Did Admiral Nelson have one eye?

Question

Assuming that

$$\neg (\text{DID NOT HAVE})= \text{HAD},$$

I get

$$\neg (\text{Admiral Nelson did not have one eye})=\text{Admiral Nelson had one eye}.$$

The statement and its negation are simultaneously true. What is the mistake here?

Answer 1

I think this has less to do with math than with English...

Let $n\in \mathbb N$ denote the number of eyes that Admiral Nelson has.

$$(n=1)=(\text{Admiral Nelson had one eye})$$ $$\neg(n=1)=(n\ne 1)=(\text{Admiral Nelson had zero, two, three...eyes})$$

Answer 2

This is just a matter of translating natural language into mathematical language. The problem is the initial statement "Nelson did no have one eye", and what exactly that sentence means. There are multiple interpretations of the statement, each with a different meaning, and also a different negation.

Interpretation 1:

"It is not true that the number of eyes that Admiral Nelson had is one".

The negation of this statement is "The number of eyes Nelson had is one".

Interpretation 2:

"There exists an eye that Nelson did not have"

The negation of this statement is "There does not exist an eye that Nelson did not have", which simplifies to "Nelson had both eyes".

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Note that the first interpretation is incorrect, since it is true that the number of Nelson's eyes is one. Your mistake is taking the second interpretation of the original statement, then negating the first interpretation.

Answer 3

$$\neg (\text{Admiral Nelson did not have one eye})=\text{Admiral Nelson had one eye}$$

The statement and its negation are simultaneously true.

I guess you are imagining a scenario where Nelson had two eyes, and thinking:

1. Nelson did not have one eye.

   (More clearly: Nelson did not have exactly one eye.)

2. Nelson had an eye (which does not preclude the fact that he had multiple eyes).

   (More clearly: Nelson had at least one eye.)

What is the mistake here?

In the above scenario, statements 1 and 2 are indeed simultaneously true. Therefore, they are not actually negations of each other!

Statement 1's negation is

3. Nelson had exactly one eye.

while statement 2's negation is

4. Nelson had no eye.

Summing up: The sentence “Nelson had one eye” is ambiguous, and your confusion stems from reading it both as “Nelson had exactly one eye” and as “Nelson had at least one eye”.

P.S. Observe that in a scenario where Nelson had no eye, statement 1 is true while statement 2 is false. The fact remains that two statements negate each other if and only if they have opposite truth values regardless of the scenario.