Consider statement for every x belonging to set consisting of {-1 , -2 } : square root of x is equal to one. Of course it is false, given that square root is not defined over negative nubers which takes real value. So, the opposite statement should be true: exists x belonging to the set {-1 , -2 } : such that square root of x is not equal to one. So it should be less or more than one. This statement is also not true. So, can we conclude that statement is meaningless?
[Math] True, false and meaningless statements in math.
logic
Best Answer
The issue here has to do with "undefined terms". Suppose that we are working with the field of real numbers (without non-real complex numbers). The signature for this field has symbols for addition and multiplication, the order relation $<$, and the equality sign.
This signature does not have a symbol for square roots. So the phrase "the square root of $-1$ is not equal to $1$" does not have any direct translation into the formal language at hand. If every number did have a unique square root, we could introduce a new function symbol $\sqrt{}$, and interpret it so that $\sqrt{x}$ is always a square root of $x$.
In contrast, every real number does have a square, so we could add a new unary function $S$ such that $S(x)$ gives $x\cdot x$. We cannot do that for square roots. In the field of real numbers, we cannot refer to $\sqrt{-1}$, as we have no symbol $\sqrt{}$.
We could translate the phrase "the square root of $-1$ is not equal to $1$" as "for all $x$, if $x^2 = -1$ then $x \not =1$". That new quoted statement can be directly translated to the language of fields, and it is true in the field of real numbers. On the other hand, the statement "the square root of $-1$ is not equal to $1$" can also be translated as "there is an $x$ with $x^2 = 1$ and $x \not = 1$". That statement is false in the field of real numbers. So the translation that we choose will affect the truth value.
In the general, an English sentence that talks about "the" object with a particular property makes an existential assumption that there is a unique object with that property. When that existential assumption is correct, it usually straightforward to translate the English sentence into a formal sentence, and the truth of the formal sentence will not depend on which reasonable translation we use. But, when the existential assumption is not satisfied, the exact translation that we use can make a difference.
Another approach to handling issues like real square roots (which don't always exist) is to introduce partial function symbols - function symbols that can be undefined on certain inputs. This is the subject of "free logic". But normal first-order logic is not a free logic: it assumes that a function symbol applied to an object always returns an object.
Yet another approach is to add a new function symbol $\sqrt{}$ and define it in some arbitrary way when it would not normally be defined. The, all we can say is that if $x$ has a square root then $\sqrt{x}$ is a square root of $x$. In that case, the statement "the square root of $-1$ is not equal to $1$" will depend on exactly how we arbitrarily assign values for numbers that don't have square roots.