Translating statements into Predicate Logic I am facing problem in translating these statements to logic statements.


*

*Some horses are gentle only if they have been well trained.

*Some horses are gentle if they have been well trained.
I am not able to differentiate the above two statements.
$Hx$: $x$ is a horse.
$Gx$: $x$ is gentle. 
$Tx$: $x$ has been well trained.
I translated the first statement as $\exists x (Hx\rightarrow (Tx \rightarrow Gx))$.

Note the following:


*

*"$p$ only if $q$" means $p\to q$.

*"$p$ if $q$" means $p\leftarrow q$, more commonly denoted $q\to p$.

*"Some foo are bar" means that there exists one or more $x$ such that $x$ is a foo that is bar, i.e., such that $x$ is both foo and bar.


With these in mind, it should hopefully be clear that the statements you've supplied can be translated as:


*

*$\exists x(Hx \land (Gx \to Tx))$

*$\exists x(Hx \land (Tx \to Gx))$

