Intuitionistic logic facts for kids
Intuitionistic logic, also known as constructive logic, is a special way of thinking about proofs in logic. It's different from the usual "classical logic" you might know. The main idea is that to say something is true, you need to have direct proof or a way to build it. You can't just say it's true because it's not false.
This type of logic doesn't use two common rules from classical logic:
- The law of the excluded middle: This rule says that a statement is either true or false, with no other options. Intuitionistic logic doesn't always agree with this.
- Double negation elimination: This rule says that if something is "not not true," then it must be true. Intuitionistic logic doesn't always accept this either.
Arend Heyting first developed intuitionistic logic to give a formal structure to the ideas of L. E. J. Brouwer, who started the idea of intuitionism. From a proof-theoretic point of view, Heyting's system is like classical logic but without those two main rules.
Scientists have looked at different ways to understand intuitionistic logic. One way uses something called Heyting algebras instead of the usual Boolean algebras. Another way uses Kripke models. These are tools to study the system, not necessarily how Brouwer first thought about it.
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Constructive Mathematics
In classical logic, every statement is either "true" or "false." This is the "law of excluded middle." For example, a number is either even or not even. We don't need direct proof to say it's one or the other.
But in intuitionistic logic, a statement is only considered "true" if we have direct proof for it. If we don't have proof, we can't say it's true. It's like saying, "I believe it's raining only if I see rain or a weather report confirming it." You wouldn't say "it's either raining or not raining" without evidence.
Intuitionistic logic is very useful in a field called constructive mathematics. This is where mathematicians try to build or find examples of things they prove exist.
Some mathematicians and philosophers have argued about using constructive logic. For example, David Hilbert, a famous mathematician, felt that taking away the law of excluded middle was like taking away a telescope from an astronomer. He thought it was essential for mathematics.
However, intuitionistic logic has practical uses. Its rules mean that if you prove something exists, your proof often gives you a way to actually find or build that thing. This is called the Curry–Howard correspondence, which links proofs to computer algorithms.
This is very helpful for proof assistants, which are computer programs that help mathematicians check and even create very complex proofs. These proofs can be so big that humans can't easily check them by hand. Tools like Agda or Coq help modern mathematicians tackle problems that were once too hard to verify. One famous example is the four color theorem, which was finally proven with computer help and later verified using Coq.
How Intuitionistic Logic Works
The way you write formulas in intuitionistic logic is similar to other types of logic. However, the basic connections between ideas (like "and," "or," "if...then," and "not") can't always be defined using each other in the same way as in classical logic. So, they are all important on their own.
Weaker Than Classical Logic
Intuitionistic logic is "weaker" than classical logic. This means it's more careful about what it allows you to conclude. Every statement that is true in intuitionistic logic is also true in classical logic. But the opposite is not true. Many statements that are considered always true in classical logic (called tautologies) are not true in intuitionistic logic.
For example, the law of the excluded middle (a statement is either true or false) is a tautology in classical logic, but not in intuitionistic logic. This is because intuitionistic logic wants to avoid "non-constructive proofs." A non-constructive proof might show that something exists without giving you an actual example of it.
Building Blocks of Intuitionistic Logic
Gerhard Gentzen found that if he made a simple change to his system for classical logic, he could create a system for intuitionistic logic. His system, called LJ, only allows one conclusion for each step in a proof, unlike classical logic which can have many.
Intuitionistic logic can also be built using a set of basic rules and starting points (called axioms). These rules help you figure out what statements are true.
- Modus Ponens: This is a basic rule. If you know "A is true" and "If A, then B is true," then you can conclude "B is true."
There are also axioms for "if...then," "and," "or," and "false." For example:
- AND-1: If "A and B" is true, then "A" is true.
- OR-1: If "A" is true, then "A or B" is true.
- FALSE: If "false" is true, then anything is true (this means "false" leads to a contradiction).
For more complex logic that talks about "all" or "some" things, there are also rules and axioms for "for all" (∀) and "there exists" (∃).
Negation
In intuitionistic logic, "not A" (¬A) is often thought of as "if A, then false." This means if A is true, it leads to a contradiction.
Adding Classical Rules
You can turn intuitionistic logic into classical logic by adding just one of these rules:
- Law of the excluded middle: "A or not A" is always true.
- Double negation elimination: If "not not A" is true, then "A" is true.
- Peirce's law: If "((A implies B) implies A)" is true, then "A" is true.
This shows how closely related the two types of logic are, even though they have different starting points.
How Operators Work
In classical logic, you can often define one logical connection (like "and") using others (like "or" and "not"). For example, "A and B" can be written as "not (not A or not B)."
But in intuitionistic logic, this isn't always possible. Because it doesn't assume the law of the excluded middle, each basic connection ("and," "or," "if...then," "not," "for all," "there exists") is needed on its own. They can't all be defined in terms of each other.
"There Exists" vs. "For All"
In intuitionistic logic, if you say "there exists an X that is not Y," it means "it's not true that all X are Y."
* This means: If there's an X that doesn't have property `\phi`, then it's not true that all X have property `\phi`.
However, the reverse isn't always true in intuitionistic logic. Just because it's not true that all X are Y, doesn't mean you can always find a specific X that is not Y.
"Or" vs. "And"
Some relationships between "or" and "and" are different in intuitionistic logic.
* This means: If "not A or not B" is true, then "not (A and B)" is true.
* This means: "Not A and not B" is true if and only if "not (A or B)" is true. This is similar to one of De Morgan's laws.
Understanding Intuitionistic Logic
It's important to know that statements that are not proven in intuitionistic logic are not given a "middle" truth value. They simply remain of unknown truth value until they are either proven true or proven false (by showing they lead to a contradiction).
Heyting Algebra Semantics
In classical logic, we often use "truth values" like "true" and "false" (which are like 1 and 0 in a Boolean algebra). A statement is true if its value is always 1.
For intuitionistic logic, we use something called a Heyting algebra. Boolean algebras are a special type of Heyting algebra. A statement is true in intuitionistic logic if its value is always the "top element" (like "true") in any Heyting algebra.
One way to understand this is by thinking about open sets on a number line.
- "False" (`\bot`) is like the empty set (`\emptyset`).
- "True" (`\top`) is like the entire number line (`\mathbf{R`).
- "A and B" (`A \land B`) is like the overlap of set A and set B (`\text{Value}[A] \cap \text{Value}[B`).
- "A or B" (`A \lor B`) is like combining set A and set B (`\text{Value}[A] \cup \text{Value}[B`).
- "Not A" (`\neg A`) is like the "interior" of everything *outside* of A.
Using this, you can show that "not (A and not A)" is always true (the entire line). But "A or not A" (the law of excluded middle) is not always true. For example, if A is the set of all positive numbers, then "A or not A" becomes the set of all numbers except zero, which is not the entire number line.
Kripke Semantics
Saul Kripke, who also worked on modal logic, created another way to understand intuitionistic logic called Kripke semantics. This uses "possible worlds" and how they relate to each other.
Relation to Other Logics
- Minimal logic is an even simpler version of intuitionistic logic, where the "false implies anything" rule is removed.
- Intermediate logics are systems that fall between intuitionistic logic and classical logic.
- Modal logic can be used to translate intuitionistic logic. There's a special translation called the Gödel–McKinsey–Tarski translation that shows a formula is true in intuitionistic logic if and only if its translated version is true in a specific modal logic called S4.
Lambda Calculus
There's a deep connection between intuitionistic logic and simply-typed lambda calculus, which is a system used in computer science to study functions. This connection is called the Curry–Howard isomorphism. It means that a proof in intuitionistic logic can be seen as a computer program, and a program can be seen as a proof.
See also
- BHK interpretation
- Constructive proof
- Curry–Howard correspondence
- Intermediate logics
- Kripke semantics
- Paraconsistent logic
Images for kids
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The Rieger–Nishimura lattice. This shows how different statements in intuitionistic logic relate to each other.
