Exclusive or Inclusive Disjunction?

Date: 06/28/2001 at 04:29:05
From: Gemma Baltazar
Subject: Worded Logic Problem - Shall I use Exclusive Disjunction or 
Inclusive Disjunction

I have difficulty interpreting this problem, especially the first 
sentence:

Tanya is either a singer or a ballerina. If she is a singer, then she 
has a lovely voice. If she is a ballerina, then she has long legs. 
Tanya has a lovely voice and long legs, so she is both.


MY SOLUTION:
let p = Tanya is a singer
    q = Tanya is a ballerina
    r = Tanya has a lovely voice
    s = Tanya has long legs

Interpretation 1 (using exclusive disjunction):
not (p iff q), if p then r, if q then s, (r and s) therefore (p and q)

Interpretation 2 (using inclusive disjunction):
p or q, if p then r, if q then s, (r and s) therefore (p and q)

Does the "either...or" statement mean an exclusive disjunction, or 
does it simply mean an (inclusive) disjunction? I am really confused 
because I have two books here that give different interpretations.

Date: 06/28/2001 at 15:08:12
From: Doctor Achilles
Subject: Re: Worded Logic Problem - Shall I use Exclusive Disjunction 
or Inclusive Disjunction

Hi Gemma,

Thanks for writing to Dr. Math.

I usually translate "either...or" as an exclusive disjunction.  
Sentences such as

  "Either I will go to a movie or I will read a book"

are best translated as exclusive.

However, there are contexts in which "either...or" can be translated 
as an inclusive disjunction. For example, if your mom tells you

  "Either do the dishes or walk the dog"

then it is understood that you must do at least one of those things, 
but you are not precluded from doing both.

In my opinion, when translating disjunctions, context is much more 
important to determining whether an "or" is inclusive or exclusive 
than the presence or absence of an "either." Compare the two examples 
above to these two sentences:

  "I will go to a movie or I will read a book."

  "Do the dishes or walk the dog."

I would argue that the first is an exclusive disjunction and the 
second is an inclusive disjunction. In other words, the "either" that 
appeared in the original two sentences is superfluous; it can be left 
out without changing the way that the sentences are translated into 
logical terms. Context is what is important here.

English is often very difficult to translate into logical sentences.  
The more context you can get for a sentence, the better.  I would 
argue that most of the time, the sample sentences above would be 
translated as exclusive and inclusive, respectively. However, there 
are some contexts in which this would not be the case. Imagine the 
following sample dialogues:

  "Either I will go to a movie or I will read a book."
  "What movie would you like to see?"
  "The new action thriller in theatres sounds good,
    and when we get back I think I'll read the book I'm 
    in the middle of."

  "Either do the dishes or walk the dog."
  "Can the I walk the dog later, after I do the dishes?"
  "No, I want your brother to help out too; don't do both."

Here the context of the sentences changes the interpretation.

What about your sentence? At first glance (before I look at the rest 
of the sentences), I prefer the exclusive interpretation for the 
sentence "Tanya is either a singer or a ballerina," but let's take a 
look at the possible translations of your problem and see what we can 
tell from the context of each. Let's look first at the inclusive 
disjunction (the second translation):

  1)  (p or q)        Tanya is a singer or she is a ballerina

  2)  (if p then r)   If she is a singer then she has a lovely voice

  3)  (if q then s)   If she is a ballerina then she has long legs

  4)  (r and s)       She has a lovely voice and she has long legs

  5)  Therefore, (p and q)

Notice first of all that this is not a valid deduction. To see why, 
let's try to deduce (5) from (1) through (4). First, let's separate 
(4) into two parts:

  4a)   r             She has a lovely voice

  4b)   s             She has long legs

Now (2) says that if we know p, then we can deduce r. However, it does 
not allow us to do the reverse: if we have r we cannot conclude p.  
The same problem holds for q and s. (This is called the fallacy of 
affirming the consequent.)

The inclusive disjunction interpretation thus seems to have a few 
problems. Let's look at the exclusive interpretation:

  1)  (not (p iff q)) Tanya is either a singer or a ballerina

  2)  (if p then r)   If she is a singer then she has a lovely voice

  3)  (if q then s)   If she is a ballerina then she has long legs

  4)  (r and s)       She has a lovely voice and she has long legs

  5)  Therefore, (p and q)

Again, if you try to deduce (5) from (1) through (4) you will have the 
same problem as above. In fact, things are substantially worse for 
this interpretation because (1) contradicts (5). Why?  (5) asserts 
that p is true and q is true. But if p and q are both true, then (p 
iff q) is true so (not (p iff q)) is false.

So if you gave me the sentence "Tanya is either a singer or a 
ballerina" without any other context, I would say it should probably 
be translated as an exclusive disjunction. However, while both 
interpretations lead to an invalid inference, the exclusive deduction 
is a flat-out contradiction, so given that I'd say the context makes 
me prefer the inclusive translation.

I hope all this helps.  If you have any other questions about this or 
if you want some help on other logic problems, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/