Brackets in $\color{red}{(}X\rightarrow (Y\rightarrow {\sim} Z)\color{red}{)}\,\&\,X$

Question

Consider the following formula:

$$\color{red}{(}X\rightarrow (Y\rightarrow {\sim} Z)\color{red}{)}\,\&\,X$$

Are the brackets in red compulsory?

Answer 1

Formal answer: This depends entirely on your notational conventions - in particular the question of how tightly $\rightarrow$ binds relative to $\&$, i.e. does $P\rightarrow Q \,\&\, R$ mean $(P\rightarrow Q)\,\&\, R$ or $P\rightarrow (Q\,\&\,R)$? If it's the former, you don't need to include the brackets in your expression. If it's the latter, you do. Notational conventions like this are far from universal, so it's impossible to answer your question definitively unless you give (a reference to) the conventions you're using.

Informal answer: Yes, these brackets are compulsory, exactly because notational conventions are not universal. If you don't include the brackets, someone is likely to be confused, while if you include them, there is no room for confusion.