[Math] Disjunction: Why did the inclusive “OR” become the convention

logicmath-history

In How to Prove it by Velleman, for defining disjunctions, he gives the difference between exclusive "OR", and inclusive "OR."

Given two events $P$ and $Q$, the disjunction is defined for them as:

  • Inclusive: One of $P$ and $Q$, or both.

  • Exclusive: One of $P$ and $Q$, but not both.

Quoting from his book:

"In mathematics, or always means inclusive or, unless specified otherwise, …" (Velleman, 2006, p.15)

My question is –

Why did the inclusive definition of disjunction become the convention?

Was it coincidental, or is there some aspect to the inclusive definition that makes it more convenient?

Best Answer

The operations logical AND $(\wedge)$, inclusive OR $(\vee)$ are dual, in the sense that the following hold.

  1. $\neg (A \wedge B) \leftrightarrow \neg A \vee \neg B$
  2. $\neg(A \vee B) \leftrightarrow \neg A \wedge \neg B.$

This means they have essentially the same properties. They're both associative, commutative, and idempotent; and they distribute over one another. So in conclusion, inclusive OR has nice properties, and it interacts nicely with logical AND.

It also seems to show up a lot more often.

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