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

  • I
  • Thread starter Thread starter The Head
  • Start date Start date
  • Tags Tags
    Conditional
AI Thread Summary
The logical statement (∃x)(∀y)(Fyx ⊃ Fyy) is being analyzed for its implications regarding relationships between entities and their self-referential actions. The discussion revolves around two interpretations: one suggesting that if all entities perform action F on a specific x, they also perform F on themselves, and the other proposing that if there exists an x that all entities perform F on, then they all perform F on themselves. Clarification is sought on whether the antecedent guarantees that every entity is acting on some x or if it applies individually to each entity. An example is provided to illustrate that the statement can be interpreted as there being someone liked only by those who also like themselves. Ultimately, the distinction between the interpretations hinges on the bracketing of the logical components.
The Head
Messages
137
Reaction score
2
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.
 
Physics news on Phys.org
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)
 
Back
Top