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Formalise the following argument, then prove or disprove its validity:

Some psychologists admire Freud. Some psychologists like no one who admires Freud. Therefore some psychologists are not liked by all psychologists.

$\exists x(Px \land Fx)$

$\exists x(Px \land \forall y(Fy \implies -Lxy))$

$\therefore$ $\exists x(Px \land \forall y(Py \implies -Lyx))$

ryang
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JCAL
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  • No. All statements in natural language involve the plural "psychologists". Your first formalization involves just one psychologist though! One does not equal more than one, and the plural form "psychologists" implies more than one. So, not this argument has not gotten symbolized correctly. – Doug Spoonwood Apr 08 '23 at 07:07
  • Wow okay ya, I’m definitely stuck then. – JCAL Apr 08 '23 at 07:56
  • @JCAL Doug is just having a bit of fun being pedantic, which is unnecessary since the ambiguity that I pointed out below shows that the exercise's author is not a particularly careful writer anyway. $\quad$ This exercise posted 12hrs ago contains a similar issue, and the first answerer Ethan has the common sense to interpret "There are quadrilaterals such that.." as "There is a quadrilateral.." based on the author's most plausible intention, even though the first sentence is technically an ungrammatical/wrong way to express the second one. – ryang Apr 09 '23 at 06:22
  • @JCAL In my opinion, the informal utterance "some people have property X" typically is intended to allow for the case "exactly one person has property X"; perhaps the speaker feels that saying "some person has property X" does not sufficiently indicate that plurality is the likely case, or perhaps the speaker isn't a particularly logical thinker so doesn't realise that "some person has property X" actually allows for plurality. – ryang Apr 09 '23 at 06:28
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    I agree with @ryang that most likely the author is looking for a simple single existential when 'some' is used. And also remember that an existential is often informally translated back to English as 'some [X]s' (i.e. plural) even though technically all we can say is 'there is at least one ..'. I think the context in which this exercise will make this clear: if the author has just covered numerical claims ('at least too, at most three, etc.) then you might indeed want to worry whether this English statement was meant as 'at least two', but I highly doubt that is what is going on here. – Bram28 Apr 10 '23 at 11:51

1 Answers1

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You have formalised the premises correctly.

The conclusion "Therefore some psychologists are not liked by all psychologists" sounds to me like it means "Therefore it's not that some psychologists are liked by all psychologists" rather than "Therefore some psychologists are disliked by all psychologists", so I would formalise it as $$\therefore \exists x(Px \land -\forall y(Py \implies Lyx))$$ instead.

P.S. Separately, observe that the argument is valid if formalised my way, and invalid if formalised your way.

Addendum

Oh wow I am wrong eh?! I was super confident, well thank you very much!

The given natural-language statement is ambiguous, so you're not really wrong; notice that if the author really intends the "disliked" interpretation then they would likely have written just that; what they wrote suggests that they mean the "not that" interpretation instead.

ryang
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