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For the statement below:

One of my two cars was stolen.

What is the negation? For me, it seems like there are two ways of interpreting this.

First, if we interpret the statement as:

$N = $ the number of my two cars which were stolen
$P(N) = $ N is equal to 1

Then the negation of the above statement would be $\sim P(N)$, or:

The number of my two cars which were stolen is not equal to one.

On the other hand, the more "intuitive" negation of the statement would be:

One of my two cars was not stolen.

And this makes sense if we interpret the original statement as actually referring to a specific car out of the two.

Which of these (if either) is correct, and why? Thanks

b_pcakes
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2 Answers2

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One of my two cars was stolen.

There is some ambiguity here. Was only one car stolen? Or was at least one car stolen?

Since you have 2 cars, you could have either 0, 1 or 2 cars stolen.

If only one car was stolen, then you did not have either 0 or 2 cars stolen. The negation would be that you had either 0 or 2 cars stolen.

If at least one car was stolen, then you did not have zero cars stolen. The negation would be that you had zero cars stolen.

Dan Christensen
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The negation is "either neither of my cars was stolen, or both of my two cars were stolen". Apologies for "either neither", which might sound funny.

Of the two alternatives you provide, the correct one is "The number of my two cars which were stolen is not equal to one." It's not ideal, because now you're primarily talking about numbers, which the original statement doesn't mention. After all, the original statement isn't "the number of cars I had was two, and the number of cars stolen from me is one", although that's equivalent in a language rich enough to talk about numbers. However, the original can be (accurately enough) expressed in first order logic without any explicit mention of "one" or "two".

BrianO
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