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

• Isn't the first one an "if and only if"? Commented Feb 25, 2013 at 7:07
• @Anon Anon Do you know there's a difference between "A if B" and "A only if B"? Commented Feb 25, 2013 at 7:36
• yes. Is my first answer correct ? Commented Feb 25, 2013 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." Commented Feb 25, 2013 at 8:12
• @Anonymous: "$p$ only if $q$" is a perfectly cromulent construct meaning "$p\to q$." Commented Feb 25, 2013 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. $\exists x(Hx \land (Tx \to Gx))$
• 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 "$\rightarrow$".
– borg
Commented Feb 25, 2013 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. Commented Feb 25, 2013 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.' Commented Feb 25, 2013 at 16:33
• @AnonAnon: That's jwodder's claim not, not mine. I would notate it $q\to p$ from the beginning. Commented Feb 25, 2013 at 16:41