# Predicate Logic with Book Genres

I need to write each English language statement as a predicate expression.

S(x) is "x is a spy novel." L(x) is "x is long." M(x) is "x is a mystery." B(x,y) "x is better than y."

• Only mysteries are long. -> how would i use predicate symbols to show "only?"
• Some spy novels are better than mysteries. -> does this mean all mysteries?
• Spy novels are better than mysteries. -> here I'm just unsure how to use the B(x,y) for better than
• Some mysteries are better than all spy novels. -> how would I write a statement that uses "some" and "all?"

Any help is appreciated!

Hint

1) "Only mysteries are long" can be rewritten as : "There are no long things that are no mysteries".

Thus : $¬∃x(Long(x)∧¬Myst(x))$, i.e. :

$∀x(Long(x) \to Myst(x))$.

2) Maybe; but if we read "Some spy novels are better than mysteries" to mean that some spy novels are better than all mysteries, than 2) is the same as 4) ...

Thus, I think that 2) and 3) must be formalized with free variables, like for 3) :

$Spy(x) \land Myst(y) \land Bet(x,y)$.

4) "Some mysteries are better than all spy novels" is :

$\exists x (Myst(x) \land \forall y (Spy(y) \to Bet(x,y))$.