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Consider the predicate L, such that L(x,y) = "x loves y"

I'm not sure about how the meaning of the predicates change by swapping the order of variables and quantifiers. I've done my best to interpret the following statements. Can someone please let me know if my interpretation is correct?

  1. ∀x$\exists$y L(x,y) = For all people x, the following is true: There exists some person y where x loves y. In plain English, this means that "every person has some person out there that they love." (They do not have to be the same person).

  2. $\exists$x∀y L(x,y) = There exists some person x, such that for all y, x loves y. In plain English, this means that "some single person loves each and every person in some group of people".

  3. ∀y$\exists$x L(x,y) = For all people y, the following is true: There exists some person x, where x loves all y. In plain English, this means that "each and every person in a group of people is loved by some single person".

  4. $\exists$y∀x L(x,y) = There exists some person y, such that all people x love y. In plain English, this means that "some single person is loved by each and every person in a group of people".

If my interpretation is correct, the I'm noticing a few things. Can someone also let me know if these statements are correct?

  1. The key difference between statements 1 and 4 is that in statement 1, all the people in x each love a potentially different person y. However, in statement 4, a single person that exists, person y, is loved by all the people in x.

  2. The key difference between statements 2 and 3 is the subject and the "phrasing", although they are essentially saying the same thing. In statement 2, the subject is a single person x, who loves all the people in y. In statement 3, the subject is a group of people y, who are all loved by a single person x.

Thanks for the help. This concept is quite confusing to me, and I want to make sure that I am understanding it correctly.

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  • $\begingroup$ Why single person? Could not the person be married? More seriously, "single person" has the connotation of exactly one person. And you don't want to assert that, it could be several. Instead of "some single person" it is better to say "some person" or more briefly someone. $\endgroup$ – André Nicolas Oct 13 '15 at 18:48
  • $\begingroup$ Sorry yes, I meant "some single person" to mean "one person", not "a person who is single". $\endgroup$ – Charles Oct 13 '15 at 18:49
  • $\begingroup$ Again "one person" has a uniqueness connotation that we want to avoid. $\endgroup$ – André Nicolas Oct 13 '15 at 18:51
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One by one:

  1. Yes: Everyone loves somebody (not the same somebody for everyone!)
  2. Yes: This is "there's someone (x) who loves everybody (y)".
  3. No: Everyone (y) is loved by somebody (x).
  4. Yes: This is "somebody (y) is loved by everyone (x)", or "there is somebody (y) whom everybody (x) loves".
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