Predicate Logic: Everybody can fool Fred

Question

Let F(x,y) denote the statement "x can fool y" and D denote the domain of all people. So far I have the quantified proposition of "Everybody can fool Fred" as, ∀x∈D ∃y∈D: F(x,y) But I know that I have to be specific when it comes to Fred. So my question is how to I specify Fred in the domain of all people. Do I have to create another variable or do I have to specify y some how? Any help would be appreciated. Thank you.

Answer 1

$\forall x\in D$ $\exists y\in D: F(x,y)$ is the statement, "Everyone can fool someone."

$\exists y\in D$ $\forall x\in D: F(x,y)$ is the statement, "There is a person everyone can fool."

Those two statements are very different, and it looks like the second statement is more along the lines of what you're going for.

As for your statement, I would write it as $\forall x\in D: F(x,Fred)$.

Answer 2

You can use an individual constant for Fred. So that's not a variable, but a constant that denotes Fred, for example $f$. Also, if the domain is all people, then there is no need to specify that in the symbolization, and you can just use:

$$\forall x \ F(x,f)$$