Distributing and over or in logic: how to do it if brackets ambiguous?

Question

I understand the rule is (A v ( B ^ C)) is equivalent to (A v B) ^ (A v C)

What happens if you have (A v B ^ C)? Or (A v B ^ C ^ D)? Is there an assumption that this is all left associative so first you do (A v B) ^ C to make it (A v B) ^ (A v C) and then look at the ^ D to make (A v B v D) ^ (A v C v D)? Not sure if this question makes sense but I'm just wondering if this would be ambiguous.

Answer 1

What happens if you have $(A \lor B \land C)$?

There is a preference for modern authors to avoid such ambiguity by using explicit parenthesis.   However, the traditional order of operations gives precedence to $\land$ over $\lor$, as analogous to $\times$ over $+$, so this would be read as $\big(A\lor(B\land C)\big)$.

$$\big(A\lor(B\land C)\big) ~=~ \big( (A\lor B)\land(A\lor C) \big)$$

Or (A v B ^ C ^ D)?

Likewise we would read this as $\big(A\lor(B\land C\land D)\big)$

$$\big(A\lor(B\land C\land D)\big)~=~\big((A\lor B)\land(A\lor C)\land(A\lor D)\big)$$