued logic in which truth values are discrete, rather than continuous. Infinite-valued logic comprises continuous fuzzy logic, though fuzzy logic in some of its forms can further encompass finite-valued logic. For example, finite-valued logic can be applied in Boolean-valued modeling, description logics, and defuzzification of fuzzy logic.

History Isaac Newton and Gottfried Wilhelm Leibniz used both infinities and infinitesimals to develop the differential and integral calculus in the late 17th century. Richard Dedekind, who defined real numbers in terms of certain sets of rational numbers in the 19th century, also developed an axiom of continuity stating that a single correct value exists at the limit of any trial and error approximation. Felix Hausdorff demonstrated the logical possibility of an absolutely continuous ordering of words comprising bivalent values, each word having absolutely infinite length, in 1938. However, the definition of a random real number, meaning a real number that has no finite description whatsoever, remains somewhat in the realm of paradox. Jan Łukasiewicz developed a system of three-valued logic in 1920. He generalized the system to many-valued logics in 1922 and went on to develop logics with ℵ 0 {\displaystyle \aleph _{0}} (infinite within a range) truth values. Kurt Gödel developed a deductive system, applicable for both finite- and infinite-valued first-order logic (a formal logic in which a predicate can refer to a single subject) as well as for intermediate logic (a formal intuitionistic logic usable to provide proofs such as a consistency proof for arithmetic), and showed in 1932 that logical intuition cannot be characterized by finite-valued logic. The concept of expressing truth values as real numbers in the range between 0 and 1 can bring to mind the possibility of using complex numbers to express truth values. These truth values would have an imaginary dimension, for example between 0 and i. Two- or higher-dimensional truth could potentially be useful in systems of paraconsistent logic. If practical applications were to arise for such systems, multidimensional infinite-valued logic could develop as a concept independent of real-valued logic. Lotfi A. Zadeh proposed a formal methodology of fuzzy logic and its applications in the early 1970s. By 1973, other researchers were applying the theory of Zadeh fuzzy controllers to various mechanical and industrial processes. The fuzzy modeling concept that evolved from this research was applied to neural networks in the 1980s and to machine learning in the 1990s. The formal methodology also led to generalizations of mathematical theories in the family of t-norm fuzzy logics.

Examples Basic fuzzy logic is the logic of continuous t-norms (binary operations on the real unit interval [0, 1]). Applications involving fuzzy logic include facial recognition systems, home appliances, anti-lock braking systems, automatic transmissions, controllers for rapid transit systems and unmanned aerial vehicles, knowledge-based and engineering optimization systems, weather forecasting, pricing, and risk assessment modeling systems, medical diagnosis and treatment planning, international trading systems, and more. Fuzzy logic is used to optimize efficiency in thermostats for control of heating and cooling, for industrial automation and process control, computer animation, signal processing, and data analysis. Fuzzy logic has made significant contributions in the fields of machine learning and data mining. In infinitary logic, degrees of provability of propositions can be expressed in terms of infinite-valued logic that can be described via evaluated formulas, written as ordered pairs each consisting of a truth degree symbol and a formula. In mathematics, number-free semantics can express facts about classical mathematical notions and make them derivable by logical deductions in infinite-valued logic. T-norm fuzzy logics can be applied to eliminate references to real numbers from definitions and theorems, in order to simplify certain mathematical concepts and facilitate certain generalizations. A framework employed for number-free formalization of mathematical concepts is known as fuzzy class theory. Philosophical questions, including the Sorites paradox, have been considered based on an infinite-valued logic known as fuzzy epistemicism. The Sorites paradox suggests that if adding a grain of sand to something that is not a heap cannot create a heap, then a heap of sand cannot be created. A stepwise approach toward a limit, in which truth is gradually "leaked", tends to refute that suggestion. In the study of logic itself, infinite-valued logic has served as an aid to understand the nature of the human understanding of logical concepts. Kurt Gödel attempted to comprehend the human ability for logical intuition in terms of finite-valued logic before concluding that the ability is based on infinite-valued logic. Open questions remain regarding the handling, in natural language semantics, of indeterminate truth values.

See also Many-valued logic Finite-valued logic Intuitionistic logic Logical intuition Fuzzy logic Fuzzy number Construction of t-norms Set theory

References

Worked examples

Example 1 — a first encounter with Infinite-valued logic

Start with the simplest possible case. Write down what Infinite-valued 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 Infinite-valued 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 Infinite-valued 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 Infinite-valued logic

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

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How to study Infinite-valued logic in 20 minutes

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

Frequently asked questions

What is Infinite-valued logic in simple terms?

Why does Infinite-valued 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 Infinite-valued 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 Infinite-valued logic.