Inclusive disjunction (logical OR)
Inclusive disjunction, commonly called OR, is the logical operation that is false only when all inputs are false; it appears in logic, mathematics, programming, and digital circuits.
The inclusive disjunction — usually written with the symbol "∨" and commonly called "or" — is a basic logical connective that combines one or more propositions and yields true except when every input is false. In propositional logic the connective takes truth values as inputs and produces a single truth value as output. The term "inclusive" emphasises that the compound statement is true when one or more of its components are true, including the case when all are true.
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Formal meaning and truth cases
For two propositions A and B, the expression A ∨ B is true in every situation except when both A and B are false. A compact way of listing the four possibilities is:
- A is false, B is false → A ∨ B is false
- A is true, B is false → A ∨ B is true
- A is false, B is true → A ∨ B is true
- A is true, B is true → A ∨ B is true
Using ordinary language: "It is raining or it is snowing" is taken as true if it is raining, or snowing, or both. In symbolic manipulations A ∨ B can also be expressed in terms of other connectives; for example it is logically equivalent to ¬A → B, and to ¬(¬A ∧ ¬B).
Key algebraic properties
- Commutative: A ∨ B = B ∨ A
- Associative: (A ∨ B) ∨ C = A ∨ (B ∨ C)
- Idempotent: A ∨ A = A
- Identity element: A ∨ false = A
- Dominating element: A ∨ true = true
- Distributivity: A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
These properties place inclusive disjunction at the centre of Boolean algebra and set-theoretic operations: logical OR corresponds to set union when propositions are interpreted as membership conditions.
Applications and examples
Inclusive OR is widely used in mathematics, computer science, philosophy, and everyday reasoning. In digital electronics, the OR gate outputs a high signal when one or more inputs are high. In programming, many languages provide a logical OR operator (often written ||) and a bitwise OR operator (often |); some languages implement short-circuit evaluation so that the second operand is only evaluated if needed.
Concrete example: let A be "the light is on" and B be "the switch is up"; A ∨ B is true if either condition holds. In query languages and search engines an OR expands results to include items matching any of several terms.
Distinctions and language notes
Inclusive disjunction differs from exclusive disjunction (XOR), which is true only when exactly one operand is true. Ordinary natural-language "or" can be ambiguous between inclusive and exclusive readings depending on context; formal logic typically treats "or" as inclusive unless specified otherwise. For more technical background on the connective and further examples see discussion of logical operations and the contrast with exclusive disjunction.
Because of its simplicity and ubiquity, inclusive disjunction is one of the first connectives introduced in logic courses and remains a fundamental building block in reasoning systems, circuit design, and programming semantics.
Non-exclusive disjunction
The non-exclusive disjunction (alternative, adjunction) is a compound statement of the type "A or B (or both)"; it states that at least one of the two statements involved is true.
Notation
In Polish notation, the capital letter A is used for disjunction:
Aab
In the notation of a conjunction of statements, the symbol 
(Unicode: U+2228, ∨) stands for the non-exclusive disjunction as a propositional junction. It resembles the character 
for the unification set, and is reminiscent of the letter "v" that begins the Latin word "vel", which stands for such a non-exclusive Or.
The truth table for the vel function (OR function of a gate) as a truth value function of the non-exclusive disjunction is thus:
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|
|
|
| true | true | true |
| true | Incorrect | true |
| Incorrect | true | true |
| Incorrect | Incorrect | Incorrect |
A disjunction is a Boolean expression, it is associative and commutative.
It follows from what has been said:
- If A is false and if B is false, the disjunction is false; in every other case it is true.
- If the disjunction is false, then both A and B are false.
- If the disjunction is true, one of the following must be true:
- both disjoints are true
- A is false and B is true or
- A is true and B is false
Example
The statement "Tom helps paint or Anna helps paint" consists of the following parts:
- the partial statement/disjunct A: "Tom helps paint".
- the disjunctive "or", here not understood as exclusive
- the partial statement/disjunct B: "Anna helps paint".
Neither partial statement here excludes the other. The statement is false if neither Tom nor Anna help paint, true otherwise. In particular, it is also true if both Tom and Anna help paint.
Excluding disjunction
→ Main article: Contravalence
The exclusionary disjunction (contravalence, XOR) is a compound statement in which two statements are linked with the phrase "either - or (but not both)", for example the statement "Anna is either studying French or she is studying Spanish (but not both)." This excludes the case where both sub-statements are true - in the example, the case where Anna studies both French and Spanish - this is precisely the difference between this and non-exclusive disjunction. The Latin expression for the exclusionary or is "aut - aut".
The truth table for the aut-function (XOR-function of a gate) as a truth value function of the exclusionary disjunction is thus:
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| true | true | Incorrect |
| true | Incorrect | true |
| Incorrect | true | true |
| Incorrect | Incorrect | Incorrect |
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Author
AlegsaOnline.com Inclusive disjunction (logical OR) Leandro Alegsa
URL: https://en.alegsaonline.com/art/47012