Is any false statement a negation of a true statement? 
*

*Consider the following example:

What is the negation of this statement? $$\forall x{\in}\mathbb{R}\;\exists y{\in}\mathbb{R}\quad x+y\geq 0.\tag P$$

I know that the standard answer is $$\exists x{\in}\mathbb{R}\;\forall y{\in}\mathbb{R}\quad x+y<0.\tag{¬P}$$
But since we know that the first statement $P$ is true in mathematics, is it correct to say that any statement that is false is a negation of $P\,?$ For example, is $Q$ also a negation of $P\,?$ $$1<0\tag Q$$ Why or why not?


*Is any contradiction a negation of a tautology?
For example, can I say that $$P \wedge \neg P$$ is a negation of $$Q\vee \neg Q\,?$$
 A: *

*In classical logic, to negate a formula (including an open formula that has no definite truth value) means to logically flip its truth value.


*Therefore, these statements are equivalent:

*

*$P$ and $Q$ are negations of each other;

*regardless of interpretation, $P$ and $Q$ have opposite truth values;

*$P↔Q$ is unsatisfiable;

*$P\land Q$ is unsatisfiable, i.e., $P$ and $Q$ are inconsistent with each other.

(A more pedantic definition of ‘negation’ than #1 considers $(¬A∨¬B)$ to be merely logically equivalent to the negation $¬(A∧B)$ of $(A∧B);$ in this case, change “$P$ and $Q$” in bullet 2 to “the truth-functional form of $P$ and $Q$”, and change “satisfiable” to “a contradiction”.)


*On the other hand, these statements are equivalent:

*

*$P$ and $Q$ are are logically equivalent;

*regardless of interpretation, $P$ and $Q$ have the same truth value;

*$P↔Q$ is valid, i.e., $P\equiv Q;$

*$P\lor \lnot Q$ is valid.



*

If statement $P$ is true in mathematics, then is any false statement in mathematics a negation of $P\,?$ For example, here, is $Q$ a negation of $P$? $$\forall x{\in}\mathbb{R}\;\exists y{\in}\mathbb{R}\quad x+y\geq 0.\tag P$$ $$1<0\tag Q$$

Two formulae with opposite truth values (in a given interpretation) do not necessarily negate each other.
For example, $(\forall x\;x^2\le0)$ and $(x=x)$ have opposite truth values in the universe $\mathbb R,$ but the same truth value in the universe of all imaginary numbers (that is, in this interpretation, the two formula are equivalent); so, they aren't mutual negations.
In your example, in an interpretation in which the binary operation + is defined x+y := -3-|xy|, where subtraction & multiplication retain their standard meanings, sentences $P$ and $Q$ are equivalent to each other, even though in mathematics they have opposite truth values.


*Notice that negation preserves logical equivalence.


*By the definition in #1:

*

*Negating a valid formula gives an unsatisfiable formula, and vice versa.

*Negating an invalid formula gives a satisfiable formula, and vice versa.

Examples:
Observe that both $\big(\forall x\,P(x)\big)$ and its negation $\big(\exists x\,¬P(x)\big)$ are satisfiable and invalid.
$(\forall x\;x^2\ge0)$ is true in the universe $\mathbb R,$ but its negation $(\exists x\;x^2<0)$ is true in the universe of all imaginary numbers so is satisfiable albeit invalid.
$(x=x)$ is valid (i.e., logically true), and its negation $(x\ne x)$ is not merely invalid, but unsatisfiable (though not a propositional-logic contradiction).


*

Is any contradiction a negation of a tautology?

Yes, by the definition in #1.
A: 
Any statement that is false is negation of some true stament?
Any contradiction is a negation of some tautology?

As long as the laws $\sim T\equiv F$ and $\sim F\equiv T$ are valid in the universe of discourse, the answers to both of your questions is YES.
