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Assume it is given $2$ predicates as below:

$A(x):$ $x$ is a horse;

$B(x,y):$ $x$ is a tail of $y$.

Then, translate the following sentence into predicate logic formula: "$w$ is a tail of horse" - where $w$ is arbitrary variable

My ideas: I am struggling on which variable out of $w$ and horse, we should pick up as existential, and universal? Would appreciate to know your ideas.

Update: Is the formula for above sentence expressed as $\forall(y)(A(y) \Rightarrow B(w,y))$?

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    $\begingroup$ The $w$ doesn't get a quantifier, it's just a constant in the statement. $\endgroup$
    – coffeemath
    Commented Sep 26, 2020 at 11:56
  • $\begingroup$ But I have been told that $w$ is regarded as a variable, meaning we should use either existential to universal quantifier for it. $\endgroup$
    – rentbuyer
    Commented Sep 26, 2020 at 11:56
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    $\begingroup$ A "constant" in first order logic is really just a variable being used to denote a specific object in the universe it is taken from. $\endgroup$
    – coffeemath
    Commented Sep 26, 2020 at 12:00
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    $\begingroup$ I agree with @coffeemath , in the statement the variable $w$ is not quantified $\endgroup$
    – Air Mike
    Commented Sep 26, 2020 at 12:00
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    $\begingroup$ What is $H$ in your formula? $\endgroup$
    – Air Mike
    Commented Sep 26, 2020 at 12:07

1 Answer 1

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Given the instructions, $w$ is supposed to be a free variable that remains unquantified. It acts as a placeholder to insert arbitrary names for, like in the definitions of the two predicates you gave, where $x$ and $y$ are not bound either.

Your proposal is incorrect. Your formula expresses "$w$ is the tail of every horse". But that's not the meaning of "a tail's horse". "horse's tail" means "a horse's tail" -- that particular tail is the tail of one horse, not all the horses in the world -- so the horse variable will be existentially, not universally quantified.
Also, as explained in the answer to your other post, "a horse's tail" means that $w$ is the tail of $y$ and $y$ is a horse -- so you need conjunction, not implication. As a rule of thumb, existential quantifiers go with conjunction and universal quantifiers go with implication.

Can you figure out the correct solution given these hints?

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  • $\begingroup$ Well, I do not agree that existential quantifier will be used. In my opinion, correct formula should be the following: $\forall(y)(A(y) \Rightarrow B(w,y))$ $\endgroup$
    – rentbuyer
    Commented Sep 26, 2020 at 12:56
  • $\begingroup$ @rentbuyer? Why do you think so? $\endgroup$ Commented Sep 26, 2020 at 13:01
  • $\begingroup$ Can you provide formula with existential quantifier? I think my formula works pretty normal. $\endgroup$
    – rentbuyer
    Commented Sep 26, 2020 at 13:02
  • $\begingroup$ Dear, I have already done that) $\endgroup$
    – rentbuyer
    Commented Sep 26, 2020 at 13:03
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    $\begingroup$ That's correct. $\endgroup$ Commented Sep 26, 2020 at 13:22

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