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Which of these does not correctly represent “Every person loves at least one corgi”?

  1. $\forall x[person(x)\rightarrow \exists y[corgi(y)\Lambda Loves(x,y)]]$
  2. $\forall x\exists y[person(x)\rightarrow corgi(y)\Lambda Loves(x,y)]$
  3. $\exists y\forall x[person(x)\rightarrow corgi(y)\Lambda Loves(x,y)]$
in Mathematical Logic by (403 points)
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b….?
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It’s a self doubt. I dont have the answer. But if you feel B is wrong, plz provide a reason.
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C doesn't seem right. It will be interpreted as, "There exists a corgi, such that every person loves that corgi".
But our implication in question goes on way only. Person loves at least corgi. There maybe corgis that no one loves. So, C seems wrong.

B seems correct, it can be interpreted as "For each person, there exists a corgi that is loved by that person."

A is correct.

What do you say?

1 Answer

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A.For all $x$ such that if $x$ is a person then there exist at least one $y$ such that $y$ is a corgi and $x$ loves $y$. It means that every person loves atleast $1$ corgi. So it is correct.

B. $\sim p(x) \vee (c(y) \wedge l(x,y)) $

It means that for all x there exist a y such that either x is not a person OR $y$ is a corgi and $x$ loves $y$. So it is also correct.

C. There exist a $y$ for all $x$  such that if $x$ is any person then $y$ is  corgi and $x$ loves $y$. It is also not correct. It means that in the universe there is only $1$ corgi and every person loves him/her/it.
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This is from Stanford slides and A represents it correctly. I have doubts about B and C.
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How can A be correct.

Question is asking about "at least one" so more than 1 should also have been expressed in the option.

It is representing "exactly one y "
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Check option A now. Bracket was misplaced.
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updated my answer please check.
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I don’t think you opened → right in option B. ^ has greater priority over → so (corgi(y) Λ Loves(x,y)) should be treated as a unit.

It should be ~person (x) ∨ (corgi(y) Λ Loves(x,y))

I’m finding difficulty in distinguishing B from A :(
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yes you are correct.

both A and B ar correct.

The difference between A and B can be seen from coding perspective.

in Option A y will only be created if x turns out to be a person. so this saves some space wasted due to creation of y unnecessarily.

in Option B y is created before we check condition. if condition is not true then y is not used and if condition is true i.e. x turns out to be a person then we check the relation between them.
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Option C means that "There exists a corgi that every person loves." Right?
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