Omitting parantheses in formulas Lately I read the following:

parentheses can always be omitted, so instead of $((\neg A)\Rightarrow B)$ we may write $(\neg A)\Rightarrow B$. But we may not write $\neg A\Rightarrow B$, because this would not distinguish the intended formula from $\neg(A\Rightarrow B)$.

I understand why the outermost parantheses may be omitted but I don't understand why the omission of the inner parantheses around $A$ does not distinguish the formula from the last formula stated in the picture. I tried to reason as to why they would not be distinguished from each other and all I could come up with was that it's probably an aesthetic rather than a structural reason. 
 A: All of this is a matter of order of operations (or operator precedence).
For instance, this is carefully defined in any programming language.
From the same wikipedia article:
In logic, $\neg$ has higher precedence than $\wedge$, $\wedge$ higher than $\vee$, and $\vee$ higher than $\rightarrow$. 
A: It ought to be clear that it's unworkable to allow omitting the parentheses in both $(\neg A)\Rightarrow B$ and $\neg(A\Rightarrow B)$. These are two different formulas with different meanings, yet each of them becomes "$\neg A\Rightarrow B$" if we omit the parentheses.
Thus, in order to be sure what we're speaking about we need either to declare that "$\neg A\Rightarrow B$" is not a legal way to write a formula, or agree on some convention for which of $(\neg A)\Rightarrow B$ or $\neg(A\Rightarrow B)$ it is going to mean.
The quote that confuses you describes situation before such a convention has been set.
Once you have decided to use the common convention, $\neg A\Rightarrow B$ is going to mean $(\neg A)\Rightarrow B$. But it is important to be aware that there is a choice being made and that we could equally well (from a formal point of view) have chosen the opposite convention for writing down things. That's just not the common choice.
A: As others have mentioned, the main issue here (one that used to bother me as well) concerns operator precedence. As Henning notes, a convention has to be established first for anything to make sense and to avoid ambiguity. Broadly speaking, the following is a little table that lists binary connectives and their corresponding ranks:
\begin{array}{ccc}
\hline
{} & \text{Rank of Operators} & {}\\
\hline
\text{Operator}& {} & \text{Rank}\\
\hline
\Large\leftrightarrow & {} & 5\\[0.5em]
\Large\rightarrow & {} & 4\\[0.5em]
\Large\lor & {} & 3\\[0.5em]
\Large\land & {} & 2\\[0.5em]
\Large\neg & {} & 1\\
\hline
\end{array}
With these rankings in mind, a simple process may be used to determine how to eliminate parentheses. The following description of this process may appear to be somewhat complicated at first, but its simplicity will be illustrated with a few examples.

Examples: As examples, we will consider
$$
Q \rightarrow R \rightarrow \neg S \land \neg T \lor U \leftrightarrow V \land W \land X \lor Y\tag{1}
$$
and
$$
[(Q \lor R) \leftrightarrow S] \rightarrow [(\neg T) \land U].\tag{2}
$$

Suppose we encounter the propositional form $\Upsilon \, \phi \, \Lambda$, where $\Upsilon$ and $\Lambda$ may be statement letters or propositional forms with several components, and $\phi$ is one of the connectives, $\leftrightarrow, \rightarrow, \lor, \text{or } \land$. The outermost parentheses, if any, are dropped from $\Upsilon$ if the rank of $\phi$ is greater than or equal to the rank of $\Upsilon$. The outermost parentheses, if any, are dropped from $\Lambda$ if the rank of $\phi$ is greater than the rank of $\Lambda$. This process is repeated until parentheses can no longer be dropped. 
Any difficulties encountered while trying to understand the finer points of how the process works should be put to rest by seeing how the process is used to add parentheses to $(1)$ and remove parentheses from $(2)$. If we let $\rho$ and $\sigma$ represent $(2)$ and $(1)$, respectively, then we may use the process to get
\begin{align}
 \rho  &\Leftrightarrow  [(Q \lor R) \leftrightarrow S] \rightarrow [(\neg T) \land U] \nonumber  \\[0.5em]
  &\Leftrightarrow [(Q \lor R) \leftrightarrow S] \rightarrow (\neg T) \land U \nonumber \\[0.5em]
  &\Leftrightarrow  [Q \lor R \leftrightarrow S] \rightarrow (\neg T) \land U \nonumber \\[0.5em]
  &\Leftrightarrow (Q \lor R \leftrightarrow S) \rightarrow \neg T \land U  \nonumber  
\end{align}
and 
\begin{align}
 \sigma  &\Leftrightarrow Q \rightarrow R \rightarrow \neg S \land \neg T \lor U \leftrightarrow V \land W \land X \lor Y   \nonumber \\[0.5em]
   &\Leftrightarrow [Q \rightarrow R \rightarrow \neg S \land \neg T \lor U] \leftrightarrow [V \land W \land X \lor Y] \nonumber \\[0.5em]
   &\Leftrightarrow [(Q \rightarrow R) \rightarrow (\neg S \land \neg T \lor U)] \leftrightarrow [V \land W \land X \lor Y]  \nonumber \\[0.5em]
   &\Leftrightarrow  [(Q \rightarrow R) \rightarrow ((\neg S \land \neg T) \lor U)] \leftrightarrow [(V \land W \land X) \lor Y] \nonumber \\[0.5em]
   &\Leftrightarrow  [(Q \rightarrow R) \rightarrow ((\neg S \land \neg T) \lor U)] \leftrightarrow [((V \land W) \land X) \lor Y] \nonumber \\[0.5em]
   &\Leftrightarrow  [(Q \rightarrow R) \rightarrow (((\neg S) \land (\neg T)) \lor U)] \leftrightarrow [((V \land W) \land X) \lor Y]. \nonumber
\end{align}

As you can see, those are rather heady examples. You can get as fine-detailed as you want, but the main point to keep in mind when communicating mathematics is to strike a balance between clarity and brevity. From the examples above, it is clear that it would be perfectly valid to simply write
$$
Q \rightarrow R \rightarrow \neg S \land \neg T \lor U \leftrightarrow V \land W \land X \lor Y,
$$
but what reader is going to be able to parse that effectively? 
Using the method outlined above, you can choose for yourself, more or less, just how many parentheses you chose to include or exclude, but always remember that clarity is a must. 
If you'd like, you can read where most of this post came from--an "article" I wrote a few years back. It may be accessed here. 
