Convert "Only Believers respect God" to predicate logic.

Question

Only Believers respect God

Convert the above given English statement to predicate logic.

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I tried it as

$$B(x) = x~\textrm{ is a believer};~~ G(x) = x~\textrm{ respects God}\\ \exists~ x : ~(B(x)~ \land~ G(x))$$

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Answer 1

OP's proposed solution is that there exists at least one believer, who also respects God. This is not equal to the original request, and indeed need not even be true. For example, if nobody respects God, then the original is true but OP's solution is false.

A correct version, using the same predicates $B(x), G(x)$, is $$\forall x, (\neg B(x))\to (\neg G(x))$$

A simpler correct version is $$\forall x, G(x)\to B(x)$$

Answer 2

Your attempt translates back into English as "there is someone who is a believer and respects God", which is not at all what you meant: this would be true if a billion nonbelievers respected God as long as one single believer did too.

To get a correct translation, think about this. First, an existential statement only talks about one person; you want a universal statement. Second, suppose $x $ respects God. What can you conclude about $x $?

Answer 3

The problem is the statement doesn't say that believers don't disrespect god as well. It says only they can respect god, but that doesn't mean they all do. You can alternatively pose logically from the alternative non-believer side. A non-believer can neither respect nor disrespect something they know not to exist. There is an ambiguity in the statement that needs to be expressed too. The other factor is that respect and disrespect is only perceived by the believing group.

Answer 4

¬∃x: ¬B(x) ∧ G(x)

If only believers respect God, then a person who's not a believe yet respects God can not exist.

UPDATE: regarding OP's question in comment, let me elaborate:

All believers respect God means there is no such believer that does not respect. ¬∃x: B(x) ∧ ¬G(x) --> ∀x: ¬(B(x) ∧ ¬G(x)) --> ∀x: B(x) → G(x) Meaning: if you're a believer, you respect God

If only believers respect God, then there's no such person that respects God while not being a believer. ¬∃x: ¬B(x) ∧ G(x) --> ∀x: G(x) → B(x) Meaning: if you respect God, you must be a believer