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The question says,

Translate the following sentence into the logical notation of propositional functions and quantifiers.

H(x): x is a horse
G(x): x is gentle.
T(x): x has been well trained.

"Only horses are gentle if they have been well trained."

I am not really sure what exactly this question means, does it mean that horses are the only thing that can be made gentle by training?

In that case my answer would be:

$$ \forall x [ (T(x) \to G(x))\to H(x)]$$

Is this correct? If not then what is the correct interpretation?

Problem source: Symbolic Logic by Irving M. Copi (problem #$38,$ page-$70$)

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2 Answers

up vote 3 down vote accepted

Compare 'Only adults are sent to prison if they commit serious offences (juveniles are sent to other institutions)'. This doesn't entail that all adults who commit serious offences are sent to prison, some might get other punishments: it just that (as it says!) only adults get sent to prison if they offend. So, if someone has offended, then if they end up in prison, they will be an adult.

We can read the claim about horses similarly. Thus consider

'What animals are gentle if well trained?'

'Only horses are gentle if they have been well trained, though not even all of them are: as for other animals, any kind of training gives them an angry disposition.'

A silly conversation, perhaps, but logically cogent. So in this context the italicized proposition says (putting it casually) if an animal has been well trained, then if it has ended up gentle, it must be a horse. Or in symbols, quantifying over animals,

$$\forall x(Tx \to (Gx \to Hx))$$

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Until you wrote this, I could not understand how you were reading the sentence. I agree that your reading is plausible. –  MJD Nov 3 '12 at 13:36
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You are right to be confused: the sentence is not clear or grammatically correct. I believe the intention is that a horse will be gentle only if it has been well-trained. I suggest that when you answer the question you clearly state what sentence you are translating. Otherwise, the grader might mark down your answer because they have a different idea of what "Only horses are gentle if they have been well trained" means.

But note that your proposed translation of "horses are the only thing that can be made gentle by training" is incorrect: it says that whenever $T(x)\to G(x)$ is true for $x$, then $x$ is a horse. $T(x)\to G(x)$ is true if $x$ is something that is both trained and gentle, but it is also true if $x$ is anything untrained. So your translation also claims that anything that has not been trained is a horse.

Note that $T(x)\to G(x)$ does not mean "$x$ becomes gentle by training". It is equivalent to $\lnot T(x)\lor G(x)$, which says that $x$ is untrained or gentle. Your translation is equivalent to:

$$ \forall x [ (\lnot T(x) \lor G(x))\to H(x)]$$

which says that any $x$ which is untrained or gentle, must be a horse. This surely wasn't what you meant to say, since it says that untrained lemurs and gentle octopuses are horses, and there was no trace of that in the English version.

I don't think it is possible to translate "horses are the only thing that can be made gentle by training" propositionally. The problem is that you don't have any relation that expresses that the gentleness is the result of the training.

I suggest you try "the only horses that are gentle are those that have been well-trained", as I did, or "nothing well-trained is gentle, except for horses" as Peter Smith did elsewhere. My own translation of "the only horses that are gentle are those that have been well-trained" is:

$$\forall x [(H(x)\land G(x))\to T(x)]$$
or equivalently
$$\forall x [H(x)\to (G(x)\to T(x))]$$

(Mouse over for spoiler.)

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MJD's suggested translation renders 'All gentle horses are well-trained" which isn't certainly a natural reading of the original. Though I agree the original has a couple of readings, this surely isn't one of them! –  Peter Smith Nov 3 '12 at 9:08
    
@Peter Smith: Then what will be the correct natural reading of the original? –  Quixotic Nov 3 '12 at 11:03
    
@PeterSmith: If you interpret it as "A horse is gentle only if it is well-trained," this is equivalent to saying "If a horse is gentle, then it is well-trained," which is equivalent to "All gentle horses are well-trained." It doesn't sound like it means the same in English, but they are all logically equivalent. Of course, if you're trying to stick to the original phrasing, then none of these makes sense. ;) –  Cameron Buie Nov 3 '12 at 18:55
    
@Cameron Thanks. I agree with your comment. As I said, I was interpreting the sentence "Only horses are gentle if they have been well trained" to mean "the only horses that are gentle are those that have been well-trained". But the original is confusing enough that I don't have high confidence that this is what the asker intended. –  MJD Nov 3 '12 at 19:07
    
That's surely just wrong. (1) "Only horses are gentle if they have been well trained" surely is never equivalent to (2) "the only horses that are gentle are those that have been well-trained". Compare: (1') "Only adults are sent to prison if they commit robbery" is plainly not equivalent to (2') "The only adults that are sent to prison are those that commit robbery". Suppose juveniles are never sent to prison for theft, while adults are also sent to prison for other offences. Then (1') could be true while (2') is false. –  Peter Smith Nov 3 '12 at 20:39
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