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Im learning about predicate logic from this textbook and I stumbled upon something that really confused me (both in phrase and in the contradiction I think I found).

On page 64 there is a sentence "There is someone who likes everyone who likes everyone that he likes". The purpose is to break it down to be put into predicate logic. Ok, confusing, but paraphrasing "There is a person who likes all individuals, such that the individual likes all the same people that he also likes". But what if the individual didn't like themselves? Everyone else the individual likes the same as the person but if they didn't like themselves this would be a contradiction. The person likes the individual, but the individual didn't like themselves, then they don't like all the same people. But if the person now doesn't like the individual, they now like all the same people so the person likes the individual. Am I not understanding something, or is this really a very confusing paradox?

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    $\begingroup$ It might be paradoxical, but even so, you should still be able to translate it. In fact, once you have that translation, you might be able to formally show that it is indeed paradoxical! So, don't worry about any paradoxes for now, just translate it. $\endgroup$ – Bram28 Jan 13 at 3:49
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What you show is that it is not possible for this person not to like themselves. That is not a paradox, however ... it just means that apparently this person must like themselves.

That is, there is indeed a contradiction if this person does not like themselves. But there is no contradiction if this person does like themselves.

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  • $\begingroup$ So If I Included as a premise that this person didn't like themselves, it would indeed lead to a contradiction? $\endgroup$ – Colin Hicks Jan 13 at 3:57
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    $\begingroup$ @ColinHicks Yes. It would be a good exercise to formally show that ... but that requires you do symbolize the statement correctly. $\endgroup$ – Bram28 Jan 13 at 3:58

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