# Predicate Logic - Formula

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 $$\forall(y)(A(y) \Rightarrow B(w,y))$$?

• The $w$ doesn't get a quantifier, it's just a constant in the statement. Commented Sep 26, 2020 at 11:56
• But I have been told that $w$ is regarded as a variable, meaning we should use either existential to universal quantifier for it. Commented Sep 26, 2020 at 11:56
• 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. Commented Sep 26, 2020 at 12:00
• I agree with @coffeemath , in the statement the variable $w$ is not quantified Commented Sep 26, 2020 at 12:00
• What is $H$ in your formula? Commented Sep 26, 2020 at 12:07

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.
• Well, I do not agree that existential quantifier will be used. In my opinion, correct formula should be the following: $\forall(y)(A(y) \Rightarrow B(w,y))$ Commented Sep 26, 2020 at 12:56