Assertion not well-defined My question is the following : does the assertion
$$x=2\implies x^2=4 \tag1$$ have sense without declaring what $x$ should be, or should I write something like
$$\forall x\in\mathbb{R},x=2\implies x^2=4\tag2$$ to be correct ? In the first one, I allow $x$ to be anything (a complex number for example, or a function to be more exotic), but perhaps that writing this way would reveal some absurdities. Thank you for your comments and answers !
 A: First, a caveat: I'm not a logician, and I'm just writing this from the point of view of a working mathematician. Logicians might have other ways of looking at these things, but I'm going to answer in a way that might give you some insight into how other mathematicians typically write and read logical statements like this.
Informally, there's no ambiguity about the statement you wrote, because the hypothesis $x=2$ makes it clear that it applies to one object and one object only, namely the integer $2$. 
But since you asked the question, it's worth pointing out that there's a convention that's almost always followed in ordinary mathematical writing (as opposed to formal symbolic logic) called implicit universal quantification. It applies in the following situation: If an implication contains a variable in both its hypothesis and conclusion, and that variable is unquantifed and otherwise undefined, then the statement is to be interpreted as if that variable were universally quantified. In other words, if $P(x)$ and $Q(x)$ are open sentences containing $x$ as a free variable, we interpret 
$$P(x)\implies Q(x)$$ 
to mean 
$$\forall x, \ P(x)\implies Q(x).$$ 
Of course, for this to be a well-defined statement, there has to be a clear understanding of the domain of the quantifier $\forall x$. In practice, this is usually determined either implicitly from the context, or explicitly by including something in the hypothesis that stipulates what set of values of $x$ are allowable. For example, the statement "If $x$ is a real number and $x\ge 2$, then $x^2 \ge 4$" is to be read as though it started with "For all $x$," and the hypothesis makes it clear that the domain of the quantifier is the set of real numbers. 
A: Let $\mathcal L$ a first-order logic language, with equality as binary relation symbol, $\times$ as binary function symbol and $2$, $4$ as constant symbols.
Then, the formula $x = 2 \to x \times x = 4$ is well-defined, where $x$ is a free variable.
A: 
My question is the following : does the assertion
  $$x=2\implies x^2=4 \tag1$$ have sense without declaring what $x$ should be

On its own, this statement is ambiguous, in part for not declaring what $x$ is.

Or should I write something like
  $$\forall x\in\mathbb{R},x=2\implies x^2=4\tag2$$ to be correct?

(2) would be less ambiguous than (1), though adding the universal quantifier changes the statement somewhat. Instead of a single number $x$, you would be talking about infinitely many values for $x$. You could have written simply:$$ x\in\mathbb{R} \land [x=2\implies x^2=4]$$
This would be a statement about a single real value for $x$.
A stickler for detail, howerver, may still legitimately ask if you are talking about equality and exponentiation on $\mathbb{R}$ in the usual sense.
A: This is way outside my field of expertise, so the details of this answer may be a little off.
To get an answer your question, you need to do two things.


*

*Decide which formal system you're working in. (If you're not using a formal system, but rather just trying to communicate with people in natural language, then your question has a straightforward answer: an assertion has sense if and only if your readers understand what you're talking about.)

*Decide exactly what you mean by "having sense." From context, I think it's likely that you mean "being a well-formed formula" or "being a theorem," but you might also mean something else.
To give an example, let's say the formal system we're working in is Hofstadter's Typographical Number Theory (TNT). The language and rules of inference for this system are described here and here, as well as in Hofstadter's book Gödel, Escher, Bach.
Having answered (1), we can now go on to answer (2) for various meanings of "having sense."

In TNT, the string

$\mathtt{x = 2 \Longrightarrow x^2 = 4}$

is not a well-formed formula. In fact, $\mathtt{x}$, $\mathtt{2}$, $\mathtt{4}$, $\Longrightarrow$, and $\mathtt{{}^2}$ aren't even symbols in the language. However,

$\mathtt{\langle a = SS0 \supset (a \cdot a) = SSSS0 \rangle}$

is a well-formed formula, and if you interpret well-formed formulas of TNT in the standard way, it means the same thing as the natural-language assertion you asked about. The string

$\mathtt{\langle a = SSS0 \supset (a \cdot a) = SSSS0 \rangle}$

is also a well-formed formula of TNT.

I'm quite sure that the formula

$\mathtt{\langle a = SS0 \supset (a \cdot a) = SSSS0 \rangle}$

is a theorem of TNT. I haven't come up with proof, but for your purposes, a proof may be a bit beside the point. What you really want to know is this. Suppose you have a proof of the formula

$\mathtt{\forall a \colon \langle a = SS0 \supset (a \cdot a) = SSSS0 \rangle},$

which I'm quite sure is also a theorem. Since $\mathtt{a}$ is not bound in the part of the formula that comes after the $\mathtt{\forall a}$, you can use the rule of specification to remove the $\mathtt{\forall a}$, giving a proof of the unquantified formula.
Nobody knows whether

$\mathtt{\langle a = SSS0 \supset (a \cdot a) = SSSS0 \rangle}$

is a theorem of TNT. If it were a theorem, TNT would be inconsistent. Most people I know really hope that TNT is consistent, but nobody has managed to prove it from within TNT. Paradoxically, that's good news for those of us who hope TNT is consistent: if TNT could prove its own consistency, it would have to be inconsistent.
A: I fail to see the problem. If the assumption of the implication $(1)$ is true, then $x$ is specified, it is the number 2. The conclusion $x^2 = 4$ is true as well, as $x$ was $2$ and the square of $2$ is $4$.
If the assumption is wrong, then we are not interested in $x$, except that we know that $x$ is not the number $2$.
