OR

Definition

OR is a logical operator that combines two propositions.

It is true when at least one proposition is true.

Example:

P OR Q

This means:

P is true, or Q is true, or both are true

In standard logic, OR is inclusive.

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Simple Example

Let:

P = "The file is saved"
Q = "The file is backed up"

Then:

P OR Q

is true if:

the file is saved

or:

the file is backed up

or both.

OR requires at least one true part.

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Truth Pattern

OR has this truth pattern:

P      Q      P OR Q
true   true   true
true   false  true
false  true   true
false  false  false

Only the last row is false.

Both parts must be false for the whole OR statement to be false.

OR is false only when all alternatives fail.

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Inclusive OR

In everyday language, "or" can sometimes mean one or the other, but not both.

In basic logic, OR usually means inclusive OR.

Example:

P OR Q

is true when:

P is true and Q is true

This matters because logical OR does not automatically exclude both being true.

Logical OR usually allows both parts to be true.

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OR As Alternatives

OR can express possible alternatives.

Example:

x = 0 OR x = 1

This says x is allowed to be either 0 or 1.

If:

x = 0

the statement is true.

If:

x = 2

the statement is false.

OR widens what can satisfy a statement.

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Final Summary

• OR combines two propositions.
• It is true when at least one proposition is true.
• Standard logical OR is inclusive.
• OR is false only when both parts are false.
• OR expresses alternatives.
• OR widens the set of satisfying cases.