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The following sentence came up in a recent conversation with a Polish national:

I don't have a wife and any children.

I corrected his sentence to:

I don't have a wife or any children.

He immediately pointed out that "or" usually means only one of the statements need be true, and that he wanted to convey that both statements were true, i.e.:

I don't have a wife and I don't have any children.

What is the rationale for using "or" in this case?

[As a side question, should we be using "nor" in this case? I'm assuming not, though again I'm unable to explain why.]

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¬[p ∨ q] ≡ ¬p ∧ ¬q The problem only arises with negation. –  John Lawler Mar 22 at 14:47

2 Answers 2

up vote 4 down vote accepted

Actually, in this case, boolean logic does apply. (It often doesn't, as language usually has a tendency to escape the grasp of logic.)

There are two statements in your example, both of which are false.

Let's call "I have a wife" P1, and "I have children" P2.

I have no wife, so NOT P1 is a true statement. Also, I have no children, so NOT P2 is also true.

Since both are true, this is true:

(NOT P1) AND (NOT P2)

This is exactly how you build up your sentence: I do not have a wife and I do not have children.

Now, logic tells us that from > (NOT P1) AND (NOT P2) we can conclude:

NOT (P1 OR P2)

Hold on... that is weird? Yups. If I know that one thing is NOT true, and another thing is also NOT true, then I can negate both at the same time if I use OR.
If I would use AND in this case: NOT (P1 AND P2), then P1 might be true, or P2 might be true, and the statement would still be true.

If I say I drink coffee AND tea, obviously I drink both. If I say I do NOT drink coffee AND tea, it means that I do NOT drink both - I may very well drink either! If I say I do NOT drink tea OR coffee, it means I drink neither of the two!

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As a programmer, I can readily appreciate this, and, indeed, it's the first thing that came to mind. However, are we merely making this example of phrasing fit the logic, or would this turn of phrase have been conceived because of the logic? (i.e. Which came first... ?) –  Richard Horrocks Mar 22 at 14:15
    
What came first - our language or our reason? I won't get in between the fighting armies on that one :) –  oerkelens Mar 22 at 14:38
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Logic is a stick-figure representation of natural language semantics. And this is a lovely example of what it's good for -- when it is any good, which it isn't always -- in analyzing natural language. Details on De Morgan's Laws available here. –  John Lawler Mar 22 at 14:43
    
Ok, thanks guys - this does make sense. –  Richard Horrocks Mar 22 at 15:19
    
+1 - well done! –  medica Mar 23 at 1:52

You should be using 'nor'.

Used positively 'or' does mean that only one of the facts is true. ' That bag contains apples or oranges'. The implication is that it doesn't contain both.

'Either that person is a man or a woman'. Clearly the person cannot be both. They are either one or the other.

When used negatively it acts as an accumulator and rules out any of the things mentioned. 'I have neither a car nor a bicycle' means I don't have either of them.

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This is often true for the word or in positive sentences. For negative sentences, the rules are different: "You can't have cake and pie" means you can have one, but not both. "You can't have cake or pie" means you can't have either. –  Peter Shor Mar 22 at 12:57

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