Correct negation of a statement (living in L.A. & winning the lottery)

[Ozma]

English is not my first language, so maybe there is a intrinsic problem here; I apologize if my doubts are not easy to understand because I am communicating them badly.

I read a problem (in English) which asks to write, in logical symbols, the negation of the statement: "Anyone living in Los Angeles who has blue eyes will win the lottery and will take their retirement before the age of 50". Denoting with [imath]X[/imath] the set of people living in Los Angeles, with [imath]x[/imath] a generic element of [imath]X[/imath], with [imath]B(x)[/imath] the predicate "[imath]x[/imath] has blue eyes", with [imath]L(x)[/imath] the predicate "[imath]x[/imath] will win the lottery" and finally with [imath]R(x)[/imath] the predicate "[imath]x[/imath] will take retirement before the age of 50", how can I distinguish, from the English written sentence, if the sentence means:

(i) [imath]\forall x \in X [B(x) \implies (L(x) \wedge R(x))][/imath]

or it means

(ii) [imath]\forall x \in X [(B(x) \implies L(x)) \wedge R(x)][/imath]?

In other words, how can I distinguish, from the English written sentence if having blue eyes implies only to win the lottery (and the part about the retirement is unrelated to the property of having blue eyes) or having blue eyes implies both to win the lottery and to retire before the age of 50?

In the first case, the negation should be [imath]\exists x_0 \in X [B(x_0) \wedge (\neg L(x_0) \lor \neg R(x_0))][/imath].

In the second case, the negation should be [imath]\exists x_0 \in X [(B(x_0) \wedge \neg L(x_0)) \lor \neg R(x_0))][/imath].

They both are phrased the same in English: "There is a person living in Los Angeles that has blue eyes and will not win the lottery or will retire after the age of 50". But the "and" and "or" in logical symbols are related in a specific order that can be deduced by distributing correctly the logical connectives, while in the common language there is ambiguity. Or am I wrong and they don't translate the same? If they don't translate the same, how can they be wrote correctly in English to distinguish them?

From the solution, the author negates the sentence as: "There is a person living in Los Angeles who has blue eyes and who will not win the lottery or retire after the age of 50". Is this a grammar rule of English language that, in phrases like this, it is intended that having blue eyes refers only to the first predicate of the conjunction "and" or is the English phrase actually ambiguous? Maybe, the correct translation of (i) in English is the phrase: "Anyone living in Los Angeles who has blue eyes will both win the lottery and retire after the age of 50", or something like this? In this latter case, I agree that the English statement is not ambiguous and I have done a mistake interpreting it uncorrectly; but I am not sure if this latter translation is indeed (i).

 

[pka]

I confess that I am not at all sure as to your problem?
From my experience I would expect that your I) is correct.
For my students I would want: The universe is [imath]\Omega=[/imath] the set of people living in Los Angeles.
The the coding: [imath]\neg\left( {\forall x} \right)\left[ {B(x) \Rightarrow \left( {L(x) \wedge R(x)} \right)} \right] \equiv \left( {\exists x_0} \right)\left[ {B(x_0) \wedge \left( {\neg L(x_0) \vee \neg R(x_0)} \right)} \right][/imath]
If that is not helpful, please reply with questions.

[imath][/imath][imath][/imath][imath][/imath]

 

[Dr.Peterson]

Your question is about translation of English, so I'll try to answer that.

"Anyone living in Los Angeles who has blue eyes will win the lottery and will take their retirement before the age of 50".

In other words, how can I distinguish, from the English written sentence if having blue eyes implies only to win the lottery (and the part about the retirement is unrelated to the property of having blue eyes) or having blue eyes implies both to win the lottery and to retire before the age of 50?

I think it is clear that both conclusions (lottery and retirement) apply only to those with blue eyes, because the subject of both verbs is the entire phrase "Anyone living in Los Angeles who has blue eyes". I can't even see a way to modify the sentence in a minor way so that the retirement clause would apply to anyone in LA, which is what your second interpretation does.

They both are phrased the same in English: "There is a person living in Los Angeles that has blue eyes and will not win the lottery or will retire after the age of 50". But the "and" and "or" in logical symbols are related in a specific order that can be deduced by distributing correctly the logical connectives, while in the common language there is ambiguity. Or am I wrong and they don't translate the same? If they don't translate the same, how can they be wrote correctly in English to distinguish them?

This sentence does not correctly represent either of your negations (the first, which is correct, or the second); you dropped a "not".

Here is how I would translate each of them to English:

In the first case, the negation should be [imath]\exists x_0 \in X [B(x_0) \wedge (\neg L(x_0) \lor \neg R(x_0))][/imath].

In the second case, the negation should be [imath]\exists x_0 \in X [(B(x_0) \wedge \neg L(x_0)) \lor \neg R(x_0))][/imath].

The first:

There is a person living in Los Angeles who has blue eyes, and either will not win the lottery, or will not retire after the age of 50.​

The second:

There is a person living in Los Angeles who either has blue eyes and will not win the lottery, or will not retire after the age of 50.​

The commas I added help clarify; the word "either" helps to group clauses (though it could be taken as making an exclusive or).

From the solution, the author negates the sentence as: "There is a person living in Los Angeles who has blue eyes and who will not win the lottery or retire after the age of 50". Is this a grammar rule of English language that, in phrases like this, it is intended that having blue eyes refers only to the first predicate of the conjunction "and" or is the English phrase actually ambiguous?

I hope you copied this sentence incorrectly! The biggest error is that "will not win the lottery or retire" would probably be understood as "will neither win the lottery nor retire" (that is, "will not win the lottery and will not retire"), which is the wrong negation. Your addition of an extra "will" in your version above corrects this, but in doing so, you need to add the extra "not" I pointed out.

I'm not sure what you mean by "having blue eyes refers only to the first predicate of the conjunction 'and'". That clause is the first predicate of the conjunction!

Maybe, the correct translation of (i) in English is the phrase: "Anyone living in Los Angeles who has blue eyes will both win the lottery and retire after the age of 50", or something like this? In this latter case, I agree that the English statement is not ambiguous and I have done a mistake interpreting it incorrectly; but I am not sure if this latter translation is indeed (i).

This ("both") is perhaps an improvement on the original, but I don't think it's necessary.

But others may have a different opinion on some of these points of grammar. English can indeed be ambiguous in expressing logic, as I think any language can. This is why we use symbols!

 

[MaxMath]

In this example, the 'standard' answer itself is quite ambiguous and is far from satisfactory in my opinion. A better answer would be (with slight changes of wording to remove the ambiguities)—

There is a person living in Los Angeles and having blue eyes who will not win the lottery or will retire after the age of 50.

Note the changes include (1) "who has" -> "having", (2) the removal of "and" before "who will not win", and (3) the addition of "will" before "retire". These are important because—
(1) this makes the qualifier "has blue eyes" tightly associated with "a person living in Los Angeles". This is necessary because in the original form, the subject is "person living in Los Angeles who has blue eyes", which should be treated as an 'atomic' element that is undividable.
(2) this separates the subject "a person living in Los Angeles and having blue eyes" from the predicates that follow, so avoids the reader from messing"having blue eyes" with those predicates in whatever logic operation.
(3) this makes it clear that "after the age of 50" is only for "will retire", not for "will [either] win the lottery or retire".

The original sentence clearly means—
[math]\forall x \in X [B(x) \implies (L(x) \wedge R(x))][/math]
This is because, as said above, the qualifier "has blue eyes" is solely for what precedes it without ambiguity. The whole phrase "[a person] living in Los Angeles who has blue eyes" should be treated as an inseparable element.

The English version of your alternative formulation is—

Anyone living in Los Angeles will either win the lottery, if he or she has blue eyes, or take their retirement before the age of 50.

This is very different from the original sentence.

From the solution, the author negates the sentence as: "There is a person living in Los Angeles who has blue eyes and who will not win the lottery or retire after the age of 50". Is this a grammar rule of English language that, in phrases like this, it is intended that having blue eyes refers only to the first predicate of the conjunction "and" or is the English phrase actually ambiguous? Maybe, the correct translation of (i) in English is the phrase: "Anyone living in Los Angeles who has blue eyes will both win the lottery and retire after the age of 50", or something like this? In this latter case, I agree that the English statement is not ambiguous and I have done a mistake interpreting it uncorrectly; but I am not sure if this latter translation is indeed (i).

Formula (i) is best written as "Anyone living in Los Angeles who has blue eyes will both win the lottery, and retire after the age of 50". I've added a comma to your "latter" version, which is not liked by the autocorrect feature of the forum, but is essential to clarity by signalling that it's not "both win the lottery and retire, after the age of 50". Notice the difference.

 

[MaxMath]

Formula (i) is best written as "Anyone living in Los Angeles who has blue eyes will both win the lottery, and retire after the age of 50". I've added a comma to your "latter" version, which is not liked by the autocorrect feature of the forum, but is essential to clarity by signalling that it's not "both win the lottery and retire, after the age of 50". Notice the difference.

I admit that to make the logic clearer, the punctuation becomes a bit weird – that's why the autocorrect feature complained. An alternative to this, which is closer to human language, is to change the order of the two arms of "both ... and ..." (actually the words "both ... and ..." are no longer necessary)—

Anyone living in Los Angeles who has blue eyes will retire after the age of 50 and will win the lottery.

(You may take the second "will" out safely.)