In a book about mathematical logic I am currently reading, there's the following lemma:
Let $\phi$ be a proposition, w a finite string. Then: If the string $\phi w$ is a proposition, w must be empty, so that $\phi w=\phi$
I don't see why that should be the case, what about $\phi=(a\land b)$ and $w=\lor(b\land c)$, then w is not empty and $\phi w= (a\land b)\lor(b\land c)$ is a proposion?
This has to do with the precise details of the formal definition of "proposition" in the system in question. In formal logic I'm surprised they're not talking about well formed formulas instead.
Since you don't give the details of the definition in force it's impossible to give a definitive answer to your question. But one possibility is this, just to give an idea of what the answer might be: It happens sometimes that wffs are defined by saying that a propositional variable is a wff, and if $a$ and $b$ are wffs then $(a\land c)$, etc, are wffs. With that definition your $(a\land b)\lor(b\land c)$ would not be a wff, because the outer parentheses are missing.