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I have a phrase that I am trying to translate into predicate logic. The phrase is as follows:

All lions except old ones roar

So far I have written down that:

$∀x((L(x) \land \lnot O(x)) \to R(x))$

Where $L(x)$ is "$x$ is a lion", $O(x)$ is "$x$ is old", and $R(x)$ is "$x$ roars". I am wondering if this is correct notation. I am mostly confused about the "except" in the phrase because as I have translated states that all lions who are not old roar.

Does any one have any thoughts about the notation for this phrase?

Andrés E. Caicedo
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user3471031
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    Do you think your sentence states that old lions don't roar, or does it simply say that young and middle-aged lions do roar (as you correctly interpret it)? – Fabio Somenzi Sep 22 '18 at 00:28
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    This is predicate logic, not propositional logic. – hmakholm left over Monica Sep 22 '18 at 00:29
  • @FabioSomenzi I think my sentence says that lions who are not old roar, which can also be lions who are young and middle aged roar – user3471031 Sep 22 '18 at 00:32
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    Yes, I was being (perhaps unsuccessfully) mildly facetious. There are two issues to be resolved. One is whether "except" implies that old lions are not guaranteed to roar, or whether they are guaranteed not to roar. Natural language is often ambiguous, but I'd vote for the latter interpretation. The other issue, raised by Henning Makholm, is that yours is a sentence of predicate logic. You don't have quantification in propositional logic. – Fabio Somenzi Sep 22 '18 at 00:38
  • @FabioSomenzi that makes sense. Thanks for your input! – user3471031 Sep 22 '18 at 00:48
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    @user3471031: your rendering says nothing about whether old lions roar, which is the first version in Fabio Somenzi's comment. He then says he votes for the second version where old lions are guaranteed not to roar. I would agree with the first and your rendering. I think the important thing is to understand the difference. – Ross Millikan Sep 22 '18 at 01:17
  • @RossMillikan - I totally agree with you. You should write this in a answer, I will upvote. The question has already been discussed here. – Taroccoesbrocco Sep 22 '18 at 05:51

2 Answers2

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∀x(L(x)∧¬O(x)→R(x))
says all lions that are not old roar.

To render the except requires more:
∀x(L(x)∧¬O(x)→R(x)) ∧ ∀x(L(x)∧O(x)→¬R(x))

William Elliot
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I think this also works (assuming that except means that old lions do not roar): $$\forall x\,\Big[L(x)\rightarrow\big(\neg O(x)\leftrightarrow R(x)\big)\Big]\,.$$

Batominovski
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