SafekipediaLogical disjunction
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Logical disjunction, often called "logical OR," is a basic idea in logic that helps us combine statements. It is usually written with the symbol ∨ or simply the word "or." For example, if we say "it is sunny or it is warm," we are using a disjunction. In logic, this means that the statement is true if either "it is sunny" is true, "it is warm" is true, or both are true.
In classical logic, a disjunction is true unless both parts are false. This makes it an "inclusive" idea because it includes the case where both parts might be true at the same time. This is different from an "exclusive" disjunction, where only one part can be true, not both. Disjunction is important in many areas, including computer science and mathematics, because it helps build more complex logical statements and arguments.
Disjunction can be studied in different ways, and people have looked at it from many angles beyond the basic rules. This helps solve problems like ancient debates about future events or modern questions about uncertainty in science. Understanding disjunction gives us a foundation for thinking clearly and making sense of how ideas connect together.
Inclusive and exclusive disjunction
The logical "or" means that a statement is true if either one part or both parts are true. This is called an inclusive disjunction. It is different from an exclusive disjunction, which is only true when one part is true, but not both—this is also called an exclusive or or XOR.
To make it clear whether both parts can be true at the same time, people sometimes use the phrase and/or. In logic, this phrase works the same as "or," but it emphasizes that both parts can be true together.
Notation
In logic, disjunction is usually shown with the symbol ∨, which is read as "or." This symbol is used in many areas, such as electronics and computer programming. Sometimes, the plus sign (+) is used instead, especially in electronics. In certain computer languages, two vertical lines (||) might be used to mean "or."
Mathematicians also have a special way to show "or" when talking about many things at once. They use a larger symbol ⋁, which stands for "or" repeated many times. For example, if you have several items a₁ through aₙ, you can write ⋁ from i=1 to n of aᵢ, which means a₁ or a₂ or … or aₙ.
Classical disjunction
In classical logic, disjunction—often called "logical or"—is a way to connect ideas. It says that if either idea is true, or both are true, then the whole statement is true. Only when both ideas are false is the whole statement false.
Disjunction can also be explained using other logic ideas. For example, "A or B" can be rewritten as "not (not A and not B)." This means that if A is not false and B is not false, then "A or B" is true. Disjunction has special rules, like switching the order (commutativity) or grouping differently (associativity), which help us understand how it works in different situations.
| A {\displaystyle A} | B {\displaystyle B} | A ∨ B {\displaystyle A\lor B} |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
| A {\displaystyle A} | B {\displaystyle B} | ¬ A {\displaystyle \neg A} | ¬ A → B {\displaystyle \neg A\rightarrow B} | A ∨ B {\displaystyle A\lor B} |
|---|---|---|---|---|
| F | F | T | F | F |
| F | T | T | T | T |
| T | F | F | T | T |
| T | T | F | T | T |
| A {\displaystyle A} | B {\displaystyle B} | A → B {\displaystyle A\rightarrow B} | ( A → B ) → B {\displaystyle (A\rightarrow B)\rightarrow B} | A ∨ B {\displaystyle A\lor B} |
|---|---|---|---|---|
| F | F | T | F | F |
| F | T | T | T | T |
| T | F | F | T | T |
| T | T | T | T | T |
Applications in computer science
Operators that work like logical "or" exist in most programming languages. In computer programs, this "or" can be used to work with individual bits of data. For example, when you combine two numbers using "or," the result keeps a bit as 1 if either of the original numbers had that bit as 1.
Many languages use different symbols for "or" when working with bits versus when making logical choices. This helps the computer know exactly what you want to do. Some languages also let the computer stop checking after finding one true condition, which makes programs run faster.
Set theory
In set theory, the idea of joining two groups together can be described using a logical "or". This means an item is in the combined group if it is in the first group or in the second group. Because of this, the way we use "or" in logic works very similarly to how we combine groups in set theory. This helps us understand rules like switching the order of groups or combining groups in different ways.
Natural language
Disjunction in natural languages, like when we say "or," doesn't always match perfectly with how it works in formal logic. For example, if someone says, "Mary is eating an apple or a pear," we often understand this to mean she is eating one of the two, not both. This difference shows how language can be more specific than logic rules.
In many languages, "or" helps us ask questions, like "Is Mary a philosopher or a linguist?" This can mean either checking if one statement is true or choosing between two options. Different languages have unique ways to express "or," such as using special word endings. For example, in the language Maricopa, disjunction is shown by adding a suffix to verbs.
This article is a child-friendly adaptation of the Wikipedia article on Logical disjunction, available under CC BY-SA 4.0.
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