Symbolizing "somebody likes everybody bob likes"

Question

I would like to know how to symbolize "somebody likes everybody bob likes"

I have tried the following: for all of x (if bob likes x, then for some y, y likes x)

which I think would look like: $∀x∃y(Lbx\Rightarrow (Lyx))$

Is this correct?

Thank you

Answer 1

Observe that the phrase, "Somebody likes everybody that Bob likes" is equivalent to the phrase, "There exists an individual (Alice) such that for every individual $I,$ if Bob likes $I,$ then Alice likes $I,$ as well."

Quantifying that gives us $(\exists A \,|\, \forall I, B \text{ likes } I \implies A \text{ likes } I).$

Answer 2

You've symbolized the statement as a universal, but the statement is actually an existential statement. It asserts (first and foremost) that there exists an individual. It's more clear if you write it more formally: There exists an individual, $x$, such that $x$ likes every individual Bob likes. Thus, we have

$$\exists x (\forall y (Lby \to Lxy))$$

Answer 3

Your formalization reads "Everyone who Bob likes is liked by someone"; but that permits for the possibility that each of Bob's liked ones is liked by someone different:

        />  Henry <- Mary
    Bob ->  Tom   <- John     ("->" = "likes")
        \>  Susan <- Josy

whereas the the original sentence asserts that there should be at least one single person who likes all of them. You therefore have to swap the existential and the universal quantifier:

$$\exists y \forall x(Lbx \to Lyx)$$

This only permits structures like the following:

        />  Henry <\ 
    Bob ->  Tom   <- John     ("->" = "likes")
        \>  Susan </ 

-- as desired.