# Division by Zero

A blog about math, puzzles, teaching, and academic technology
Posted by: Dave Richeson | September 27, 2009

## Is or an inclusive or or an exclusive or?

(That was a fun title to write!)

At the start of our discrete mathematics course we talk about symbolic logic. Students are often confused by the logical operator “OR.”

If p and q are statements then p OR q is true if either p is true or q is true or if both p and q are true. This is easily expressed in a truth table:

|    |    |        |
|----|----|--------|
| p  | q  | p OR q |
| T  | T  | T      |
| T  | F  | T      |
| F  | T  | T      |
| F  | F  | F      |

The reason this confuses students is that sometimes when we say “or” in everyday conversation we mean p is true or q is true, but p and q are not both true. (For example, “the door is open or the door is closed.”)

This brings to mind the logical operation exclusive or, “XOR” (the usual “or” is inclusive or). The truth table for XOR is shown below.

|    |    |         |
|----|----|---------|
| p  | q  | p XOR q |
| T  | T  | F       |
| T  | F  | T       |
| F  | T  | T       |
| F  | F  | F       |

It seems like we use “or” as exclusive sometimes and inclusive other times.

My colleagues and I were talking about this at the lunch table the other day. One of my colleagues presented a simple example that illustrates this confusion.

Waiter: “Would you like tea or coffee?” (exclusive or)

Waiter: “Would you like cream or sugar?” (inclusive or)

Patron: “I’d like both, thank you.”

I thought this was a great example.

Another one of my lunchmates is a linguist and he asserted that when we say “or” we always mean inclusive or—even though it seems that we’re using exclusive or. For example, the patron above could have asked for both coffee and tea, it is just that that isn’t usually done.

What about the door being open/closed example? A door can’t be both open and closed. In particular, ”the door is open” and “the door is closed” can’t both be true at the same time. But this doesn’t mean that that use of “or” is exclusive or. Exclusive or means that when both statements p and q are true, p XOR q is false. In the door example, we never encounter the “true or true” situation!

According to Wikipedia the source of this argument is a 1971 article by Barrett and Stenner called “The Myth of the Exclusive ‘Or’” (Mind, 80 (317), 116–121).

No author has produced an example of an English or-sentence that appears to be false because both of its inputs are true. Certainly there are many or-sentences such as “The light bulb is either on or off” in which it is obvious that both disjuncts cannot be true. But it is not obvious that this is due to the nature of the word “or” rather than to particular facts about the world.

Update: I had another example to illustrate this misconception. The sentence

$x<0\text{ OR } x\ge 0$,

is true for all $x$, right? And this is the logical (inclusive) OR, right? But this is exactly the same as “the door is open or the door is closed.” Just as the door is either open or closed, but can’t be both open and closed, one of the two inequalities $x<0$ and $x\ge 0$ must be true, but both can’t be true simultaneously. The fact that both halves can’t be true at the same time (mathematically) doesn’t mean that two trues joined by this “or” is false.

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Posted in Math, Teaching | Tags: discrete mathematics, exclusive or, inclusive or, logic

## Responses

1. Reblogged this on imasciencegeek and commented:
A fresh recap on logical use of inclusive and exclusive or. Been thinking about this recently in association with other systems.

By: imasciencegeek on July 9, 2012
at 7:47 pm

2. Thank you for addressing this topic. Most people seem to take the simplistic interpretation: a or b implies if a then not b. But as you point out, both common situations like the waiter, and set theory illustrated by Venn diagrams imply that, in my words “or includes and”.

By: Thomas Swanson on April 1, 2013
at 5:25 pm
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