[Math] How to convert an English sentence that contains ‘all and only’ into predicate calculus sentence

logicpredicate-logic

I have the the following English sentence that I need to convert into a sentence in predicate logic:

All and only spartans are bold

I'm not sure if it is simply: $∃x [B(x) ↔ S(x)]$ or whether this doesn't entail that 'all' spartans are bold, just that 'only' spartans are bold.

Best Answer

I would rephrase it in English like this: For all individuals x, the individual is Spartan if and only if the individual is bold. So in logic, it would be: $\forall x [B(x) \Leftrightarrow S(x)]$.

Your statement says something only about a single $x$ existing, not about all of them.