How to paraphrase this sentence into quantificational notation

Question

Sentence: If a tenor respects all sopranos who respect him, then that tenor is respected by all sopranos.

Using 'Tx' to symbolize as 'x is a tenor'; 'Sx' to symbolize 'x is a soprano'; and 'Rxy' to symbolize 'x respects y'

The sentence seems to be a conditional, so I broke it up as follows:

(a tenor respects all sopranos who respect him) ⊃ (that tenor is respected by all sopranos)

I schematized the antecedent as: (∃x)(Tx & (∀y)(Sy & Ryx ⊃ Rxy)

I schematized the consequent as: (∃x)(Tx & (∀y)(Sy ⊃ Ryz)

So I have the whole thing as:

(∃x)(Tx & (∀y)(Sy & Ryx ⊃ Rxy) ⊃ (∃x)(Tx & (∀y)(Sy ⊃ Ryz)

But this seems to say: "If there is a tenor that respects all sopranos that respect him, then there is a soprano that is respected by all sopranos". The main feature this sentence seems to leave out is the anaphoric use of 'that tenor' in the consequent. In other words, this sentence seems to be true even when the tenor at issue in the antecedent is different than the one in the consequent. The anaphoric use of 'that tenor' seems to rule such dislocations out.

So then I tried letting the first existential have scope over the conditional like this:

(∃x)[Tx & ((∀y)(Sy & Ryx ⊃ Rxy) ⊃ (∀z)(Sz ⊃ Rzx))]

But this seems to say: "There is a tenor respected by all sopranos if they respect all sopranos that respect them."

First question: Is this sentence (the first one) a conditional or an existential?

Second question: Am I on the right track with the second schematization?

This is practice question IIIa2f from Goldfarb's Deductive Logic.

Thanks so much!

Answer 1

Our outermost semantic element is an if-then statement. So the quantification begins like this: $$\forall x(Tx\land\dots\implies \forall y(Sy\implies\dots))$$ The right blank is easy to fill. The left blank needs another variable just for the sopranos who respect tenor $x$: $$\forall x(Tx\land\forall z(Sz\land Rzx\implies Rxz)\implies\forall y(Sy\implies Ryx))$$