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

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In summary, the logical statement (∃x)(∀y) (Fyx ⊃ Fyy) can be interpreted as either (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." The conditional in the statement may be causing confusion, but it essentially means that if someone is liked by everyone, then everyone likes themselves.
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The Head
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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.
 
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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)
 

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

1. What is the meaning of a conditional statement with quantifiers?

A conditional statement with quantifiers is a statement that uses the symbols (∃x) and (∀y) to express the existence of a variable x and the universal quantification of a variable y. This means that the statement is true for all possible values of y if a specific value of x exists.

2. How do you read a conditional statement with quantifiers?

A conditional statement with quantifiers is read as "for some x, for all y, if Fyx is true, then Fyy is also true." This can also be written as "there exists an x such that for all y, if Fyx is true, then Fyy is also true."

3. What is the difference between (∃x)(∀y)(Fyx ⊃ Fyy) and (∀y)(∃x)(Fyx ⊃ Fyy)?

The first statement, (∃x)(∀y)(Fyx ⊃ Fyy), means that there exists at least one x for which the statement (∀y)(Fyx ⊃ Fyy) is true. The second statement, (∀y)(∃x)(Fyx ⊃ Fyy), means that for all possible values of y, there exists at least one x for which the statement (Fyx ⊃ Fyy) is true. In other words, the first statement focuses on the existence of a specific x, while the second statement focuses on the existence of at least one x for every y.

4. What is the role of quantifiers in a conditional statement?

The quantifiers (∃x) and (∀y) in a conditional statement indicate the scope of the variables x and y. The existential quantifier (∃x) means "there exists" and the universal quantifier (∀y) means "for all." Together, they determine the conditions under which the statement is true.

5. Can you give an example of a conditional statement with quantifiers?

One example of a conditional statement with quantifiers is (∃x)(∀y)(x + y = 10 ⊃ x > y). This statement can be read as "there exists a number x such that for all numbers y, if x + y equals 10, then x is greater than y." In simpler terms, this means that there is at least one number that, when added to any other number to equal 10, will be greater than that number.

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