Propositional form of sentence

Question

What will be propositional logic form of this sentence:

If the budget is not cut, then prices remain stable if and only if taxes will be raised.

My Approach: suppose

$p$=the budget is not cut

$q$=prices remain stable

$r$=taxes will be raised.

So it should be $p\implies (q \Leftrightarrow r)$.

but I am confused between $p\implies (q \Leftrightarrow r)$. and $(p\implies q )\Leftrightarrow r$.

so what would be correct answer?

Answer 1

Natural language, in this case, the English language, is inherently ambiguous, and hence its translation into non-ambiguous logic isn't always straight foward.

In this case, I believe your first translation is the most correct, $$p\rightarrow(q\iff r)$$ and I say that primarily because of the placement of the comma within the statement. The first clause reads $p$, and depending on $p$, it follows that $q\iff r$.

If the comma were placed as follows:

If the budget is not cut then prices remain stable, if and only if taxes will be raised.

This second sentence is best represented by $$(p\rightarrow q) \iff r$$