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Let be:

p: is A a rational number.

Not p: A is not a rational number.

Hence, A can be a tree, the moon, etc...

But I would like to obtain something like this:

p: A is a rational number.
Not p: A is an irrational number.

Now:

p: A is a number and A is a rational number
Not p: A is not a rational number or A is not a number.

In this case A still can be a tree or the moon.

In general, does the negation of a sentence like "X is Y" always implies that "X" can be everything in the universe except Y?

How do we restrict a negation so it includes only a subset of the universe?

I mean in the sense expressed above. I know that the negation of "X is not Y" gives that "X is Y".

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    You might find this useful: en.wikipedia.org/wiki/Many-sorted_logic Commented Mar 11, 2022 at 5:19
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    There’s a bit of rhetorical trickery going on here. In terms of “restricting a logical negation”, it’s not too tricky to include an additional predicate domain and to define the outcome of an operation relative to that domain. However, you’re defining a new operation - what comes out doesn’t have the same logical force as Negation as such. This is important to appreciate when considering the application of logical methods, because it risks invoking distribution fallacies (en.m.wikipedia.org/wiki/Categorical_proposition#Distributivity)
    – Paul Ross
    Commented Mar 11, 2022 at 8:51

1 Answer 1

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First one defines the universe of discourse, i.e. one defines the set A of all elements to be considered.

Then one can define propositions about the universe of discourse: For example „All elements from A satisfy property P“. The proposition negates as „There exists at least one element of A which violates property P.“

Your example: A = the set of real numbers. P = the property of being rational.

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