Translating statements into Predicate Logic

Question

I am facing problem in translating these statements to logic statements.

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

2. 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 $\exists x (Hx\rightarrow (Tx \rightarrow Gx))$.

Answer 1

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:

1. $\exists x(Hx \land (Gx \to Tx))$
2. $\exists x(Hx \land (Tx \to Gx))$