# Translating statements into Predicate Logic

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

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Isn't the first one an "if and only if"? – Anonymous Feb 25 '13 at 7:07
@Anon Anon Do you know there's a difference between "A if B" and "A only if B"? – Git Gud Feb 25 '13 at 7:36
yes. Is my first answer correct ? – Anon Anon Feb 25 '13 at 7:53
@GitGud Do you mean there is no difference between statement 1 and this statement : "Some horses are gentle if only if they have been well trained." – Anon Anon Feb 25 '13 at 8:12
@Anonymous: "$p$ only if $q$" is a perfectly cromulent construct meaning "$p\to q$." – jwodder Feb 25 '13 at 8:38

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. $\exists x(Hx \land (Tx \to Gx))$
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Your distinction between "$p$ only if $q$" and "$p$ if $q$" is wrong. I can't see how "only" would make any difference in the "direction" of implication. At most it can make "$\leftrightarrow$" from "$\rightarrow$". – data Feb 25 '13 at 8:47
@m.woj: No, $\leftrightarrow$ is conventionally pronounced as "if and only if" exactly because "if" expresses implication in one direction and "only if" expresses implication in the other direction. The idiom would perhaps have been clearer if it had been "only when" instead of "only if", but jwodder has the established semantics completely right. – Henning Makholm Feb 25 '13 at 12:42
@HenningMakholm Could you please provide any link/website supporting your claim that ' "p if q" means p←q, more commonly denoted q→p.' – Anon Anon Feb 25 '13 at 16:33
@AnonAnon: That's jwodder's claim not, not mine. I would notate it $q\to p$ from the beginning. – Henning Makholm Feb 25 '13 at 16:41