Inclusive disjunction

Author: Leandro Alegsa Creation date: November 13, 2022

Inclusive disjunction (also called or) is a logic operation. It normally takes two truth values as inputs and returns one truth value as output. It is false when both inputs are false, but is true otherwise. It is written with the symbol $\lor$ .

In general, given two propositions $A$ and $B$ , $A\lor B$ is true if $A$ is true, or if $B$ is true, or if both $A$ and $B$ are true.

This is different from the exclusive disjunction, which asserts "either x or y, but not both".

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

$A\lor B$

In Polish notation, the capital letter A is used for disjunction:

Aab

In the notation $A\lor B$ of a conjunction of statements, the symbol $\lor$ (Unicode: U+2228, ∨) stands for the non-exclusive disjunction as a propositional junction. It resembles the character $\textstyle \cup$ 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:

		$A\lor B$
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:

		$A~{\dot {\lor }}~B$
true	true	Incorrect
true	Incorrect	true
Incorrect	true	true
Incorrect	Incorrect	Incorrect