The Meaning of 'Or' in Logic Statements

Date: 12/19/2003 at 13:02:01
From: steven
Subject: logic

This question was recently given on a logic test.  I am still having a
hard time understanding why the answer is C.

Assume the statement "James is taking fencing or algebra" is true. 
Which of the following statements must be false?

A.  James is taking only fencing
B.  James is taking both fencing and algebra.
C.  James is taking neither fencing, nor algebra.
D.  James is taking only algebra.

I think that "James is taking fencing or algebra" implies that he is
in only one of the classes, but we don't know which one.  Thus B is
the correct answer to me.

Date: 12/19/2003 at 14:11:59
From: Doctor Paul
Subject: Re: logic

Hi Steven -

At issue here is the meaning of the word "or".  Consider these two 
events:

Event P: James is taking fencing
Event Q: James is taking algebra

When we want to inquire about the truthfulness of a statement such as
"P AND Q", it's often easiest to draw a "truth table".

The statement "P AND Q" is true if and only if P is true and Q is 
true.  If either statement is false, then the statement is false.  So 
the truth table looks like this:

  P  |  Q  |  P AND Q
-------------------
  T  |  T  |  T
  T  |  F  |  F
  F  |  T  |  F
  F  |  F  |  F

So when P is True and Q is True, then the statement "P AND Q" is 
true.  In all other possibilities, the statement "P AND Q" is false.

Now, the statement "P OR Q" is true if and only if either P is true 
or Q is true.  So the truth table looks like this:

  P  |  Q  |  P OR Q
-------------------
  T  |  T  |  T
  T  |  F  |  T
  F  |  T  |  T
  F  |  F  |  F

The only way that the statement "P OR Q" can be false is if neither P 
nor Q is true.  In particular, notice that the statement "P OR Q" is 
true when both P and Q are true.  So if James is taking both fencing 
and algebra, then the statement "James is taking fencing or algebra" 
is true.  This explains why the answer to your original question 
is "C".

But it doesn't settle the debate about what the word "or" means.  At 
issue here is the fact that our use of the word "or" in common 
English language is not consistent with the way we defined "P OR Q" 
above.

If I write the logic statement "James is taking fencing or algebra",
it doesn't mean that James could not be taking both fencing and
algebra.  But this is certainly implied when I write it or say it in
the English language.  You stated above that you thought that the use
of the word "or" in the sentence about James meant that James was only
taking one of the classes.  That would be the correct way to interpret
the statement in English, but the use of the word "OR" in logic has a
different meaning.

To help eliminate some of the confusion, logic has introduced the
"exclusive or" which is written as XOR.  So the event "P XOR Q" is
true if and only if either P is true or Q is true and it is also the
case that P and Q are not both true.  So the truth table looks like this:

  P  |  Q  |  P XOR Q
-------------------
  T  |  T  |  F
  T  |  F  |  T
  F  |  T  |  T
  F  |  F  |  F

It is unfortunate that our use of the word "or" in the English
language pretty much always refers to the logic XOR.

Questions such as the one you asked about are confusing--you don't 
know whether the "or" in the sentence refers to the logic OR or the 
logic XOR.  If you can ask the instructor for clarification, do so.  
If you can't ask for clarification (for instance, if this question 
appears on a standardized test), you have no choice but to assume 
that the word "or" in the sentence refers to the logic OR (I will 
agree that it is seemingly counterintuitive to think this way--
basically I'm telling you that "or" means one thing in your logic 
class and another thing everywhere else).  If the sentence wanted you 
to assume that the word "or" referred to the logic XOR, then the 
sentence would have read:

"James is taking fencing or algebra but not both."

I hope this helps.  Please write back if you'd like to talk about 
this some more.

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