Clarifying Meaning of a Conditional w/ Quantifiers (∃x)(∀y)(Fyx ⊃ Fyy)

[The Head]

TL;DR Summary

Interpreting: (∃x)(∀y) (Fyx ⊃ Fyy)

I've been reading a logic book and saw the logical statement below and have been trying to consider its meaning:

(∃x)(∀y) (Fyx ⊃ Fyy)

I keep going back and forth whether this statement is implying:
a) For all things, if they do F to x, then they do F to themselves
-OR-
b) If there's some x that all things do F to, they all do F to themselves

Appreciate any clarification. The conditional is just throwing me off because I'm not sure if the antecedent is guarantees everything is doing an action to some x or if it's just "for each thing" that is doing an action to x.

 

[yossell]

Interesting question.

I take it to be something like (a).

Consider: (Ay)(Fyc -> Fyy): For any y, if y likes John, then y likes himself. 'likes' is 'F', John is a constant -- c. Informally, this statement says: 'anyone who likes John, also likes himself.' There's no implication that ALL people like John though.

Then your statement is just the existential generalisation on c of the above: there is someone who is such that, anybody who likes him also likes himself. Or 'somebody is liked by only people who like themselves.'

(b) sounds as if you've got a different bracketing in mind and the statement is in fact a conditional: if someone is liked by everyone, then everyone likes themselves:

[(Ex)(Ay)(Fyx)] -> Ay(Fyy)