Interpretation of negation when translating a sentence to propositional logic (outside bracket vs. applied to each proposition inside bracket)

Question

Assume we have the following propositions:

• John was victorious: J
• Robert was victorious: R
• Dan was victorious: D

Now we are given the sentence "If John was defeated, Robert and Dan suffered the same fate". What would be the correct way of translating this to a propositional logic formula? My first instinct is to say $\neg J \implies \neg R \wedge \neg D$, however I think it would also be possible to interpret this as $\neg J \implies \neg(R \wedge D)$

The two options yield different clauses when converting to conjunctive normal form and therefore one of them won't be right and won't allow for correct deductions in a knowledge base resolution for example. What is the correct way of interpreting the sentence? Is there a rule of thumb to disambiguate these kinds of statements?

Answer 1

The sentence may be parsed in two plausible ways:

• Robert and Dan lost (i.e. shared the same fate with John). This is $\neg J\implies(\neg R\land\neg D)$ (your "first instinct") and may be simplified as $(R\lor D)\implies J$.
• Robert and Dan shared the same fate with each other. This is $\neg J\implies(R\iff D)$.

However, intuitively the first interpretation is more "natural": "shared the same fate" suggests a long-range association, which points to the clause about John. "Suffered" also hints at Robert and Dan losing.